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<h1 class="ltx_title ltx_title_document">Diffusion Forcing: Next-token Prediction
<br class="ltx_break">Meets Full-Sequence Diffusion</h1>
<div class="ltx_authors">
<span class="ltx_creator ltx_role_author">
<span class="ltx_personname">Boyuan Chen 
<br class="ltx_break">MIT CSAIL
<br class="ltx_break"><span id="id1.1.id1" class="ltx_text ltx_font_typewriter">boyuanc@mit.edu</span> 
<br class="ltx_break">&amp;Diego Martí Monsó 
<br class="ltx_break">Technical University of Munich
<br class="ltx_break"><span id="id2.2.id2" class="ltx_text ltx_font_typewriter">diego.marti@tum.de</span> 
<br class="ltx_break">&amp;Yilun Du 
<br class="ltx_break">MIT CSAIL
<br class="ltx_break"><span id="id3.3.id3" class="ltx_text ltx_font_typewriter">yilundu@mit.edu</span> 
<br class="ltx_break">&amp;Max Simchowitz 
<br class="ltx_break">MIT CSAIL
<br class="ltx_break"><span id="id4.4.id4" class="ltx_text ltx_font_typewriter">msimchow@mit.edu</span> 
<br class="ltx_break">&amp;Russ Tedrake 
<br class="ltx_break">MIT CSAIL
<br class="ltx_break"><span id="id5.5.id5" class="ltx_text ltx_font_typewriter">russt@mit.edu</span> 
<br class="ltx_break">&amp;Vincent Sitzmann 
<br class="ltx_break">MIT CSAIL
<br class="ltx_break"><span id="id6.6.id6" class="ltx_text ltx_font_typewriter">sitzmann@mit.edu</span> 
<br class="ltx_break">
</span><span class="ltx_author_notes">Work done as a visiting student at MIT.</span></span>
</div>

<div class="ltx_abstract">
<h6 class="ltx_title ltx_title_abstract">Abstract</h6>
<p id="id7.id1" class="ltx_p">This paper presents Diffusion Forcing, a new training paradigm where a diffusion model is trained to denoise a set of tokens with <em id="id7.id1.1" class="ltx_emph ltx_font_italic">independent</em> per-token noise levels.
We apply Diffusion Forcing to sequence generative modeling by training a causal next-token prediction model to generate one or several future tokens without fully diffusing past ones. Our approach is shown to combine the strengths of next-token prediction models, such as variable-length generation, with the strengths of full-sequence diffusion models, such as the ability to guide sampling to desirable trajectories. Our method offers a range of additional capabilities, such as (1) rolling-out sequences of continuous tokens, such as video, with lengths past the training horizon, where baselines diverge and (2) new sampling and guiding schemes that uniquely profit from Diffusion Forcing’s variable-horizon and causal architecture, and which lead to marked performance gains in decision-making and planning tasks. In addition to its empirical success, our method is proven to optimize a variational lower bound on the likelihoods of all subsequences of tokens drawn from the true joint distribution. Project website: <a target="_blank" href="https://boyuan.space/diffusion-forcing" title="" class="ltx_ref ltx_url ltx_font_typewriter">https://boyuan.space/diffusion-forcing</a></p>
</div>
<section id="S1" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2>

<div id="S1.p1" class="ltx_para">
<p id="S1.p1.1" class="ltx_p">Probabilistic sequence modeling plays a crucial role in diverse machine learning applications including natural language processing <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib6" title="" class="ltx_ref">6</a>, <a href="#bib.bib46" title="" class="ltx_ref">46</a>]</cite>, video prediction <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib31" title="" class="ltx_ref">31</a>]</cite> and decision making <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib3" title="" class="ltx_ref">3</a>, <a href="#bib.bib22" title="" class="ltx_ref">22</a>]</cite>.
Next-token prediction models in particular have a number of desirable properties. They enable the generation of sequences with varying length <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib32" title="" class="ltx_ref">32</a>, <a href="#bib.bib21" title="" class="ltx_ref">21</a>, <a href="#bib.bib37" title="" class="ltx_ref">37</a>]</cite> (generating only a single token or an “infinite” number of tokens via auto-regressive sampling), can be conditioned on varying amounts of history <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib21" title="" class="ltx_ref">21</a>, <a href="#bib.bib37" title="" class="ltx_ref">37</a>]</cite>, support efficient tree search<cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib66" title="" class="ltx_ref">66</a>, <a href="#bib.bib23" title="" class="ltx_ref">23</a>, <a href="#bib.bib25" title="" class="ltx_ref">25</a>]</cite>, and can be used for online feedback control <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib22" title="" class="ltx_ref">22</a>, <a href="#bib.bib3" title="" class="ltx_ref">3</a>]</cite>.</p>
</div>
<div id="S1.p2" class="ltx_para">
<p id="S1.p2.1" class="ltx_p">Current next-token prediction models are trained via <em id="S1.p2.1.1" class="ltx_emph ltx_font_italic">teacher forcing</em> <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib62" title="" class="ltx_ref">62</a>]</cite>, where the model predicts the immediate next token based on a ground truth history of previous tokens.
This results in two limitations: (1) there is no mechanism by which one can guide the sampling of a sequence to minimize a certain objective, and
(2) current next-token models easily become <em id="S1.p2.1.2" class="ltx_emph ltx_font_italic">unstable</em> on continuous data. For example, when attempting to auto-regressively generate a video (as opposed to text <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib6" title="" class="ltx_ref">6</a>]</cite> or vector-quantized latents <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib33" title="" class="ltx_ref">33</a>]</cite>) past the training horizon, slight errors in frame-to-frame predictions accumulate and the model diverges.</p>
</div>
<div id="S1.p3" class="ltx_para">
<p id="S1.p3.1" class="ltx_p"><em id="S1.p3.1.1" class="ltx_emph ltx_font_italic">Full-sequence diffusion</em> seemingly offers a solution.
Commonly used in video generation and long-horizon planning, one directly models the joint distribution of a fixed number of tokens by diffusing their concatenation <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib31" title="" class="ltx_ref">31</a>, <a href="#bib.bib1" title="" class="ltx_ref">1</a>]</cite>, where the noise level is identical across all tokens.
They offer <em id="S1.p3.1.2" class="ltx_emph ltx_font_italic">diffusion guidance</em> <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib30" title="" class="ltx_ref">30</a>, <a href="#bib.bib16" title="" class="ltx_ref">16</a>]</cite> to guide sampling to a desirable sequence, invaluable in decision-making (planning) applications <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib36" title="" class="ltx_ref">36</a>, <a href="#bib.bib34" title="" class="ltx_ref">34</a>]</cite>.
They further excel at generating continuous signals such as video <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib31" title="" class="ltx_ref">31</a>]</cite>.
However, full-sequence diffusion is universally parameterized via non-causal, unmasked architectures.
In addition to restricting sampling to full sequences, as opposed to variable length generation, we show that this limits the possibilities for both guidance and subsequence generation (<a href="#S1.F1" title="In 1 Introduction ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">1</span></a>).
Further, we demonstrate that a naive attempt at combining the best of both worlds by training a next-token prediction model for full-sequence diffusion leads to poor generations, intuitively because it does not model the fact that small uncertainty in an early token necessitates high uncertainty in a later one.</p>
</div>
<div id="S1.p4" class="ltx_para">
<p id="S1.p4.1" class="ltx_p">In this paper, we introduce <em id="S1.p4.1.1" class="ltx_emph ltx_font_italic">Diffusion Forcing</em> (DF), a training and sampling paradigm where each token is associated with a <em id="S1.p4.1.2" class="ltx_emph ltx_font_italic">random, independent</em> noise level, and where tokens can be denoised according to arbitrary, independent, per-token schedules through a shared next-or-next-few-token prediction model. Our approach is motivated by the observation that noising tokens is a form of <em id="S1.p4.1.3" class="ltx_emph ltx_font_italic">partial masking</em>—zero noise means a token is unmasked, and complete noise fully masks out a token. Thus, DF forces the model to learn to “unmask” any collection of variably noised tokens (<a href="#S2.F2" title="In Bayesian Filtering. ‣ 2 Related Work and Preliminaries ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">2</span></a>). Simultaneously, by parameterizing predictions as a composition of next-token prediction models, our system can flexibly generate varying length sequences as well as compositionally generalize to new trajectories (Figure <a href="#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>).</p>
</div>
<div id="S1.p5" class="ltx_para">
<p id="S1.p5.1" class="ltx_p">We implement DF for sequence generation as <em id="S1.p5.1.1" class="ltx_emph ltx_font_italic">Causal Diffusion Forcing</em> (CDF), in which future tokens depend on past ones via a causal architecture.
We train the model to denoise all tokens of a sequence at once, with an independent noise level per token.
During sampling, CDF gradually denoises a sequence of Gaussian noise frames into clean samples where different frames may have different noise levels at each denoising step.
Like next-token prediction models, CDF can generate variable-length sequences; unlike next-token prediction, it does so stabily from the immediate next token to thousands of tokens in the future – even for continuous tokens. Moreover, like full-sequence diffusion it accepts guidance towards high-reward generations. Synergistically leveraging causality, flexible horizon, and variable noise schedules, CDF enables a new capability,
Monte Carlo Tree Guidance (MCTG), that dramatically improves the sampling of high-reward generations compared to non-causal full-sequence diffusion models. Fig. <a href="#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> overviews these capabilities.</p>
</div>
<figure id="S1.F1" class="ltx_figure"><img src="" id="S1.F1.g1" class="ltx_graphics ltx_missing ltx_missing_image" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="S1.F1.5.1.1" class="ltx_text" style="font-size:90%;">Figure 1</span>: </span><span id="S1.F1.6.2" class="ltx_text ltx_font_bold" style="font-size:90%;">Diffusion Forcing capabilities.<span id="S1.F1.6.2.1" class="ltx_text ltx_font_medium"> Today, different applications such as language modeling <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib6" title="" class="ltx_ref">6</a>]</cite>, planning <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib36" title="" class="ltx_ref">36</a>]</cite>, or video generation <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib31" title="" class="ltx_ref">31</a>]</cite> rely on <em id="S1.F1.6.2.1.1" class="ltx_emph ltx_font_italic">either</em> auto-regressive next-token prediction <em id="S1.F1.6.2.1.2" class="ltx_emph ltx_font_italic">or</em> full-sequence diffusion, according to their respective unique capabilities. The proposed Diffusion Forcing is a novel sequence generative model that enjoys key strengths of both model types.</span></span></figcaption>
</figure>
<div id="S1.p6" class="ltx_para">
<p id="S1.p6.1" class="ltx_p">In summary, our contributions are:
(1) We propose Diffusion Forcing, a new probabilistic sequence model that has the flexibility of next-token prediction models while being able to perform long-horizon guidance like full-sequence diffusion models.
(2) Taking advantage of Diffusion Forcing’s unique capabilities, we introduce a novel decision-making framework that allows us to use Diffusion Forcing as simultaneously a <em id="S1.p6.1.1" class="ltx_emph ltx_font_italic">policy</em> (<cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib10" title="" class="ltx_ref">10</a>]</cite>) and as a <em id="S1.p6.1.2" class="ltx_emph ltx_font_italic">planner</em> (<cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib36" title="" class="ltx_ref">36</a>]</cite>).
(3) We formally prove that, under appropriate conditions, optimizing our proposed training objective maximizes a lower bound on the likelihood of the joint distribution of <em id="S1.p6.1.3" class="ltx_emph ltx_font_italic">all sub-sequences</em> observed at training time.
(4) We empirically evaluate CDF across diverse domains such as video generation, model-based planning, visual imitation learning, and time series prediction, and demonstrate CDF’s unique capabilities, such as stabilizing long-rollout autoregressive video generation, composing sub-sequences of those observed at training time with user-determined memory horizon, Monte Carlo Tree Guidance, and more.</p>
</div>
</section>
<section id="S2" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">2 </span>Related Work and Preliminaries</h2>

<div id="S2.p1" class="ltx_para">
<p id="S2.p1.1" class="ltx_p">We discuss related work and preliminaries for our core application, sequence generative modeling; see <a href="#A3" title="Appendix C Extended Related Work ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">C</span></a> for further literature review.</p>
</div>
<div id="S2.p2" class="ltx_para">
<p id="S2.p2.4" class="ltx_p">Our method unifies two perspectives on sequence modeling: Bayesian filtering along the time axis, denoted by subscript <math id="S2.p2.1.m1.1" class="ltx_Math" alttext="t" display="inline"><semantics id="S2.p2.1.m1.1a"><mi id="S2.p2.1.m1.1.1" xref="S2.p2.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.p2.1.m1.1b"><ci id="S2.p2.1.m1.1.1.cmml" xref="S2.p2.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.1.m1.1c">t</annotation></semantics></math>, and diffusion along an “uncertainty” (or noise level) axis denoted by superscript <math id="S2.p2.2.m2.1" class="ltx_Math" alttext="k" display="inline"><semantics id="S2.p2.2.m2.1a"><mi id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.p2.2.m2.1b"><ci id="S2.p2.2.m2.1.1.cmml" xref="S2.p2.2.m2.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m2.1c">k</annotation></semantics></math>. In the following, we denote observations as <math id="S2.p2.3.m3.1" class="ltx_Math" alttext="\mathbf{x}\in\mathcal{X}" display="inline"><semantics id="S2.p2.3.m3.1a"><mrow id="S2.p2.3.m3.1.1" xref="S2.p2.3.m3.1.1.cmml"><mi id="S2.p2.3.m3.1.1.2" xref="S2.p2.3.m3.1.1.2.cmml">𝐱</mi><mo id="S2.p2.3.m3.1.1.1" xref="S2.p2.3.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.p2.3.m3.1.1.3" xref="S2.p2.3.m3.1.1.3.cmml">𝒳</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.3.m3.1b"><apply id="S2.p2.3.m3.1.1.cmml" xref="S2.p2.3.m3.1.1"><in id="S2.p2.3.m3.1.1.1.cmml" xref="S2.p2.3.m3.1.1.1"></in><ci id="S2.p2.3.m3.1.1.2.cmml" xref="S2.p2.3.m3.1.1.2">𝐱</ci><ci id="S2.p2.3.m3.1.1.3.cmml" xref="S2.p2.3.m3.1.1.3">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m3.1c">\mathbf{x}\in\mathcal{X}</annotation></semantics></math> and latent states as <math id="S2.p2.4.m4.1" class="ltx_Math" alttext="\mathbf{z}\in\mathcal{Z}" display="inline"><semantics id="S2.p2.4.m4.1a"><mrow id="S2.p2.4.m4.1.1" xref="S2.p2.4.m4.1.1.cmml"><mi id="S2.p2.4.m4.1.1.2" xref="S2.p2.4.m4.1.1.2.cmml">𝐳</mi><mo id="S2.p2.4.m4.1.1.1" xref="S2.p2.4.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.p2.4.m4.1.1.3" xref="S2.p2.4.m4.1.1.3.cmml">𝒵</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.4.m4.1b"><apply id="S2.p2.4.m4.1.1.cmml" xref="S2.p2.4.m4.1.1"><in id="S2.p2.4.m4.1.1.1.cmml" xref="S2.p2.4.m4.1.1.1"></in><ci id="S2.p2.4.m4.1.1.2.cmml" xref="S2.p2.4.m4.1.1.2">𝐳</ci><ci id="S2.p2.4.m4.1.1.3.cmml" xref="S2.p2.4.m4.1.1.3">𝒵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.m4.1c">\mathbf{z}\in\mathcal{Z}</annotation></semantics></math>.</p>
</div>
<section id="S2.SS0.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Bayesian Filtering.</h5>

<div id="S2.SS0.SSS0.Px1.p1" class="ltx_para">
<p id="S2.SS0.SSS0.Px1.p1.7" class="ltx_p">Given a Hidden Markov Model (HMM) defined by latent states <math id="S2.SS0.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{z}_{t}" display="inline"><semantics id="S2.SS0.SSS0.Px1.p1.1.m1.1a"><msub id="S2.SS0.SSS0.Px1.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.1.m1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.1.m1.1.1.2.cmml">𝐳</mi><mi id="S2.SS0.SSS0.Px1.p1.1.m1.1.1.3" xref="S2.SS0.SSS0.Px1.p1.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.1.m1.1b"><apply id="S2.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.1.m1.1.1.2">𝐳</ci><ci id="S2.SS0.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.1.m1.1c">\mathbf{z}_{t}</annotation></semantics></math> and observations <math id="S2.SS0.SSS0.Px1.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{t}" display="inline"><semantics id="S2.SS0.SSS0.Px1.p1.2.m2.1a"><msub id="S2.SS0.SSS0.Px1.p1.2.m2.1.1" xref="S2.SS0.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.2.m2.1.1.2" xref="S2.SS0.SSS0.Px1.p1.2.m2.1.1.2.cmml">𝐱</mi><mi id="S2.SS0.SSS0.Px1.p1.2.m2.1.1.3" xref="S2.SS0.SSS0.Px1.p1.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.2.m2.1b"><apply id="S2.SS0.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.1.1.2">𝐱</ci><ci id="S2.SS0.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.2.m2.1c">\mathbf{x}_{t}</annotation></semantics></math>, a Bayes filter is a probabilistic method for estimating latent states recursively over time from incoming observations. A prior model <math id="S2.SS0.SSS0.Px1.p1.3.m3.1" class="ltx_Math" alttext="p(\mathbf{z}_{t+1}|\mathbf{z}_{t})" display="inline"><semantics id="S2.SS0.SSS0.Px1.p1.3.m3.1a"><mrow id="S2.SS0.SSS0.Px1.p1.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.3" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.2" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.2.cmml">​</mo><mrow id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2.2.cmml">𝐳</mi><mrow id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2.3" 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id="S2.SS0.SSS0.Px1.p1.3.m3.1b"><apply id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1"><times id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.2"></times><ci id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.3">𝑝</ci><apply id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.1">conditional</csymbol><apply id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2.2">𝐳</ci><apply id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.1.1.1.2.3"><plus 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state given only the current state, and an observation model infers a belief over the next observation given the current latent state <math id="S2.SS0.SSS0.Px1.p1.4.m4.1" class="ltx_Math" alttext="p(\mathbf{x}_{t}|\mathbf{z}_{t})" display="inline"><semantics id="S2.SS0.SSS0.Px1.p1.4.m4.1a"><mrow id="S2.SS0.SSS0.Px1.p1.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.3" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.2" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.2.cmml">​</mo><mrow id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.cmml"><mi 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xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.2"></times><ci id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.3">𝑝</ci><apply id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.1">conditional</csymbol><apply id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.2">𝐱</ci><ci id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.1.1.1.3.1.cmml" 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When a new observation is made, a posterior model <math id="S2.SS0.SSS0.Px1.p1.5.m5.1" class="ltx_Math" alttext="p(\mathbf{z}_{t+1}|\mathbf{z}_{t},\mathbf{x}_{t+1})" display="inline"><semantics id="S2.SS0.SSS0.Px1.p1.5.m5.1a"><mrow id="S2.SS0.SSS0.Px1.p1.5.m5.1.1" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.3" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.2" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.2.cmml">​</mo><mrow id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.cmml"><mi id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.2" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.2.cmml">𝐳</mi><mrow id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.cmml"><mi id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.2" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.2.cmml">t</mi><mo id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.1" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.1.cmml">+</mo><mn id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.3" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.3.cmml">1</mn></mrow></msub><mo fence="false" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.3.cmml">|</mo><mrow id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.3.cmml"><msub id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.2.cmml">𝐳</mi><mi id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.3" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.2" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.2.cmml">𝐱</mi><mrow id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.2" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.2.cmml">t</mi><mo id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.1" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.1.cmml">+</mo><mn id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.3" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mo stretchy="false" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.5.m5.1b"><apply id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1"><times id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.2"></times><ci id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.3">𝑝</ci><apply id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.3">conditional</csymbol><apply id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.1.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.2.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.2">𝐳</ci><apply id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3"><plus id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.1.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.1"></plus><ci id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.2.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.2">𝑡</ci><cn type="integer" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.4.3.3">1</cn></apply></apply><list id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2"><apply id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.2">𝐳</ci><ci id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.2">𝐱</ci><apply id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3"><plus id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.1"></plus><ci id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.2">𝑡</ci><cn type="integer" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.3">1</cn></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.5.m5.1c">p(\mathbf{z}_{t+1}|\mathbf{z}_{t},\mathbf{x}_{t+1})</annotation></semantics></math> provides an updated estimation of the next latent state <math id="S2.SS0.SSS0.Px1.p1.6.m6.1" class="ltx_Math" alttext="\mathbf{z}_{t+1}" display="inline"><semantics id="S2.SS0.SSS0.Px1.p1.6.m6.1a"><msub id="S2.SS0.SSS0.Px1.p1.6.m6.1.1" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.2" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.2.cmml">𝐳</mi><mrow id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.2" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.2.cmml">t</mi><mo id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.1" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.1.cmml">+</mo><mn id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.3" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.6.m6.1b"><apply id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.2">𝐳</ci><apply id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3"><plus id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.1"></plus><ci id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.2">𝑡</ci><cn type="integer" id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.6.m6.1c">\mathbf{z}_{t+1}</annotation></semantics></math>. When trained end-to-end with neural networks <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib22" title="" class="ltx_ref">22</a>, <a href="#bib.bib23" title="" class="ltx_ref">23</a>]</cite>, latent states are not an estimate of any physical quantity, but a sufficiently expressive latent that summarizes past observations for predicting future observations <math id="S2.SS0.SSS0.Px1.p1.7.m7.1" class="ltx_Math" alttext="(\mathbf{x}_{t^{\prime}})_{t^{\prime}&gt;t}" display="inline"><semantics id="S2.SS0.SSS0.Px1.p1.7.m7.1a"><msub id="S2.SS0.SSS0.Px1.p1.7.m7.1.1" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.cmml"><mrow id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.2.cmml">𝐱</mi><msup id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.2" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.2.cmml">t</mi><mo id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.3" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.3.cmml">′</mo></msup></msub><mo stretchy="false" id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.cmml"><msup id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.cmml"><mi id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.2" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.2.cmml">t</mi><mo id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.3" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.3.cmml">′</mo></msup><mo id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.1" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.1.cmml">&gt;</mo><mi id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.3" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.7.m7.1b"><apply id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1">subscript</csymbol><apply id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.2">𝐱</ci><apply id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.2">𝑡</ci><ci id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.1.1.1.3.3">′</ci></apply></apply><apply id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3"><gt id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.1"></gt><apply id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.2">𝑡</ci><ci id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.3">′</ci></apply><ci id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.7.m7.1c">(\mathbf{x}_{t^{\prime}})_{t^{\prime}&gt;t}</annotation></semantics></math> in the sequence.</p>
</div>
<figure id="S2.F2" class="ltx_figure"><img src="/html/2407.01392/assets/x2.png" id="S2.F2.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="461" height="162" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="S2.F2.7.1.1" class="ltx_text" style="font-size:90%;">Figure 2</span>: </span><span id="S2.F2.8.2" class="ltx_text ltx_font_bold" style="font-size:90%;">Method Overview.<span id="S2.F2.8.2.1" class="ltx_text ltx_font_medium">
Diffusion Forcing trains causal sequence neural networks (such as an RNN or a masked transformer) to denoise flexible-length sequences where each frame of the sequence can have a <em id="S2.F2.8.2.1.1" class="ltx_emph ltx_font_italic">different</em> noise level.
In contrast, next-token prediction models, common in language modeling, are trained to predict a single next token from a <em id="S2.F2.8.2.1.2" class="ltx_emph ltx_font_italic">ground-truth</em> sequence (teacher forcing <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib62" title="" class="ltx_ref">62</a>]</cite>), and full-sequence diffusion, common in video generation, train non-causal architectures to denoise all frames in a sequence at once with the <em id="S2.F2.8.2.1.3" class="ltx_emph ltx_font_italic">same</em> noise level.
Diffusion Forcing thus <em id="S2.F2.8.2.1.4" class="ltx_emph ltx_font_italic">interleaves</em> the time axis of the sequence and the noise axis of diffusion, unifying strengths of both alternatives and enabling completely new capabilities (see Secs. <a href="#S3.SS1" title="3.1 Diffusion Forcing Sampling and Resulting Capabilities ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3.1</span></a>,<a href="#S3.SS2" title="3.2 Diffusion Forcing for Flexible Sequential Decision Making ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3.2</span></a>).
</span></span></figcaption>
</figure>
</section>
<section id="S2.SS0.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Diffusion Models.</h5>

<div id="S2.SS0.SSS0.Px2.p1" class="ltx_para">
<p id="S2.SS0.SSS0.Px2.p1.3" class="ltx_p">Diffusion models <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib54" title="" class="ltx_ref">54</a>, <a href="#bib.bib28" title="" class="ltx_ref">28</a>]</cite> have proven to be highly expressive and reliable generative models.
We review their essentials here. Let <math id="S2.SS0.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="q(\mathbf{x})" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.1.m1.1a"><mrow id="S2.SS0.SSS0.Px2.p1.1.m1.1.2" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.1.m1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.2.2.cmml">q</mi><mo lspace="0em" rspace="0em" id="S2.SS0.SSS0.Px2.p1.1.m1.1.2.1" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.2.1.cmml">​</mo><mrow id="S2.SS0.SSS0.Px2.p1.1.m1.1.2.3.2" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.2.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.1.m1.1.2.3.2.1" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px2.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.1.cmml">𝐱</mi><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.1.m1.1.2.3.2.2" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.1.m1.1b"><apply id="S2.SS0.SSS0.Px2.p1.1.m1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.2"><times id="S2.SS0.SSS0.Px2.p1.1.m1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.2.1"></times><ci id="S2.SS0.SSS0.Px2.p1.1.m1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.2.2">𝑞</ci><ci id="S2.SS0.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.1.m1.1c">q(\mathbf{x})</annotation></semantics></math> denote a data distribution of interest, and let <math id="S2.SS0.SSS0.Px2.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{x}^{0}\equiv\mathbf{x}\sim q" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.2.m2.1a"><mrow id="S2.SS0.SSS0.Px2.p1.2.m2.1.1" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.cmml"><msup id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.2.cmml">𝐱</mi><mn id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.3.cmml">0</mn></msup><mo id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.3" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.3.cmml">≡</mo><mi id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.4" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.4.cmml">𝐱</mi><mo id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.5" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.5.cmml">∼</mo><mi id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.6" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.6.cmml">q</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.2.m2.1b"><apply id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1"><and id="S2.SS0.SSS0.Px2.p1.2.m2.1.1a.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1"></and><apply id="S2.SS0.SSS0.Px2.p1.2.m2.1.1b.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1"><equivalent id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.3"></equivalent><apply id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2">superscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.2">𝐱</ci><cn type="integer" id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.2.3">0</cn></apply><ci id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.4.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.4">𝐱</ci></apply><apply id="S2.SS0.SSS0.Px2.p1.2.m2.1.1c.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.5.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.5">similar-to</csymbol><share href="#S2.SS0.SSS0.Px2.p1.2.m2.1.1.4.cmml" id="S2.SS0.SSS0.Px2.p1.2.m2.1.1d.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1"></share><ci id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.6.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.6">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.2.m2.1c">\mathbf{x}^{0}\equiv\mathbf{x}\sim q</annotation></semantics></math>. We consider a forward diffusion process that gradually adds Gaussian noise to a data point over a series of time steps. This process is modeled as a Markov chain, where the data at each step <math id="S2.SS0.SSS0.Px2.p1.3.m3.1" class="ltx_Math" alttext="k" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.3.m3.1a"><mi id="S2.SS0.SSS0.Px2.p1.3.m3.1.1" xref="S2.SS0.SSS0.Px2.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.3.m3.1b"><ci id="S2.SS0.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.3.m3.1c">k</annotation></semantics></math> is noised incrementally:</p>
<table id="S2.E1" class="ltx_equation ltx_eqn_table">

<tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math id="S2.E1.m1.4" class="ltx_Math" alttext="q(\mathbf{x}^{k}|\mathbf{x}^{k-1})=\mathcal{N}(\mathbf{x}^{k};\sqrt{1-\beta_{k}}\mathbf{x}^{k-1},\beta_{k}\mathbf{I})" display="block"><semantics id="S2.E1.m1.4a"><mrow id="S2.E1.m1.4.4" xref="S2.E1.m1.4.4.cmml"><mrow id="S2.E1.m1.1.1.1" xref="S2.E1.m1.1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.3" xref="S2.E1.m1.1.1.1.3.cmml">q</mi><mo lspace="0em" rspace="0em" id="S2.E1.m1.1.1.1.2" xref="S2.E1.m1.1.1.1.2.cmml">​</mo><mrow id="S2.E1.m1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E1.m1.1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.cmml"><msup id="S2.E1.m1.1.1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.1.1.2.cmml"><mi id="S2.E1.m1.1.1.1.1.1.1.2.2" xref="S2.E1.m1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi id="S2.E1.m1.1.1.1.1.1.1.2.3" xref="S2.E1.m1.1.1.1.1.1.1.2.3.cmml">k</mi></msup><mo fence="false" id="S2.E1.m1.1.1.1.1.1.1.1" 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lspace="0em" rspace="0em" id="S2.E1.m1.3.3.3.2.2.2.1" xref="S2.E1.m1.3.3.3.2.2.2.1.cmml">​</mo><msup id="S2.E1.m1.3.3.3.2.2.2.3" xref="S2.E1.m1.3.3.3.2.2.2.3.cmml"><mi id="S2.E1.m1.3.3.3.2.2.2.3.2" xref="S2.E1.m1.3.3.3.2.2.2.3.2.cmml">𝐱</mi><mrow id="S2.E1.m1.3.3.3.2.2.2.3.3" xref="S2.E1.m1.3.3.3.2.2.2.3.3.cmml"><mi id="S2.E1.m1.3.3.3.2.2.2.3.3.2" xref="S2.E1.m1.3.3.3.2.2.2.3.3.2.cmml">k</mi><mo id="S2.E1.m1.3.3.3.2.2.2.3.3.1" xref="S2.E1.m1.3.3.3.2.2.2.3.3.1.cmml">−</mo><mn id="S2.E1.m1.3.3.3.2.2.2.3.3.3" xref="S2.E1.m1.3.3.3.2.2.2.3.3.3.cmml">1</mn></mrow></msup></mrow><mo id="S2.E1.m1.4.4.4.3.3.6" xref="S2.E1.m1.4.4.4.3.4.cmml">,</mo><mrow id="S2.E1.m1.4.4.4.3.3.3" xref="S2.E1.m1.4.4.4.3.3.3.cmml"><msub id="S2.E1.m1.4.4.4.3.3.3.2" xref="S2.E1.m1.4.4.4.3.3.3.2.cmml"><mi id="S2.E1.m1.4.4.4.3.3.3.2.2" xref="S2.E1.m1.4.4.4.3.3.3.2.2.cmml">β</mi><mi id="S2.E1.m1.4.4.4.3.3.3.2.3" xref="S2.E1.m1.4.4.4.3.3.3.2.3.cmml">k</mi></msub><mo lspace="0em" rspace="0em" id="S2.E1.m1.4.4.4.3.3.3.1" xref="S2.E1.m1.4.4.4.3.3.3.1.cmml">​</mo><mi id="S2.E1.m1.4.4.4.3.3.3.3" xref="S2.E1.m1.4.4.4.3.3.3.3.cmml">𝐈</mi></mrow><mo stretchy="false" id="S2.E1.m1.4.4.4.3.3.7" xref="S2.E1.m1.4.4.4.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.m1.4b"><apply id="S2.E1.m1.4.4.cmml" xref="S2.E1.m1.4.4"><eq id="S2.E1.m1.4.4.5.cmml" xref="S2.E1.m1.4.4.5"></eq><apply id="S2.E1.m1.1.1.1.cmml" xref="S2.E1.m1.1.1.1"><times id="S2.E1.m1.1.1.1.2.cmml" xref="S2.E1.m1.1.1.1.2"></times><ci id="S2.E1.m1.1.1.1.3.cmml" xref="S2.E1.m1.1.1.1.3">𝑞</ci><apply id="S2.E1.m1.1.1.1.1.1.1.cmml" xref="S2.E1.m1.1.1.1.1.1"><csymbol cd="latexml" id="S2.E1.m1.1.1.1.1.1.1.1.cmml" xref="S2.E1.m1.1.1.1.1.1.1.1">conditional</csymbol><apply id="S2.E1.m1.1.1.1.1.1.1.2.cmml" xref="S2.E1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E1.m1.1.1.1.1.1.1.2.1.cmml" xref="S2.E1.m1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.E1.m1.1.1.1.1.1.1.2.2.cmml" 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id="S2.E1.m1.4.4.4.3.3.3.1.cmml" xref="S2.E1.m1.4.4.4.3.3.3.1"></times><apply id="S2.E1.m1.4.4.4.3.3.3.2.cmml" xref="S2.E1.m1.4.4.4.3.3.3.2"><csymbol cd="ambiguous" id="S2.E1.m1.4.4.4.3.3.3.2.1.cmml" xref="S2.E1.m1.4.4.4.3.3.3.2">subscript</csymbol><ci id="S2.E1.m1.4.4.4.3.3.3.2.2.cmml" xref="S2.E1.m1.4.4.4.3.3.3.2.2">𝛽</ci><ci id="S2.E1.m1.4.4.4.3.3.3.2.3.cmml" xref="S2.E1.m1.4.4.4.3.3.3.2.3">𝑘</ci></apply><ci id="S2.E1.m1.4.4.4.3.3.3.3.cmml" xref="S2.E1.m1.4.4.4.3.3.3.3">𝐈</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.4c">q(\mathbf{x}^{k}|\mathbf{x}^{k-1})=\mathcal{N}(\mathbf{x}^{k};\sqrt{1-\beta_{k}}\mathbf{x}^{k-1},\beta_{k}\mathbf{I})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.1)</span></td>
</tr></tbody>
</table>
<p id="S2.SS0.SSS0.Px2.p1.8" class="ltx_p">where <math id="S2.SS0.SSS0.Px2.p1.4.m1.1" class="ltx_Math" alttext="\mathcal{N}" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.4.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS0.SSS0.Px2.p1.4.m1.1.1" xref="S2.SS0.SSS0.Px2.p1.4.m1.1.1.cmml">𝒩</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.4.m1.1b"><ci id="S2.SS0.SSS0.Px2.p1.4.m1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.4.m1.1.1">𝒩</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.4.m1.1c">\mathcal{N}</annotation></semantics></math> is the normal distribution and <math id="S2.SS0.SSS0.Px2.p1.5.m2.1" class="ltx_Math" alttext="\beta_{k}" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.5.m2.1a"><msub id="S2.SS0.SSS0.Px2.p1.5.m2.1.1" xref="S2.SS0.SSS0.Px2.p1.5.m2.1.1.cmml"><mi id="S2.SS0.SSS0.Px2.p1.5.m2.1.1.2" xref="S2.SS0.SSS0.Px2.p1.5.m2.1.1.2.cmml">β</mi><mi id="S2.SS0.SSS0.Px2.p1.5.m2.1.1.3" xref="S2.SS0.SSS0.Px2.p1.5.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.5.m2.1b"><apply id="S2.SS0.SSS0.Px2.p1.5.m2.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.5.m2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.5.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.5.m2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.5.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.5.m2.1.1.2">𝛽</ci><ci id="S2.SS0.SSS0.Px2.p1.5.m2.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.5.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.5.m2.1c">\beta_{k}</annotation></semantics></math> is the variance of the noise added at each step controlled by a schedule <math id="S2.SS0.SSS0.Px2.p1.6.m3.3" class="ltx_Math" alttext="\{\beta_{k}\in(0,1)\}_{k=1}^{K}" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.6.m3.3a"><msubsup id="S2.SS0.SSS0.Px2.p1.6.m3.3.3" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.cmml"><mrow id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.2.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.2" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.2.cmml">{</mo><mrow id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.2.cmml">β</mi><mi id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.1.cmml">∈</mo><mrow id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.3.2" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.3.1.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.3.2.1" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.3.1.cmml">(</mo><mn id="S2.SS0.SSS0.Px2.p1.6.m3.1.1" xref="S2.SS0.SSS0.Px2.p1.6.m3.1.1.cmml">0</mn><mo id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.3.2.2" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.3.1.cmml">,</mo><mn id="S2.SS0.SSS0.Px2.p1.6.m3.2.2" xref="S2.SS0.SSS0.Px2.p1.6.m3.2.2.cmml">1</mn><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.3.2.3" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.3" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.2.cmml">}</mo></mrow><mrow id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.2" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.2.cmml">k</mi><mo id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.1" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.1.cmml">=</mo><mn id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.3" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.3.cmml">1</mn></mrow><mi id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.3" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.3.cmml">K</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.6.m3.3b"><apply id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3">superscript</csymbol><apply id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3">subscript</csymbol><set id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1"><apply id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1"><in id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.1"></in><apply id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.2">𝛽</ci><ci id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.2.3">𝑘</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.1.1.1.3.2"><cn type="integer" id="S2.SS0.SSS0.Px2.p1.6.m3.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.1.1">0</cn><cn type="integer" id="S2.SS0.SSS0.Px2.p1.6.m3.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.2.2">1</cn></interval></apply></set><apply id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3"><eq id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.1"></eq><ci id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.2">𝑘</ci><cn type="integer" id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.3.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.1.3.3">1</cn></apply></apply><ci id="S2.SS0.SSS0.Px2.p1.6.m3.3.3.3.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m3.3.3.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.6.m3.3c">\{\beta_{k}\in(0,1)\}_{k=1}^{K}</annotation></semantics></math>. The process continues until the data is converted into pure noise at <math id="S2.SS0.SSS0.Px2.p1.7.m4.1" class="ltx_Math" alttext="\mathbf{x}^{K}" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.7.m4.1a"><msup id="S2.SS0.SSS0.Px2.p1.7.m4.1.1" xref="S2.SS0.SSS0.Px2.p1.7.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px2.p1.7.m4.1.1.2" xref="S2.SS0.SSS0.Px2.p1.7.m4.1.1.2.cmml">𝐱</mi><mi id="S2.SS0.SSS0.Px2.p1.7.m4.1.1.3" xref="S2.SS0.SSS0.Px2.p1.7.m4.1.1.3.cmml">K</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.7.m4.1b"><apply id="S2.SS0.SSS0.Px2.p1.7.m4.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m4.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.7.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m4.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.7.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m4.1.1.2">𝐱</ci><ci id="S2.SS0.SSS0.Px2.p1.7.m4.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m4.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.7.m4.1c">\mathbf{x}^{K}</annotation></semantics></math>.
The reverse process is also a Markov chain and attempts to recreate the original data from the noise with a parameterized model <math id="S2.SS0.SSS0.Px2.p1.8.m5.1" class="ltx_Math" alttext="p_{\theta}" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.8.m5.1a"><msub id="S2.SS0.SSS0.Px2.p1.8.m5.1.1" xref="S2.SS0.SSS0.Px2.p1.8.m5.1.1.cmml"><mi id="S2.SS0.SSS0.Px2.p1.8.m5.1.1.2" xref="S2.SS0.SSS0.Px2.p1.8.m5.1.1.2.cmml">p</mi><mi id="S2.SS0.SSS0.Px2.p1.8.m5.1.1.3" xref="S2.SS0.SSS0.Px2.p1.8.m5.1.1.3.cmml">θ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.8.m5.1b"><apply id="S2.SS0.SSS0.Px2.p1.8.m5.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m5.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.8.m5.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m5.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.8.m5.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m5.1.1.2">𝑝</ci><ci id="S2.SS0.SSS0.Px2.p1.8.m5.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m5.1.1.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.8.m5.1c">p_{\theta}</annotation></semantics></math>:</p>
<table id="S2.E2" class="ltx_equation ltx_eqn_table">

<tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math id="S2.E2.m1.2" class="ltx_Math" alttext="p_{\theta}(\mathbf{x}^{k-1}|\mathbf{x}^{k})=\mathcal{N}(\mathbf{x}^{k-1};\bm{\mu}(\mathbf{x}^{k},k),\gamma_{k}\mathbf{I})," display="block"><semantics id="S2.E2.m1.2a"><mrow id="S2.E2.m1.2.2.1" xref="S2.E2.m1.2.2.1.1.cmml"><mrow id="S2.E2.m1.2.2.1.1" xref="S2.E2.m1.2.2.1.1.cmml"><mrow id="S2.E2.m1.2.2.1.1.1" xref="S2.E2.m1.2.2.1.1.1.cmml"><msub id="S2.E2.m1.2.2.1.1.1.3" xref="S2.E2.m1.2.2.1.1.1.3.cmml"><mi id="S2.E2.m1.2.2.1.1.1.3.2" xref="S2.E2.m1.2.2.1.1.1.3.2.cmml">p</mi><mi id="S2.E2.m1.2.2.1.1.1.3.3" xref="S2.E2.m1.2.2.1.1.1.3.3.cmml">θ</mi></msub><mo lspace="0em" rspace="0em" id="S2.E2.m1.2.2.1.1.1.2" xref="S2.E2.m1.2.2.1.1.1.2.cmml">​</mo><mrow id="S2.E2.m1.2.2.1.1.1.1.1" xref="S2.E2.m1.2.2.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E2.m1.2.2.1.1.1.1.1.2" xref="S2.E2.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E2.m1.2.2.1.1.1.1.1.1" xref="S2.E2.m1.2.2.1.1.1.1.1.1.cmml"><msup 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id="S2.E2.m1.2.2.1.1.5" xref="S2.E2.m1.2.2.1.1.5.cmml">=</mo><mrow id="S2.E2.m1.2.2.1.1.4" xref="S2.E2.m1.2.2.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E2.m1.2.2.1.1.4.5" xref="S2.E2.m1.2.2.1.1.4.5.cmml">𝒩</mi><mo lspace="0em" rspace="0em" id="S2.E2.m1.2.2.1.1.4.4" xref="S2.E2.m1.2.2.1.1.4.4.cmml">​</mo><mrow id="S2.E2.m1.2.2.1.1.4.3.3" xref="S2.E2.m1.2.2.1.1.4.3.4.cmml"><mo stretchy="false" id="S2.E2.m1.2.2.1.1.4.3.3.4" xref="S2.E2.m1.2.2.1.1.4.3.4.cmml">(</mo><msup id="S2.E2.m1.2.2.1.1.2.1.1.1" xref="S2.E2.m1.2.2.1.1.2.1.1.1.cmml"><mi id="S2.E2.m1.2.2.1.1.2.1.1.1.2" xref="S2.E2.m1.2.2.1.1.2.1.1.1.2.cmml">𝐱</mi><mrow id="S2.E2.m1.2.2.1.1.2.1.1.1.3" xref="S2.E2.m1.2.2.1.1.2.1.1.1.3.cmml"><mi id="S2.E2.m1.2.2.1.1.2.1.1.1.3.2" xref="S2.E2.m1.2.2.1.1.2.1.1.1.3.2.cmml">k</mi><mo id="S2.E2.m1.2.2.1.1.2.1.1.1.3.1" xref="S2.E2.m1.2.2.1.1.2.1.1.1.3.1.cmml">−</mo><mn id="S2.E2.m1.2.2.1.1.2.1.1.1.3.3" xref="S2.E2.m1.2.2.1.1.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo 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xref="S2.E2.m1.2.2.1.1.3.2.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.2.2.1.1.4.3.3.6" xref="S2.E2.m1.2.2.1.1.4.3.4.cmml">,</mo><mrow id="S2.E2.m1.2.2.1.1.4.3.3.3" xref="S2.E2.m1.2.2.1.1.4.3.3.3.cmml"><msub id="S2.E2.m1.2.2.1.1.4.3.3.3.2" xref="S2.E2.m1.2.2.1.1.4.3.3.3.2.cmml"><mi id="S2.E2.m1.2.2.1.1.4.3.3.3.2.2" xref="S2.E2.m1.2.2.1.1.4.3.3.3.2.2.cmml">γ</mi><mi id="S2.E2.m1.2.2.1.1.4.3.3.3.2.3" xref="S2.E2.m1.2.2.1.1.4.3.3.3.2.3.cmml">k</mi></msub><mo lspace="0em" rspace="0em" id="S2.E2.m1.2.2.1.1.4.3.3.3.1" xref="S2.E2.m1.2.2.1.1.4.3.3.3.1.cmml">​</mo><mi id="S2.E2.m1.2.2.1.1.4.3.3.3.3" xref="S2.E2.m1.2.2.1.1.4.3.3.3.3.cmml">𝐈</mi></mrow><mo stretchy="false" id="S2.E2.m1.2.2.1.1.4.3.3.7" xref="S2.E2.m1.2.2.1.1.4.3.4.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E2.m1.2.2.1.2" xref="S2.E2.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.m1.2b"><apply id="S2.E2.m1.2.2.1.1.cmml" xref="S2.E2.m1.2.2.1"><eq id="S2.E2.m1.2.2.1.1.5.cmml" xref="S2.E2.m1.2.2.1.1.5"></eq><apply id="S2.E2.m1.2.2.1.1.1.cmml" xref="S2.E2.m1.2.2.1.1.1"><times id="S2.E2.m1.2.2.1.1.1.2.cmml" xref="S2.E2.m1.2.2.1.1.1.2"></times><apply id="S2.E2.m1.2.2.1.1.1.3.cmml" xref="S2.E2.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.E2.m1.2.2.1.1.1.3.1.cmml" xref="S2.E2.m1.2.2.1.1.1.3">subscript</csymbol><ci id="S2.E2.m1.2.2.1.1.1.3.2.cmml" xref="S2.E2.m1.2.2.1.1.1.3.2">𝑝</ci><ci id="S2.E2.m1.2.2.1.1.1.3.3.cmml" xref="S2.E2.m1.2.2.1.1.1.3.3">𝜃</ci></apply><apply id="S2.E2.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.E2.m1.2.2.1.1.1.1.1"><csymbol cd="latexml" id="S2.E2.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S2.E2.m1.2.2.1.1.1.1.1.1.1">conditional</csymbol><apply id="S2.E2.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S2.E2.m1.2.2.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E2.m1.2.2.1.1.1.1.1.1.2.1.cmml" xref="S2.E2.m1.2.2.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.E2.m1.2.2.1.1.1.1.1.1.2.2.cmml" xref="S2.E2.m1.2.2.1.1.1.1.1.1.2.2">𝐱</ci><apply 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id="S2.E2.m1.2.2.1.1.3.2.2.2.1.2.cmml" xref="S2.E2.m1.2.2.1.1.3.2.2.2.1.1"><apply id="S2.E2.m1.2.2.1.1.3.2.2.2.1.1.1.cmml" xref="S2.E2.m1.2.2.1.1.3.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.E2.m1.2.2.1.1.3.2.2.2.1.1.1.1.cmml" xref="S2.E2.m1.2.2.1.1.3.2.2.2.1.1.1">superscript</csymbol><ci id="S2.E2.m1.2.2.1.1.3.2.2.2.1.1.1.2.cmml" xref="S2.E2.m1.2.2.1.1.3.2.2.2.1.1.1.2">𝐱</ci><ci id="S2.E2.m1.2.2.1.1.3.2.2.2.1.1.1.3.cmml" xref="S2.E2.m1.2.2.1.1.3.2.2.2.1.1.1.3">𝑘</ci></apply><ci id="S2.E2.m1.1.1.cmml" xref="S2.E2.m1.1.1">𝑘</ci></interval></apply><apply id="S2.E2.m1.2.2.1.1.4.3.3.3.cmml" xref="S2.E2.m1.2.2.1.1.4.3.3.3"><times id="S2.E2.m1.2.2.1.1.4.3.3.3.1.cmml" xref="S2.E2.m1.2.2.1.1.4.3.3.3.1"></times><apply id="S2.E2.m1.2.2.1.1.4.3.3.3.2.cmml" xref="S2.E2.m1.2.2.1.1.4.3.3.3.2"><csymbol cd="ambiguous" id="S2.E2.m1.2.2.1.1.4.3.3.3.2.1.cmml" xref="S2.E2.m1.2.2.1.1.4.3.3.3.2">subscript</csymbol><ci id="S2.E2.m1.2.2.1.1.4.3.3.3.2.2.cmml" xref="S2.E2.m1.2.2.1.1.4.3.3.3.2.2">𝛾</ci><ci id="S2.E2.m1.2.2.1.1.4.3.3.3.2.3.cmml" xref="S2.E2.m1.2.2.1.1.4.3.3.3.2.3">𝑘</ci></apply><ci id="S2.E2.m1.2.2.1.1.4.3.3.3.3.cmml" xref="S2.E2.m1.2.2.1.1.4.3.3.3.3">𝐈</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.2c">p_{\theta}(\mathbf{x}^{k-1}|\mathbf{x}^{k})=\mathcal{N}(\mathbf{x}^{k-1};\bm{\mu}(\mathbf{x}^{k},k),\gamma_{k}\mathbf{I}),</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.2)</span></td>
</tr></tbody>
</table>
<p id="S2.SS0.SSS0.Px2.p1.13" class="ltx_p">where the mean <math id="S2.SS0.SSS0.Px2.p1.9.m1.1" class="ltx_Math" alttext="\bm{\mu}" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.9.m1.1a"><mi id="S2.SS0.SSS0.Px2.p1.9.m1.1.1" xref="S2.SS0.SSS0.Px2.p1.9.m1.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.9.m1.1b"><ci id="S2.SS0.SSS0.Px2.p1.9.m1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m1.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.9.m1.1c">\bm{\mu}</annotation></semantics></math> is model with a neural network, and where it is shown <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib29" title="" class="ltx_ref">29</a>]</cite> that one can set the covariance to the identity scaled by a fixed constant <math id="S2.SS0.SSS0.Px2.p1.10.m2.1" class="ltx_Math" alttext="\gamma_{k}" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.10.m2.1a"><msub id="S2.SS0.SSS0.Px2.p1.10.m2.1.1" xref="S2.SS0.SSS0.Px2.p1.10.m2.1.1.cmml"><mi id="S2.SS0.SSS0.Px2.p1.10.m2.1.1.2" xref="S2.SS0.SSS0.Px2.p1.10.m2.1.1.2.cmml">γ</mi><mi id="S2.SS0.SSS0.Px2.p1.10.m2.1.1.3" xref="S2.SS0.SSS0.Px2.p1.10.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.10.m2.1b"><apply id="S2.SS0.SSS0.Px2.p1.10.m2.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.10.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.10.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m2.1.1.2">𝛾</ci><ci id="S2.SS0.SSS0.Px2.p1.10.m2.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.10.m2.1c">\gamma_{k}</annotation></semantics></math> depending on <math id="S2.SS0.SSS0.Px2.p1.11.m3.1" class="ltx_Math" alttext="k" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.11.m3.1a"><mi id="S2.SS0.SSS0.Px2.p1.11.m3.1.1" xref="S2.SS0.SSS0.Px2.p1.11.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.11.m3.1b"><ci id="S2.SS0.SSS0.Px2.p1.11.m3.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.11.m3.1c">k</annotation></semantics></math>. Adopting the standard exposition, we reparametrize the mean <math id="S2.SS0.SSS0.Px2.p1.12.m4.1" class="ltx_Math" alttext="\bm{\mu}" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.12.m4.1a"><mi id="S2.SS0.SSS0.Px2.p1.12.m4.1.1" xref="S2.SS0.SSS0.Px2.p1.12.m4.1.1.cmml">𝝁</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.12.m4.1b"><ci id="S2.SS0.SSS0.Px2.p1.12.m4.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.12.m4.1.1">𝝁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.12.m4.1c">\bm{\mu}</annotation></semantics></math> in terms of noise prediction <math id="S2.SS0.SSS0.Px2.p1.13.m5.1" class="ltx_Math" alttext="\bm{\epsilon}=(\sqrt{1-\bar{\alpha}_{t}})^{-1}\mathbf{x}_{t}^{k_{t}}-\sqrt{\bar{\alpha}_{t}}\bm{\mu}" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.13.m5.1a"><mrow id="S2.SS0.SSS0.Px2.p1.13.m5.1.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.cmml"><mi class="ltx_mathvariant_bold-italic" mathvariant="bold-italic" id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.2.cmml">ϵ</mi><mo id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.1" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.1.cmml">=</mo><mrow id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.cmml"><mrow id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.cmml"><msup id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.cmml"><mrow id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.2.2.1" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.cmml">(</mo><msqrt id="S2.SS0.SSS0.Px2.p1.13.m5.1.1" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.cmml"><mrow id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.cmml"><mn id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.2.cmml">1</mn><mo id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.1" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.1.cmml">−</mo><msub id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.cmml"><mover accent="true" id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2.2.cmml">α</mi><mo id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2.1" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2.1.cmml">¯</mo></mover><mi id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.3.cmml">t</mi></msub></mrow></msqrt><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.2.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.cmml">)</mo></mrow><mrow id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.3.cmml"><mo id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.3a" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.3.cmml">−</mo><mn id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.3.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.3.2.cmml">1</mn></mrow></msup><mo lspace="0em" rspace="0em" id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.1" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.1.cmml">​</mo><msubsup id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.2.2.cmml">𝐱</mi><mi id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.2.3.cmml">t</mi><msub id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.3.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.3.2.cmml">k</mi><mi id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.3.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.3.3.3.cmml">t</mi></msub></msubsup></mrow><mo id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.1" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.1.cmml">−</mo><mrow id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.cmml"><msqrt id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.cmml"><msub id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.2.cmml"><mover accent="true" id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.2.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.2.2.2.cmml">α</mi><mo id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.2.2.1" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.2.2.1.cmml">¯</mo></mover><mi id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.2.2.3.cmml">t</mi></msub></msqrt><mo lspace="0em" rspace="0em" id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.1" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.1.cmml">​</mo><mi id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.3" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.3.3.cmml">𝝁</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.13.m5.1b"><apply id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2"><eq id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.1"></eq><ci id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.2">bold-italic-ϵ</ci><apply id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3"><minus id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.1"></minus><apply id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2"><times id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.1"></times><apply id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2">superscript</csymbol><apply id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.2.2"><root id="S2.SS0.SSS0.Px2.p1.13.m5.1.1a.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.2.3.2.2.2.2"></root><apply id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2"><minus id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.1"></minus><cn type="integer" id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.2">1</cn><apply id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3">subscript</csymbol><apply id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2"><ci id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2.1">¯</ci><ci id="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m5.1.1.2.3.2.2">𝛼</ci></apply><ci 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id="S2.SS0.SSS0.Px2.p1.13.m5.1c">\bm{\epsilon}=(\sqrt{1-\bar{\alpha}_{t}})^{-1}\mathbf{x}_{t}^{k_{t}}-\sqrt{\bar{\alpha}_{t}}\bm{\mu}</annotation></semantics></math>. This leads <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib28" title="" class="ltx_ref">28</a>]</cite> to the following least squares objective:</p>
<table id="S2.E3" class="ltx_equation ltx_eqn_table">

<tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.3.2">bold-italic-ϵ</ci><ci id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.3.3">𝑘</ci></apply><apply id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1"><times id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.2"></times><apply id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.2">bold-italic-ϵ</ci><ci id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.3">𝜃</ci></apply><interval closure="open" id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1"><apply id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2">𝐱</ci><ci id="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply><ci id="S2.E3.m1.5.5.cmml" xref="S2.E3.m1.5.5">𝑘</ci></interval></apply></apply></apply><cn type="integer" id="S2.E3.m1.6.6.1.1.1.1.1.1.3.cmml" xref="S2.E3.m1.6.6.1.1.1.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.6c">\mathcal{L}(\theta)=\mathbb{E}_{k,\mathbf{x}^{0},\epsilon}\left[\|\bm{\epsilon}^{k}-\bm{\epsilon}_{\theta}(\mathbf{x}^{k},k)\|^{2}\right],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.3)</span></td>
</tr></tbody>
</table>
<p id="S2.SS0.SSS0.Px2.p1.16" class="ltx_p">where <math id="S2.SS0.SSS0.Px2.p1.14.m1.1" class="ltx_Math" alttext="\mathbf{x}^{k}=\sqrt{\bar{\alpha_{t}}}\mathbf{x}^{0}+\sqrt{1-\bar{\alpha_{t}}}\epsilon^{k}" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.14.m1.1a"><mrow id="S2.SS0.SSS0.Px2.p1.14.m1.1.1" xref="S2.SS0.SSS0.Px2.p1.14.m1.1.1.cmml"><msup id="S2.SS0.SSS0.Px2.p1.14.m1.1.1.2" xref="S2.SS0.SSS0.Px2.p1.14.m1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.14.m1.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.14.m1.1.1.2.2.cmml">𝐱</mi><mi id="S2.SS0.SSS0.Px2.p1.14.m1.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.14.m1.1.1.2.3.cmml">k</mi></msup><mo id="S2.SS0.SSS0.Px2.p1.14.m1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.14.m1.1.1.1.cmml">=</mo><mrow id="S2.SS0.SSS0.Px2.p1.14.m1.1.1.3" xref="S2.SS0.SSS0.Px2.p1.14.m1.1.1.3.cmml"><mrow id="S2.SS0.SSS0.Px2.p1.14.m1.1.1.3.2" xref="S2.SS0.SSS0.Px2.p1.14.m1.1.1.3.2.cmml"><msqrt id="S2.SS0.SSS0.Px2.p1.14.m1.1.1.3.2.2" xref="S2.SS0.SSS0.Px2.p1.14.m1.1.1.3.2.2.cmml"><mover 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id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.2" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.2.cmml">𝒩</mi><mo lspace="0em" rspace="0em" id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.1" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.1.cmml">​</mo><mrow id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.3.2" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.3.1.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.3.2.1" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.3.1.cmml">(</mo><mn id="S2.SS0.SSS0.Px2.p1.15.m2.1.1" xref="S2.SS0.SSS0.Px2.p1.15.m2.1.1.cmml">0</mn><mo id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.3.2.2" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px2.p1.15.m2.2.2" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.2.cmml">𝐈</mi><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.3.2.3" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.15.m2.2b"><apply id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3"><csymbol cd="latexml" id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.1">similar-to</csymbol><apply id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.2">superscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.2.2">bold-italic-ϵ</ci><ci id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.2.3">𝑘</ci></apply><apply id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3"><times id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.1"></times><ci id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.2">𝒩</ci><interval closure="open" id="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.3.3.3.2"><cn type="integer" id="S2.SS0.SSS0.Px2.p1.15.m2.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.1.1">0</cn><ci id="S2.SS0.SSS0.Px2.p1.15.m2.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m2.2.2">𝐈</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.15.m2.2c">\bm{\epsilon}^{k}\sim\mathcal{N}(0,\mathbf{I})</annotation></semantics></math> . One can then sample from this model via Langevin dynamics <math id="S2.SS0.SSS0.Px2.p1.16.m3.1" class="ltx_math_unparsed" alttext="\mathbf{x}^{k-1}\leftarrow\frac{1}{\sqrt{\alpha_{k}}}(\mathbf{x}_{t}^{k}-\frac{1-\alpha_{k}}{\sqrt{1-\bar{\alpha}_{k}}}\bm{\epsilon}_{\theta}(\mathbf{x}_{t}^{k},k))+\sigma_{k}\mathbf{w})" display="inline"><semantics id="S2.SS0.SSS0.Px2.p1.16.m3.1a"><mrow id="S2.SS0.SSS0.Px2.p1.16.m3.1b"><msup id="S2.SS0.SSS0.Px2.p1.16.m3.1.2"><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.2.2">𝐱</mi><mrow id="S2.SS0.SSS0.Px2.p1.16.m3.1.2.3"><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.2.3.2">k</mi><mo id="S2.SS0.SSS0.Px2.p1.16.m3.1.2.3.1">−</mo><mn id="S2.SS0.SSS0.Px2.p1.16.m3.1.2.3.3">1</mn></mrow></msup><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.16.m3.1.3">←</mo><mfrac id="S2.SS0.SSS0.Px2.p1.16.m3.1.4"><mn id="S2.SS0.SSS0.Px2.p1.16.m3.1.4.2">1</mn><msqrt id="S2.SS0.SSS0.Px2.p1.16.m3.1.4.3"><msub id="S2.SS0.SSS0.Px2.p1.16.m3.1.4.3.2"><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.4.3.2.2">α</mi><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.4.3.2.3">k</mi></msub></msqrt></mfrac><mrow id="S2.SS0.SSS0.Px2.p1.16.m3.1.5"><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.1">(</mo><msubsup id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.2"><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.2.2.2">𝐱</mi><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.2.2.3">t</mi><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.2.3">k</mi></msubsup><mo id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.3">−</mo><mfrac id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4"><mrow id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.2"><mn id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.2.2">1</mn><mo id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.2.1">−</mo><msub id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.2.3"><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.2.3.2">α</mi><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.2.3.3">k</mi></msub></mrow><msqrt id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.3"><mrow id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.3.2"><mn id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.3.2.2">1</mn><mo id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.3.2.1">−</mo><msub id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.3.2.3"><mover accent="true" id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.3.2.3.2"><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.3.2.3.2.2">α</mi><mo id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.3.2.3.2.1">¯</mo></mover><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.4.3.2.3.3">k</mi></msub></mrow></msqrt></mfrac><msub id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.5"><mi class="ltx_mathvariant_bold-italic" mathvariant="bold-italic" id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.5.2">ϵ</mi><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.5.3">θ</mi></msub><mrow id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.6"><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.6.1">(</mo><msubsup id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.6.2"><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.6.2.2.2">𝐱</mi><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.6.2.2.3">t</mi><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.6.2.3">k</mi></msubsup><mo id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.6.3">,</mo><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.1">k</mi><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.6.4">)</mo></mrow><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.16.m3.1.5.7">)</mo></mrow><mo id="S2.SS0.SSS0.Px2.p1.16.m3.1.6">+</mo><msub id="S2.SS0.SSS0.Px2.p1.16.m3.1.7"><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.7.2">σ</mi><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.7.3">k</mi></msub><mi id="S2.SS0.SSS0.Px2.p1.16.m3.1.8">𝐰</mi><mo stretchy="false" id="S2.SS0.SSS0.Px2.p1.16.m3.1.9">)</mo></mrow><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.16.m3.1c">\mathbf{x}^{k-1}\leftarrow\frac{1}{\sqrt{\alpha_{k}}}(\mathbf{x}_{t}^{k}-\frac{1-\alpha_{k}}{\sqrt{1-\bar{\alpha}_{k}}}\bm{\epsilon}_{\theta}(\mathbf{x}_{t}^{k},k))+\sigma_{k}\mathbf{w})</annotation></semantics></math>  <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib28" title="" class="ltx_ref">28</a>]</cite>.</p>
</div>
</section>
<section id="S2.SS0.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Guidance of Diffusion Models.</h5>

<div id="S2.SS0.SSS0.Px3.p1" class="ltx_para">
<p id="S2.SS0.SSS0.Px3.p1.6" class="ltx_p">Guidance <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib30" title="" class="ltx_ref">30</a>, <a href="#bib.bib16" title="" class="ltx_ref">16</a>]</cite> allows biasing diffusion generation towards desirable predictions at sampling time.
We focus on classifier guidance <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib16" title="" class="ltx_ref">16</a>]</cite>: given a classifier <math id="S2.SS0.SSS0.Px3.p1.1.m1.1" class="ltx_Math" alttext="c(y|\mathbf{x}^{k})" display="inline"><semantics id="S2.SS0.SSS0.Px3.p1.1.m1.1a"><mrow id="S2.SS0.SSS0.Px3.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.3" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.3.cmml">c</mi><mo lspace="0em" rspace="0em" id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.2" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.2.cmml">​</mo><mrow id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.2" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.2.cmml">y</mi><mo fence="false" id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.1.cmml">|</mo><msup id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.2" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.2.cmml">𝐱</mi><mi id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.3" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.3.cmml">k</mi></msup></mrow><mo stretchy="false" id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.3" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.1.m1.1b"><apply id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1"><times id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.2"></times><ci id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.3">𝑐</ci><apply id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.1">conditional</csymbol><ci id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.2">𝑦</ci><apply id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.2">𝐱</ci><ci id="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px3.p1.1.m1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.1.m1.1c">c(y|\mathbf{x}^{k})</annotation></semantics></math> of some desired <math id="S2.SS0.SSS0.Px3.p1.2.m2.1" class="ltx_Math" alttext="y" display="inline"><semantics id="S2.SS0.SSS0.Px3.p1.2.m2.1a"><mi id="S2.SS0.SSS0.Px3.p1.2.m2.1.1" xref="S2.SS0.SSS0.Px3.p1.2.m2.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.2.m2.1b"><ci id="S2.SS0.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.2.m2.1c">y</annotation></semantics></math> (e.g. class or success indicator), one modifies the Langevin sampling  <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib29" title="" class="ltx_ref">29</a>]</cite> gradient <math id="S2.SS0.SSS0.Px3.p1.3.m3.2" class="ltx_Math" alttext="\bm{\epsilon}_{\theta}(\mathbf{x}^{k},k)" display="inline"><semantics id="S2.SS0.SSS0.Px3.p1.3.m3.2a"><mrow id="S2.SS0.SSS0.Px3.p1.3.m3.2.2" xref="S2.SS0.SSS0.Px3.p1.3.m3.2.2.cmml"><msub id="S2.SS0.SSS0.Px3.p1.3.m3.2.2.3" xref="S2.SS0.SSS0.Px3.p1.3.m3.2.2.3.cmml"><mi class="ltx_mathvariant_bold-italic" mathvariant="bold-italic" id="S2.SS0.SSS0.Px3.p1.3.m3.2.2.3.2" xref="S2.SS0.SSS0.Px3.p1.3.m3.2.2.3.2.cmml">ϵ</mi><mi 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This allows sampling from the joint distribution of <math id="S2.SS0.SSS0.Px3.p1.5.m5.1" class="ltx_Math" alttext="\mathbf{x}" display="inline"><semantics id="S2.SS0.SSS0.Px3.p1.5.m5.1a"><mi id="S2.SS0.SSS0.Px3.p1.5.m5.1.1" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.5.m5.1b"><ci id="S2.SS0.SSS0.Px3.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.5.m5.1c">\mathbf{x}</annotation></semantics></math> and class label <math id="S2.SS0.SSS0.Px3.p1.6.m6.1" class="ltx_Math" alttext="y" display="inline"><semantics id="S2.SS0.SSS0.Px3.p1.6.m6.1a"><mi id="S2.SS0.SSS0.Px3.p1.6.m6.1.1" xref="S2.SS0.SSS0.Px3.p1.6.m6.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.6.m6.1b"><ci id="S2.SS0.SSS0.Px3.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.6.m6.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.6.m6.1c">y</annotation></semantics></math> without the need to train a conditional model. Other energies such as a least-squares objective comparing the model output to a desirable ground-truth have been explored in applications such as decision making <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib16" title="" class="ltx_ref">16</a>, <a href="#bib.bib36" title="" class="ltx_ref">36</a>]</cite>.</p>
</div>
</section>
<section id="S2.SS0.SSS0.Px4" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Next-Token Prediction Models.</h5>

<div id="S2.SS0.SSS0.Px4.p1" class="ltx_para">
<p id="S2.SS0.SSS0.Px4.p1.9" class="ltx_p">Next-token prediction models are sequence models that predict the next frame <math id="S2.SS0.SSS0.Px4.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}_{t+1}" display="inline"><semantics id="S2.SS0.SSS0.Px4.p1.1.m1.1a"><msub id="S2.SS0.SSS0.Px4.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.cmml"><mi id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.2" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.2.cmml">𝐱</mi><mrow id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.2" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.2.cmml">t</mi><mo id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.1" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.1.cmml">+</mo><mn id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.3" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.1.m1.1b"><apply id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.2">𝐱</ci><apply id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3"><plus id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.1"></plus><ci id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.2">𝑡</ci><cn type="integer" id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.1.m1.1c">\mathbf{x}_{t+1}</annotation></semantics></math> given past frames <math id="S2.SS0.SSS0.Px4.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{1:t}" display="inline"><semantics id="S2.SS0.SSS0.Px4.p1.2.m2.1a"><msub id="S2.SS0.SSS0.Px4.p1.2.m2.1.1" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.cmml"><mi id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.2" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.2.cmml">𝐱</mi><mrow id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.cmml"><mn id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.2" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.1" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.1.cmml">:</mo><mi id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.3" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.2.m2.1b"><apply id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.2">𝐱</ci><apply id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3"><ci id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.1">:</ci><cn type="integer" id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.2">1</cn><ci id="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.2.m2.1c">\mathbf{x}_{1:t}</annotation></semantics></math>. At training time, one feeds a neural network with <math id="S2.SS0.SSS0.Px4.p1.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{1:t}" display="inline"><semantics id="S2.SS0.SSS0.Px4.p1.3.m3.1a"><msub id="S2.SS0.SSS0.Px4.p1.3.m3.1.1" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.cmml"><mi id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2.cmml">𝐱</mi><mrow id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.cmml"><mn id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.2" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.1" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.1.cmml">:</mo><mi id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.3" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.3.m3.1b"><apply id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2">𝐱</ci><apply id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3"><ci id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.1">:</ci><cn type="integer" id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.2">1</cn><ci id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.3.m3.1c">\mathbf{x}_{1:t}</annotation></semantics></math> and minimizes <math id="S2.SS0.SSS0.Px4.p1.4.m4.1" class="ltx_Math" alttext="||\hat{\mathbf{x}}-\mathbf{x}||^{2}" display="inline"><semantics id="S2.SS0.SSS0.Px4.p1.4.m4.1a"><msup id="S2.SS0.SSS0.Px4.p1.4.m4.1.1" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.cmml"><mrow id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.2.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.2" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.2.1.cmml">‖</mo><mrow id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.cmml"><mover accent="true" id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2.2.cmml">𝐱</mi><mo id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2.1" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2.1.cmml">^</mo></mover><mo id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.1.cmml">−</mo><mi id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.3.cmml">𝐱</mi></mrow><mo stretchy="false" id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.3" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.3" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.4.m4.1b"><apply id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1">superscript</csymbol><apply id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.2">norm</csymbol><apply id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1"><minus id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.1"></minus><apply id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2"><ci id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2.1">^</ci><ci id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.2.2">𝐱</ci></apply><ci id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.1.1.1.3">𝐱</ci></apply></apply><cn type="integer" id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.4.m4.1c">||\hat{\mathbf{x}}-\mathbf{x}||^{2}</annotation></semantics></math> for continuous data or a cross-entropy loss for discrete data <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib62" title="" class="ltx_ref">62</a>]</cite>. At sampling time, one samples the next frame <math id="S2.SS0.SSS0.Px4.p1.5.m5.1" class="ltx_Math" alttext="\hat{\mathbf{x}}_{t+1}" display="inline"><semantics id="S2.SS0.SSS0.Px4.p1.5.m5.1a"><msub id="S2.SS0.SSS0.Px4.p1.5.m5.1.1" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.cmml"><mover accent="true" id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2.2" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2.2.cmml">𝐱</mi><mo id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2.1" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2.1.cmml">^</mo></mover><mrow id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.2" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.2.cmml">t</mi><mo id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.1" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.1.cmml">+</mo><mn id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.3" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.5.m5.1b"><apply id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1">subscript</csymbol><apply id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2"><ci id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2.1">^</ci><ci id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.2.2">𝐱</ci></apply><apply id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3"><plus id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.1"></plus><ci id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.2">𝑡</ci><cn type="integer" id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.5.m5.1c">\hat{\mathbf{x}}_{t+1}</annotation></semantics></math> following <math id="S2.SS0.SSS0.Px4.p1.6.m6.1" class="ltx_Math" alttext="p(\mathbf{x}_{t+1}|\mathbf{x}_{1:t})" display="inline"><semantics id="S2.SS0.SSS0.Px4.p1.6.m6.1a"><mrow id="S2.SS0.SSS0.Px4.p1.6.m6.1.1" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.3" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.2" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.2.cmml">​</mo><mrow id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.2" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.2" 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xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.1" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.1.cmml">:</mo><mi id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.3" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.3.cmml">t</mi></mrow></msub></mrow><mo stretchy="false" id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.3" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.6.m6.1b"><apply id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1"><times id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.2"></times><ci id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.3">𝑝</ci><apply id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.1">conditional</csymbol><apply id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.2">𝐱</ci><apply id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.3"><plus id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.3.1"></plus><ci id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.3.2">𝑡</ci><cn type="integer" id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.2">𝐱</ci><apply id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3"><ci id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.1">:</ci><cn type="integer" id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.2">1</cn><ci id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.6.m6.1c">p(\mathbf{x}_{t+1}|\mathbf{x}_{1:t})</annotation></semantics></math>. If one treats <math id="S2.SS0.SSS0.Px4.p1.7.m7.1" class="ltx_Math" alttext="\hat{\mathbf{x}}_{t+1}" display="inline"><semantics id="S2.SS0.SSS0.Px4.p1.7.m7.1a"><msub id="S2.SS0.SSS0.Px4.p1.7.m7.1.1" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.cmml"><mover accent="true" id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2.2" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2.2.cmml">𝐱</mi><mo id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2.1" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2.1.cmml">^</mo></mover><mrow id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.2" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.2.cmml">t</mi><mo id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.1" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.1.cmml">+</mo><mn id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.3" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.7.m7.1b"><apply id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1">subscript</csymbol><apply id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2"><ci id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2.1">^</ci><ci id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.2.2">𝐱</ci></apply><apply id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3"><plus id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.1"></plus><ci id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.2">𝑡</ci><cn type="integer" id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.7.m7.1c">\hat{\mathbf{x}}_{t+1}</annotation></semantics></math> as <math id="S2.SS0.SSS0.Px4.p1.8.m8.1" class="ltx_Math" alttext="\mathbf{x}_{t+1}" display="inline"><semantics id="S2.SS0.SSS0.Px4.p1.8.m8.1a"><msub id="S2.SS0.SSS0.Px4.p1.8.m8.1.1" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.cmml"><mi id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.2" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.2.cmml">𝐱</mi><mrow id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.2" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.2.cmml">t</mi><mo id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.1" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.1.cmml">+</mo><mn id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.3" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.8.m8.1b"><apply id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.2">𝐱</ci><apply id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3"><plus id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.1"></plus><ci id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.2">𝑡</ci><cn type="integer" id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.8.m8.1c">\mathbf{x}_{t+1}</annotation></semantics></math>, one can use the same model to predict <math id="S2.SS0.SSS0.Px4.p1.9.m9.1" class="ltx_Math" alttext="\mathbf{x}_{t+2}" display="inline"><semantics id="S2.SS0.SSS0.Px4.p1.9.m9.1a"><msub id="S2.SS0.SSS0.Px4.p1.9.m9.1.1" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.cmml"><mi id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.2" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.2.cmml">𝐱</mi><mrow id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.2" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.2.cmml">t</mi><mo id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.1" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.1.cmml">+</mo><mn id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.3" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.3.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.9.m9.1b"><apply id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.2">𝐱</ci><apply id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3"><plus id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.1"></plus><ci id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.2">𝑡</ci><cn type="integer" id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.9.m9.1c">\mathbf{x}_{t+2}</annotation></semantics></math> and repeat until a full sequence is sampled. Unlike full-sequence diffusion models, next-token models do not accept multi-step guidance, as prior frames must be fully determined to sample future frames.</p>
</div>
</section>
<section id="S2.SS0.SSS0.Px5" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Diffusion Sequence Models.</h5>

<div id="S2.SS0.SSS0.Px5.p1" class="ltx_para">
<p id="S2.SS0.SSS0.Px5.p1.1" class="ltx_p">Diffusion has been widely used in sequence modeling. <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib42" title="" class="ltx_ref">42</a>]</cite> use full-sequence diffusion models to achieve controllable text generation via guidance, such as generating text following specified parts of speech. <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib31" title="" class="ltx_ref">31</a>]</cite> trains full-sequence diffusion models to synthesize short videos and uses a sliding window to roll out longer conditioned on previously generated frames. <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib36" title="" class="ltx_ref">36</a>]</cite> uses full-sequence diffusion models as planners in offline reinforcement learning. This is achieved by training on a dataset of interaction trajectories with the environment and using classifier guidance at sampling time to sample trajectories with high rewards towards a chosen goal. <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib48" title="" class="ltx_ref">48</a>]</cite> modifies auto-regressive models to denoise the next token conditioned on previous tokens. It trains with teacher forcing <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib62" title="" class="ltx_ref">62</a>]</cite> and samples next-token auto-regressively for time series data. Most similar to our work is AR-Diffusion <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib63" title="" class="ltx_ref">63</a>]</cite>, which trains full-sequence text diffusion with a causal architecture with linearly dependent noise level along the time axis. We provide a detailed comparision between this approach and ours in <a href="#A3" title="Appendix C Extended Related Work ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Appendix</span> <span class="ltx_text ltx_ref_tag">C</span></a>.</p>
</div>
</section>
</section>
<section id="S3" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">3 </span>Diffusion Forcing</h2>

<section id="S3.SS0.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Noising as partial masking.</h5>

<div id="S3.SS0.SSS0.Px1.p1" class="ltx_para">
<p id="S3.SS0.SSS0.Px1.p1.7" class="ltx_p">Recall that <em id="S3.SS0.SSS0.Px1.p1.7.1" class="ltx_emph ltx_font_italic">masking</em> is the practice of occluding a subset of data, such as patches of an image <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib26" title="" class="ltx_ref">26</a>]</cite> or timesteps in a sequence <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib15" title="" class="ltx_ref">15</a>, <a href="#bib.bib47" title="" class="ltx_ref">47</a>]</cite>, and training a model to recover unmasked portions.
Without loss of generality, we can view any collection of tokens, sequential or not, as an ordered set indexed by <math id="S3.SS0.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="t" display="inline"><semantics id="S3.SS0.SSS0.Px1.p1.1.m1.1a"><mi id="S3.SS0.SSS0.Px1.p1.1.m1.1.1" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.1.m1.1b"><ci id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.1.m1.1c">t</annotation></semantics></math>.
Training next-token prediction with teacher forcing can then be interpreted as masking each token <math id="S3.SS0.SSS0.Px1.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{t}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p1.2.m2.1a"><msub id="S3.SS0.SSS0.Px1.p1.2.m2.1.1" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.2" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.3" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.2.m2.1b"><apply id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1.2">𝐱</ci><ci id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.2.m2.1c">\mathbf{x}_{t}</annotation></semantics></math> at time <math id="S3.SS0.SSS0.Px1.p1.3.m3.1" class="ltx_Math" alttext="t" display="inline"><semantics id="S3.SS0.SSS0.Px1.p1.3.m3.1a"><mi id="S3.SS0.SSS0.Px1.p1.3.m3.1.1" xref="S3.SS0.SSS0.Px1.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.3.m3.1b"><ci id="S3.SS0.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.3.m3.1c">t</annotation></semantics></math> and making predictions from the past <math id="S3.SS0.SSS0.Px1.p1.4.m4.1" class="ltx_Math" alttext="\mathbf{x}_{1:t-1}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p1.4.m4.1a"><msub id="S3.SS0.SSS0.Px1.p1.4.m4.1.1" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.2" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.2.cmml">𝐱</mi><mrow id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.cmml"><mn id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.2" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.1" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.1.cmml">:</mo><mrow id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.cmml"><mi id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.2" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.2.cmml">t</mi><mo id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.1" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.1.cmml">−</mo><mn id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.3" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.4.m4.1b"><apply id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.2">𝐱</ci><apply id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3"><ci id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.1">:</ci><cn type="integer" id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.2">1</cn><apply id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3"><minus id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.1.cmml" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.1"></minus><ci id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.2.cmml" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.2">𝑡</ci><cn type="integer" id="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.3.cmml" xref="S3.SS0.SSS0.Px1.p1.4.m4.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.4.m4.1c">\mathbf{x}_{1:t-1}</annotation></semantics></math>. Restricted to sequences, we refer to all these practices as <em id="S3.SS0.SSS0.Px1.p1.7.2" class="ltx_emph ltx_font_italic">masking along the time axis</em>. We can also view full-sequence forward diffusion, i.e., gradually adding noise to the data <math id="S3.SS0.SSS0.Px1.p1.5.m5.1" class="ltx_Math" alttext="\mathbf{x}^{0}_{1:T}\equiv\mathbf{x}_{1:T}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p1.5.m5.1a"><mrow id="S3.SS0.SSS0.Px1.p1.5.m5.1.1" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.cmml"><msubsup id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.2.2" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.2.2.cmml">𝐱</mi><mrow id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.cmml"><mn id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.2" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.1" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.1.cmml">:</mo><mi id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.3" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.3.cmml">T</mi></mrow><mn id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.2.3" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.2.3.cmml">0</mn></msubsup><mo id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.1" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.1.cmml">≡</mo><msub id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.2" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.2.cmml">𝐱</mi><mrow id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.cmml"><mn id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.2" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.1" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.1.cmml">:</mo><mi id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.3" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.3.cmml">T</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.5.m5.1b"><apply id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1"><equivalent id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.1"></equivalent><apply id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2">subscript</csymbol><apply id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.2.1.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2">superscript</csymbol><ci id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.2.2.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.2.2">𝐱</ci><cn type="integer" id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.2.3.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.2.3">0</cn></apply><apply id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3"><ci id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.1.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.1">:</ci><cn type="integer" id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.2.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.2">1</cn><ci id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.3.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.2.3.3">𝑇</ci></apply></apply><apply id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.2">𝐱</ci><apply id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3"><ci id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.1.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.1">:</ci><cn type="integer" id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.2.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.2">1</cn><ci id="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.3.cmml" xref="S3.SS0.SSS0.Px1.p1.5.m5.1.1.3.3.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.5.m5.1c">\mathbf{x}^{0}_{1:T}\equiv\mathbf{x}_{1:T}</annotation></semantics></math>, as a form of <em id="S3.SS0.SSS0.Px1.p1.7.3" class="ltx_emph ltx_font_italic">partial masking</em>, which we refer to as <em id="S3.SS0.SSS0.Px1.p1.7.4" class="ltx_emph ltx_font_italic">masking along the noise axis</em>. Indeed, after <math id="S3.SS0.SSS0.Px1.p1.6.m6.1" class="ltx_Math" alttext="K" display="inline"><semantics id="S3.SS0.SSS0.Px1.p1.6.m6.1a"><mi id="S3.SS0.SSS0.Px1.p1.6.m6.1.1" xref="S3.SS0.SSS0.Px1.p1.6.m6.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.6.m6.1b"><ci id="S3.SS0.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.6.m6.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.6.m6.1c">K</annotation></semantics></math> steps of noising, <math id="S3.SS0.SSS0.Px1.p1.7.m7.1" class="ltx_Math" alttext="\mathbf{x}^{K}_{1:T}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p1.7.m7.1a"><msubsup id="S3.SS0.SSS0.Px1.p1.7.m7.1.1" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.2.2" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.2.2.cmml">𝐱</mi><mrow id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.cmml"><mn id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.2" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.1" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.1.cmml">:</mo><mi id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.3" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.3.cmml">T</mi></mrow><mi id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.2.3" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.2.3.cmml">K</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.7.m7.1b"><apply id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1">subscript</csymbol><apply id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1">superscript</csymbol><ci id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.2.3">𝐾</ci></apply><apply id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3"><ci id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.1">:</ci><cn type="integer" id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.2">1</cn><ci id="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px1.p1.7.m7.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.7.m7.1c">\mathbf{x}^{K}_{1:T}</annotation></semantics></math> is (approximately) pure white noise without information about the original data.</p>
</div>
<div id="S3.SS0.SSS0.Px1.p2" class="ltx_para">
<p id="S3.SS0.SSS0.Px1.p2.9" class="ltx_p">We establish a unified view along both axes of masking (see Fig. <a href="#S2.F2" title="Figure 2 ‣ Bayesian Filtering. ‣ 2 Related Work and Preliminaries ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>). We denote <math id="S3.SS0.SSS0.Px1.p2.1.m1.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p2.1.m1.1a"><msub id="S3.SS0.SSS0.Px1.p2.1.m1.1.1" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.2" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.2.cmml">𝐱</mi><mrow id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.cmml"><mn id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.2" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.1" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.1.cmml">:</mo><mi id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.3" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.1.m1.1b"><apply id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.2">𝐱</ci><apply id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3"><ci id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.1">:</ci><cn type="integer" id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.2">1</cn><ci id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.1.m1.1c">\mathbf{x}_{1:T}</annotation></semantics></math> for a sequence of tokens, where the subscript indicates the time axis. As above, <math id="S3.SS0.SSS0.Px1.p2.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p2.2.m2.1a"><msubsup id="S3.SS0.SSS0.Px1.p2.2.m2.1.1" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.2.2" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.2.3" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.2.3.cmml">t</mi><msub id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.2" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.3" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.2.m2.1b"><apply id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1">superscript</csymbol><apply id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.2.3">𝑡</ci></apply><apply id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.2">𝑘</ci><ci id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.2.m2.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math> denotes <math id="S3.SS0.SSS0.Px1.p2.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{t}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p2.3.m3.1a"><msub id="S3.SS0.SSS0.Px1.p2.3.m3.1.1" xref="S3.SS0.SSS0.Px1.p2.3.m3.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p2.3.m3.1.1.2" xref="S3.SS0.SSS0.Px1.p2.3.m3.1.1.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px1.p2.3.m3.1.1.3" xref="S3.SS0.SSS0.Px1.p2.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.3.m3.1b"><apply id="S3.SS0.SSS0.Px1.p2.3.m3.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.3.m3.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.3.m3.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m3.1.1.2">𝐱</ci><ci id="S3.SS0.SSS0.Px1.p2.3.m3.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.3.m3.1c">\mathbf{x}_{t}</annotation></semantics></math> at noise level <math id="S3.SS0.SSS0.Px1.p2.4.m4.1" class="ltx_Math" alttext="k_{t}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p2.4.m4.1a"><msub id="S3.SS0.SSS0.Px1.p2.4.m4.1.1" xref="S3.SS0.SSS0.Px1.p2.4.m4.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p2.4.m4.1.1.2" xref="S3.SS0.SSS0.Px1.p2.4.m4.1.1.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px1.p2.4.m4.1.1.3" xref="S3.SS0.SSS0.Px1.p2.4.m4.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.4.m4.1b"><apply id="S3.SS0.SSS0.Px1.p2.4.m4.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.4.m4.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.4.m4.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.4.m4.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.4.m4.1.1.2">𝑘</ci><ci id="S3.SS0.SSS0.Px1.p2.4.m4.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.4.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.4.m4.1c">k_{t}</annotation></semantics></math> under the forward diffusion process (<a href="#S2.E1" title="Equation 2.1 ‣ Diffusion Models. ‣ 2 Related Work and Preliminaries ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2.1</span></a>); <math id="S3.SS0.SSS0.Px1.p2.5.m5.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{0}=\mathbf{x}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p2.5.m5.1a"><mrow id="S3.SS0.SSS0.Px1.p2.5.m5.1.1" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.cmml"><msubsup id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.2.2" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.2.3" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.2.3.cmml">t</mi><mn id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.3" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.3.cmml">0</mn></msubsup><mo id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.1" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.1.cmml">=</mo><mi id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.3" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.3.cmml">𝐱</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.5.m5.1b"><apply id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1"><eq id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.1"></eq><apply id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2">superscript</csymbol><apply id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.2.1.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.2.2.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.2.3.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.2.3">𝑡</ci></apply><cn type="integer" id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.2.3">0</cn></apply><ci id="S3.SS0.SSS0.Px1.p2.5.m5.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m5.1.1.3">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.5.m5.1c">\mathbf{x}_{t}^{0}=\mathbf{x}</annotation></semantics></math> is the unnoised token, and <math id="S3.SS0.SSS0.Px1.p2.6.m6.1" class="ltx_Math" alttext="\mathbf{x}^{K}_{t}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p2.6.m6.1a"><msubsup id="S3.SS0.SSS0.Px1.p2.6.m6.1.1" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p2.6.m6.1.1.2.2" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px1.p2.6.m6.1.1.3" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1.3.cmml">t</mi><mi id="S3.SS0.SSS0.Px1.p2.6.m6.1.1.2.3" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1.2.3.cmml">K</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.6.m6.1b"><apply id="S3.SS0.SSS0.Px1.p2.6.m6.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.6.m6.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1">subscript</csymbol><apply id="S3.SS0.SSS0.Px1.p2.6.m6.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.6.m6.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1">superscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.6.m6.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px1.p2.6.m6.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1.2.3">𝐾</ci></apply><ci id="S3.SS0.SSS0.Px1.p2.6.m6.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.6.m6.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.6.m6.1c">\mathbf{x}^{K}_{t}</annotation></semantics></math> is white noise <math id="S3.SS0.SSS0.Px1.p2.7.m7.2" class="ltx_Math" alttext="\mathcal{N}(0,\mathbf{I})" display="inline"><semantics id="S3.SS0.SSS0.Px1.p2.7.m7.2a"><mrow id="S3.SS0.SSS0.Px1.p2.7.m7.2.3" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS0.SSS0.Px1.p2.7.m7.2.3.2" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3.2.cmml">𝒩</mi><mo lspace="0em" rspace="0em" id="S3.SS0.SSS0.Px1.p2.7.m7.2.3.1" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3.1.cmml">​</mo><mrow id="S3.SS0.SSS0.Px1.p2.7.m7.2.3.3.2" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3.3.1.cmml"><mo stretchy="false" id="S3.SS0.SSS0.Px1.p2.7.m7.2.3.3.2.1" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3.3.1.cmml">(</mo><mn id="S3.SS0.SSS0.Px1.p2.7.m7.1.1" xref="S3.SS0.SSS0.Px1.p2.7.m7.1.1.cmml">0</mn><mo id="S3.SS0.SSS0.Px1.p2.7.m7.2.3.3.2.2" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3.3.1.cmml">,</mo><mi id="S3.SS0.SSS0.Px1.p2.7.m7.2.2" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.2.cmml">𝐈</mi><mo stretchy="false" id="S3.SS0.SSS0.Px1.p2.7.m7.2.3.3.2.3" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.7.m7.2b"><apply id="S3.SS0.SSS0.Px1.p2.7.m7.2.3.cmml" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3"><times id="S3.SS0.SSS0.Px1.p2.7.m7.2.3.1.cmml" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3.1"></times><ci id="S3.SS0.SSS0.Px1.p2.7.m7.2.3.2.cmml" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3.2">𝒩</ci><interval closure="open" id="S3.SS0.SSS0.Px1.p2.7.m7.2.3.3.1.cmml" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.3.3.2"><cn type="integer" id="S3.SS0.SSS0.Px1.p2.7.m7.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.7.m7.1.1">0</cn><ci id="S3.SS0.SSS0.Px1.p2.7.m7.2.2.cmml" xref="S3.SS0.SSS0.Px1.p2.7.m7.2.2">𝐈</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.7.m7.2c">\mathcal{N}(0,\mathbf{I})</annotation></semantics></math>. Thus, <math id="S3.SS0.SSS0.Px1.p2.8.m8.1" class="ltx_Math" alttext="(\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p2.8.m8.1a"><msub id="S3.SS0.SSS0.Px1.p2.8.m8.1.1" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.cmml"><mrow id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.2" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.cmml">(</mo><msubsup id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.2.2" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.2.3" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.2.3.cmml">t</mi><msub id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.2" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.3" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.3.cmml">t</mi></msub></msubsup><mo stretchy="false" id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.3" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.cmml"><mn id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.2" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.2.cmml">1</mn><mo id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.3" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.3.cmml">≤</mo><mi id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.4" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.4.cmml">t</mi><mo id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.5" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.5.cmml">≤</mo><mi id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.6" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.6.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.8.m8.1b"><apply id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1">subscript</csymbol><apply id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1">superscript</csymbol><apply id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.2">𝑘</ci><ci id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.1.1.1.3.3">𝑡</ci></apply></apply><apply id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3"><and id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3a.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3"></and><apply id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3b.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3"><leq id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.3"></leq><cn type="integer" id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.2">1</cn><ci id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.4.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.4">𝑡</ci></apply><apply id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3c.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3"><leq id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.5.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.5"></leq><share href="#S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.4.cmml" id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3d.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3"></share><ci id="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.6.cmml" xref="S3.SS0.SSS0.Px1.p2.8.m8.1.1.3.6">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.8.m8.1c">(\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T}</annotation></semantics></math> denotes a sequence of noisy observations where each token has a <em id="S3.SS0.SSS0.Px1.p2.9.1" class="ltx_emph ltx_font_italic">different</em> noise level <math id="S3.SS0.SSS0.Px1.p2.9.m9.1" class="ltx_Math" alttext="k_{t}" display="inline"><semantics id="S3.SS0.SSS0.Px1.p2.9.m9.1a"><msub id="S3.SS0.SSS0.Px1.p2.9.m9.1.1" xref="S3.SS0.SSS0.Px1.p2.9.m9.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p2.9.m9.1.1.2" xref="S3.SS0.SSS0.Px1.p2.9.m9.1.1.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px1.p2.9.m9.1.1.3" xref="S3.SS0.SSS0.Px1.p2.9.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.9.m9.1b"><apply id="S3.SS0.SSS0.Px1.p2.9.m9.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.9.m9.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.9.m9.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.9.m9.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.9.m9.1.1.2">𝑘</ci><ci id="S3.SS0.SSS0.Px1.p2.9.m9.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.9.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.9.m9.1c">k_{t}</annotation></semantics></math>, which can be seen as the degree of <em id="S3.SS0.SSS0.Px1.p2.9.2" class="ltx_emph ltx_font_italic">partial masking</em> applied to each token through noising.</p>
</div>
</section>
<section id="S3.SS0.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Diffusion Forcing: Different noise nevels for different tokens.</h5>

<div id="S3.SS0.SSS0.Px2.p1" class="ltx_para">
<p id="S3.SS0.SSS0.Px2.p1.3" class="ltx_p"><em id="S3.SS0.SSS0.Px2.p1.3.2" class="ltx_emph ltx_font_italic">Diffusion Forcing</em> (DF) is a framework for training and sampling arbitrary sequence lengths of noisy tokens <math id="S3.SS0.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="(\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p1.1.m1.1a"><msub id="S3.SS0.SSS0.Px2.p1.1.m1.1.1" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.cmml"><mrow id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.2" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.2" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.3" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.3.cmml">t</mi><msub id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.2" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.3" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.3.cmml">t</mi></msub></msubsup><mo stretchy="false" id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.3" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.cmml"><mn id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.2" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.2.cmml">1</mn><mo id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.3" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.3.cmml">≤</mo><mi id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.4" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.4.cmml">t</mi><mo id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.5" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.5.cmml">≤</mo><mi id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.6" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.6.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.1.m1.1b"><apply id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1">subscript</csymbol><apply id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1">superscript</csymbol><apply id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.3">𝑡</ci></apply></apply><apply id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3"><and id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3a.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3"></and><apply id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3b.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3"><leq id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.3"></leq><cn type="integer" id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.2">1</cn><ci id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.4.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.4">𝑡</ci></apply><apply id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3c.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3"><leq id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.5.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.5"></leq><share href="#S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.4.cmml" id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3d.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3"></share><ci id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.6.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.6">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.1.m1.1c">(\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T}</annotation></semantics></math>, where critically, <em id="S3.SS0.SSS0.Px2.p1.2.1" class="ltx_emph ltx_font_italic">the noise level <math id="S3.SS0.SSS0.Px2.p1.2.1.m1.1" class="ltx_Math" alttext="k_{t}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p1.2.1.m1.1a"><msub id="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1" xref="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.2" xref="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.3" xref="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.2.1.m1.1b"><apply id="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p1.2.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.2.1.m1.1c">k_{t}</annotation></semantics></math> of each token can vary by time step</em>.
In this paper, we focus on time series data, and thus instantiate Diffusion Forcing with causal architectures (where <math id="S3.SS0.SSS0.Px2.p1.3.m2.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p1.3.m2.1a"><msubsup id="S3.SS0.SSS0.Px2.p1.3.m2.1.1" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.2.2" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.2.3" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.2.3.cmml">t</mi><msub id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.2" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.3" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.3.m2.1b"><apply id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1">superscript</csymbol><apply id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.2.3">𝑡</ci></apply><apply id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m2.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.3.m2.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math> depends only on past noisy tokens), which we call <em id="S3.SS0.SSS0.Px2.p1.3.3" class="ltx_emph ltx_font_italic">Causal Diffusion Forcing</em> (CDF).
For simplicity, we focus on a minimal implementation with a vanilla Recurrent Neural Network (RNN)  <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib11" title="" class="ltx_ref">11</a>]</cite>.</p>
</div>
<figure id="S3.SS0.SSS0.Px2.fig1" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_2">
<figure id="alg1" class="ltx_float ltx_figure_panel ltx_minipage ltx_align_top ltx_framed ltx_framed_top" style="width:216.8pt;">
<figcaption class="ltx_caption" style="font-size:80%;"><span class="ltx_tag ltx_tag_float"><span id="alg1.4.1.1" class="ltx_text ltx_font_bold">Algorithm 1</span> </span> Diffusion Forcing Training</figcaption>
<div id="alg1.5" class="ltx_listing ltx_listing">
<div id="alg1.l1" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l1.1.1.1" class="ltx_text" style="font-size:80%;">1:</span></span><span id="alg1.l1.2" class="ltx_text" style="font-size:80%;">  </span><span id="alg1.l1.3" class="ltx_text ltx_font_bold" style="font-size:80%;">loop</span><span id="alg1.l1.4" class="ltx_text" style="font-size:80%;">

</span>
</div>
<div id="alg1.l2" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l2.1.1.1" class="ltx_text" style="font-size:80%;">2:</span></span><span id="alg1.l2.2" class="ltx_text" style="font-size:80%;">    Sample tajectory of observations </span><math id="alg1.l2.m1.3" class="ltx_Math" alttext="(\mathbf{x}_{1},...,\mathbf{x}_{T})" display="inline"><semantics id="alg1.l2.m1.3a"><mrow id="alg1.l2.m1.3.3.2" xref="alg1.l2.m1.3.3.3.cmml"><mo maxsize="80%" minsize="80%" id="alg1.l2.m1.3.3.2.3" xref="alg1.l2.m1.3.3.3.cmml">(</mo><msub id="alg1.l2.m1.2.2.1.1" xref="alg1.l2.m1.2.2.1.1.cmml"><mi mathsize="80%" id="alg1.l2.m1.2.2.1.1.2" xref="alg1.l2.m1.2.2.1.1.2.cmml">𝐱</mi><mn mathsize="80%" id="alg1.l2.m1.2.2.1.1.3" xref="alg1.l2.m1.2.2.1.1.3.cmml">1</mn></msub><mo mathsize="80%" id="alg1.l2.m1.3.3.2.4" xref="alg1.l2.m1.3.3.3.cmml">,</mo><mi mathsize="80%" mathvariant="normal" id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml">…</mi><mo mathsize="80%" id="alg1.l2.m1.3.3.2.5" xref="alg1.l2.m1.3.3.3.cmml">,</mo><msub id="alg1.l2.m1.3.3.2.2" xref="alg1.l2.m1.3.3.2.2.cmml"><mi mathsize="80%" id="alg1.l2.m1.3.3.2.2.2" xref="alg1.l2.m1.3.3.2.2.2.cmml">𝐱</mi><mi mathsize="80%" id="alg1.l2.m1.3.3.2.2.3" xref="alg1.l2.m1.3.3.2.2.3.cmml">T</mi></msub><mo maxsize="80%" minsize="80%" id="alg1.l2.m1.3.3.2.6" xref="alg1.l2.m1.3.3.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.3b"><vector id="alg1.l2.m1.3.3.3.cmml" xref="alg1.l2.m1.3.3.2"><apply id="alg1.l2.m1.2.2.1.1.cmml" xref="alg1.l2.m1.2.2.1.1"><csymbol cd="ambiguous" id="alg1.l2.m1.2.2.1.1.1.cmml" xref="alg1.l2.m1.2.2.1.1">subscript</csymbol><ci id="alg1.l2.m1.2.2.1.1.2.cmml" xref="alg1.l2.m1.2.2.1.1.2">𝐱</ci><cn type="integer" id="alg1.l2.m1.2.2.1.1.3.cmml" xref="alg1.l2.m1.2.2.1.1.3">1</cn></apply><ci id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1">…</ci><apply id="alg1.l2.m1.3.3.2.2.cmml" xref="alg1.l2.m1.3.3.2.2"><csymbol cd="ambiguous" id="alg1.l2.m1.3.3.2.2.1.cmml" xref="alg1.l2.m1.3.3.2.2">subscript</csymbol><ci id="alg1.l2.m1.3.3.2.2.2.cmml" xref="alg1.l2.m1.3.3.2.2.2">𝐱</ci><ci id="alg1.l2.m1.3.3.2.2.3.cmml" xref="alg1.l2.m1.3.3.2.2.3">𝑇</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.3c">(\mathbf{x}_{1},...,\mathbf{x}_{T})</annotation></semantics></math><span id="alg1.l2.3" class="ltx_text" style="font-size:80%;">.
</span>
</div>
<div id="alg1.l3" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l3.1.1.1" class="ltx_text" style="font-size:80%;">3:</span></span><span id="alg1.l3.2" class="ltx_text" style="font-size:80%;">    </span><span id="alg1.l3.3" class="ltx_text ltx_font_bold" style="font-size:80%;">for</span><span id="alg1.l3.4" class="ltx_text" style="font-size:80%;"> </span><math id="alg1.l3.m1.3" class="ltx_Math" alttext="t=1,...,T" display="inline"><semantics id="alg1.l3.m1.3a"><mrow id="alg1.l3.m1.3.4" xref="alg1.l3.m1.3.4.cmml"><mi mathsize="80%" id="alg1.l3.m1.3.4.2" xref="alg1.l3.m1.3.4.2.cmml">t</mi><mo mathsize="80%" id="alg1.l3.m1.3.4.1" xref="alg1.l3.m1.3.4.1.cmml">=</mo><mrow id="alg1.l3.m1.3.4.3.2" xref="alg1.l3.m1.3.4.3.1.cmml"><mn mathsize="80%" id="alg1.l3.m1.1.1" xref="alg1.l3.m1.1.1.cmml">1</mn><mo mathsize="80%" id="alg1.l3.m1.3.4.3.2.1" xref="alg1.l3.m1.3.4.3.1.cmml">,</mo><mi mathsize="80%" mathvariant="normal" id="alg1.l3.m1.2.2" xref="alg1.l3.m1.2.2.cmml">…</mi><mo mathsize="80%" id="alg1.l3.m1.3.4.3.2.2" xref="alg1.l3.m1.3.4.3.1.cmml">,</mo><mi mathsize="80%" id="alg1.l3.m1.3.3" xref="alg1.l3.m1.3.3.cmml">T</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.3b"><apply id="alg1.l3.m1.3.4.cmml" xref="alg1.l3.m1.3.4"><eq id="alg1.l3.m1.3.4.1.cmml" xref="alg1.l3.m1.3.4.1"></eq><ci id="alg1.l3.m1.3.4.2.cmml" xref="alg1.l3.m1.3.4.2">𝑡</ci><list id="alg1.l3.m1.3.4.3.1.cmml" xref="alg1.l3.m1.3.4.3.2"><cn type="integer" id="alg1.l3.m1.1.1.cmml" xref="alg1.l3.m1.1.1">1</cn><ci id="alg1.l3.m1.2.2.cmml" xref="alg1.l3.m1.2.2">…</ci><ci id="alg1.l3.m1.3.3.cmml" xref="alg1.l3.m1.3.3">𝑇</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.3c">t=1,...,T</annotation></semantics></math><span id="alg1.l3.5" class="ltx_text" style="font-size:80%;"> </span><span id="alg1.l3.6" class="ltx_text ltx_font_bold" style="font-size:80%;">do</span><span id="alg1.l3.7" class="ltx_text" style="font-size:80%;">


</span>
</div>
<div id="alg1.l4" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l4.1.1.1" class="ltx_text" style="font-size:80%;">4:</span></span><span id="alg1.l4.2" class="ltx_text" style="font-size:80%;">      Sample independent noise level </span><math id="alg1.l4.m1.4" class="ltx_Math" alttext="k_{t}\in\{0,1,...,K\}" display="inline"><semantics id="alg1.l4.m1.4a"><mrow id="alg1.l4.m1.4.5" xref="alg1.l4.m1.4.5.cmml"><msub id="alg1.l4.m1.4.5.2" xref="alg1.l4.m1.4.5.2.cmml"><mi mathsize="80%" id="alg1.l4.m1.4.5.2.2" xref="alg1.l4.m1.4.5.2.2.cmml">k</mi><mi mathsize="80%" id="alg1.l4.m1.4.5.2.3" xref="alg1.l4.m1.4.5.2.3.cmml">t</mi></msub><mo mathsize="80%" id="alg1.l4.m1.4.5.1" xref="alg1.l4.m1.4.5.1.cmml">∈</mo><mrow id="alg1.l4.m1.4.5.3.2" xref="alg1.l4.m1.4.5.3.1.cmml"><mo maxsize="80%" minsize="80%" id="alg1.l4.m1.4.5.3.2.1" xref="alg1.l4.m1.4.5.3.1.cmml">{</mo><mn mathsize="80%" id="alg1.l4.m1.1.1" xref="alg1.l4.m1.1.1.cmml">0</mn><mo mathsize="80%" id="alg1.l4.m1.4.5.3.2.2" xref="alg1.l4.m1.4.5.3.1.cmml">,</mo><mn mathsize="80%" id="alg1.l4.m1.2.2" xref="alg1.l4.m1.2.2.cmml">1</mn><mo mathsize="80%" id="alg1.l4.m1.4.5.3.2.3" xref="alg1.l4.m1.4.5.3.1.cmml">,</mo><mi mathsize="80%" mathvariant="normal" id="alg1.l4.m1.3.3" xref="alg1.l4.m1.3.3.cmml">…</mi><mo mathsize="80%" id="alg1.l4.m1.4.5.3.2.4" xref="alg1.l4.m1.4.5.3.1.cmml">,</mo><mi mathsize="80%" id="alg1.l4.m1.4.4" xref="alg1.l4.m1.4.4.cmml">K</mi><mo maxsize="80%" minsize="80%" id="alg1.l4.m1.4.5.3.2.5" xref="alg1.l4.m1.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.4b"><apply id="alg1.l4.m1.4.5.cmml" xref="alg1.l4.m1.4.5"><in id="alg1.l4.m1.4.5.1.cmml" xref="alg1.l4.m1.4.5.1"></in><apply id="alg1.l4.m1.4.5.2.cmml" xref="alg1.l4.m1.4.5.2"><csymbol cd="ambiguous" id="alg1.l4.m1.4.5.2.1.cmml" xref="alg1.l4.m1.4.5.2">subscript</csymbol><ci id="alg1.l4.m1.4.5.2.2.cmml" xref="alg1.l4.m1.4.5.2.2">𝑘</ci><ci id="alg1.l4.m1.4.5.2.3.cmml" xref="alg1.l4.m1.4.5.2.3">𝑡</ci></apply><set id="alg1.l4.m1.4.5.3.1.cmml" xref="alg1.l4.m1.4.5.3.2"><cn type="integer" id="alg1.l4.m1.1.1.cmml" xref="alg1.l4.m1.1.1">0</cn><cn type="integer" id="alg1.l4.m1.2.2.cmml" xref="alg1.l4.m1.2.2">1</cn><ci id="alg1.l4.m1.3.3.cmml" xref="alg1.l4.m1.3.3">…</ci><ci id="alg1.l4.m1.4.4.cmml" xref="alg1.l4.m1.4.4">𝐾</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.4c">k_{t}\in\{0,1,...,K\}</annotation></semantics></math><span id="alg1.l4.3" class="ltx_text" style="font-size:80%;">
</span>
</div>
<div id="alg1.l5" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l5.1.1.1" class="ltx_text" style="font-size:80%;">5:</span></span><span id="alg1.l5.2" class="ltx_text" style="font-size:80%;">      </span><math id="alg1.l5.m1.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}=\ " display="inline"><semantics id="alg1.l5.m1.1a"><mrow id="alg1.l5.m1.1.1" xref="alg1.l5.m1.1.1.cmml"><msubsup id="alg1.l5.m1.1.1.2" xref="alg1.l5.m1.1.1.2.cmml"><mi mathsize="80%" id="alg1.l5.m1.1.1.2.2.2" xref="alg1.l5.m1.1.1.2.2.2.cmml">𝐱</mi><mi mathsize="80%" id="alg1.l5.m1.1.1.2.2.3" xref="alg1.l5.m1.1.1.2.2.3.cmml">t</mi><msub id="alg1.l5.m1.1.1.2.3" xref="alg1.l5.m1.1.1.2.3.cmml"><mi mathsize="80%" id="alg1.l5.m1.1.1.2.3.2" xref="alg1.l5.m1.1.1.2.3.2.cmml">k</mi><mi mathsize="80%" id="alg1.l5.m1.1.1.2.3.3" xref="alg1.l5.m1.1.1.2.3.3.cmml">t</mi></msub></msubsup><mo mathsize="80%" rspace="0.678em" id="alg1.l5.m1.1.1.1" xref="alg1.l5.m1.1.1.1.cmml">=</mo><mi id="alg1.l5.m1.1.1.3" xref="alg1.l5.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.m1.1b"><apply id="alg1.l5.m1.1.1.cmml" xref="alg1.l5.m1.1.1"><eq id="alg1.l5.m1.1.1.1.cmml" xref="alg1.l5.m1.1.1.1"></eq><apply id="alg1.l5.m1.1.1.2.cmml" xref="alg1.l5.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l5.m1.1.1.2.1.cmml" xref="alg1.l5.m1.1.1.2">superscript</csymbol><apply id="alg1.l5.m1.1.1.2.2.cmml" xref="alg1.l5.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l5.m1.1.1.2.2.1.cmml" xref="alg1.l5.m1.1.1.2">subscript</csymbol><ci id="alg1.l5.m1.1.1.2.2.2.cmml" xref="alg1.l5.m1.1.1.2.2.2">𝐱</ci><ci id="alg1.l5.m1.1.1.2.2.3.cmml" xref="alg1.l5.m1.1.1.2.2.3">𝑡</ci></apply><apply id="alg1.l5.m1.1.1.2.3.cmml" xref="alg1.l5.m1.1.1.2.3"><csymbol cd="ambiguous" id="alg1.l5.m1.1.1.2.3.1.cmml" xref="alg1.l5.m1.1.1.2.3">subscript</csymbol><ci id="alg1.l5.m1.1.1.2.3.2.cmml" xref="alg1.l5.m1.1.1.2.3.2">𝑘</ci><ci id="alg1.l5.m1.1.1.2.3.3.cmml" xref="alg1.l5.m1.1.1.2.3.3">𝑡</ci></apply></apply><csymbol cd="latexml" id="alg1.l5.m1.1.1.3.cmml" xref="alg1.l5.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.1c">\mathbf{x}_{t}^{k_{t}}=\ </annotation></semantics></math><span id="alg1.l5.3" class="ltx_text" style="font-size:80%;">ForwardDiffuse</span><math id="alg1.l5.m2.2" class="ltx_Math" alttext="(\mathbf{x}_{t},k_{t})" display="inline"><semantics id="alg1.l5.m2.2a"><mrow id="alg1.l5.m2.2.2.2" xref="alg1.l5.m2.2.2.3.cmml"><mo maxsize="80%" minsize="80%" id="alg1.l5.m2.2.2.2.3" xref="alg1.l5.m2.2.2.3.cmml">(</mo><msub id="alg1.l5.m2.1.1.1.1" xref="alg1.l5.m2.1.1.1.1.cmml"><mi mathsize="80%" id="alg1.l5.m2.1.1.1.1.2" xref="alg1.l5.m2.1.1.1.1.2.cmml">𝐱</mi><mi mathsize="80%" id="alg1.l5.m2.1.1.1.1.3" xref="alg1.l5.m2.1.1.1.1.3.cmml">t</mi></msub><mo mathsize="80%" id="alg1.l5.m2.2.2.2.4" xref="alg1.l5.m2.2.2.3.cmml">,</mo><msub id="alg1.l5.m2.2.2.2.2" xref="alg1.l5.m2.2.2.2.2.cmml"><mi mathsize="80%" id="alg1.l5.m2.2.2.2.2.2" xref="alg1.l5.m2.2.2.2.2.2.cmml">k</mi><mi mathsize="80%" id="alg1.l5.m2.2.2.2.2.3" xref="alg1.l5.m2.2.2.2.2.3.cmml">t</mi></msub><mo maxsize="80%" minsize="80%" id="alg1.l5.m2.2.2.2.5" xref="alg1.l5.m2.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.m2.2b"><interval closure="open" id="alg1.l5.m2.2.2.3.cmml" xref="alg1.l5.m2.2.2.2"><apply id="alg1.l5.m2.1.1.1.1.cmml" xref="alg1.l5.m2.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l5.m2.1.1.1.1.1.cmml" xref="alg1.l5.m2.1.1.1.1">subscript</csymbol><ci id="alg1.l5.m2.1.1.1.1.2.cmml" xref="alg1.l5.m2.1.1.1.1.2">𝐱</ci><ci id="alg1.l5.m2.1.1.1.1.3.cmml" xref="alg1.l5.m2.1.1.1.1.3">𝑡</ci></apply><apply id="alg1.l5.m2.2.2.2.2.cmml" xref="alg1.l5.m2.2.2.2.2"><csymbol cd="ambiguous" id="alg1.l5.m2.2.2.2.2.1.cmml" xref="alg1.l5.m2.2.2.2.2">subscript</csymbol><ci id="alg1.l5.m2.2.2.2.2.2.cmml" xref="alg1.l5.m2.2.2.2.2.2">𝑘</ci><ci id="alg1.l5.m2.2.2.2.2.3.cmml" xref="alg1.l5.m2.2.2.2.2.3">𝑡</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m2.2c">(\mathbf{x}_{t},k_{t})</annotation></semantics></math><span id="alg1.l5.4" class="ltx_text" style="font-size:80%;">
</span>
</div>
<div id="alg1.l6" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l6.1.1.1" class="ltx_text" style="font-size:80%;">6:</span></span><span id="alg1.l6.2" class="ltx_text" style="font-size:80%;">      Define </span><math id="alg1.l6.m1.1" class="ltx_Math" alttext="\epsilon_{t}=\frac{\mathbf{x}_{t}^{k_{t}}-\sqrt{\bar{\alpha}_{k_{t}}}\mathbf{x}_{t}}{\sqrt{1-\bar{\alpha}_{k_{t}}}}" display="inline"><semantics id="alg1.l6.m1.1a"><mrow id="alg1.l6.m1.1.1" xref="alg1.l6.m1.1.1.cmml"><msub id="alg1.l6.m1.1.1.2" xref="alg1.l6.m1.1.1.2.cmml"><mi mathsize="80%" id="alg1.l6.m1.1.1.2.2" xref="alg1.l6.m1.1.1.2.2.cmml">ϵ</mi><mi mathsize="80%" id="alg1.l6.m1.1.1.2.3" xref="alg1.l6.m1.1.1.2.3.cmml">t</mi></msub><mo mathsize="80%" id="alg1.l6.m1.1.1.1" xref="alg1.l6.m1.1.1.1.cmml">=</mo><mfrac id="alg1.l6.m1.1.1.3" xref="alg1.l6.m1.1.1.3.cmml"><mrow id="alg1.l6.m1.1.1.3.2" xref="alg1.l6.m1.1.1.3.2.cmml"><msubsup id="alg1.l6.m1.1.1.3.2.2" xref="alg1.l6.m1.1.1.3.2.2.cmml"><mi mathsize="80%" id="alg1.l6.m1.1.1.3.2.2.2.2" xref="alg1.l6.m1.1.1.3.2.2.2.2.cmml">𝐱</mi><mi mathsize="80%" id="alg1.l6.m1.1.1.3.2.2.2.3" xref="alg1.l6.m1.1.1.3.2.2.2.3.cmml">t</mi><msub id="alg1.l6.m1.1.1.3.2.2.3" xref="alg1.l6.m1.1.1.3.2.2.3.cmml"><mi mathsize="80%" id="alg1.l6.m1.1.1.3.2.2.3.2" xref="alg1.l6.m1.1.1.3.2.2.3.2.cmml">k</mi><mi mathsize="80%" id="alg1.l6.m1.1.1.3.2.2.3.3" xref="alg1.l6.m1.1.1.3.2.2.3.3.cmml">t</mi></msub></msubsup><mo mathsize="80%" id="alg1.l6.m1.1.1.3.2.1" xref="alg1.l6.m1.1.1.3.2.1.cmml">−</mo><mrow id="alg1.l6.m1.1.1.3.2.3" xref="alg1.l6.m1.1.1.3.2.3.cmml"><msqrt id="alg1.l6.m1.1.1.3.2.3.2" xref="alg1.l6.m1.1.1.3.2.3.2.cmml"><msub id="alg1.l6.m1.1.1.3.2.3.2.2" xref="alg1.l6.m1.1.1.3.2.3.2.2.cmml"><mover accent="true" id="alg1.l6.m1.1.1.3.2.3.2.2.2" xref="alg1.l6.m1.1.1.3.2.3.2.2.2.cmml"><mi mathsize="80%" id="alg1.l6.m1.1.1.3.2.3.2.2.2.2" xref="alg1.l6.m1.1.1.3.2.3.2.2.2.2.cmml">α</mi><mo mathsize="80%" id="alg1.l6.m1.1.1.3.2.3.2.2.2.1" xref="alg1.l6.m1.1.1.3.2.3.2.2.2.1.cmml">¯</mo></mover><msub id="alg1.l6.m1.1.1.3.2.3.2.2.3" xref="alg1.l6.m1.1.1.3.2.3.2.2.3.cmml"><mi mathsize="80%" id="alg1.l6.m1.1.1.3.2.3.2.2.3.2" xref="alg1.l6.m1.1.1.3.2.3.2.2.3.2.cmml">k</mi><mi mathsize="80%" id="alg1.l6.m1.1.1.3.2.3.2.2.3.3" xref="alg1.l6.m1.1.1.3.2.3.2.2.3.3.cmml">t</mi></msub></msub></msqrt><mo lspace="0em" rspace="0em" id="alg1.l6.m1.1.1.3.2.3.1" xref="alg1.l6.m1.1.1.3.2.3.1.cmml">​</mo><msub id="alg1.l6.m1.1.1.3.2.3.3" xref="alg1.l6.m1.1.1.3.2.3.3.cmml"><mi mathsize="80%" id="alg1.l6.m1.1.1.3.2.3.3.2" xref="alg1.l6.m1.1.1.3.2.3.3.2.cmml">𝐱</mi><mi mathsize="80%" id="alg1.l6.m1.1.1.3.2.3.3.3" xref="alg1.l6.m1.1.1.3.2.3.3.3.cmml">t</mi></msub></mrow></mrow><msqrt id="alg1.l6.m1.1.1.3.3" xref="alg1.l6.m1.1.1.3.3.cmml"><mrow id="alg1.l6.m1.1.1.3.3.2" xref="alg1.l6.m1.1.1.3.3.2.cmml"><mn mathsize="80%" id="alg1.l6.m1.1.1.3.3.2.2" xref="alg1.l6.m1.1.1.3.3.2.2.cmml">1</mn><mo mathsize="80%" id="alg1.l6.m1.1.1.3.3.2.1" xref="alg1.l6.m1.1.1.3.3.2.1.cmml">−</mo><msub id="alg1.l6.m1.1.1.3.3.2.3" xref="alg1.l6.m1.1.1.3.3.2.3.cmml"><mover accent="true" id="alg1.l6.m1.1.1.3.3.2.3.2" xref="alg1.l6.m1.1.1.3.3.2.3.2.cmml"><mi mathsize="80%" id="alg1.l6.m1.1.1.3.3.2.3.2.2" xref="alg1.l6.m1.1.1.3.3.2.3.2.2.cmml">α</mi><mo mathsize="80%" id="alg1.l6.m1.1.1.3.3.2.3.2.1" xref="alg1.l6.m1.1.1.3.3.2.3.2.1.cmml">¯</mo></mover><msub id="alg1.l6.m1.1.1.3.3.2.3.3" xref="alg1.l6.m1.1.1.3.3.2.3.3.cmml"><mi mathsize="80%" id="alg1.l6.m1.1.1.3.3.2.3.3.2" xref="alg1.l6.m1.1.1.3.3.2.3.3.2.cmml">k</mi><mi mathsize="80%" id="alg1.l6.m1.1.1.3.3.2.3.3.3" xref="alg1.l6.m1.1.1.3.3.2.3.3.3.cmml">t</mi></msub></msub></mrow></msqrt></mfrac></mrow><annotation-xml encoding="MathML-Content" id="alg1.l6.m1.1b"><apply id="alg1.l6.m1.1.1.cmml" xref="alg1.l6.m1.1.1"><eq id="alg1.l6.m1.1.1.1.cmml" xref="alg1.l6.m1.1.1.1"></eq><apply id="alg1.l6.m1.1.1.2.cmml" xref="alg1.l6.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.2.1.cmml" xref="alg1.l6.m1.1.1.2">subscript</csymbol><ci id="alg1.l6.m1.1.1.2.2.cmml" xref="alg1.l6.m1.1.1.2.2">italic-ϵ</ci><ci id="alg1.l6.m1.1.1.2.3.cmml" xref="alg1.l6.m1.1.1.2.3">𝑡</ci></apply><apply id="alg1.l6.m1.1.1.3.cmml" xref="alg1.l6.m1.1.1.3"><divide id="alg1.l6.m1.1.1.3.1.cmml" xref="alg1.l6.m1.1.1.3"></divide><apply id="alg1.l6.m1.1.1.3.2.cmml" xref="alg1.l6.m1.1.1.3.2"><minus id="alg1.l6.m1.1.1.3.2.1.cmml" xref="alg1.l6.m1.1.1.3.2.1"></minus><apply id="alg1.l6.m1.1.1.3.2.2.cmml" xref="alg1.l6.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.3.2.2.1.cmml" xref="alg1.l6.m1.1.1.3.2.2">superscript</csymbol><apply id="alg1.l6.m1.1.1.3.2.2.2.cmml" xref="alg1.l6.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.3.2.2.2.1.cmml" xref="alg1.l6.m1.1.1.3.2.2">subscript</csymbol><ci id="alg1.l6.m1.1.1.3.2.2.2.2.cmml" xref="alg1.l6.m1.1.1.3.2.2.2.2">𝐱</ci><ci id="alg1.l6.m1.1.1.3.2.2.2.3.cmml" xref="alg1.l6.m1.1.1.3.2.2.2.3">𝑡</ci></apply><apply id="alg1.l6.m1.1.1.3.2.2.3.cmml" xref="alg1.l6.m1.1.1.3.2.2.3"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.3.2.2.3.1.cmml" xref="alg1.l6.m1.1.1.3.2.2.3">subscript</csymbol><ci id="alg1.l6.m1.1.1.3.2.2.3.2.cmml" xref="alg1.l6.m1.1.1.3.2.2.3.2">𝑘</ci><ci id="alg1.l6.m1.1.1.3.2.2.3.3.cmml" xref="alg1.l6.m1.1.1.3.2.2.3.3">𝑡</ci></apply></apply><apply id="alg1.l6.m1.1.1.3.2.3.cmml" xref="alg1.l6.m1.1.1.3.2.3"><times id="alg1.l6.m1.1.1.3.2.3.1.cmml" xref="alg1.l6.m1.1.1.3.2.3.1"></times><apply id="alg1.l6.m1.1.1.3.2.3.2.cmml" xref="alg1.l6.m1.1.1.3.2.3.2"><root id="alg1.l6.m1.1.1.3.2.3.2a.cmml" xref="alg1.l6.m1.1.1.3.2.3.2"></root><apply id="alg1.l6.m1.1.1.3.2.3.2.2.cmml" xref="alg1.l6.m1.1.1.3.2.3.2.2"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.3.2.3.2.2.1.cmml" xref="alg1.l6.m1.1.1.3.2.3.2.2">subscript</csymbol><apply id="alg1.l6.m1.1.1.3.2.3.2.2.2.cmml" xref="alg1.l6.m1.1.1.3.2.3.2.2.2"><ci id="alg1.l6.m1.1.1.3.2.3.2.2.2.1.cmml" xref="alg1.l6.m1.1.1.3.2.3.2.2.2.1">¯</ci><ci id="alg1.l6.m1.1.1.3.2.3.2.2.2.2.cmml" xref="alg1.l6.m1.1.1.3.2.3.2.2.2.2">𝛼</ci></apply><apply id="alg1.l6.m1.1.1.3.2.3.2.2.3.cmml" xref="alg1.l6.m1.1.1.3.2.3.2.2.3"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.3.2.3.2.2.3.1.cmml" xref="alg1.l6.m1.1.1.3.2.3.2.2.3">subscript</csymbol><ci id="alg1.l6.m1.1.1.3.2.3.2.2.3.2.cmml" xref="alg1.l6.m1.1.1.3.2.3.2.2.3.2">𝑘</ci><ci id="alg1.l6.m1.1.1.3.2.3.2.2.3.3.cmml" xref="alg1.l6.m1.1.1.3.2.3.2.2.3.3">𝑡</ci></apply></apply></apply><apply id="alg1.l6.m1.1.1.3.2.3.3.cmml" xref="alg1.l6.m1.1.1.3.2.3.3"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.3.2.3.3.1.cmml" xref="alg1.l6.m1.1.1.3.2.3.3">subscript</csymbol><ci id="alg1.l6.m1.1.1.3.2.3.3.2.cmml" xref="alg1.l6.m1.1.1.3.2.3.3.2">𝐱</ci><ci id="alg1.l6.m1.1.1.3.2.3.3.3.cmml" xref="alg1.l6.m1.1.1.3.2.3.3.3">𝑡</ci></apply></apply></apply><apply id="alg1.l6.m1.1.1.3.3.cmml" xref="alg1.l6.m1.1.1.3.3"><root id="alg1.l6.m1.1.1.3.3a.cmml" xref="alg1.l6.m1.1.1.3.3"></root><apply id="alg1.l6.m1.1.1.3.3.2.cmml" xref="alg1.l6.m1.1.1.3.3.2"><minus id="alg1.l6.m1.1.1.3.3.2.1.cmml" xref="alg1.l6.m1.1.1.3.3.2.1"></minus><cn type="integer" id="alg1.l6.m1.1.1.3.3.2.2.cmml" xref="alg1.l6.m1.1.1.3.3.2.2">1</cn><apply id="alg1.l6.m1.1.1.3.3.2.3.cmml" xref="alg1.l6.m1.1.1.3.3.2.3"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.3.3.2.3.1.cmml" xref="alg1.l6.m1.1.1.3.3.2.3">subscript</csymbol><apply id="alg1.l6.m1.1.1.3.3.2.3.2.cmml" xref="alg1.l6.m1.1.1.3.3.2.3.2"><ci id="alg1.l6.m1.1.1.3.3.2.3.2.1.cmml" xref="alg1.l6.m1.1.1.3.3.2.3.2.1">¯</ci><ci id="alg1.l6.m1.1.1.3.3.2.3.2.2.cmml" xref="alg1.l6.m1.1.1.3.3.2.3.2.2">𝛼</ci></apply><apply id="alg1.l6.m1.1.1.3.3.2.3.3.cmml" xref="alg1.l6.m1.1.1.3.3.2.3.3"><csymbol cd="ambiguous" id="alg1.l6.m1.1.1.3.3.2.3.3.1.cmml" xref="alg1.l6.m1.1.1.3.3.2.3.3">subscript</csymbol><ci id="alg1.l6.m1.1.1.3.3.2.3.3.2.cmml" xref="alg1.l6.m1.1.1.3.3.2.3.3.2">𝑘</ci><ci id="alg1.l6.m1.1.1.3.3.2.3.3.3.cmml" xref="alg1.l6.m1.1.1.3.3.2.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m1.1c">\epsilon_{t}=\frac{\mathbf{x}_{t}^{k_{t}}-\sqrt{\bar{\alpha}_{k_{t}}}\mathbf{x}_{t}}{\sqrt{1-\bar{\alpha}_{k_{t}}}}</annotation></semantics></math><span id="alg1.l6.3" class="ltx_text" style="font-size:80%;">
</span>
</div>
<div id="alg1.l7" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l7.1.1.1" class="ltx_text" style="font-size:80%;">7:</span></span><span id="alg1.l7.2" class="ltx_text" style="font-size:80%;">      Update </span><math id="alg1.l7.m1.1" class="ltx_Math" alttext="\mathbf{z}_{t}\sim p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})" display="inline"><semantics id="alg1.l7.m1.1a"><mrow id="alg1.l7.m1.1.1" xref="alg1.l7.m1.1.1.cmml"><msub id="alg1.l7.m1.1.1.3" xref="alg1.l7.m1.1.1.3.cmml"><mi mathsize="80%" id="alg1.l7.m1.1.1.3.2" xref="alg1.l7.m1.1.1.3.2.cmml">𝐳</mi><mi mathsize="80%" id="alg1.l7.m1.1.1.3.3" xref="alg1.l7.m1.1.1.3.3.cmml">t</mi></msub><mo mathsize="80%" id="alg1.l7.m1.1.1.2" xref="alg1.l7.m1.1.1.2.cmml">∼</mo><mrow id="alg1.l7.m1.1.1.1" xref="alg1.l7.m1.1.1.1.cmml"><msub id="alg1.l7.m1.1.1.1.3" xref="alg1.l7.m1.1.1.1.3.cmml"><mi mathsize="80%" id="alg1.l7.m1.1.1.1.3.2" xref="alg1.l7.m1.1.1.1.3.2.cmml">p</mi><mi mathsize="80%" id="alg1.l7.m1.1.1.1.3.3" xref="alg1.l7.m1.1.1.1.3.3.cmml">θ</mi></msub><mo lspace="0em" rspace="0em" id="alg1.l7.m1.1.1.1.2" xref="alg1.l7.m1.1.1.1.2.cmml">​</mo><mrow id="alg1.l7.m1.1.1.1.1.1" xref="alg1.l7.m1.1.1.1.1.1.1.cmml"><mo maxsize="80%" minsize="80%" id="alg1.l7.m1.1.1.1.1.1.2" xref="alg1.l7.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="alg1.l7.m1.1.1.1.1.1.1" xref="alg1.l7.m1.1.1.1.1.1.1.cmml"><msub id="alg1.l7.m1.1.1.1.1.1.1.5" xref="alg1.l7.m1.1.1.1.1.1.1.5.cmml"><mi mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.5.2" xref="alg1.l7.m1.1.1.1.1.1.1.5.2.cmml">𝐳</mi><mi mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.5.3" xref="alg1.l7.m1.1.1.1.1.1.1.5.3.cmml">t</mi></msub><mo fence="false" mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.4" xref="alg1.l7.m1.1.1.1.1.1.1.4.cmml">|</mo><mrow id="alg1.l7.m1.1.1.1.1.1.1.3.3" xref="alg1.l7.m1.1.1.1.1.1.1.3.4.cmml"><msub id="alg1.l7.m1.1.1.1.1.1.1.1.1.1" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.cmml"><mi mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.2" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.2" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">t</mi><mo mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.1" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.3" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.3.3.4" xref="alg1.l7.m1.1.1.1.1.1.1.3.4.cmml">,</mo><msubsup id="alg1.l7.m1.1.1.1.1.1.1.2.2.2" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.cmml"><mi mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.2.2" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mi mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.2.3" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.2.3.cmml">t</mi><msub id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.cmml"><mi mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.2" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.2.cmml">k</mi><mi mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.3" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.3.cmml">t</mi></msub></msubsup><mo mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.3.3.5" xref="alg1.l7.m1.1.1.1.1.1.1.3.4.cmml">,</mo><msub id="alg1.l7.m1.1.1.1.1.1.1.3.3.3" xref="alg1.l7.m1.1.1.1.1.1.1.3.3.3.cmml"><mi mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.3.3.3.2" xref="alg1.l7.m1.1.1.1.1.1.1.3.3.3.2.cmml">k</mi><mi mathsize="80%" id="alg1.l7.m1.1.1.1.1.1.1.3.3.3.3" xref="alg1.l7.m1.1.1.1.1.1.1.3.3.3.3.cmml">t</mi></msub></mrow></mrow><mo maxsize="80%" minsize="80%" id="alg1.l7.m1.1.1.1.1.1.3" xref="alg1.l7.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l7.m1.1b"><apply id="alg1.l7.m1.1.1.cmml" xref="alg1.l7.m1.1.1"><csymbol cd="latexml" id="alg1.l7.m1.1.1.2.cmml" xref="alg1.l7.m1.1.1.2">similar-to</csymbol><apply id="alg1.l7.m1.1.1.3.cmml" xref="alg1.l7.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.3.1.cmml" xref="alg1.l7.m1.1.1.3">subscript</csymbol><ci id="alg1.l7.m1.1.1.3.2.cmml" xref="alg1.l7.m1.1.1.3.2">𝐳</ci><ci id="alg1.l7.m1.1.1.3.3.cmml" xref="alg1.l7.m1.1.1.3.3">𝑡</ci></apply><apply id="alg1.l7.m1.1.1.1.cmml" xref="alg1.l7.m1.1.1.1"><times id="alg1.l7.m1.1.1.1.2.cmml" xref="alg1.l7.m1.1.1.1.2"></times><apply id="alg1.l7.m1.1.1.1.3.cmml" xref="alg1.l7.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.1.3.1.cmml" xref="alg1.l7.m1.1.1.1.3">subscript</csymbol><ci id="alg1.l7.m1.1.1.1.3.2.cmml" xref="alg1.l7.m1.1.1.1.3.2">𝑝</ci><ci id="alg1.l7.m1.1.1.1.3.3.cmml" xref="alg1.l7.m1.1.1.1.3.3">𝜃</ci></apply><apply id="alg1.l7.m1.1.1.1.1.1.1.cmml" xref="alg1.l7.m1.1.1.1.1.1"><csymbol cd="latexml" id="alg1.l7.m1.1.1.1.1.1.1.4.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.4">conditional</csymbol><apply id="alg1.l7.m1.1.1.1.1.1.1.5.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.5"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.1.1.1.1.5.1.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.5">subscript</csymbol><ci id="alg1.l7.m1.1.1.1.1.1.1.5.2.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.5.2">𝐳</ci><ci id="alg1.l7.m1.1.1.1.1.1.1.5.3.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.5.3">𝑡</ci></apply><list id="alg1.l7.m1.1.1.1.1.1.1.3.4.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.3.3"><apply id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.2">𝐳</ci><apply id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3"><minus id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.1"></minus><ci id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.2">𝑡</ci><cn type="integer" id="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.1.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2">superscript</csymbol><apply id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.2.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.2.1.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.2.2.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.2.2">𝐱</ci><ci id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.2.3.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.2.3">𝑡</ci></apply><apply id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.1.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3">subscript</csymbol><ci id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.2.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.2">𝑘</ci><ci id="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.3.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.2.2.2.3.3">𝑡</ci></apply></apply><apply id="alg1.l7.m1.1.1.1.1.1.1.3.3.3.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="alg1.l7.m1.1.1.1.1.1.1.3.3.3.1.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.3.3.3">subscript</csymbol><ci id="alg1.l7.m1.1.1.1.1.1.1.3.3.3.2.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.3.3.3.2">𝑘</ci><ci id="alg1.l7.m1.1.1.1.1.1.1.3.3.3.3.cmml" xref="alg1.l7.m1.1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m1.1c">\mathbf{z}_{t}\sim p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})</annotation></semantics></math><span id="alg1.l7.3" class="ltx_text" style="font-size:80%;">.
</span>
</div>
<div id="alg1.l8" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l8.1.1.1" class="ltx_text" style="font-size:80%;">8:</span></span><span id="alg1.l8.2" class="ltx_text" style="font-size:80%;">      Set </span><math id="alg1.l8.m1.3" class="ltx_Math" alttext="\hat{\epsilon}_{t}=\epsilon_{\theta}(\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})" display="inline"><semantics id="alg1.l8.m1.3a"><mrow id="alg1.l8.m1.3.3" xref="alg1.l8.m1.3.3.cmml"><msub id="alg1.l8.m1.3.3.5" xref="alg1.l8.m1.3.3.5.cmml"><mover accent="true" id="alg1.l8.m1.3.3.5.2" xref="alg1.l8.m1.3.3.5.2.cmml"><mi mathsize="80%" id="alg1.l8.m1.3.3.5.2.2" xref="alg1.l8.m1.3.3.5.2.2.cmml">ϵ</mi><mo mathsize="80%" id="alg1.l8.m1.3.3.5.2.1" xref="alg1.l8.m1.3.3.5.2.1.cmml">^</mo></mover><mi mathsize="80%" id="alg1.l8.m1.3.3.5.3" xref="alg1.l8.m1.3.3.5.3.cmml">t</mi></msub><mo mathsize="80%" id="alg1.l8.m1.3.3.4" xref="alg1.l8.m1.3.3.4.cmml">=</mo><mrow id="alg1.l8.m1.3.3.3" xref="alg1.l8.m1.3.3.3.cmml"><msub id="alg1.l8.m1.3.3.3.5" xref="alg1.l8.m1.3.3.3.5.cmml"><mi mathsize="80%" id="alg1.l8.m1.3.3.3.5.2" xref="alg1.l8.m1.3.3.3.5.2.cmml">ϵ</mi><mi mathsize="80%" id="alg1.l8.m1.3.3.3.5.3" xref="alg1.l8.m1.3.3.3.5.3.cmml">θ</mi></msub><mo lspace="0em" rspace="0em" id="alg1.l8.m1.3.3.3.4" xref="alg1.l8.m1.3.3.3.4.cmml">​</mo><mrow id="alg1.l8.m1.3.3.3.3.3" xref="alg1.l8.m1.3.3.3.3.4.cmml"><mo maxsize="80%" minsize="80%" id="alg1.l8.m1.3.3.3.3.3.4" xref="alg1.l8.m1.3.3.3.3.4.cmml">(</mo><msub id="alg1.l8.m1.1.1.1.1.1.1" xref="alg1.l8.m1.1.1.1.1.1.1.cmml"><mi mathsize="80%" id="alg1.l8.m1.1.1.1.1.1.1.2" xref="alg1.l8.m1.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="alg1.l8.m1.1.1.1.1.1.1.3" xref="alg1.l8.m1.1.1.1.1.1.1.3.cmml"><mi mathsize="80%" id="alg1.l8.m1.1.1.1.1.1.1.3.2" xref="alg1.l8.m1.1.1.1.1.1.1.3.2.cmml">t</mi><mo mathsize="80%" id="alg1.l8.m1.1.1.1.1.1.1.3.1" xref="alg1.l8.m1.1.1.1.1.1.1.3.1.cmml">−</mo><mn mathsize="80%" id="alg1.l8.m1.1.1.1.1.1.1.3.3" xref="alg1.l8.m1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo mathsize="80%" id="alg1.l8.m1.3.3.3.3.3.5" xref="alg1.l8.m1.3.3.3.3.4.cmml">,</mo><msubsup id="alg1.l8.m1.2.2.2.2.2.2" xref="alg1.l8.m1.2.2.2.2.2.2.cmml"><mi mathsize="80%" id="alg1.l8.m1.2.2.2.2.2.2.2.2" xref="alg1.l8.m1.2.2.2.2.2.2.2.2.cmml">𝐱</mi><mi mathsize="80%" id="alg1.l8.m1.2.2.2.2.2.2.2.3" xref="alg1.l8.m1.2.2.2.2.2.2.2.3.cmml">t</mi><msub id="alg1.l8.m1.2.2.2.2.2.2.3" xref="alg1.l8.m1.2.2.2.2.2.2.3.cmml"><mi mathsize="80%" id="alg1.l8.m1.2.2.2.2.2.2.3.2" xref="alg1.l8.m1.2.2.2.2.2.2.3.2.cmml">k</mi><mi mathsize="80%" id="alg1.l8.m1.2.2.2.2.2.2.3.3" xref="alg1.l8.m1.2.2.2.2.2.2.3.3.cmml">t</mi></msub></msubsup><mo mathsize="80%" id="alg1.l8.m1.3.3.3.3.3.6" xref="alg1.l8.m1.3.3.3.3.4.cmml">,</mo><msub id="alg1.l8.m1.3.3.3.3.3.3" xref="alg1.l8.m1.3.3.3.3.3.3.cmml"><mi mathsize="80%" id="alg1.l8.m1.3.3.3.3.3.3.2" xref="alg1.l8.m1.3.3.3.3.3.3.2.cmml">k</mi><mi mathsize="80%" id="alg1.l8.m1.3.3.3.3.3.3.3" xref="alg1.l8.m1.3.3.3.3.3.3.3.cmml">t</mi></msub><mo maxsize="80%" minsize="80%" id="alg1.l8.m1.3.3.3.3.3.7" xref="alg1.l8.m1.3.3.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l8.m1.3b"><apply id="alg1.l8.m1.3.3.cmml" xref="alg1.l8.m1.3.3"><eq id="alg1.l8.m1.3.3.4.cmml" xref="alg1.l8.m1.3.3.4"></eq><apply id="alg1.l8.m1.3.3.5.cmml" xref="alg1.l8.m1.3.3.5"><csymbol cd="ambiguous" id="alg1.l8.m1.3.3.5.1.cmml" xref="alg1.l8.m1.3.3.5">subscript</csymbol><apply id="alg1.l8.m1.3.3.5.2.cmml" xref="alg1.l8.m1.3.3.5.2"><ci id="alg1.l8.m1.3.3.5.2.1.cmml" xref="alg1.l8.m1.3.3.5.2.1">^</ci><ci id="alg1.l8.m1.3.3.5.2.2.cmml" xref="alg1.l8.m1.3.3.5.2.2">italic-ϵ</ci></apply><ci id="alg1.l8.m1.3.3.5.3.cmml" xref="alg1.l8.m1.3.3.5.3">𝑡</ci></apply><apply id="alg1.l8.m1.3.3.3.cmml" xref="alg1.l8.m1.3.3.3"><times id="alg1.l8.m1.3.3.3.4.cmml" xref="alg1.l8.m1.3.3.3.4"></times><apply id="alg1.l8.m1.3.3.3.5.cmml" xref="alg1.l8.m1.3.3.3.5"><csymbol cd="ambiguous" id="alg1.l8.m1.3.3.3.5.1.cmml" xref="alg1.l8.m1.3.3.3.5">subscript</csymbol><ci id="alg1.l8.m1.3.3.3.5.2.cmml" xref="alg1.l8.m1.3.3.3.5.2">italic-ϵ</ci><ci id="alg1.l8.m1.3.3.3.5.3.cmml" xref="alg1.l8.m1.3.3.3.5.3">𝜃</ci></apply><vector id="alg1.l8.m1.3.3.3.3.4.cmml" xref="alg1.l8.m1.3.3.3.3.3"><apply id="alg1.l8.m1.1.1.1.1.1.1.cmml" xref="alg1.l8.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l8.m1.1.1.1.1.1.1.1.cmml" xref="alg1.l8.m1.1.1.1.1.1.1">subscript</csymbol><ci id="alg1.l8.m1.1.1.1.1.1.1.2.cmml" xref="alg1.l8.m1.1.1.1.1.1.1.2">𝐳</ci><apply id="alg1.l8.m1.1.1.1.1.1.1.3.cmml" xref="alg1.l8.m1.1.1.1.1.1.1.3"><minus id="alg1.l8.m1.1.1.1.1.1.1.3.1.cmml" xref="alg1.l8.m1.1.1.1.1.1.1.3.1"></minus><ci id="alg1.l8.m1.1.1.1.1.1.1.3.2.cmml" xref="alg1.l8.m1.1.1.1.1.1.1.3.2">𝑡</ci><cn type="integer" id="alg1.l8.m1.1.1.1.1.1.1.3.3.cmml" xref="alg1.l8.m1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="alg1.l8.m1.2.2.2.2.2.2.cmml" xref="alg1.l8.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="alg1.l8.m1.2.2.2.2.2.2.1.cmml" xref="alg1.l8.m1.2.2.2.2.2.2">superscript</csymbol><apply id="alg1.l8.m1.2.2.2.2.2.2.2.cmml" xref="alg1.l8.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="alg1.l8.m1.2.2.2.2.2.2.2.1.cmml" xref="alg1.l8.m1.2.2.2.2.2.2">subscript</csymbol><ci id="alg1.l8.m1.2.2.2.2.2.2.2.2.cmml" xref="alg1.l8.m1.2.2.2.2.2.2.2.2">𝐱</ci><ci id="alg1.l8.m1.2.2.2.2.2.2.2.3.cmml" xref="alg1.l8.m1.2.2.2.2.2.2.2.3">𝑡</ci></apply><apply id="alg1.l8.m1.2.2.2.2.2.2.3.cmml" xref="alg1.l8.m1.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="alg1.l8.m1.2.2.2.2.2.2.3.1.cmml" xref="alg1.l8.m1.2.2.2.2.2.2.3">subscript</csymbol><ci id="alg1.l8.m1.2.2.2.2.2.2.3.2.cmml" xref="alg1.l8.m1.2.2.2.2.2.2.3.2">𝑘</ci><ci id="alg1.l8.m1.2.2.2.2.2.2.3.3.cmml" xref="alg1.l8.m1.2.2.2.2.2.2.3.3">𝑡</ci></apply></apply><apply id="alg1.l8.m1.3.3.3.3.3.3.cmml" xref="alg1.l8.m1.3.3.3.3.3.3"><csymbol cd="ambiguous" id="alg1.l8.m1.3.3.3.3.3.3.1.cmml" xref="alg1.l8.m1.3.3.3.3.3.3">subscript</csymbol><ci id="alg1.l8.m1.3.3.3.3.3.3.2.cmml" xref="alg1.l8.m1.3.3.3.3.3.3.2">𝑘</ci><ci id="alg1.l8.m1.3.3.3.3.3.3.3.cmml" xref="alg1.l8.m1.3.3.3.3.3.3.3">𝑡</ci></apply></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l8.m1.3c">\hat{\epsilon}_{t}=\epsilon_{\theta}(\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})</annotation></semantics></math><span id="alg1.l8.3" class="ltx_text" style="font-size:80%;">
</span>
</div>
<div id="alg1.l9" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l9.1.1.1" class="ltx_text" style="font-size:80%;">9:</span></span><span id="alg1.l9.2" class="ltx_text" style="font-size:80%;">    </span><span id="alg1.l9.3" class="ltx_text ltx_font_bold" style="font-size:80%;">end</span><span id="alg1.l9.4" class="ltx_text" style="font-size:80%;"> </span><span id="alg1.l9.5" class="ltx_text ltx_font_bold" style="font-size:80%;">for</span>
</div>
<div id="alg1.l10" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l10.1.1.1" class="ltx_text" style="font-size:80%;">10:</span></span><span id="alg1.l10.2" class="ltx_text" style="font-size:80%;">    </span><math id="alg1.l10.m1.1" class="ltx_Math" alttext="L=" display="inline"><semantics id="alg1.l10.m1.1a"><mrow id="alg1.l10.m1.1.1" xref="alg1.l10.m1.1.1.cmml"><mi mathsize="80%" id="alg1.l10.m1.1.1.2" xref="alg1.l10.m1.1.1.2.cmml">L</mi><mo mathsize="80%" id="alg1.l10.m1.1.1.1" xref="alg1.l10.m1.1.1.1.cmml">=</mo><mi id="alg1.l10.m1.1.1.3" xref="alg1.l10.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l10.m1.1b"><apply id="alg1.l10.m1.1.1.cmml" xref="alg1.l10.m1.1.1"><eq id="alg1.l10.m1.1.1.1.cmml" xref="alg1.l10.m1.1.1.1"></eq><ci id="alg1.l10.m1.1.1.2.cmml" xref="alg1.l10.m1.1.1.2">𝐿</ci><csymbol cd="latexml" id="alg1.l10.m1.1.1.3.cmml" xref="alg1.l10.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m1.1c">L=</annotation></semantics></math><span id="alg1.l10.3" class="ltx_text" style="font-size:80%;">MSELoss</span><math id="alg1.l10.m2.4" class="ltx_Math" alttext="(\left[\hat{\epsilon}_{1},...,\hat{\epsilon}_{n}\right],\left[\epsilon_{1},...,\epsilon_{n}\right])" display="inline"><semantics id="alg1.l10.m2.4a"><mrow id="alg1.l10.m2.4.4.2" xref="alg1.l10.m2.4.4.3.cmml"><mo maxsize="80%" minsize="80%" id="alg1.l10.m2.4.4.2.3" xref="alg1.l10.m2.4.4.3.cmml">(</mo><mrow id="alg1.l10.m2.3.3.1.1.2" xref="alg1.l10.m2.3.3.1.1.3.cmml"><mo id="alg1.l10.m2.3.3.1.1.2.3" xref="alg1.l10.m2.3.3.1.1.3.cmml">[</mo><msub id="alg1.l10.m2.3.3.1.1.1.1" xref="alg1.l10.m2.3.3.1.1.1.1.cmml"><mover accent="true" id="alg1.l10.m2.3.3.1.1.1.1.2" xref="alg1.l10.m2.3.3.1.1.1.1.2.cmml"><mi mathsize="80%" id="alg1.l10.m2.3.3.1.1.1.1.2.2" xref="alg1.l10.m2.3.3.1.1.1.1.2.2.cmml">ϵ</mi><mo mathsize="80%" id="alg1.l10.m2.3.3.1.1.1.1.2.1" xref="alg1.l10.m2.3.3.1.1.1.1.2.1.cmml">^</mo></mover><mn mathsize="80%" id="alg1.l10.m2.3.3.1.1.1.1.3" xref="alg1.l10.m2.3.3.1.1.1.1.3.cmml">1</mn></msub><mo mathsize="80%" id="alg1.l10.m2.3.3.1.1.2.4" xref="alg1.l10.m2.3.3.1.1.3.cmml">,</mo><mi mathsize="80%" mathvariant="normal" id="alg1.l10.m2.1.1" xref="alg1.l10.m2.1.1.cmml">…</mi><mo mathsize="80%" id="alg1.l10.m2.3.3.1.1.2.5" xref="alg1.l10.m2.3.3.1.1.3.cmml">,</mo><msub id="alg1.l10.m2.3.3.1.1.2.2" xref="alg1.l10.m2.3.3.1.1.2.2.cmml"><mover accent="true" id="alg1.l10.m2.3.3.1.1.2.2.2" xref="alg1.l10.m2.3.3.1.1.2.2.2.cmml"><mi mathsize="80%" id="alg1.l10.m2.3.3.1.1.2.2.2.2" xref="alg1.l10.m2.3.3.1.1.2.2.2.2.cmml">ϵ</mi><mo mathsize="80%" id="alg1.l10.m2.3.3.1.1.2.2.2.1" xref="alg1.l10.m2.3.3.1.1.2.2.2.1.cmml">^</mo></mover><mi mathsize="80%" id="alg1.l10.m2.3.3.1.1.2.2.3" xref="alg1.l10.m2.3.3.1.1.2.2.3.cmml">n</mi></msub><mo id="alg1.l10.m2.3.3.1.1.2.6" xref="alg1.l10.m2.3.3.1.1.3.cmml">]</mo></mrow><mo mathsize="80%" id="alg1.l10.m2.4.4.2.4" xref="alg1.l10.m2.4.4.3.cmml">,</mo><mrow id="alg1.l10.m2.4.4.2.2.2" xref="alg1.l10.m2.4.4.2.2.3.cmml"><mo id="alg1.l10.m2.4.4.2.2.2.3" xref="alg1.l10.m2.4.4.2.2.3.cmml">[</mo><msub id="alg1.l10.m2.4.4.2.2.1.1" xref="alg1.l10.m2.4.4.2.2.1.1.cmml"><mi mathsize="80%" id="alg1.l10.m2.4.4.2.2.1.1.2" xref="alg1.l10.m2.4.4.2.2.1.1.2.cmml">ϵ</mi><mn mathsize="80%" id="alg1.l10.m2.4.4.2.2.1.1.3" xref="alg1.l10.m2.4.4.2.2.1.1.3.cmml">1</mn></msub><mo mathsize="80%" id="alg1.l10.m2.4.4.2.2.2.4" xref="alg1.l10.m2.4.4.2.2.3.cmml">,</mo><mi mathsize="80%" mathvariant="normal" id="alg1.l10.m2.2.2" xref="alg1.l10.m2.2.2.cmml">…</mi><mo mathsize="80%" id="alg1.l10.m2.4.4.2.2.2.5" xref="alg1.l10.m2.4.4.2.2.3.cmml">,</mo><msub id="alg1.l10.m2.4.4.2.2.2.2" xref="alg1.l10.m2.4.4.2.2.2.2.cmml"><mi mathsize="80%" id="alg1.l10.m2.4.4.2.2.2.2.2" xref="alg1.l10.m2.4.4.2.2.2.2.2.cmml">ϵ</mi><mi mathsize="80%" id="alg1.l10.m2.4.4.2.2.2.2.3" xref="alg1.l10.m2.4.4.2.2.2.2.3.cmml">n</mi></msub><mo id="alg1.l10.m2.4.4.2.2.2.6" xref="alg1.l10.m2.4.4.2.2.3.cmml">]</mo></mrow><mo maxsize="80%" minsize="80%" id="alg1.l10.m2.4.4.2.5" xref="alg1.l10.m2.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="alg1.l10.m2.4b"><interval closure="open" id="alg1.l10.m2.4.4.3.cmml" xref="alg1.l10.m2.4.4.2"><list id="alg1.l10.m2.3.3.1.1.3.cmml" xref="alg1.l10.m2.3.3.1.1.2"><apply id="alg1.l10.m2.3.3.1.1.1.1.cmml" xref="alg1.l10.m2.3.3.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l10.m2.3.3.1.1.1.1.1.cmml" xref="alg1.l10.m2.3.3.1.1.1.1">subscript</csymbol><apply id="alg1.l10.m2.3.3.1.1.1.1.2.cmml" xref="alg1.l10.m2.3.3.1.1.1.1.2"><ci id="alg1.l10.m2.3.3.1.1.1.1.2.1.cmml" xref="alg1.l10.m2.3.3.1.1.1.1.2.1">^</ci><ci id="alg1.l10.m2.3.3.1.1.1.1.2.2.cmml" xref="alg1.l10.m2.3.3.1.1.1.1.2.2">italic-ϵ</ci></apply><cn type="integer" id="alg1.l10.m2.3.3.1.1.1.1.3.cmml" xref="alg1.l10.m2.3.3.1.1.1.1.3">1</cn></apply><ci id="alg1.l10.m2.1.1.cmml" xref="alg1.l10.m2.1.1">…</ci><apply id="alg1.l10.m2.3.3.1.1.2.2.cmml" xref="alg1.l10.m2.3.3.1.1.2.2"><csymbol cd="ambiguous" id="alg1.l10.m2.3.3.1.1.2.2.1.cmml" xref="alg1.l10.m2.3.3.1.1.2.2">subscript</csymbol><apply id="alg1.l10.m2.3.3.1.1.2.2.2.cmml" xref="alg1.l10.m2.3.3.1.1.2.2.2"><ci id="alg1.l10.m2.3.3.1.1.2.2.2.1.cmml" xref="alg1.l10.m2.3.3.1.1.2.2.2.1">^</ci><ci id="alg1.l10.m2.3.3.1.1.2.2.2.2.cmml" xref="alg1.l10.m2.3.3.1.1.2.2.2.2">italic-ϵ</ci></apply><ci id="alg1.l10.m2.3.3.1.1.2.2.3.cmml" xref="alg1.l10.m2.3.3.1.1.2.2.3">𝑛</ci></apply></list><list id="alg1.l10.m2.4.4.2.2.3.cmml" xref="alg1.l10.m2.4.4.2.2.2"><apply id="alg1.l10.m2.4.4.2.2.1.1.cmml" xref="alg1.l10.m2.4.4.2.2.1.1"><csymbol cd="ambiguous" id="alg1.l10.m2.4.4.2.2.1.1.1.cmml" xref="alg1.l10.m2.4.4.2.2.1.1">subscript</csymbol><ci id="alg1.l10.m2.4.4.2.2.1.1.2.cmml" xref="alg1.l10.m2.4.4.2.2.1.1.2">italic-ϵ</ci><cn type="integer" id="alg1.l10.m2.4.4.2.2.1.1.3.cmml" xref="alg1.l10.m2.4.4.2.2.1.1.3">1</cn></apply><ci id="alg1.l10.m2.2.2.cmml" xref="alg1.l10.m2.2.2">…</ci><apply id="alg1.l10.m2.4.4.2.2.2.2.cmml" xref="alg1.l10.m2.4.4.2.2.2.2"><csymbol cd="ambiguous" id="alg1.l10.m2.4.4.2.2.2.2.1.cmml" xref="alg1.l10.m2.4.4.2.2.2.2">subscript</csymbol><ci id="alg1.l10.m2.4.4.2.2.2.2.2.cmml" xref="alg1.l10.m2.4.4.2.2.2.2.2">italic-ϵ</ci><ci id="alg1.l10.m2.4.4.2.2.2.2.3.cmml" xref="alg1.l10.m2.4.4.2.2.2.2.3">𝑛</ci></apply></list></interval></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m2.4c">(\left[\hat{\epsilon}_{1},...,\hat{\epsilon}_{n}\right],\left[\epsilon_{1},...,\epsilon_{n}\right])</annotation></semantics></math><span id="alg1.l10.4" class="ltx_text" style="font-size:80%;">
</span>
</div>
<div id="alg1.l11" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l11.1.1.1" class="ltx_text" style="font-size:80%;">11:</span></span><span id="alg1.l11.2" class="ltx_text" style="font-size:80%;">    Backprop with </span><math id="alg1.l11.m1.1" class="ltx_Math" alttext="L" display="inline"><semantics id="alg1.l11.m1.1a"><mi mathsize="80%" id="alg1.l11.m1.1.1" xref="alg1.l11.m1.1.1.cmml">L</mi><annotation-xml encoding="MathML-Content" id="alg1.l11.m1.1b"><ci id="alg1.l11.m1.1.1.cmml" xref="alg1.l11.m1.1.1">𝐿</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.m1.1c">L</annotation></semantics></math><span id="alg1.l11.3" class="ltx_text" style="font-size:80%;"> and update </span><math id="alg1.l11.m2.1" class="ltx_Math" alttext="\theta" display="inline"><semantics id="alg1.l11.m2.1a"><mi mathsize="80%" id="alg1.l11.m2.1.1" xref="alg1.l11.m2.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="alg1.l11.m2.1b"><ci id="alg1.l11.m2.1.1.cmml" xref="alg1.l11.m2.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.m2.1c">\theta</annotation></semantics></math><span id="alg1.l11.4" class="ltx_text" style="font-size:80%;">
</span>
</div>
<div id="alg1.l12" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg1.l12.1.1.1" class="ltx_text" style="font-size:80%;">12:</span></span><span id="alg1.l12.2" class="ltx_text" style="font-size:80%;">  </span><span id="alg1.l12.3" class="ltx_text ltx_font_bold" style="font-size:80%;">end</span><span id="alg1.l12.4" class="ltx_text" style="font-size:80%;"> </span><span id="alg1.l12.5" class="ltx_text ltx_font_bold" style="font-size:80%;">loop</span>
</div>
</div>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_2">
<figure id="alg2" class="ltx_float ltx_figure_panel ltx_minipage ltx_align_top ltx_framed ltx_framed_top" style="width:216.8pt;">
<figcaption class="ltx_caption" style="font-size:80%;"><span class="ltx_tag ltx_tag_float"><span id="alg2.4.1.1" class="ltx_text ltx_font_bold">Algorithm 2</span> </span> DF Sampling with Guidance</figcaption>
<div id="alg2.5" class="ltx_listing ltx_listing">
<div id="alg2.l1" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l1.1.1.1" class="ltx_text" style="font-size:80%;">1:</span></span><span id="alg2.l1.2" class="ltx_text" style="font-size:80%;">  </span><span id="alg2.l1.3" class="ltx_text ltx_font_bold" style="font-size:80%;">Input:</span><span id="alg2.l1.4" class="ltx_text" style="font-size:80%;"> Model </span><math id="alg2.l1.m1.1" class="ltx_Math" alttext="\theta" display="inline"><semantics id="alg2.l1.m1.1a"><mi mathsize="80%" id="alg2.l1.m1.1.1" xref="alg2.l1.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="alg2.l1.m1.1b"><ci id="alg2.l1.m1.1.1.cmml" xref="alg2.l1.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m1.1c">\theta</annotation></semantics></math><span id="alg2.l1.5" class="ltx_text" style="font-size:80%;">, scheduling matrix </span><math id="alg2.l1.m2.1" class="ltx_Math" alttext="\mathcal{K}" display="inline"><semantics id="alg2.l1.m2.1a"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="alg2.l1.m2.1.1" xref="alg2.l1.m2.1.1.cmml">𝒦</mi><annotation-xml encoding="MathML-Content" id="alg2.l1.m2.1b"><ci id="alg2.l1.m2.1.1.cmml" xref="alg2.l1.m2.1.1">𝒦</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m2.1c">\mathcal{K}</annotation></semantics></math><span id="alg2.l1.6" class="ltx_text" style="font-size:80%;">, initial latent </span><math id="alg2.l1.m3.1" class="ltx_Math" alttext="\mathbf{z}_{0}" display="inline"><semantics id="alg2.l1.m3.1a"><msub id="alg2.l1.m3.1.1" xref="alg2.l1.m3.1.1.cmml"><mi mathsize="80%" id="alg2.l1.m3.1.1.2" xref="alg2.l1.m3.1.1.2.cmml">𝐳</mi><mn mathsize="80%" id="alg2.l1.m3.1.1.3" xref="alg2.l1.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="alg2.l1.m3.1b"><apply id="alg2.l1.m3.1.1.cmml" xref="alg2.l1.m3.1.1"><csymbol cd="ambiguous" id="alg2.l1.m3.1.1.1.cmml" xref="alg2.l1.m3.1.1">subscript</csymbol><ci id="alg2.l1.m3.1.1.2.cmml" xref="alg2.l1.m3.1.1.2">𝐳</ci><cn type="integer" id="alg2.l1.m3.1.1.3.cmml" xref="alg2.l1.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m3.1c">\mathbf{z}_{0}</annotation></semantics></math><span id="alg2.l1.7" class="ltx_text" style="font-size:80%;">, guidance cost </span><math id="alg2.l1.m4.1" class="ltx_Math" alttext="c(\cdot)" display="inline"><semantics id="alg2.l1.m4.1a"><mrow id="alg2.l1.m4.1.2" xref="alg2.l1.m4.1.2.cmml"><mi mathsize="80%" id="alg2.l1.m4.1.2.2" xref="alg2.l1.m4.1.2.2.cmml">c</mi><mo lspace="0em" rspace="0em" id="alg2.l1.m4.1.2.1" xref="alg2.l1.m4.1.2.1.cmml">​</mo><mrow id="alg2.l1.m4.1.2.3.2" xref="alg2.l1.m4.1.2.cmml"><mo maxsize="80%" minsize="80%" id="alg2.l1.m4.1.2.3.2.1" xref="alg2.l1.m4.1.2.cmml">(</mo><mo lspace="0em" mathsize="80%" rspace="0em" id="alg2.l1.m4.1.1" xref="alg2.l1.m4.1.1.cmml">⋅</mo><mo maxsize="80%" minsize="80%" id="alg2.l1.m4.1.2.3.2.2" xref="alg2.l1.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m4.1b"><apply id="alg2.l1.m4.1.2.cmml" xref="alg2.l1.m4.1.2"><times id="alg2.l1.m4.1.2.1.cmml" xref="alg2.l1.m4.1.2.1"></times><ci id="alg2.l1.m4.1.2.2.cmml" xref="alg2.l1.m4.1.2.2">𝑐</ci><ci id="alg2.l1.m4.1.1.cmml" xref="alg2.l1.m4.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m4.1c">c(\cdot)</annotation></semantics></math><span id="alg2.l1.8" class="ltx_text" style="font-size:80%;">.
</span>
</div>
<div id="alg2.l2" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l2.1.1.1" class="ltx_text" style="font-size:80%;">2:</span></span><span id="alg2.l2.2" class="ltx_text" style="font-size:80%;">  </span><span id="alg2.l2.3" class="ltx_text ltx_font_bold" style="font-size:80%;">Initialize</span><span id="alg2.l2.4" class="ltx_text" style="font-size:80%;"> </span><math id="alg2.l2.m1.5" class="ltx_Math" alttext="\mathbf{x}_{1},\dots,\mathbf{x}_{T}\sim\mathcal{N}(0,\sigma_{K}^{2}I)" display="inline"><semantics id="alg2.l2.m1.5a"><mrow id="alg2.l2.m1.5.5" xref="alg2.l2.m1.5.5.cmml"><mrow id="alg2.l2.m1.4.4.2.2" xref="alg2.l2.m1.4.4.2.3.cmml"><msub id="alg2.l2.m1.3.3.1.1.1" xref="alg2.l2.m1.3.3.1.1.1.cmml"><mi mathsize="80%" id="alg2.l2.m1.3.3.1.1.1.2" xref="alg2.l2.m1.3.3.1.1.1.2.cmml">𝐱</mi><mn mathsize="80%" id="alg2.l2.m1.3.3.1.1.1.3" xref="alg2.l2.m1.3.3.1.1.1.3.cmml">1</mn></msub><mo mathsize="80%" id="alg2.l2.m1.4.4.2.2.3" xref="alg2.l2.m1.4.4.2.3.cmml">,</mo><mi mathsize="80%" mathvariant="normal" id="alg2.l2.m1.2.2" xref="alg2.l2.m1.2.2.cmml">…</mi><mo mathsize="80%" id="alg2.l2.m1.4.4.2.2.4" xref="alg2.l2.m1.4.4.2.3.cmml">,</mo><msub id="alg2.l2.m1.4.4.2.2.2" xref="alg2.l2.m1.4.4.2.2.2.cmml"><mi mathsize="80%" id="alg2.l2.m1.4.4.2.2.2.2" xref="alg2.l2.m1.4.4.2.2.2.2.cmml">𝐱</mi><mi mathsize="80%" id="alg2.l2.m1.4.4.2.2.2.3" xref="alg2.l2.m1.4.4.2.2.2.3.cmml">T</mi></msub></mrow><mo mathsize="80%" id="alg2.l2.m1.5.5.4" xref="alg2.l2.m1.5.5.4.cmml">∼</mo><mrow id="alg2.l2.m1.5.5.3" xref="alg2.l2.m1.5.5.3.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="alg2.l2.m1.5.5.3.3" xref="alg2.l2.m1.5.5.3.3.cmml">𝒩</mi><mo lspace="0em" rspace="0em" id="alg2.l2.m1.5.5.3.2" xref="alg2.l2.m1.5.5.3.2.cmml">​</mo><mrow id="alg2.l2.m1.5.5.3.1.1" xref="alg2.l2.m1.5.5.3.1.2.cmml"><mo maxsize="80%" minsize="80%" id="alg2.l2.m1.5.5.3.1.1.2" xref="alg2.l2.m1.5.5.3.1.2.cmml">(</mo><mn mathsize="80%" id="alg2.l2.m1.1.1" xref="alg2.l2.m1.1.1.cmml">0</mn><mo mathsize="80%" id="alg2.l2.m1.5.5.3.1.1.3" xref="alg2.l2.m1.5.5.3.1.2.cmml">,</mo><mrow id="alg2.l2.m1.5.5.3.1.1.1" xref="alg2.l2.m1.5.5.3.1.1.1.cmml"><msubsup id="alg2.l2.m1.5.5.3.1.1.1.2" xref="alg2.l2.m1.5.5.3.1.1.1.2.cmml"><mi mathsize="80%" id="alg2.l2.m1.5.5.3.1.1.1.2.2.2" xref="alg2.l2.m1.5.5.3.1.1.1.2.2.2.cmml">σ</mi><mi mathsize="80%" id="alg2.l2.m1.5.5.3.1.1.1.2.2.3" xref="alg2.l2.m1.5.5.3.1.1.1.2.2.3.cmml">K</mi><mn mathsize="80%" id="alg2.l2.m1.5.5.3.1.1.1.2.3" xref="alg2.l2.m1.5.5.3.1.1.1.2.3.cmml">2</mn></msubsup><mo lspace="0em" rspace="0em" id="alg2.l2.m1.5.5.3.1.1.1.1" xref="alg2.l2.m1.5.5.3.1.1.1.1.cmml">​</mo><mi mathsize="80%" id="alg2.l2.m1.5.5.3.1.1.1.3" xref="alg2.l2.m1.5.5.3.1.1.1.3.cmml">I</mi></mrow><mo maxsize="80%" minsize="80%" id="alg2.l2.m1.5.5.3.1.1.4" xref="alg2.l2.m1.5.5.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l2.m1.5b"><apply id="alg2.l2.m1.5.5.cmml" xref="alg2.l2.m1.5.5"><csymbol cd="latexml" id="alg2.l2.m1.5.5.4.cmml" xref="alg2.l2.m1.5.5.4">similar-to</csymbol><list id="alg2.l2.m1.4.4.2.3.cmml" xref="alg2.l2.m1.4.4.2.2"><apply id="alg2.l2.m1.3.3.1.1.1.cmml" xref="alg2.l2.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="alg2.l2.m1.3.3.1.1.1.1.cmml" xref="alg2.l2.m1.3.3.1.1.1">subscript</csymbol><ci id="alg2.l2.m1.3.3.1.1.1.2.cmml" xref="alg2.l2.m1.3.3.1.1.1.2">𝐱</ci><cn type="integer" id="alg2.l2.m1.3.3.1.1.1.3.cmml" xref="alg2.l2.m1.3.3.1.1.1.3">1</cn></apply><ci id="alg2.l2.m1.2.2.cmml" xref="alg2.l2.m1.2.2">…</ci><apply id="alg2.l2.m1.4.4.2.2.2.cmml" xref="alg2.l2.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="alg2.l2.m1.4.4.2.2.2.1.cmml" xref="alg2.l2.m1.4.4.2.2.2">subscript</csymbol><ci id="alg2.l2.m1.4.4.2.2.2.2.cmml" xref="alg2.l2.m1.4.4.2.2.2.2">𝐱</ci><ci id="alg2.l2.m1.4.4.2.2.2.3.cmml" xref="alg2.l2.m1.4.4.2.2.2.3">𝑇</ci></apply></list><apply id="alg2.l2.m1.5.5.3.cmml" xref="alg2.l2.m1.5.5.3"><times id="alg2.l2.m1.5.5.3.2.cmml" xref="alg2.l2.m1.5.5.3.2"></times><ci id="alg2.l2.m1.5.5.3.3.cmml" xref="alg2.l2.m1.5.5.3.3">𝒩</ci><interval closure="open" id="alg2.l2.m1.5.5.3.1.2.cmml" xref="alg2.l2.m1.5.5.3.1.1"><cn type="integer" id="alg2.l2.m1.1.1.cmml" xref="alg2.l2.m1.1.1">0</cn><apply id="alg2.l2.m1.5.5.3.1.1.1.cmml" xref="alg2.l2.m1.5.5.3.1.1.1"><times id="alg2.l2.m1.5.5.3.1.1.1.1.cmml" xref="alg2.l2.m1.5.5.3.1.1.1.1"></times><apply id="alg2.l2.m1.5.5.3.1.1.1.2.cmml" xref="alg2.l2.m1.5.5.3.1.1.1.2"><csymbol cd="ambiguous" id="alg2.l2.m1.5.5.3.1.1.1.2.1.cmml" xref="alg2.l2.m1.5.5.3.1.1.1.2">superscript</csymbol><apply id="alg2.l2.m1.5.5.3.1.1.1.2.2.cmml" xref="alg2.l2.m1.5.5.3.1.1.1.2"><csymbol cd="ambiguous" id="alg2.l2.m1.5.5.3.1.1.1.2.2.1.cmml" xref="alg2.l2.m1.5.5.3.1.1.1.2">subscript</csymbol><ci id="alg2.l2.m1.5.5.3.1.1.1.2.2.2.cmml" xref="alg2.l2.m1.5.5.3.1.1.1.2.2.2">𝜎</ci><ci id="alg2.l2.m1.5.5.3.1.1.1.2.2.3.cmml" xref="alg2.l2.m1.5.5.3.1.1.1.2.2.3">𝐾</ci></apply><cn type="integer" id="alg2.l2.m1.5.5.3.1.1.1.2.3.cmml" xref="alg2.l2.m1.5.5.3.1.1.1.2.3">2</cn></apply><ci id="alg2.l2.m1.5.5.3.1.1.1.3.cmml" xref="alg2.l2.m1.5.5.3.1.1.1.3">𝐼</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l2.m1.5c">\mathbf{x}_{1},\dots,\mathbf{x}_{T}\sim\mathcal{N}(0,\sigma_{K}^{2}I)</annotation></semantics></math><span id="alg2.l2.5" class="ltx_text" style="font-size:80%;">.
</span>
</div>
<div id="alg2.l3" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l3.1.1.1" class="ltx_text" style="font-size:80%;">3:</span></span><span id="alg2.l3.2" class="ltx_text" style="font-size:80%;">  </span><span id="alg2.l3.3" class="ltx_text ltx_font_bold" style="font-size:80%;">for</span><span id="alg2.l3.4" class="ltx_text" style="font-size:80%;"> row </span><math id="alg2.l3.m1.3" class="ltx_Math" alttext="m=M-1,...,0" display="inline"><semantics id="alg2.l3.m1.3a"><mrow id="alg2.l3.m1.3.3" xref="alg2.l3.m1.3.3.cmml"><mi mathsize="80%" id="alg2.l3.m1.3.3.3" xref="alg2.l3.m1.3.3.3.cmml">m</mi><mo mathsize="80%" id="alg2.l3.m1.3.3.2" xref="alg2.l3.m1.3.3.2.cmml">=</mo><mrow id="alg2.l3.m1.3.3.1.1" xref="alg2.l3.m1.3.3.1.2.cmml"><mrow id="alg2.l3.m1.3.3.1.1.1" xref="alg2.l3.m1.3.3.1.1.1.cmml"><mi mathsize="80%" id="alg2.l3.m1.3.3.1.1.1.2" xref="alg2.l3.m1.3.3.1.1.1.2.cmml">M</mi><mo mathsize="80%" id="alg2.l3.m1.3.3.1.1.1.1" xref="alg2.l3.m1.3.3.1.1.1.1.cmml">−</mo><mn mathsize="80%" id="alg2.l3.m1.3.3.1.1.1.3" xref="alg2.l3.m1.3.3.1.1.1.3.cmml">1</mn></mrow><mo mathsize="80%" id="alg2.l3.m1.3.3.1.1.2" xref="alg2.l3.m1.3.3.1.2.cmml">,</mo><mi mathsize="80%" mathvariant="normal" id="alg2.l3.m1.1.1" xref="alg2.l3.m1.1.1.cmml">…</mi><mo mathsize="80%" id="alg2.l3.m1.3.3.1.1.3" xref="alg2.l3.m1.3.3.1.2.cmml">,</mo><mn mathsize="80%" id="alg2.l3.m1.2.2" xref="alg2.l3.m1.2.2.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l3.m1.3b"><apply id="alg2.l3.m1.3.3.cmml" xref="alg2.l3.m1.3.3"><eq id="alg2.l3.m1.3.3.2.cmml" xref="alg2.l3.m1.3.3.2"></eq><ci id="alg2.l3.m1.3.3.3.cmml" xref="alg2.l3.m1.3.3.3">𝑚</ci><list id="alg2.l3.m1.3.3.1.2.cmml" xref="alg2.l3.m1.3.3.1.1"><apply id="alg2.l3.m1.3.3.1.1.1.cmml" xref="alg2.l3.m1.3.3.1.1.1"><minus id="alg2.l3.m1.3.3.1.1.1.1.cmml" xref="alg2.l3.m1.3.3.1.1.1.1"></minus><ci id="alg2.l3.m1.3.3.1.1.1.2.cmml" xref="alg2.l3.m1.3.3.1.1.1.2">𝑀</ci><cn type="integer" id="alg2.l3.m1.3.3.1.1.1.3.cmml" xref="alg2.l3.m1.3.3.1.1.1.3">1</cn></apply><ci id="alg2.l3.m1.1.1.cmml" xref="alg2.l3.m1.1.1">…</ci><cn type="integer" id="alg2.l3.m1.2.2.cmml" xref="alg2.l3.m1.2.2">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m1.3c">m=M-1,...,0</annotation></semantics></math><span id="alg2.l3.5" class="ltx_text" style="font-size:80%;"> </span><span id="alg2.l3.6" class="ltx_text ltx_font_bold" style="font-size:80%;">do</span><span id="alg2.l3.7" class="ltx_text" style="font-size:80%;">


</span>
</div>
<div id="alg2.l4" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l4.1.1.1" class="ltx_text" style="font-size:80%;">4:</span></span><span id="alg2.l4.2" class="ltx_text" style="font-size:80%;">    </span><span id="alg2.l4.3" class="ltx_text ltx_font_bold" style="font-size:80%;">for</span><span id="alg2.l4.4" class="ltx_text" style="font-size:80%;"> </span><math id="alg2.l4.m1.3" class="ltx_Math" alttext="t=1,\dots,T" display="inline"><semantics id="alg2.l4.m1.3a"><mrow id="alg2.l4.m1.3.4" xref="alg2.l4.m1.3.4.cmml"><mi mathsize="80%" id="alg2.l4.m1.3.4.2" xref="alg2.l4.m1.3.4.2.cmml">t</mi><mo mathsize="80%" id="alg2.l4.m1.3.4.1" xref="alg2.l4.m1.3.4.1.cmml">=</mo><mrow id="alg2.l4.m1.3.4.3.2" xref="alg2.l4.m1.3.4.3.1.cmml"><mn mathsize="80%" id="alg2.l4.m1.1.1" xref="alg2.l4.m1.1.1.cmml">1</mn><mo mathsize="80%" id="alg2.l4.m1.3.4.3.2.1" xref="alg2.l4.m1.3.4.3.1.cmml">,</mo><mi mathsize="80%" mathvariant="normal" id="alg2.l4.m1.2.2" xref="alg2.l4.m1.2.2.cmml">…</mi><mo mathsize="80%" id="alg2.l4.m1.3.4.3.2.2" xref="alg2.l4.m1.3.4.3.1.cmml">,</mo><mi mathsize="80%" id="alg2.l4.m1.3.3" xref="alg2.l4.m1.3.3.cmml">T</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l4.m1.3b"><apply id="alg2.l4.m1.3.4.cmml" xref="alg2.l4.m1.3.4"><eq id="alg2.l4.m1.3.4.1.cmml" xref="alg2.l4.m1.3.4.1"></eq><ci id="alg2.l4.m1.3.4.2.cmml" xref="alg2.l4.m1.3.4.2">𝑡</ci><list id="alg2.l4.m1.3.4.3.1.cmml" xref="alg2.l4.m1.3.4.3.2"><cn type="integer" id="alg2.l4.m1.1.1.cmml" xref="alg2.l4.m1.1.1">1</cn><ci id="alg2.l4.m1.2.2.cmml" xref="alg2.l4.m1.2.2">…</ci><ci id="alg2.l4.m1.3.3.cmml" xref="alg2.l4.m1.3.3">𝑇</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l4.m1.3c">t=1,\dots,T</annotation></semantics></math><span id="alg2.l4.5" class="ltx_text" style="font-size:80%;"> </span><span id="alg2.l4.6" class="ltx_text ltx_font_bold" style="font-size:80%;">do</span><span id="alg2.l4.7" class="ltx_text" style="font-size:80%;">


</span>
</div>
<div id="alg2.l5" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l5.1.1.1" class="ltx_text" style="font-size:80%;">5:</span></span><span id="alg2.l5.2" class="ltx_text" style="font-size:80%;">      </span><math id="alg2.l5.m1.3" class="ltx_Math" alttext="\mathbf{z}_{t}^{\mathrm{new}}\sim p_{\theta}(\mathbf{z}_{t}\mid\mathbf{z}_{t-1},\mathbf{x}_{t},\mathcal{K}_{m+1,t})" display="inline"><semantics id="alg2.l5.m1.3a"><mrow id="alg2.l5.m1.3.3" xref="alg2.l5.m1.3.3.cmml"><msubsup id="alg2.l5.m1.3.3.3" xref="alg2.l5.m1.3.3.3.cmml"><mi mathsize="80%" id="alg2.l5.m1.3.3.3.2.2" xref="alg2.l5.m1.3.3.3.2.2.cmml">𝐳</mi><mi mathsize="80%" id="alg2.l5.m1.3.3.3.2.3" xref="alg2.l5.m1.3.3.3.2.3.cmml">t</mi><mi mathsize="80%" id="alg2.l5.m1.3.3.3.3" xref="alg2.l5.m1.3.3.3.3.cmml">new</mi></msubsup><mo mathsize="80%" id="alg2.l5.m1.3.3.2" xref="alg2.l5.m1.3.3.2.cmml">∼</mo><mrow id="alg2.l5.m1.3.3.1" xref="alg2.l5.m1.3.3.1.cmml"><msub id="alg2.l5.m1.3.3.1.3" xref="alg2.l5.m1.3.3.1.3.cmml"><mi mathsize="80%" id="alg2.l5.m1.3.3.1.3.2" xref="alg2.l5.m1.3.3.1.3.2.cmml">p</mi><mi mathsize="80%" id="alg2.l5.m1.3.3.1.3.3" xref="alg2.l5.m1.3.3.1.3.3.cmml">θ</mi></msub><mo lspace="0em" rspace="0em" id="alg2.l5.m1.3.3.1.2" xref="alg2.l5.m1.3.3.1.2.cmml">​</mo><mrow id="alg2.l5.m1.3.3.1.1.1" xref="alg2.l5.m1.3.3.1.1.1.1.cmml"><mo maxsize="80%" minsize="80%" id="alg2.l5.m1.3.3.1.1.1.2" xref="alg2.l5.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="alg2.l5.m1.3.3.1.1.1.1" xref="alg2.l5.m1.3.3.1.1.1.1.cmml"><msub id="alg2.l5.m1.3.3.1.1.1.1.5" xref="alg2.l5.m1.3.3.1.1.1.1.5.cmml"><mi mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.5.2" xref="alg2.l5.m1.3.3.1.1.1.1.5.2.cmml">𝐳</mi><mi mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.5.3" xref="alg2.l5.m1.3.3.1.1.1.1.5.3.cmml">t</mi></msub><mo mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.4" xref="alg2.l5.m1.3.3.1.1.1.1.4.cmml">∣</mo><mrow id="alg2.l5.m1.3.3.1.1.1.1.3.3" xref="alg2.l5.m1.3.3.1.1.1.1.3.4.cmml"><msub id="alg2.l5.m1.3.3.1.1.1.1.1.1.1" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.cmml"><mi mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.2" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.cmml"><mi mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.2" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.2.cmml">t</mi><mo mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.1" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.3" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.3.3.4" xref="alg2.l5.m1.3.3.1.1.1.1.3.4.cmml">,</mo><msub id="alg2.l5.m1.3.3.1.1.1.1.2.2.2" xref="alg2.l5.m1.3.3.1.1.1.1.2.2.2.cmml"><mi mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.2.2.2.2" xref="alg2.l5.m1.3.3.1.1.1.1.2.2.2.2.cmml">𝐱</mi><mi mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.2.2.2.3" xref="alg2.l5.m1.3.3.1.1.1.1.2.2.2.3.cmml">t</mi></msub><mo mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.3.3.5" xref="alg2.l5.m1.3.3.1.1.1.1.3.4.cmml">,</mo><msub id="alg2.l5.m1.3.3.1.1.1.1.3.3.3" xref="alg2.l5.m1.3.3.1.1.1.1.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="alg2.l5.m1.3.3.1.1.1.1.3.3.3.2" xref="alg2.l5.m1.3.3.1.1.1.1.3.3.3.2.cmml">𝒦</mi><mrow id="alg2.l5.m1.2.2.2.2" xref="alg2.l5.m1.2.2.2.3.cmml"><mrow id="alg2.l5.m1.2.2.2.2.1" xref="alg2.l5.m1.2.2.2.2.1.cmml"><mi mathsize="80%" id="alg2.l5.m1.2.2.2.2.1.2" xref="alg2.l5.m1.2.2.2.2.1.2.cmml">m</mi><mo mathsize="80%" id="alg2.l5.m1.2.2.2.2.1.1" xref="alg2.l5.m1.2.2.2.2.1.1.cmml">+</mo><mn mathsize="80%" id="alg2.l5.m1.2.2.2.2.1.3" xref="alg2.l5.m1.2.2.2.2.1.3.cmml">1</mn></mrow><mo mathsize="80%" id="alg2.l5.m1.2.2.2.2.2" xref="alg2.l5.m1.2.2.2.3.cmml">,</mo><mi mathsize="80%" id="alg2.l5.m1.1.1.1.1" xref="alg2.l5.m1.1.1.1.1.cmml">t</mi></mrow></msub></mrow></mrow><mo maxsize="80%" minsize="80%" id="alg2.l5.m1.3.3.1.1.1.3" xref="alg2.l5.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l5.m1.3b"><apply id="alg2.l5.m1.3.3.cmml" xref="alg2.l5.m1.3.3"><csymbol cd="latexml" id="alg2.l5.m1.3.3.2.cmml" xref="alg2.l5.m1.3.3.2">similar-to</csymbol><apply id="alg2.l5.m1.3.3.3.cmml" xref="alg2.l5.m1.3.3.3"><csymbol cd="ambiguous" id="alg2.l5.m1.3.3.3.1.cmml" xref="alg2.l5.m1.3.3.3">superscript</csymbol><apply id="alg2.l5.m1.3.3.3.2.cmml" xref="alg2.l5.m1.3.3.3"><csymbol cd="ambiguous" id="alg2.l5.m1.3.3.3.2.1.cmml" xref="alg2.l5.m1.3.3.3">subscript</csymbol><ci id="alg2.l5.m1.3.3.3.2.2.cmml" xref="alg2.l5.m1.3.3.3.2.2">𝐳</ci><ci id="alg2.l5.m1.3.3.3.2.3.cmml" xref="alg2.l5.m1.3.3.3.2.3">𝑡</ci></apply><ci id="alg2.l5.m1.3.3.3.3.cmml" xref="alg2.l5.m1.3.3.3.3">new</ci></apply><apply id="alg2.l5.m1.3.3.1.cmml" xref="alg2.l5.m1.3.3.1"><times id="alg2.l5.m1.3.3.1.2.cmml" xref="alg2.l5.m1.3.3.1.2"></times><apply id="alg2.l5.m1.3.3.1.3.cmml" xref="alg2.l5.m1.3.3.1.3"><csymbol cd="ambiguous" id="alg2.l5.m1.3.3.1.3.1.cmml" xref="alg2.l5.m1.3.3.1.3">subscript</csymbol><ci id="alg2.l5.m1.3.3.1.3.2.cmml" xref="alg2.l5.m1.3.3.1.3.2">𝑝</ci><ci id="alg2.l5.m1.3.3.1.3.3.cmml" xref="alg2.l5.m1.3.3.1.3.3">𝜃</ci></apply><apply id="alg2.l5.m1.3.3.1.1.1.1.cmml" xref="alg2.l5.m1.3.3.1.1.1"><csymbol cd="latexml" id="alg2.l5.m1.3.3.1.1.1.1.4.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.4">conditional</csymbol><apply id="alg2.l5.m1.3.3.1.1.1.1.5.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.5"><csymbol cd="ambiguous" id="alg2.l5.m1.3.3.1.1.1.1.5.1.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.5">subscript</csymbol><ci id="alg2.l5.m1.3.3.1.1.1.1.5.2.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.5.2">𝐳</ci><ci id="alg2.l5.m1.3.3.1.1.1.1.5.3.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.5.3">𝑡</ci></apply><list id="alg2.l5.m1.3.3.1.1.1.1.3.4.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.3.3"><apply id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.2">𝐳</ci><apply id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3"><minus id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.1.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.1"></minus><ci id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.2.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.2">𝑡</ci><cn type="integer" id="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.3.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="alg2.l5.m1.3.3.1.1.1.1.2.2.2.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="alg2.l5.m1.3.3.1.1.1.1.2.2.2.1.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.2.2.2">subscript</csymbol><ci id="alg2.l5.m1.3.3.1.1.1.1.2.2.2.2.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.2.2.2.2">𝐱</ci><ci id="alg2.l5.m1.3.3.1.1.1.1.2.2.2.3.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.2.2.2.3">𝑡</ci></apply><apply id="alg2.l5.m1.3.3.1.1.1.1.3.3.3.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="alg2.l5.m1.3.3.1.1.1.1.3.3.3.1.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.3.3.3">subscript</csymbol><ci id="alg2.l5.m1.3.3.1.1.1.1.3.3.3.2.cmml" xref="alg2.l5.m1.3.3.1.1.1.1.3.3.3.2">𝒦</ci><list id="alg2.l5.m1.2.2.2.3.cmml" xref="alg2.l5.m1.2.2.2.2"><apply id="alg2.l5.m1.2.2.2.2.1.cmml" xref="alg2.l5.m1.2.2.2.2.1"><plus id="alg2.l5.m1.2.2.2.2.1.1.cmml" xref="alg2.l5.m1.2.2.2.2.1.1"></plus><ci id="alg2.l5.m1.2.2.2.2.1.2.cmml" xref="alg2.l5.m1.2.2.2.2.1.2">𝑚</ci><cn type="integer" id="alg2.l5.m1.2.2.2.2.1.3.cmml" xref="alg2.l5.m1.2.2.2.2.1.3">1</cn></apply><ci id="alg2.l5.m1.1.1.1.1.cmml" xref="alg2.l5.m1.1.1.1.1">𝑡</ci></list></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m1.3c">\mathbf{z}_{t}^{\mathrm{new}}\sim p_{\theta}(\mathbf{z}_{t}\mid\mathbf{z}_{t-1},\mathbf{x}_{t},\mathcal{K}_{m+1,t})</annotation></semantics></math><span id="alg2.l5.3" class="ltx_text" style="font-size:80%;">. 
</span>
</div>
<div id="alg2.l6" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l6.1.1.1" class="ltx_text" style="font-size:80%;">6:</span></span><span id="alg2.l6.2" class="ltx_text" style="font-size:80%;">      </span><math id="alg2.l6.m1.2" class="ltx_Math" alttext="k\leftarrow\mathcal{K}_{m,t}" display="inline"><semantics id="alg2.l6.m1.2a"><mrow id="alg2.l6.m1.2.3" xref="alg2.l6.m1.2.3.cmml"><mi mathsize="80%" id="alg2.l6.m1.2.3.2" xref="alg2.l6.m1.2.3.2.cmml">k</mi><mo mathsize="80%" stretchy="false" id="alg2.l6.m1.2.3.1" xref="alg2.l6.m1.2.3.1.cmml">←</mo><msub id="alg2.l6.m1.2.3.3" xref="alg2.l6.m1.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="alg2.l6.m1.2.3.3.2" xref="alg2.l6.m1.2.3.3.2.cmml">𝒦</mi><mrow id="alg2.l6.m1.2.2.2.4" xref="alg2.l6.m1.2.2.2.3.cmml"><mi mathsize="80%" id="alg2.l6.m1.1.1.1.1" xref="alg2.l6.m1.1.1.1.1.cmml">m</mi><mo mathsize="80%" id="alg2.l6.m1.2.2.2.4.1" xref="alg2.l6.m1.2.2.2.3.cmml">,</mo><mi mathsize="80%" id="alg2.l6.m1.2.2.2.2" xref="alg2.l6.m1.2.2.2.2.cmml">t</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="alg2.l6.m1.2b"><apply id="alg2.l6.m1.2.3.cmml" xref="alg2.l6.m1.2.3"><ci id="alg2.l6.m1.2.3.1.cmml" xref="alg2.l6.m1.2.3.1">←</ci><ci id="alg2.l6.m1.2.3.2.cmml" xref="alg2.l6.m1.2.3.2">𝑘</ci><apply id="alg2.l6.m1.2.3.3.cmml" xref="alg2.l6.m1.2.3.3"><csymbol cd="ambiguous" id="alg2.l6.m1.2.3.3.1.cmml" xref="alg2.l6.m1.2.3.3">subscript</csymbol><ci id="alg2.l6.m1.2.3.3.2.cmml" xref="alg2.l6.m1.2.3.3.2">𝒦</ci><list id="alg2.l6.m1.2.2.2.3.cmml" xref="alg2.l6.m1.2.2.2.4"><ci id="alg2.l6.m1.1.1.1.1.cmml" xref="alg2.l6.m1.1.1.1.1">𝑚</ci><ci id="alg2.l6.m1.2.2.2.2.cmml" xref="alg2.l6.m1.2.2.2.2">𝑡</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l6.m1.2c">k\leftarrow\mathcal{K}_{m,t}</annotation></semantics></math><span id="alg2.l6.3" class="ltx_text" style="font-size:80%;">, </span><math id="alg2.l6.m2.2" class="ltx_Math" alttext="\mathbf{w}\sim\mathcal{N}(0,\mathbf{I})" display="inline"><semantics id="alg2.l6.m2.2a"><mrow id="alg2.l6.m2.2.3" xref="alg2.l6.m2.2.3.cmml"><mi mathsize="80%" id="alg2.l6.m2.2.3.2" xref="alg2.l6.m2.2.3.2.cmml">𝐰</mi><mo mathsize="80%" id="alg2.l6.m2.2.3.1" xref="alg2.l6.m2.2.3.1.cmml">∼</mo><mrow id="alg2.l6.m2.2.3.3" xref="alg2.l6.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="alg2.l6.m2.2.3.3.2" xref="alg2.l6.m2.2.3.3.2.cmml">𝒩</mi><mo lspace="0em" rspace="0em" id="alg2.l6.m2.2.3.3.1" xref="alg2.l6.m2.2.3.3.1.cmml">​</mo><mrow id="alg2.l6.m2.2.3.3.3.2" xref="alg2.l6.m2.2.3.3.3.1.cmml"><mo maxsize="80%" minsize="80%" id="alg2.l6.m2.2.3.3.3.2.1" xref="alg2.l6.m2.2.3.3.3.1.cmml">(</mo><mn mathsize="80%" id="alg2.l6.m2.1.1" xref="alg2.l6.m2.1.1.cmml">0</mn><mo mathsize="80%" id="alg2.l6.m2.2.3.3.3.2.2" xref="alg2.l6.m2.2.3.3.3.1.cmml">,</mo><mi mathsize="80%" id="alg2.l6.m2.2.2" xref="alg2.l6.m2.2.2.cmml">𝐈</mi><mo maxsize="80%" minsize="80%" id="alg2.l6.m2.2.3.3.3.2.3" xref="alg2.l6.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l6.m2.2b"><apply id="alg2.l6.m2.2.3.cmml" xref="alg2.l6.m2.2.3"><csymbol cd="latexml" id="alg2.l6.m2.2.3.1.cmml" xref="alg2.l6.m2.2.3.1">similar-to</csymbol><ci id="alg2.l6.m2.2.3.2.cmml" xref="alg2.l6.m2.2.3.2">𝐰</ci><apply id="alg2.l6.m2.2.3.3.cmml" xref="alg2.l6.m2.2.3.3"><times id="alg2.l6.m2.2.3.3.1.cmml" xref="alg2.l6.m2.2.3.3.1"></times><ci id="alg2.l6.m2.2.3.3.2.cmml" xref="alg2.l6.m2.2.3.3.2">𝒩</ci><interval closure="open" id="alg2.l6.m2.2.3.3.3.1.cmml" xref="alg2.l6.m2.2.3.3.3.2"><cn type="integer" id="alg2.l6.m2.1.1.cmml" xref="alg2.l6.m2.1.1">0</cn><ci id="alg2.l6.m2.2.2.cmml" xref="alg2.l6.m2.2.2">𝐈</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l6.m2.2c">\mathbf{w}\sim\mathcal{N}(0,\mathbf{I})</annotation></semantics></math><span id="alg2.l6.4" class="ltx_text" style="font-size:80%;">.
</span>
</div>
<div id="alg2.l7" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l7.1.1.1" class="ltx_text" style="font-size:80%;">7:</span></span><span id="alg2.l7.2" class="ltx_text" style="font-size:80%;">      </span><math id="alg2.l7.m1.2" class="ltx_Math" alttext="\mathbf{x}_{t}^{\mathrm{new}}\leftarrow\frac{1}{\sqrt{\alpha_{k}}}(\mathbf{x}_{t}-\frac{1-\alpha_{k}}{\sqrt{1-\bar{\alpha}_{k}}}\epsilon_{\theta}(\mathbf{z}_{t}^{\mathrm{new}},\mathbf{x}_{t},k))+\sigma_{k}\mathbf{w}" display="inline"><semantics id="alg2.l7.m1.2a"><mrow id="alg2.l7.m1.2.2" xref="alg2.l7.m1.2.2.cmml"><msubsup id="alg2.l7.m1.2.2.3" xref="alg2.l7.m1.2.2.3.cmml"><mi mathsize="80%" id="alg2.l7.m1.2.2.3.2.2" xref="alg2.l7.m1.2.2.3.2.2.cmml">𝐱</mi><mi mathsize="80%" id="alg2.l7.m1.2.2.3.2.3" xref="alg2.l7.m1.2.2.3.2.3.cmml">t</mi><mi mathsize="80%" id="alg2.l7.m1.2.2.3.3" xref="alg2.l7.m1.2.2.3.3.cmml">new</mi></msubsup><mo mathsize="80%" stretchy="false" id="alg2.l7.m1.2.2.2" xref="alg2.l7.m1.2.2.2.cmml">←</mo><mrow 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id="alg2.l7.m1.2.2.1.1.1.1.1.4.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.4.2">𝐱</ci><ci id="alg2.l7.m1.2.2.1.1.1.1.1.4.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.4.3">𝑡</ci></apply><apply id="alg2.l7.m1.2.2.1.1.1.1.1.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2"><times id="alg2.l7.m1.2.2.1.1.1.1.1.2.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.3"></times><apply id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4"><divide id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4"></divide><apply id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2"><minus id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.1"></minus><cn type="integer" id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.2">1</cn><apply id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.3"><csymbol cd="ambiguous" id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.3.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.3">subscript</csymbol><ci id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.3.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.3.2">𝛼</ci><ci id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.3.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.2.3.3">𝑘</ci></apply></apply><apply id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3"><root id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3a.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3"></root><apply id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2"><minus id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.1"></minus><cn type="integer" id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.2">1</cn><apply id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3"><csymbol cd="ambiguous" id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3">subscript</csymbol><apply id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3.2"><ci id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3.2.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3.2.1">¯</ci><ci id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3.2.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3.2.2">𝛼</ci></apply><ci id="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.4.3.2.3.3">𝑘</ci></apply></apply></apply></apply><apply id="alg2.l7.m1.2.2.1.1.1.1.1.2.5.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.5"><csymbol cd="ambiguous" id="alg2.l7.m1.2.2.1.1.1.1.1.2.5.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.5">subscript</csymbol><ci id="alg2.l7.m1.2.2.1.1.1.1.1.2.5.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.5.2">italic-ϵ</ci><ci id="alg2.l7.m1.2.2.1.1.1.1.1.2.5.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.5.3">𝜃</ci></apply><vector id="alg2.l7.m1.2.2.1.1.1.1.1.2.2.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.2.2"><apply id="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1">superscript</csymbol><apply id="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1.2.2">𝐳</ci><ci id="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1.2.3">𝑡</ci></apply><ci id="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.1.1.1.1.3">new</ci></apply><apply id="alg2.l7.m1.2.2.1.1.1.1.1.2.2.2.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="alg2.l7.m1.2.2.1.1.1.1.1.2.2.2.2.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.2.2.2">subscript</csymbol><ci id="alg2.l7.m1.2.2.1.1.1.1.1.2.2.2.2.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.2.2.2.2">𝐱</ci><ci id="alg2.l7.m1.2.2.1.1.1.1.1.2.2.2.2.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.2.2.2.3">𝑡</ci></apply><ci id="alg2.l7.m1.1.1.cmml" xref="alg2.l7.m1.1.1">𝑘</ci></vector></apply></apply></apply><apply id="alg2.l7.m1.2.2.1.3.cmml" xref="alg2.l7.m1.2.2.1.3"><times id="alg2.l7.m1.2.2.1.3.1.cmml" xref="alg2.l7.m1.2.2.1.3.1"></times><apply id="alg2.l7.m1.2.2.1.3.2.cmml" xref="alg2.l7.m1.2.2.1.3.2"><csymbol cd="ambiguous" id="alg2.l7.m1.2.2.1.3.2.1.cmml" xref="alg2.l7.m1.2.2.1.3.2">subscript</csymbol><ci id="alg2.l7.m1.2.2.1.3.2.2.cmml" xref="alg2.l7.m1.2.2.1.3.2.2">𝜎</ci><ci id="alg2.l7.m1.2.2.1.3.2.3.cmml" xref="alg2.l7.m1.2.2.1.3.2.3">𝑘</ci></apply><ci id="alg2.l7.m1.2.2.1.3.3.cmml" xref="alg2.l7.m1.2.2.1.3.3">𝐰</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l7.m1.2c">\mathbf{x}_{t}^{\mathrm{new}}\leftarrow\frac{1}{\sqrt{\alpha_{k}}}(\mathbf{x}_{t}-\frac{1-\alpha_{k}}{\sqrt{1-\bar{\alpha}_{k}}}\epsilon_{\theta}(\mathbf{z}_{t}^{\mathrm{new}},\mathbf{x}_{t},k))+\sigma_{k}\mathbf{w}</annotation></semantics></math><span id="alg2.l7.3" class="ltx_text" style="font-size:80%;">

</span>
</div>
<div id="alg2.l8" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l8.1.1.1" class="ltx_text" style="font-size:80%;">8:</span></span><span id="alg2.l8.2" class="ltx_text" style="font-size:80%;">      </span><span id="alg2.l8.3" class="ltx_text ltx_font_bold" style="font-size:80%;">Update</span><span id="alg2.l8.4" class="ltx_text" style="font-size:80%;"> </span><math id="alg2.l8.m1.1" class="ltx_Math" alttext="\mathbf{z}_{t}\leftarrow\mathbf{z}_{t}^{\mathrm{new}}" display="inline"><semantics id="alg2.l8.m1.1a"><mrow id="alg2.l8.m1.1.1" xref="alg2.l8.m1.1.1.cmml"><msub id="alg2.l8.m1.1.1.2" xref="alg2.l8.m1.1.1.2.cmml"><mi mathsize="80%" id="alg2.l8.m1.1.1.2.2" xref="alg2.l8.m1.1.1.2.2.cmml">𝐳</mi><mi mathsize="80%" id="alg2.l8.m1.1.1.2.3" xref="alg2.l8.m1.1.1.2.3.cmml">t</mi></msub><mo mathsize="80%" stretchy="false" id="alg2.l8.m1.1.1.1" xref="alg2.l8.m1.1.1.1.cmml">←</mo><msubsup id="alg2.l8.m1.1.1.3" xref="alg2.l8.m1.1.1.3.cmml"><mi mathsize="80%" id="alg2.l8.m1.1.1.3.2.2" xref="alg2.l8.m1.1.1.3.2.2.cmml">𝐳</mi><mi mathsize="80%" id="alg2.l8.m1.1.1.3.2.3" xref="alg2.l8.m1.1.1.3.2.3.cmml">t</mi><mi mathsize="80%" id="alg2.l8.m1.1.1.3.3" xref="alg2.l8.m1.1.1.3.3.cmml">new</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="alg2.l8.m1.1b"><apply id="alg2.l8.m1.1.1.cmml" xref="alg2.l8.m1.1.1"><ci id="alg2.l8.m1.1.1.1.cmml" xref="alg2.l8.m1.1.1.1">←</ci><apply id="alg2.l8.m1.1.1.2.cmml" xref="alg2.l8.m1.1.1.2"><csymbol cd="ambiguous" id="alg2.l8.m1.1.1.2.1.cmml" xref="alg2.l8.m1.1.1.2">subscript</csymbol><ci id="alg2.l8.m1.1.1.2.2.cmml" xref="alg2.l8.m1.1.1.2.2">𝐳</ci><ci id="alg2.l8.m1.1.1.2.3.cmml" xref="alg2.l8.m1.1.1.2.3">𝑡</ci></apply><apply id="alg2.l8.m1.1.1.3.cmml" xref="alg2.l8.m1.1.1.3"><csymbol cd="ambiguous" id="alg2.l8.m1.1.1.3.1.cmml" xref="alg2.l8.m1.1.1.3">superscript</csymbol><apply id="alg2.l8.m1.1.1.3.2.cmml" xref="alg2.l8.m1.1.1.3"><csymbol cd="ambiguous" id="alg2.l8.m1.1.1.3.2.1.cmml" xref="alg2.l8.m1.1.1.3">subscript</csymbol><ci id="alg2.l8.m1.1.1.3.2.2.cmml" xref="alg2.l8.m1.1.1.3.2.2">𝐳</ci><ci id="alg2.l8.m1.1.1.3.2.3.cmml" xref="alg2.l8.m1.1.1.3.2.3">𝑡</ci></apply><ci id="alg2.l8.m1.1.1.3.3.cmml" xref="alg2.l8.m1.1.1.3.3">new</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l8.m1.1c">\mathbf{z}_{t}\leftarrow\mathbf{z}_{t}^{\mathrm{new}}</annotation></semantics></math><span id="alg2.l8.5" class="ltx_text" style="font-size:80%;">.
</span>
</div>
<div id="alg2.l9" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l9.1.1.1" class="ltx_text" style="font-size:80%;">9:</span></span><span id="alg2.l9.2" class="ltx_text" style="font-size:80%;">    </span><span id="alg2.l9.3" class="ltx_text ltx_font_bold" style="font-size:80%;">end</span><span id="alg2.l9.4" class="ltx_text" style="font-size:80%;"> </span><span id="alg2.l9.5" class="ltx_text ltx_font_bold" style="font-size:80%;">for</span>
</div>
<div id="alg2.l10" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l10.1.1.1" class="ltx_text" style="font-size:80%;">10:</span></span><span id="alg2.l10.2" class="ltx_text" style="font-size:80%;">    </span><math id="alg2.l10.m1.1" class="ltx_Math" alttext="\mathbf{x}_{1:H}\leftarrow" display="inline"><semantics id="alg2.l10.m1.1a"><mrow id="alg2.l10.m1.1.1" xref="alg2.l10.m1.1.1.cmml"><msub id="alg2.l10.m1.1.1.2" xref="alg2.l10.m1.1.1.2.cmml"><mi mathsize="80%" id="alg2.l10.m1.1.1.2.2" xref="alg2.l10.m1.1.1.2.2.cmml">𝐱</mi><mrow id="alg2.l10.m1.1.1.2.3" xref="alg2.l10.m1.1.1.2.3.cmml"><mn mathsize="80%" id="alg2.l10.m1.1.1.2.3.2" xref="alg2.l10.m1.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="80%" rspace="0.278em" id="alg2.l10.m1.1.1.2.3.1" xref="alg2.l10.m1.1.1.2.3.1.cmml">:</mo><mi mathsize="80%" id="alg2.l10.m1.1.1.2.3.3" xref="alg2.l10.m1.1.1.2.3.3.cmml">H</mi></mrow></msub><mo mathsize="80%" stretchy="false" id="alg2.l10.m1.1.1.1" xref="alg2.l10.m1.1.1.1.cmml">←</mo><mi id="alg2.l10.m1.1.1.3" xref="alg2.l10.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="alg2.l10.m1.1b"><apply id="alg2.l10.m1.1.1.cmml" xref="alg2.l10.m1.1.1"><ci id="alg2.l10.m1.1.1.1.cmml" xref="alg2.l10.m1.1.1.1">←</ci><apply id="alg2.l10.m1.1.1.2.cmml" xref="alg2.l10.m1.1.1.2"><csymbol cd="ambiguous" id="alg2.l10.m1.1.1.2.1.cmml" xref="alg2.l10.m1.1.1.2">subscript</csymbol><ci id="alg2.l10.m1.1.1.2.2.cmml" xref="alg2.l10.m1.1.1.2.2">𝐱</ci><apply id="alg2.l10.m1.1.1.2.3.cmml" xref="alg2.l10.m1.1.1.2.3"><ci id="alg2.l10.m1.1.1.2.3.1.cmml" xref="alg2.l10.m1.1.1.2.3.1">:</ci><cn type="integer" id="alg2.l10.m1.1.1.2.3.2.cmml" xref="alg2.l10.m1.1.1.2.3.2">1</cn><ci id="alg2.l10.m1.1.1.2.3.3.cmml" xref="alg2.l10.m1.1.1.2.3.3">𝐻</ci></apply></apply><csymbol cd="latexml" id="alg2.l10.m1.1.1.3.cmml" xref="alg2.l10.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l10.m1.1c">\mathbf{x}_{1:H}\leftarrow</annotation></semantics></math><span id="alg2.l10.3" class="ltx_text" style="font-size:80%;">AddGuidance</span><math id="alg2.l10.m2.2" class="ltx_Math" alttext="(\mathbf{x}_{1:H}^{\mathrm{new}},\nabla_{\mathbf{x}}\log c({\mathbf{x}}_{1:H}^{\mathrm{new}}))" display="inline"><semantics id="alg2.l10.m2.2a"><mrow id="alg2.l10.m2.2.2.2" xref="alg2.l10.m2.2.2.3.cmml"><mo maxsize="80%" minsize="80%" id="alg2.l10.m2.2.2.2.3" xref="alg2.l10.m2.2.2.3.cmml">(</mo><msubsup id="alg2.l10.m2.1.1.1.1" xref="alg2.l10.m2.1.1.1.1.cmml"><mi mathsize="80%" id="alg2.l10.m2.1.1.1.1.2.2" xref="alg2.l10.m2.1.1.1.1.2.2.cmml">𝐱</mi><mrow id="alg2.l10.m2.1.1.1.1.2.3" xref="alg2.l10.m2.1.1.1.1.2.3.cmml"><mn mathsize="80%" id="alg2.l10.m2.1.1.1.1.2.3.2" xref="alg2.l10.m2.1.1.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="80%" rspace="0.278em" id="alg2.l10.m2.1.1.1.1.2.3.1" xref="alg2.l10.m2.1.1.1.1.2.3.1.cmml">:</mo><mi mathsize="80%" id="alg2.l10.m2.1.1.1.1.2.3.3" xref="alg2.l10.m2.1.1.1.1.2.3.3.cmml">H</mi></mrow><mi mathsize="80%" id="alg2.l10.m2.1.1.1.1.3" xref="alg2.l10.m2.1.1.1.1.3.cmml">new</mi></msubsup><mo mathsize="80%" id="alg2.l10.m2.2.2.2.4" xref="alg2.l10.m2.2.2.3.cmml">,</mo><mrow id="alg2.l10.m2.2.2.2.2" xref="alg2.l10.m2.2.2.2.2.cmml"><mrow id="alg2.l10.m2.2.2.2.2.3" xref="alg2.l10.m2.2.2.2.2.3.cmml"><mrow id="alg2.l10.m2.2.2.2.2.3.1" xref="alg2.l10.m2.2.2.2.2.3.1.cmml"><msub id="alg2.l10.m2.2.2.2.2.3.1.1" xref="alg2.l10.m2.2.2.2.2.3.1.1.cmml"><mo mathsize="80%" rspace="0.167em" id="alg2.l10.m2.2.2.2.2.3.1.1.2" xref="alg2.l10.m2.2.2.2.2.3.1.1.2.cmml">∇</mo><mi mathsize="80%" id="alg2.l10.m2.2.2.2.2.3.1.1.3" xref="alg2.l10.m2.2.2.2.2.3.1.1.3.cmml">𝐱</mi></msub><mi mathsize="80%" id="alg2.l10.m2.2.2.2.2.3.1.2" xref="alg2.l10.m2.2.2.2.2.3.1.2.cmml">log</mi></mrow><mo lspace="0.167em" id="alg2.l10.m2.2.2.2.2.3a" xref="alg2.l10.m2.2.2.2.2.3.cmml">⁡</mo><mi mathsize="80%" id="alg2.l10.m2.2.2.2.2.3.2" xref="alg2.l10.m2.2.2.2.2.3.2.cmml">c</mi></mrow><mo lspace="0em" rspace="0em" id="alg2.l10.m2.2.2.2.2.2" xref="alg2.l10.m2.2.2.2.2.2.cmml">​</mo><mrow id="alg2.l10.m2.2.2.2.2.1.1" xref="alg2.l10.m2.2.2.2.2.1.1.1.cmml"><mo maxsize="80%" minsize="80%" id="alg2.l10.m2.2.2.2.2.1.1.2" xref="alg2.l10.m2.2.2.2.2.1.1.1.cmml">(</mo><msubsup id="alg2.l10.m2.2.2.2.2.1.1.1" xref="alg2.l10.m2.2.2.2.2.1.1.1.cmml"><mi mathsize="80%" id="alg2.l10.m2.2.2.2.2.1.1.1.2.2" xref="alg2.l10.m2.2.2.2.2.1.1.1.2.2.cmml">𝐱</mi><mrow id="alg2.l10.m2.2.2.2.2.1.1.1.2.3" xref="alg2.l10.m2.2.2.2.2.1.1.1.2.3.cmml"><mn mathsize="80%" id="alg2.l10.m2.2.2.2.2.1.1.1.2.3.2" xref="alg2.l10.m2.2.2.2.2.1.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="80%" rspace="0.278em" id="alg2.l10.m2.2.2.2.2.1.1.1.2.3.1" xref="alg2.l10.m2.2.2.2.2.1.1.1.2.3.1.cmml">:</mo><mi mathsize="80%" id="alg2.l10.m2.2.2.2.2.1.1.1.2.3.3" xref="alg2.l10.m2.2.2.2.2.1.1.1.2.3.3.cmml">H</mi></mrow><mi mathsize="80%" id="alg2.l10.m2.2.2.2.2.1.1.1.3" xref="alg2.l10.m2.2.2.2.2.1.1.1.3.cmml">new</mi></msubsup><mo maxsize="80%" minsize="80%" id="alg2.l10.m2.2.2.2.2.1.1.3" xref="alg2.l10.m2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo maxsize="80%" minsize="80%" id="alg2.l10.m2.2.2.2.5" xref="alg2.l10.m2.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="alg2.l10.m2.2b"><interval closure="open" id="alg2.l10.m2.2.2.3.cmml" xref="alg2.l10.m2.2.2.2"><apply id="alg2.l10.m2.1.1.1.1.cmml" xref="alg2.l10.m2.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l10.m2.1.1.1.1.1.cmml" xref="alg2.l10.m2.1.1.1.1">superscript</csymbol><apply id="alg2.l10.m2.1.1.1.1.2.cmml" xref="alg2.l10.m2.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l10.m2.1.1.1.1.2.1.cmml" xref="alg2.l10.m2.1.1.1.1">subscript</csymbol><ci id="alg2.l10.m2.1.1.1.1.2.2.cmml" xref="alg2.l10.m2.1.1.1.1.2.2">𝐱</ci><apply id="alg2.l10.m2.1.1.1.1.2.3.cmml" xref="alg2.l10.m2.1.1.1.1.2.3"><ci id="alg2.l10.m2.1.1.1.1.2.3.1.cmml" xref="alg2.l10.m2.1.1.1.1.2.3.1">:</ci><cn type="integer" id="alg2.l10.m2.1.1.1.1.2.3.2.cmml" xref="alg2.l10.m2.1.1.1.1.2.3.2">1</cn><ci id="alg2.l10.m2.1.1.1.1.2.3.3.cmml" xref="alg2.l10.m2.1.1.1.1.2.3.3">𝐻</ci></apply></apply><ci id="alg2.l10.m2.1.1.1.1.3.cmml" xref="alg2.l10.m2.1.1.1.1.3">new</ci></apply><apply id="alg2.l10.m2.2.2.2.2.cmml" xref="alg2.l10.m2.2.2.2.2"><times id="alg2.l10.m2.2.2.2.2.2.cmml" xref="alg2.l10.m2.2.2.2.2.2"></times><apply id="alg2.l10.m2.2.2.2.2.3.cmml" xref="alg2.l10.m2.2.2.2.2.3"><apply id="alg2.l10.m2.2.2.2.2.3.1.cmml" xref="alg2.l10.m2.2.2.2.2.3.1"><apply id="alg2.l10.m2.2.2.2.2.3.1.1.cmml" xref="alg2.l10.m2.2.2.2.2.3.1.1"><csymbol cd="ambiguous" id="alg2.l10.m2.2.2.2.2.3.1.1.1.cmml" xref="alg2.l10.m2.2.2.2.2.3.1.1">subscript</csymbol><ci id="alg2.l10.m2.2.2.2.2.3.1.1.2.cmml" xref="alg2.l10.m2.2.2.2.2.3.1.1.2">∇</ci><ci id="alg2.l10.m2.2.2.2.2.3.1.1.3.cmml" xref="alg2.l10.m2.2.2.2.2.3.1.1.3">𝐱</ci></apply><log id="alg2.l10.m2.2.2.2.2.3.1.2.cmml" xref="alg2.l10.m2.2.2.2.2.3.1.2"></log></apply><ci id="alg2.l10.m2.2.2.2.2.3.2.cmml" xref="alg2.l10.m2.2.2.2.2.3.2">𝑐</ci></apply><apply id="alg2.l10.m2.2.2.2.2.1.1.1.cmml" xref="alg2.l10.m2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="alg2.l10.m2.2.2.2.2.1.1.1.1.cmml" xref="alg2.l10.m2.2.2.2.2.1.1">superscript</csymbol><apply id="alg2.l10.m2.2.2.2.2.1.1.1.2.cmml" xref="alg2.l10.m2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="alg2.l10.m2.2.2.2.2.1.1.1.2.1.cmml" xref="alg2.l10.m2.2.2.2.2.1.1">subscript</csymbol><ci id="alg2.l10.m2.2.2.2.2.1.1.1.2.2.cmml" xref="alg2.l10.m2.2.2.2.2.1.1.1.2.2">𝐱</ci><apply id="alg2.l10.m2.2.2.2.2.1.1.1.2.3.cmml" xref="alg2.l10.m2.2.2.2.2.1.1.1.2.3"><ci id="alg2.l10.m2.2.2.2.2.1.1.1.2.3.1.cmml" xref="alg2.l10.m2.2.2.2.2.1.1.1.2.3.1">:</ci><cn type="integer" id="alg2.l10.m2.2.2.2.2.1.1.1.2.3.2.cmml" xref="alg2.l10.m2.2.2.2.2.1.1.1.2.3.2">1</cn><ci id="alg2.l10.m2.2.2.2.2.1.1.1.2.3.3.cmml" xref="alg2.l10.m2.2.2.2.2.1.1.1.2.3.3">𝐻</ci></apply></apply><ci id="alg2.l10.m2.2.2.2.2.1.1.1.3.cmml" xref="alg2.l10.m2.2.2.2.2.1.1.1.3">new</ci></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="alg2.l10.m2.2c">(\mathbf{x}_{1:H}^{\mathrm{new}},\nabla_{\mathbf{x}}\log c({\mathbf{x}}_{1:H}^{\mathrm{new}}))</annotation></semantics></math><span id="alg2.l10.4" class="ltx_text" style="font-size:80%;">
</span>
</div>
<div id="alg2.l11" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l11.1.1.1" class="ltx_text" style="font-size:80%;">11:</span></span><span id="alg2.l11.2" class="ltx_text" style="font-size:80%;">  </span><span id="alg2.l11.3" class="ltx_text ltx_font_bold" style="font-size:80%;">end</span><span id="alg2.l11.4" class="ltx_text" style="font-size:80%;"> </span><span id="alg2.l11.5" class="ltx_text ltx_font_bold" style="font-size:80%;">for</span>
</div>
<div id="alg2.l12" class="ltx_listingline">
<span class="ltx_tag ltx_tag_listingline"><span id="alg2.l12.1.1.1" class="ltx_text" style="font-size:80%;">12:</span></span><span id="alg2.l12.2" class="ltx_text" style="font-size:80%;">  </span><span id="alg2.l12.3" class="ltx_text ltx_font_bold" style="font-size:80%;">Return</span><span id="alg2.l12.4" class="ltx_text" style="font-size:80%;"> </span><math id="alg2.l12.m1.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}" display="inline"><semantics id="alg2.l12.m1.1a"><msub id="alg2.l12.m1.1.1" xref="alg2.l12.m1.1.1.cmml"><mi mathsize="80%" id="alg2.l12.m1.1.1.2" xref="alg2.l12.m1.1.1.2.cmml">𝐱</mi><mrow id="alg2.l12.m1.1.1.3" xref="alg2.l12.m1.1.1.3.cmml"><mn mathsize="80%" id="alg2.l12.m1.1.1.3.2" xref="alg2.l12.m1.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="80%" rspace="0.278em" id="alg2.l12.m1.1.1.3.1" xref="alg2.l12.m1.1.1.3.1.cmml">:</mo><mi mathsize="80%" id="alg2.l12.m1.1.1.3.3" xref="alg2.l12.m1.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="alg2.l12.m1.1b"><apply id="alg2.l12.m1.1.1.cmml" xref="alg2.l12.m1.1.1"><csymbol cd="ambiguous" id="alg2.l12.m1.1.1.1.cmml" xref="alg2.l12.m1.1.1">subscript</csymbol><ci id="alg2.l12.m1.1.1.2.cmml" xref="alg2.l12.m1.1.1.2">𝐱</ci><apply id="alg2.l12.m1.1.1.3.cmml" xref="alg2.l12.m1.1.1.3"><ci id="alg2.l12.m1.1.1.3.1.cmml" xref="alg2.l12.m1.1.1.3.1">:</ci><cn type="integer" id="alg2.l12.m1.1.1.3.2.cmml" xref="alg2.l12.m1.1.1.3.2">1</cn><ci id="alg2.l12.m1.1.1.3.3.cmml" xref="alg2.l12.m1.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l12.m1.1c">\mathbf{x}_{1:T}</annotation></semantics></math><span id="alg2.l12.5" class="ltx_text" style="font-size:80%;">.

</span>
</div>
</div>
</figure>
</div>
</div>
</figure>
<div id="S3.SS0.SSS0.Px2.p2" class="ltx_para">
<p id="S3.SS0.SSS0.Px2.p2.18" class="ltx_p">The RNN with weights <math id="S3.SS0.SSS0.Px2.p2.1.m1.1" class="ltx_Math" alttext="\theta" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.1.m1.1a"><mi id="S3.SS0.SSS0.Px2.p2.1.m1.1.1" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.1.m1.1b"><ci id="S3.SS0.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.1.m1.1c">\theta</annotation></semantics></math> maintains latents <math id="S3.SS0.SSS0.Px2.p2.2.m2.1" class="ltx_Math" alttext="\mathbf{z}_{t}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.2.m2.1a"><msub id="S3.SS0.SSS0.Px2.p2.2.m2.1.1" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.2" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.2.cmml">𝐳</mi><mi id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.3" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.2.m2.1b"><apply id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.2">𝐳</ci><ci id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.2.m2.1c">\mathbf{z}_{t}</annotation></semantics></math> capturing the influence of past tokens, and these evolve via dynamics <math id="S3.SS0.SSS0.Px2.p2.3.m3.1" class="ltx_Math" alttext="\mathbf{z}_{t}\sim p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.3.m3.1a"><mrow id="S3.SS0.SSS0.Px2.p2.3.m3.1.1" xref="S3.SS0.SSS0.Px2.p2.3.m3.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p2.3.m3.1.1.3" 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When an incoming noisy observation <math id="S3.SS0.SSS0.Px2.p2.4.m4.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.4.m4.1a"><msubsup id="S3.SS0.SSS0.Px2.p2.4.m4.1.1" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.2.2" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.2.3" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.2.3.cmml">t</mi><msub id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.3" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.3.2" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.3.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.3.3" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.4.m4.1b"><apply id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.1.cmml" 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made, the hidden state is updated in a Markovian fashion <math id="S3.SS0.SSS0.Px2.p2.5.m5.1" class="ltx_Math" alttext="\mathbf{z}_{t}\sim p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.5.m5.1a"><mrow id="S3.SS0.SSS0.Px2.p2.5.m5.1.1" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.3" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.3.2" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.3.2.cmml">𝐳</mi><mi id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.3.3" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.3.3.cmml">t</mi></msub><mo id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.2" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.2.cmml">∼</mo><mrow id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.3" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.3.2" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.3.2.cmml">p</mi><mi 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xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.2.2.2.2.3">𝑡</ci></apply><apply id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.2.2.2.3.cmml" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.2.2.2.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.2.2.2.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.2.2.2.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.2.2.2.3.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.2.2.2.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.2.2.2.3.3">𝑡</ci></apply></apply><apply id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.3.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.3.3.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.3.3.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.3.3.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.3.3.3.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.3.3.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.5.m5.1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.5.m5.1c">\mathbf{z}_{t}\sim p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})</annotation></semantics></math><span id="footnote1" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>We implement <math id="footnote1.m1.1" class="ltx_Math" alttext="\mathbf{z}_{t}=p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})" display="inline"><semantics id="footnote1.m1.1b"><mrow id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml"><msub id="footnote1.m1.1.1.3" xref="footnote1.m1.1.1.3.cmml"><mi id="footnote1.m1.1.1.3.2" xref="footnote1.m1.1.1.3.2.cmml">𝐳</mi><mi id="footnote1.m1.1.1.3.3" xref="footnote1.m1.1.1.3.3.cmml">t</mi></msub><mo id="footnote1.m1.1.1.2" xref="footnote1.m1.1.1.2.cmml">=</mo><mrow id="footnote1.m1.1.1.1" xref="footnote1.m1.1.1.1.cmml"><msub id="footnote1.m1.1.1.1.3" xref="footnote1.m1.1.1.1.3.cmml"><mi id="footnote1.m1.1.1.1.3.2" xref="footnote1.m1.1.1.1.3.2.cmml">p</mi><mi id="footnote1.m1.1.1.1.3.3" xref="footnote1.m1.1.1.1.3.3.cmml">θ</mi></msub><mo lspace="0em" rspace="0em" id="footnote1.m1.1.1.1.2" xref="footnote1.m1.1.1.1.2.cmml">​</mo><mrow id="footnote1.m1.1.1.1.1.1" xref="footnote1.m1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="footnote1.m1.1.1.1.1.1.2" xref="footnote1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="footnote1.m1.1.1.1.1.1.1" xref="footnote1.m1.1.1.1.1.1.1.cmml"><msub id="footnote1.m1.1.1.1.1.1.1.5" xref="footnote1.m1.1.1.1.1.1.1.5.cmml"><mi id="footnote1.m1.1.1.1.1.1.1.5.2" xref="footnote1.m1.1.1.1.1.1.1.5.2.cmml">𝐳</mi><mi id="footnote1.m1.1.1.1.1.1.1.5.3" xref="footnote1.m1.1.1.1.1.1.1.5.3.cmml">t</mi></msub><mo fence="false" id="footnote1.m1.1.1.1.1.1.1.4" xref="footnote1.m1.1.1.1.1.1.1.4.cmml">|</mo><mrow id="footnote1.m1.1.1.1.1.1.1.3.3" xref="footnote1.m1.1.1.1.1.1.1.3.4.cmml"><msub id="footnote1.m1.1.1.1.1.1.1.1.1.1" xref="footnote1.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="footnote1.m1.1.1.1.1.1.1.1.1.1.2" xref="footnote1.m1.1.1.1.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="footnote1.m1.1.1.1.1.1.1.1.1.1.3" xref="footnote1.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="footnote1.m1.1.1.1.1.1.1.1.1.1.3.2" xref="footnote1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">t</mi><mo id="footnote1.m1.1.1.1.1.1.1.1.1.1.3.1" xref="footnote1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="footnote1.m1.1.1.1.1.1.1.1.1.1.3.3" xref="footnote1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="footnote1.m1.1.1.1.1.1.1.3.3.4" xref="footnote1.m1.1.1.1.1.1.1.3.4.cmml">,</mo><msubsup id="footnote1.m1.1.1.1.1.1.1.2.2.2" xref="footnote1.m1.1.1.1.1.1.1.2.2.2.cmml"><mi id="footnote1.m1.1.1.1.1.1.1.2.2.2.2.2" xref="footnote1.m1.1.1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mi id="footnote1.m1.1.1.1.1.1.1.2.2.2.2.3" xref="footnote1.m1.1.1.1.1.1.1.2.2.2.2.3.cmml">t</mi><msub id="footnote1.m1.1.1.1.1.1.1.2.2.2.3" xref="footnote1.m1.1.1.1.1.1.1.2.2.2.3.cmml"><mi id="footnote1.m1.1.1.1.1.1.1.2.2.2.3.2" xref="footnote1.m1.1.1.1.1.1.1.2.2.2.3.2.cmml">k</mi><mi id="footnote1.m1.1.1.1.1.1.1.2.2.2.3.3" xref="footnote1.m1.1.1.1.1.1.1.2.2.2.3.3.cmml">t</mi></msub></msubsup><mo id="footnote1.m1.1.1.1.1.1.1.3.3.5" xref="footnote1.m1.1.1.1.1.1.1.3.4.cmml">,</mo><msub id="footnote1.m1.1.1.1.1.1.1.3.3.3" xref="footnote1.m1.1.1.1.1.1.1.3.3.3.cmml"><mi id="footnote1.m1.1.1.1.1.1.1.3.3.3.2" xref="footnote1.m1.1.1.1.1.1.1.3.3.3.2.cmml">k</mi><mi id="footnote1.m1.1.1.1.1.1.1.3.3.3.3" xref="footnote1.m1.1.1.1.1.1.1.3.3.3.3.cmml">t</mi></msub></mrow></mrow><mo stretchy="false" id="footnote1.m1.1.1.1.1.1.3" xref="footnote1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><apply id="footnote1.m1.1.1.cmml" xref="footnote1.m1.1.1"><eq id="footnote1.m1.1.1.2.cmml" xref="footnote1.m1.1.1.2"></eq><apply id="footnote1.m1.1.1.3.cmml" xref="footnote1.m1.1.1.3"><csymbol cd="ambiguous" id="footnote1.m1.1.1.3.1.cmml" xref="footnote1.m1.1.1.3">subscript</csymbol><ci id="footnote1.m1.1.1.3.2.cmml" xref="footnote1.m1.1.1.3.2">𝐳</ci><ci id="footnote1.m1.1.1.3.3.cmml" xref="footnote1.m1.1.1.3.3">𝑡</ci></apply><apply id="footnote1.m1.1.1.1.cmml" xref="footnote1.m1.1.1.1"><times id="footnote1.m1.1.1.1.2.cmml" xref="footnote1.m1.1.1.1.2"></times><apply id="footnote1.m1.1.1.1.3.cmml" xref="footnote1.m1.1.1.1.3"><csymbol cd="ambiguous" id="footnote1.m1.1.1.1.3.1.cmml" xref="footnote1.m1.1.1.1.3">subscript</csymbol><ci id="footnote1.m1.1.1.1.3.2.cmml" xref="footnote1.m1.1.1.1.3.2">𝑝</ci><ci id="footnote1.m1.1.1.1.3.3.cmml" xref="footnote1.m1.1.1.1.3.3">𝜃</ci></apply><apply id="footnote1.m1.1.1.1.1.1.1.cmml" 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xref="footnote1.m1.1.1.1.1.1.1.2.2.2.3"><csymbol cd="ambiguous" id="footnote1.m1.1.1.1.1.1.1.2.2.2.3.1.cmml" xref="footnote1.m1.1.1.1.1.1.1.2.2.2.3">subscript</csymbol><ci id="footnote1.m1.1.1.1.1.1.1.2.2.2.3.2.cmml" xref="footnote1.m1.1.1.1.1.1.1.2.2.2.3.2">𝑘</ci><ci id="footnote1.m1.1.1.1.1.1.1.2.2.2.3.3.cmml" xref="footnote1.m1.1.1.1.1.1.1.2.2.2.3.3">𝑡</ci></apply></apply><apply id="footnote1.m1.1.1.1.1.1.1.3.3.3.cmml" xref="footnote1.m1.1.1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="footnote1.m1.1.1.1.1.1.1.3.3.3.1.cmml" xref="footnote1.m1.1.1.1.1.1.1.3.3.3">subscript</csymbol><ci id="footnote1.m1.1.1.1.1.1.1.3.3.3.2.cmml" xref="footnote1.m1.1.1.1.1.1.1.3.3.3.2">𝑘</ci><ci id="footnote1.m1.1.1.1.1.1.1.3.3.3.3.cmml" xref="footnote1.m1.1.1.1.1.1.1.3.3.3.3">𝑡</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m1.1d">\mathbf{z}_{t}=p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})</annotation></semantics></math> to be deterministic, with <math id="footnote1.m2.1" class="ltx_Math" alttext="\mathbf{z}_{t}" display="inline"><semantics id="footnote1.m2.1b"><msub id="footnote1.m2.1.1" xref="footnote1.m2.1.1.cmml"><mi id="footnote1.m2.1.1.2" xref="footnote1.m2.1.1.2.cmml">𝐳</mi><mi id="footnote1.m2.1.1.3" xref="footnote1.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="footnote1.m2.1c"><apply id="footnote1.m2.1.1.cmml" xref="footnote1.m2.1.1"><csymbol cd="ambiguous" id="footnote1.m2.1.1.1.cmml" xref="footnote1.m2.1.1">subscript</csymbol><ci id="footnote1.m2.1.1.2.cmml" xref="footnote1.m2.1.1.2">𝐳</ci><ci id="footnote1.m2.1.1.3.cmml" xref="footnote1.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m2.1d">\mathbf{z}_{t}</annotation></semantics></math> representing a distribution over beliefs rather than a sample from it. This allows training by backpropogating through the latent dynamics in Eq.(<a href="#S3.E1" title="Equation 3.1 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3.1</span></a>).</span></span></span>. When <math id="S3.SS0.SSS0.Px2.p2.6.m6.1" class="ltx_Math" alttext="k_{t}=0" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.6.m6.1a"><mrow id="S3.SS0.SSS0.Px2.p2.6.m6.1.1" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.2" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.3" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.3.cmml">t</mi></msub><mo id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.1" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.1.cmml">=</mo><mn id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.3" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.6.m6.1b"><apply id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1"><eq id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.1"></eq><apply id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.2.3">𝑡</ci></apply><cn type="integer" id="S3.SS0.SSS0.Px2.p2.6.m6.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.6.m6.1c">k_{t}=0</annotation></semantics></math>, this is the posterior update in Bayes filtering; whereas when <math id="S3.SS0.SSS0.Px2.p2.7.m7.1" class="ltx_Math" alttext="k_{t}=K" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.7.m7.1a"><mrow id="S3.SS0.SSS0.Px2.p2.7.m7.1.1" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.2" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.3" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.3.cmml">t</mi></msub><mo id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.1" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.1.cmml">=</mo><mi id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.3" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.7.m7.1b"><apply id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1"><eq id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.1"></eq><apply id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.2.3">𝑡</ci></apply><ci id="S3.SS0.SSS0.Px2.p2.7.m7.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.7.m7.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.7.m7.1c">k_{t}=K</annotation></semantics></math> (and <math id="S3.SS0.SSS0.Px2.p2.8.m8.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{K}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.8.m8.1a"><msubsup id="S3.SS0.SSS0.Px2.p2.8.m8.1.1" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.8.m8.1.1.2.2" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px2.p2.8.m8.1.1.2.3" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1.2.3.cmml">t</mi><mi id="S3.SS0.SSS0.Px2.p2.8.m8.1.1.3" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1.3.cmml">K</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.8.m8.1b"><apply id="S3.SS0.SSS0.Px2.p2.8.m8.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.8.m8.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1">superscript</csymbol><apply id="S3.SS0.SSS0.Px2.p2.8.m8.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.8.m8.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.8.m8.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px2.p2.8.m8.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1.2.3">𝑡</ci></apply><ci id="S3.SS0.SSS0.Px2.p2.8.m8.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.8.m8.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.8.m8.1c">\mathbf{x}_{t}^{K}</annotation></semantics></math> is pure noise and thus uninformative), this is equivalent to modeling the “prior distribution” <math id="S3.SS0.SSS0.Px2.p2.9.m9.1" class="ltx_Math" alttext="p_{\theta}(\mathbf{z}_{t}\mid\mathbf{z}_{t-1})" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.9.m9.1a"><mrow id="S3.SS0.SSS0.Px2.p2.9.m9.1.1" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.2" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.2.cmml">p</mi><mi id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.3" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.3.cmml">θ</mi></msub><mo lspace="0em" rspace="0em" id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.2" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.2.cmml">​</mo><mrow id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.2" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.2" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.2.cmml">𝐳</mi><mi id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.3" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.3.cmml">t</mi></msub><mo id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.1.cmml">∣</mo><msub id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.2" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.2.cmml">𝐳</mi><mrow id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.2" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.2.cmml">t</mi><mo id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.1" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.1.cmml">−</mo><mn id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.3" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><mo stretchy="false" id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.3" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.9.m9.1b"><apply id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1"><times id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.2"></times><apply id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.2">𝑝</ci><ci id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.3.3">𝜃</ci></apply><apply id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1"><csymbol cd="latexml" id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.1">conditional</csymbol><apply id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.2">𝐳</ci><ci id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.2">𝐳</ci><apply id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3"><minus id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.1"></minus><ci id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.2">𝑡</ci><cn type="integer" id="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.9.m9.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.9.m9.1c">p_{\theta}(\mathbf{z}_{t}\mid\mathbf{z}_{t-1})</annotation></semantics></math> in Bayes filtering. Given latent <math id="S3.SS0.SSS0.Px2.p2.10.m10.1" class="ltx_Math" alttext="\mathbf{z}_{t}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.10.m10.1a"><msub id="S3.SS0.SSS0.Px2.p2.10.m10.1.1" xref="S3.SS0.SSS0.Px2.p2.10.m10.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.10.m10.1.1.2" xref="S3.SS0.SSS0.Px2.p2.10.m10.1.1.2.cmml">𝐳</mi><mi id="S3.SS0.SSS0.Px2.p2.10.m10.1.1.3" xref="S3.SS0.SSS0.Px2.p2.10.m10.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.10.m10.1b"><apply id="S3.SS0.SSS0.Px2.p2.10.m10.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.10.m10.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.10.m10.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.10.m10.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.10.m10.1.1.2">𝐳</ci><ci id="S3.SS0.SSS0.Px2.p2.10.m10.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.10.m10.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.10.m10.1c">\mathbf{z}_{t}</annotation></semantics></math>, an observation model <math id="S3.SS0.SSS0.Px2.p2.11.m11.1" class="ltx_Math" alttext="p_{\theta}(\mathbf{x}_{t}^{0}|\mathbf{z}_{t})" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.11.m11.1a"><mrow id="S3.SS0.SSS0.Px2.p2.11.m11.1.1" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.2" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.2.cmml">p</mi><mi id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.3" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.3.cmml">θ</mi></msub><mo lspace="0em" rspace="0em" id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.2" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.2.cmml">​</mo><mrow id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.2" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.cmml"><msubsup id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.2.2" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.2.3" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.2.3.cmml">t</mi><mn id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.3" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.3.cmml">0</mn></msubsup><mo fence="false" id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.1.cmml">|</mo><msub id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.2" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.2.cmml">𝐳</mi><mi id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.3" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.3.cmml">t</mi></msub></mrow><mo stretchy="false" id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.3" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.11.m11.1b"><apply id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1"><times id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.2"></times><apply id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.2">𝑝</ci><ci id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.3.3">𝜃</ci></apply><apply id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1"><csymbol cd="latexml" id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.1">conditional</csymbol><apply id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2">superscript</csymbol><apply id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.2.1.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.2.2.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.2.3.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.2.3">𝑡</ci></apply><cn type="integer" id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.2.3">0</cn></apply><apply id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.2">𝐳</ci><ci id="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.11.m11.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.11.m11.1c">p_{\theta}(\mathbf{x}_{t}^{0}|\mathbf{z}_{t})</annotation></semantics></math> predicts <math id="S3.SS0.SSS0.Px2.p2.12.m12.1" class="ltx_Math" alttext="\mathbf{x}_{t}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.12.m12.1a"><msub id="S3.SS0.SSS0.Px2.p2.12.m12.1.1" xref="S3.SS0.SSS0.Px2.p2.12.m12.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.12.m12.1.1.2" xref="S3.SS0.SSS0.Px2.p2.12.m12.1.1.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px2.p2.12.m12.1.1.3" xref="S3.SS0.SSS0.Px2.p2.12.m12.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.12.m12.1b"><apply id="S3.SS0.SSS0.Px2.p2.12.m12.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.12.m12.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.12.m12.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.12.m12.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.12.m12.1.1.2">𝐱</ci><ci id="S3.SS0.SSS0.Px2.p2.12.m12.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.12.m12.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.12.m12.1c">\mathbf{x}_{t}</annotation></semantics></math>;
this unit has the same input-output behavior as a standard conditional diffusion model, using a conditioning variable <math id="S3.SS0.SSS0.Px2.p2.13.m13.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.13.m13.1a"><msub id="S3.SS0.SSS0.Px2.p2.13.m13.1.1" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.2" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.2.cmml">𝐳</mi><mrow id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.2" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.2.cmml">t</mi><mo id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.1" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.1.cmml">−</mo><mn id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.3" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.13.m13.1b"><apply id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.2">𝐳</ci><apply id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3"><minus id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.1"></minus><ci id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.2">𝑡</ci><cn type="integer" id="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.13.m13.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.13.m13.1c">\mathbf{z}_{t-1}</annotation></semantics></math> and a noisy token <math id="S3.SS0.SSS0.Px2.p2.14.m14.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.14.m14.1a"><msubsup id="S3.SS0.SSS0.Px2.p2.14.m14.1.1" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.2.2" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.2.3" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.2.3.cmml">t</mi><msub id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.2" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.3" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.14.m14.1b"><apply id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1">superscript</csymbol><apply id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.2.3">𝑡</ci></apply><apply id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.14.m14.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.14.m14.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math> as input to predict the unnoised <math id="S3.SS0.SSS0.Px2.p2.15.m15.1" class="ltx_Math" alttext="\mathbf{x}_{t}=\mathbf{x}_{t}^{0}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.15.m15.1a"><mrow id="S3.SS0.SSS0.Px2.p2.15.m15.1.1" xref="S3.SS0.SSS0.Px2.p2.15.m15.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p2.15.m15.1.1.2" xref="S3.SS0.SSS0.Px2.p2.15.m15.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px2.p2.15.m15.1.1.2.2" xref="S3.SS0.SSS0.Px2.p2.15.m15.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px2.p2.15.m15.1.1.2.3" xref="S3.SS0.SSS0.Px2.p2.15.m15.1.1.2.3.cmml">t</mi></msub><mo id="S3.SS0.SSS0.Px2.p2.15.m15.1.1.1" xref="S3.SS0.SSS0.Px2.p2.15.m15.1.1.1.cmml">=</mo><msubsup id="S3.SS0.SSS0.Px2.p2.15.m15.1.1.3" xref="S3.SS0.SSS0.Px2.p2.15.m15.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p2.15.m15.1.1.3.2.2" xref="S3.SS0.SSS0.Px2.p2.15.m15.1.1.3.2.2.cmml">𝐱</mi><mi id="S3.SS0.SSS0.Px2.p2.15.m15.1.1.3.2.3" xref="S3.SS0.SSS0.Px2.p2.15.m15.1.1.3.2.3.cmml">t</mi><mn id="S3.SS0.SSS0.Px2.p2.15.m15.1.1.3.3" xref="S3.SS0.SSS0.Px2.p2.15.m15.1.1.3.3.cmml">0</mn></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.15.m15.1b"><apply 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xref="S3.SS0.SSS0.Px2.p2.16.m16.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.16.m16.1c">\epsilon^{k_{t}}</annotation></semantics></math> via affine reparametrization <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib29" title="" class="ltx_ref">29</a>]</cite>.
We can thus directly train (Causal) Diffusion Forcing with the conventional diffusion training objective. We parameterize the aforementioned unit in terms of noise prediction <math id="S3.SS0.SSS0.Px2.p2.17.m17.3" class="ltx_Math" alttext="\bm{\epsilon}_{\theta}(\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.17.m17.3a"><mrow id="S3.SS0.SSS0.Px2.p2.17.m17.3.3" xref="S3.SS0.SSS0.Px2.p2.17.m17.3.3.cmml"><msub id="S3.SS0.SSS0.Px2.p2.17.m17.3.3.5" xref="S3.SS0.SSS0.Px2.p2.17.m17.3.3.5.cmml"><mi class="ltx_mathvariant_bold-italic" mathvariant="bold-italic" id="S3.SS0.SSS0.Px2.p2.17.m17.3.3.5.2" xref="S3.SS0.SSS0.Px2.p2.17.m17.3.3.5.2.cmml">ϵ</mi><mi id="S3.SS0.SSS0.Px2.p2.17.m17.3.3.5.3" xref="S3.SS0.SSS0.Px2.p2.17.m17.3.3.5.3.cmml">θ</mi></msub><mo lspace="0em" rspace="0em" id="S3.SS0.SSS0.Px2.p2.17.m17.3.3.4" xref="S3.SS0.SSS0.Px2.p2.17.m17.3.3.4.cmml">​</mo><mrow id="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.3" xref="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.4.cmml"><mo stretchy="false" id="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.3.4" 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xref="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.2.2.cmml" xref="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.2.2">𝐱</ci><ci id="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.2.3.cmml" xref="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.2.3">𝑡</ci></apply><apply id="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.3.cmml" xref="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.3.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.17.m17.2.2.2.2.2.3.3">𝑡</ci></apply></apply><apply id="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.3.3.1.cmml" xref="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.3.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.3.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.3.3.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.3.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.17.m17.3.3.3.3.3.3">𝑡</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.17.m17.3c">\bm{\epsilon}_{\theta}(\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})</annotation></semantics></math>.
We then find parameters <math id="S3.SS0.SSS0.Px2.p2.18.m18.1" class="ltx_Math" alttext="\theta" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.18.m18.1a"><mi id="S3.SS0.SSS0.Px2.p2.18.m18.1.1" xref="S3.SS0.SSS0.Px2.p2.18.m18.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.18.m18.1b"><ci id="S3.SS0.SSS0.Px2.p2.18.m18.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.18.m18.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.18.m18.1c">\theta</annotation></semantics></math> by minimizing the loss</p>
<table id="S3.E1" class="ltx_equation ltx_eqn_table">

<tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math id="S3.E1.m1.2" class="ltx_Math" alttext="\mathop{{\mathbb{E}}}_{\begin{subarray}{c}k_{t},\mathbf{x}_{t},\bm{\epsilon}_{t}\\
\mathbf{z}_{t}\sim p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})\end{subarray}}{\textstyle{\sum}_{t=1}^{T}}\Big{[}\|\bm{\epsilon}_{t}-\bm{\epsilon}_{\theta}(\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})\|^{2}\Big{]}," display="block"><semantics id="S3.E1.m1.2a"><mrow id="S3.E1.m1.2.2.1" xref="S3.E1.m1.2.2.1.1.cmml"><mrow id="S3.E1.m1.2.2.1.1" xref="S3.E1.m1.2.2.1.1.cmml"><munder id="S3.E1.m1.2.2.1.1.2" xref="S3.E1.m1.2.2.1.1.2.cmml"><mo mathsize="120%" movablelimits="false" id="S3.E1.m1.2.2.1.1.2.2" xref="S3.E1.m1.2.2.1.1.2.2.cmml">𝔼</mo><mtable rowspacing="0pt" id="S3.E1.m1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.2.cmml"><mtr id="S3.E1.m1.1.1.1.1.1.1a" xref="S3.E1.m1.1.1.1.2.cmml"><mtd id="S3.E1.m1.1.1.1.1.1.1b" xref="S3.E1.m1.1.1.1.2.cmml"><mrow id="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.3" xref="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.4.cmml"><msub id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">k</mi><mi id="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S3.E1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.3.4" xref="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.4.cmml">,</mo><msub id="S3.E1.m1.1.1.1.1.1.1.2.2.2.2.2.2" xref="S3.E1.m1.1.1.1.1.1.1.2.2.2.2.2.2.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.2.2.2.2.2.2.2" xref="S3.E1.m1.1.1.1.1.1.1.2.2.2.2.2.2.2.cmml">𝐱</mi><mi id="S3.E1.m1.1.1.1.1.1.1.2.2.2.2.2.2.3" xref="S3.E1.m1.1.1.1.1.1.1.2.2.2.2.2.2.3.cmml">t</mi></msub><mo id="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.3.5" xref="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.4.cmml">,</mo><msub id="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.3.3" xref="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.3.3.cmml"><mi class="ltx_mathvariant_bold-italic" mathvariant="bold-italic" id="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.3.3.2" xref="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.3.3.2.cmml">ϵ</mi><mi id="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.3.3.3" xref="S3.E1.m1.1.1.1.1.1.1.3.3.3.3.3.3.3.cmml">t</mi></msub></mrow></mtd></mtr><mtr id="S3.E1.m1.1.1.1.1.1.1c" xref="S3.E1.m1.1.1.1.2.cmml"><mtd id="S3.E1.m1.1.1.1.1.1.1d" xref="S3.E1.m1.1.1.1.2.cmml"><mrow id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.cmml"><msub id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.3.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.3.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.3.2.cmml">𝐳</mi><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.3.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.3.3.cmml">t</mi></msub><mo id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.2.cmml">∼</mo><mrow id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.cmml"><msub id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.3.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.3.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.3.2.cmml">p</mi><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.3.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.3.3.cmml">θ</mi></msub><mo lspace="0em" rspace="0em" id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.2.cmml">​</mo><mrow id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.cmml"><msub id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.5" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.5.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.5.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.5.2.cmml">𝐳</mi><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.5.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.5.3.cmml">t</mi></msub><mo fence="false" id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.4" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.4.cmml">|</mo><mrow id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.4.cmml"><msub id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.3.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.3.2.cmml">t</mi><mo id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.3.1" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.3.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.3.4" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.4.cmml">,</mo><msubsup id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.2.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.2.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.2.3.cmml">t</mi><msub id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.3.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.3.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.3.2.cmml">k</mi><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.3.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.2.2.2.3.3.cmml">t</mi></msub></msubsup><mo id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.3.5" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.4.cmml">,</mo><msub id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.3.3" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.3.3.cmml"><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.3.3.2" xref="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.3.3.2.cmml">k</mi><mi id="S3.E1.m1.1.1.1.1.1.1.4.4.1.1.1.1.1.1.3.3.3.3" 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xref="S3.E1.m1.2.2.1.1.1.1.1.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E1.m1.2c">\mathop{{\mathbb{E}}}_{\begin{subarray}{c}k_{t},\mathbf{x}_{t},\bm{\epsilon}_{t}\\
\mathbf{z}_{t}\sim p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})\end{subarray}}{\textstyle{\sum}_{t=1}^{T}}\Big{[}\|\bm{\epsilon}_{t}-\bm{\epsilon}_{\theta}(\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}},k_{t})\|^{2}\Big{]},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3.1)</span></td>
</tr></tbody>
</table>
<p id="S3.SS0.SSS0.Px2.p2.22" class="ltx_p">where we sample <math id="S3.SS0.SSS0.Px2.p2.19.m1.1" class="ltx_Math" alttext="k_{1:T}" display="inline"><semantics id="S3.SS0.SSS0.Px2.p2.19.m1.1a"><msub id="S3.SS0.SSS0.Px2.p2.19.m1.1.1" xref="S3.SS0.SSS0.Px2.p2.19.m1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.19.m1.1.1.2" xref="S3.SS0.SSS0.Px2.p2.19.m1.1.1.2.cmml">k</mi><mrow id="S3.SS0.SSS0.Px2.p2.19.m1.1.1.3" xref="S3.SS0.SSS0.Px2.p2.19.m1.1.1.3.cmml"><mn id="S3.SS0.SSS0.Px2.p2.19.m1.1.1.3.2" xref="S3.SS0.SSS0.Px2.p2.19.m1.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S3.SS0.SSS0.Px2.p2.19.m1.1.1.3.1" xref="S3.SS0.SSS0.Px2.p2.19.m1.1.1.3.1.cmml">:</mo><mi id="S3.SS0.SSS0.Px2.p2.19.m1.1.1.3.3" xref="S3.SS0.SSS0.Px2.p2.19.m1.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.19.m1.1b"><apply id="S3.SS0.SSS0.Px2.p2.19.m1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.19.m1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.19.m1.1.1.1.cmml" 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xref="S3.SS0.SSS0.Px2.p2.22.m4.2.2.1.1.1.1.2.3">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.22.m4.2.2.1.1.1.1.2.3.2.cmml" xref="S3.SS0.SSS0.Px2.p2.22.m4.2.2.1.1.1.1.2.3.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p2.22.m4.2.2.1.1.1.1.2.3.3.cmml" xref="S3.SS0.SSS0.Px2.p2.22.m4.2.2.1.1.1.1.2.3.3">𝑡</ci></apply></apply><ci id="S3.SS0.SSS0.Px2.p2.22.m4.2.2.1.1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.22.m4.2.2.1.1.1.1.3">𝐼</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.22.m4.2c">\epsilon_{t}\sim\mathcal{N}(0,\sigma^{2}_{k_{t}}I)</annotation></semantics></math> in accordance with the forward diffusion process (see Algorithm <a href="#alg1" title="Algorithm 1 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> for pseudocode). Importantly, the loss (<a href="#S3.E1" title="Equation 3.1 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3.1</span></a>) captures essential elements of Bayesian filtering and conditional diffusion.
In Appendix <a href="#A4.SS3" title="D.3 SNR reweighting ‣ Appendix D Additional Method Details ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">D.3</span></a>, we further re-derive common techniques in diffusion model training for Diffusion Forcing. Finally, we prove the validity of this objective stated informally in the following Theorem <a href="#S3.Thmtheorem1" title="Theorem 3.1 (Informal). ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3.1</span></a> in Appendix <a href="#A1" title="Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">A</span></a>.</p>
</div>
<div id="S3.Thmtheorem1" class="ltx_theorem ltx_theorem_theorem">
<h6 class="ltx_title ltx_runin ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem"><span id="S3.Thmtheorem1.1.1.1" class="ltx_text ltx_font_bold">Theorem 3.1</span></span><span id="S3.Thmtheorem1.2.2" class="ltx_text ltx_font_bold"> </span>(Informal)<span id="S3.Thmtheorem1.3.3" class="ltx_text ltx_font_bold">.</span>
</h6>
<div id="S3.Thmtheorem1.p1" class="ltx_para">
<p id="S3.Thmtheorem1.p1.3" class="ltx_p"><span id="S3.Thmtheorem1.p1.3.3" class="ltx_text ltx_font_italic">The Diffusion Forcing training procedure (<a href="#alg1" title="In Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a>) optimizes a reweighting of an Evidence Lower Bound (ELBO) on the expected log-likelihoods <math id="S3.Thmtheorem1.p1.1.1.m1.1" class="ltx_Math" alttext="\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})" display="inline"><semantics id="S3.Thmtheorem1.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem1.p1.1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mrow id="S3.Thmtheorem1.p1.1.1.m1.1.1.3" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.1.1.m1.1.1.3.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.3.1.cmml">ln</mi><mo 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xref="S3.Thmtheorem1.p1.2.2.m2.1.2.2.3.3">𝑇</ci></apply></apply><apply id="S3.Thmtheorem1.p1.2.2.m2.1.2.3.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.2.2.m2.1.2.3.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.2.3">superscript</csymbol><apply id="S3.Thmtheorem1.p1.2.2.m2.1.2.3.2.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.2.3.2.2"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.2.2.m2.1.2.3.2.1.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.2.3.2.2.1">delimited-[]</csymbol><ci id="S3.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1">𝐾</ci></apply><ci id="S3.Thmtheorem1.p1.2.2.m2.1.2.3.3.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.2.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.2.m2.1c">k_{1:T}\sim[K]^{T}</annotation></semantics></math> and <math id="S3.Thmtheorem1.p1.3.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="S3.Thmtheorem1.p1.3.3.m3.1a"><msubsup id="S3.Thmtheorem1.p1.3.3.m3.1.1" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem1.p1.3.3.m3.1.1.2.2" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.2.2.cmml">𝐱</mi><mi id="S3.Thmtheorem1.p1.3.3.m3.1.1.2.3" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.2.3.cmml">t</mi><msub id="S3.Thmtheorem1.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3.2.cmml">k</mi><mi id="S3.Thmtheorem1.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.3.m3.1b"><apply id="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1">superscript</csymbol><apply id="S3.Thmtheorem1.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.3.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.2.2">𝐱</ci><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.2.3">𝑡</ci></apply><apply id="S3.Thmtheorem1.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3.2">𝑘</ci><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.3.m3.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math> noised according to the forward process. Moreover, under appropriate conditions, optimizing (<a href="#S3.E1" title="Equation 3.1 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3.1</span></a>) also maximizes a lower bound on the likelihood for <em id="S3.Thmtheorem1.p1.3.3.1" class="ltx_emph ltx_font_upright">all sequences of noise levels, simultaneously</em>.</span></p>
</div>
</div>
<div id="S3.SS0.SSS0.Px2.p3" class="ltx_para">
<p id="S3.SS0.SSS0.Px2.p3.2" class="ltx_p">We remark that a special case of ‘all sequences of noise levels’ are those for which either <math id="S3.SS0.SSS0.Px2.p3.1.m1.1" class="ltx_Math" alttext="k_{t}=0" display="inline"><semantics id="S3.SS0.SSS0.Px2.p3.1.m1.1a"><mrow id="S3.SS0.SSS0.Px2.p3.1.m1.1.1" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.2" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.3" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.3.cmml">t</mi></msub><mo id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.1" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p3.1.m1.1b"><apply id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1"><eq id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.1"></eq><apply id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.2.3">𝑡</ci></apply><cn type="integer" id="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p3.1.m1.1c">k_{t}=0</annotation></semantics></math> or <math id="S3.SS0.SSS0.Px2.p3.2.m2.1" class="ltx_Math" alttext="k_{t}=K" display="inline"><semantics id="S3.SS0.SSS0.Px2.p3.2.m2.1a"><mrow id="S3.SS0.SSS0.Px2.p3.2.m2.1.1" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.cmml"><msub id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.cmml"><mi id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.2" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.2.cmml">k</mi><mi id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.3" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.3.cmml">t</mi></msub><mo id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.1" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.1.cmml">=</mo><mi id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.3" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p3.2.m2.1b"><apply id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1"><eq id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.1"></eq><apply id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.2">𝑘</ci><ci id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.3.cmml" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.2.3">𝑡</ci></apply><ci id="S3.SS0.SSS0.Px2.p3.2.m2.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p3.2.m2.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p3.2.m2.1c">k_{t}=K</annotation></semantics></math>; thus, one can mask out <em id="S3.SS0.SSS0.Px2.p3.2.1" class="ltx_emph ltx_font_italic">any prior token</em> and DF will learn to sample from the correct conditional distribution, modeling the distribution of all possible sub-sequences of the training set.</p>
</div>
</section>
<section id="S3.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.1 </span>Diffusion Forcing Sampling and Resulting Capabilities</h3>

<div id="S3.SS1.p1" class="ltx_para">
<p id="S3.SS1.p1.13" class="ltx_p">Sampling is depicted in <a href="#alg2" title="In Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a> and is defined by prescribing a noise schedule on a 2D <math id="S3.SS1.p1.1.m1.1" class="ltx_Math" alttext="M\times T" display="inline"><semantics id="S3.SS1.p1.1.m1.1a"><mrow id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.p1.1.m1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.2.cmml">M</mi><mo lspace="0.222em" rspace="0.222em" id="S3.SS1.p1.1.m1.1.1.1" xref="S3.SS1.p1.1.m1.1.1.1.cmml">×</mo><mi id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1"><times id="S3.SS1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1.1"></times><ci id="S3.SS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2">𝑀</ci><ci id="S3.SS1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.p1.1.m1.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">M\times T</annotation></semantics></math> grid <math id="S3.SS1.p1.2.m2.1" class="ltx_Math" alttext="\mathcal{K}\in[K]^{M\times T}" display="inline"><semantics id="S3.SS1.p1.2.m2.1a"><mrow id="S3.SS1.p1.2.m2.1.2" xref="S3.SS1.p1.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.2.m2.1.2.2" xref="S3.SS1.p1.2.m2.1.2.2.cmml">𝒦</mi><mo id="S3.SS1.p1.2.m2.1.2.1" xref="S3.SS1.p1.2.m2.1.2.1.cmml">∈</mo><msup id="S3.SS1.p1.2.m2.1.2.3" xref="S3.SS1.p1.2.m2.1.2.3.cmml"><mrow id="S3.SS1.p1.2.m2.1.2.3.2.2" xref="S3.SS1.p1.2.m2.1.2.3.2.1.cmml"><mo stretchy="false" id="S3.SS1.p1.2.m2.1.2.3.2.2.1" xref="S3.SS1.p1.2.m2.1.2.3.2.1.1.cmml">[</mo><mi id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml">K</mi><mo stretchy="false" id="S3.SS1.p1.2.m2.1.2.3.2.2.2" xref="S3.SS1.p1.2.m2.1.2.3.2.1.1.cmml">]</mo></mrow><mrow id="S3.SS1.p1.2.m2.1.2.3.3" xref="S3.SS1.p1.2.m2.1.2.3.3.cmml"><mi id="S3.SS1.p1.2.m2.1.2.3.3.2" xref="S3.SS1.p1.2.m2.1.2.3.3.2.cmml">M</mi><mo lspace="0.222em" rspace="0.222em" id="S3.SS1.p1.2.m2.1.2.3.3.1" xref="S3.SS1.p1.2.m2.1.2.3.3.1.cmml">×</mo><mi id="S3.SS1.p1.2.m2.1.2.3.3.3" xref="S3.SS1.p1.2.m2.1.2.3.3.3.cmml">T</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.2.cmml" xref="S3.SS1.p1.2.m2.1.2"><in id="S3.SS1.p1.2.m2.1.2.1.cmml" xref="S3.SS1.p1.2.m2.1.2.1"></in><ci id="S3.SS1.p1.2.m2.1.2.2.cmml" xref="S3.SS1.p1.2.m2.1.2.2">𝒦</ci><apply id="S3.SS1.p1.2.m2.1.2.3.cmml" xref="S3.SS1.p1.2.m2.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.2.3.1.cmml" xref="S3.SS1.p1.2.m2.1.2.3">superscript</csymbol><apply id="S3.SS1.p1.2.m2.1.2.3.2.1.cmml" xref="S3.SS1.p1.2.m2.1.2.3.2.2"><csymbol cd="latexml" id="S3.SS1.p1.2.m2.1.2.3.2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.2.3.2.2.1">delimited-[]</csymbol><ci id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">𝐾</ci></apply><apply id="S3.SS1.p1.2.m2.1.2.3.3.cmml" xref="S3.SS1.p1.2.m2.1.2.3.3"><times id="S3.SS1.p1.2.m2.1.2.3.3.1.cmml" xref="S3.SS1.p1.2.m2.1.2.3.3.1"></times><ci id="S3.SS1.p1.2.m2.1.2.3.3.2.cmml" xref="S3.SS1.p1.2.m2.1.2.3.3.2">𝑀</ci><ci id="S3.SS1.p1.2.m2.1.2.3.3.3.cmml" xref="S3.SS1.p1.2.m2.1.2.3.3.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">\mathcal{K}\in[K]^{M\times T}</annotation></semantics></math>; columns correspond to time step <math id="S3.SS1.p1.3.m3.1" class="ltx_Math" alttext="t" display="inline"><semantics id="S3.SS1.p1.3.m3.1a"><mi id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><ci id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">t</annotation></semantics></math> and rows indexed by <math id="S3.SS1.p1.4.m4.1" class="ltx_Math" alttext="m" display="inline"><semantics id="S3.SS1.p1.4.m4.1a"><mi id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><ci id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">m</annotation></semantics></math> determine noise-level. <math id="S3.SS1.p1.5.m5.2" class="ltx_Math" alttext="\mathcal{K}_{m,t}" display="inline"><semantics id="S3.SS1.p1.5.m5.2a"><msub id="S3.SS1.p1.5.m5.2.3" xref="S3.SS1.p1.5.m5.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.5.m5.2.3.2" xref="S3.SS1.p1.5.m5.2.3.2.cmml">𝒦</mi><mrow id="S3.SS1.p1.5.m5.2.2.2.4" xref="S3.SS1.p1.5.m5.2.2.2.3.cmml"><mi id="S3.SS1.p1.5.m5.1.1.1.1" xref="S3.SS1.p1.5.m5.1.1.1.1.cmml">m</mi><mo id="S3.SS1.p1.5.m5.2.2.2.4.1" xref="S3.SS1.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p1.5.m5.2.2.2.2" xref="S3.SS1.p1.5.m5.2.2.2.2.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.2b"><apply id="S3.SS1.p1.5.m5.2.3.cmml" xref="S3.SS1.p1.5.m5.2.3"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m5.2.3.1.cmml" xref="S3.SS1.p1.5.m5.2.3">subscript</csymbol><ci id="S3.SS1.p1.5.m5.2.3.2.cmml" xref="S3.SS1.p1.5.m5.2.3.2">𝒦</ci><list id="S3.SS1.p1.5.m5.2.2.2.3.cmml" xref="S3.SS1.p1.5.m5.2.2.2.4"><ci id="S3.SS1.p1.5.m5.1.1.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1.1.1">𝑚</ci><ci id="S3.SS1.p1.5.m5.2.2.2.2.cmml" xref="S3.SS1.p1.5.m5.2.2.2.2">𝑡</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.2c">\mathcal{K}_{m,t}</annotation></semantics></math> represents the desired noise level of the time-step <math id="S3.SS1.p1.6.m6.1" class="ltx_Math" alttext="t" display="inline"><semantics id="S3.SS1.p1.6.m6.1a"><mi id="S3.SS1.p1.6.m6.1.1" xref="S3.SS1.p1.6.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m6.1b"><ci id="S3.SS1.p1.6.m6.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m6.1c">t</annotation></semantics></math> token for row <math id="S3.SS1.p1.7.m7.1" class="ltx_Math" alttext="m" display="inline"><semantics id="S3.SS1.p1.7.m7.1a"><mi id="S3.SS1.p1.7.m7.1.1" xref="S3.SS1.p1.7.m7.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.7.m7.1b"><ci id="S3.SS1.p1.7.m7.1.1.cmml" xref="S3.SS1.p1.7.m7.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.7.m7.1c">m</annotation></semantics></math>.
To generate a whole sequence of length <math id="S3.SS1.p1.8.m8.1" class="ltx_Math" alttext="T" display="inline"><semantics id="S3.SS1.p1.8.m8.1a"><mi id="S3.SS1.p1.8.m8.1.1" xref="S3.SS1.p1.8.m8.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.8.m8.1b"><ci id="S3.SS1.p1.8.m8.1.1.cmml" xref="S3.SS1.p1.8.m8.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.8.m8.1c">T</annotation></semantics></math>, initialize the tokens <math id="S3.SS1.p1.9.m9.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}" display="inline"><semantics id="S3.SS1.p1.9.m9.1a"><msub id="S3.SS1.p1.9.m9.1.1" xref="S3.SS1.p1.9.m9.1.1.cmml"><mi id="S3.SS1.p1.9.m9.1.1.2" xref="S3.SS1.p1.9.m9.1.1.2.cmml">𝐱</mi><mrow id="S3.SS1.p1.9.m9.1.1.3" xref="S3.SS1.p1.9.m9.1.1.3.cmml"><mn id="S3.SS1.p1.9.m9.1.1.3.2" xref="S3.SS1.p1.9.m9.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S3.SS1.p1.9.m9.1.1.3.1" xref="S3.SS1.p1.9.m9.1.1.3.1.cmml">:</mo><mi id="S3.SS1.p1.9.m9.1.1.3.3" xref="S3.SS1.p1.9.m9.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.9.m9.1b"><apply id="S3.SS1.p1.9.m9.1.1.cmml" xref="S3.SS1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.9.m9.1.1.1.cmml" xref="S3.SS1.p1.9.m9.1.1">subscript</csymbol><ci id="S3.SS1.p1.9.m9.1.1.2.cmml" xref="S3.SS1.p1.9.m9.1.1.2">𝐱</ci><apply id="S3.SS1.p1.9.m9.1.1.3.cmml" xref="S3.SS1.p1.9.m9.1.1.3"><ci id="S3.SS1.p1.9.m9.1.1.3.1.cmml" xref="S3.SS1.p1.9.m9.1.1.3.1">:</ci><cn type="integer" id="S3.SS1.p1.9.m9.1.1.3.2.cmml" xref="S3.SS1.p1.9.m9.1.1.3.2">1</cn><ci id="S3.SS1.p1.9.m9.1.1.3.3.cmml" xref="S3.SS1.p1.9.m9.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.9.m9.1c">\mathbf{x}_{1:T}</annotation></semantics></math> to be white noise, corresponding to noise level <math id="S3.SS1.p1.10.m10.1" class="ltx_Math" alttext="k=K" display="inline"><semantics id="S3.SS1.p1.10.m10.1a"><mrow id="S3.SS1.p1.10.m10.1.1" xref="S3.SS1.p1.10.m10.1.1.cmml"><mi id="S3.SS1.p1.10.m10.1.1.2" xref="S3.SS1.p1.10.m10.1.1.2.cmml">k</mi><mo id="S3.SS1.p1.10.m10.1.1.1" xref="S3.SS1.p1.10.m10.1.1.1.cmml">=</mo><mi id="S3.SS1.p1.10.m10.1.1.3" xref="S3.SS1.p1.10.m10.1.1.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.10.m10.1b"><apply id="S3.SS1.p1.10.m10.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1"><eq id="S3.SS1.p1.10.m10.1.1.1.cmml" xref="S3.SS1.p1.10.m10.1.1.1"></eq><ci id="S3.SS1.p1.10.m10.1.1.2.cmml" xref="S3.SS1.p1.10.m10.1.1.2">𝑘</ci><ci id="S3.SS1.p1.10.m10.1.1.3.cmml" xref="S3.SS1.p1.10.m10.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.10.m10.1c">k=K</annotation></semantics></math>. We iterate down the grid row-by-row, denoising left-to-right across columns to the noise levels prescribed by <math id="S3.SS1.p1.11.m11.1" class="ltx_Math" alttext="\mathcal{K}" display="inline"><semantics id="S3.SS1.p1.11.m11.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.11.m11.1.1" xref="S3.SS1.p1.11.m11.1.1.cmml">𝒦</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.11.m11.1b"><ci id="S3.SS1.p1.11.m11.1.1.cmml" xref="S3.SS1.p1.11.m11.1.1">𝒦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.11.m11.1c">\mathcal{K}</annotation></semantics></math>. By the last row <math id="S3.SS1.p1.12.m12.1" class="ltx_Math" alttext="m=0" display="inline"><semantics id="S3.SS1.p1.12.m12.1a"><mrow id="S3.SS1.p1.12.m12.1.1" xref="S3.SS1.p1.12.m12.1.1.cmml"><mi id="S3.SS1.p1.12.m12.1.1.2" xref="S3.SS1.p1.12.m12.1.1.2.cmml">m</mi><mo id="S3.SS1.p1.12.m12.1.1.1" xref="S3.SS1.p1.12.m12.1.1.1.cmml">=</mo><mn id="S3.SS1.p1.12.m12.1.1.3" xref="S3.SS1.p1.12.m12.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.12.m12.1b"><apply id="S3.SS1.p1.12.m12.1.1.cmml" xref="S3.SS1.p1.12.m12.1.1"><eq id="S3.SS1.p1.12.m12.1.1.1.cmml" xref="S3.SS1.p1.12.m12.1.1.1"></eq><ci id="S3.SS1.p1.12.m12.1.1.2.cmml" xref="S3.SS1.p1.12.m12.1.1.2">𝑚</ci><cn type="integer" id="S3.SS1.p1.12.m12.1.1.3.cmml" xref="S3.SS1.p1.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.12.m12.1c">m=0</annotation></semantics></math>, the tokens are clean, i.e. their noise level is <math id="S3.SS1.p1.13.m13.2" class="ltx_Math" alttext="\mathcal{K}_{0,t}\equiv 0" display="inline"><semantics id="S3.SS1.p1.13.m13.2a"><mrow id="S3.SS1.p1.13.m13.2.3" xref="S3.SS1.p1.13.m13.2.3.cmml"><msub id="S3.SS1.p1.13.m13.2.3.2" xref="S3.SS1.p1.13.m13.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.13.m13.2.3.2.2" xref="S3.SS1.p1.13.m13.2.3.2.2.cmml">𝒦</mi><mrow id="S3.SS1.p1.13.m13.2.2.2.4" xref="S3.SS1.p1.13.m13.2.2.2.3.cmml"><mn id="S3.SS1.p1.13.m13.1.1.1.1" xref="S3.SS1.p1.13.m13.1.1.1.1.cmml">0</mn><mo id="S3.SS1.p1.13.m13.2.2.2.4.1" xref="S3.SS1.p1.13.m13.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p1.13.m13.2.2.2.2" xref="S3.SS1.p1.13.m13.2.2.2.2.cmml">t</mi></mrow></msub><mo id="S3.SS1.p1.13.m13.2.3.1" xref="S3.SS1.p1.13.m13.2.3.1.cmml">≡</mo><mn id="S3.SS1.p1.13.m13.2.3.3" xref="S3.SS1.p1.13.m13.2.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.13.m13.2b"><apply id="S3.SS1.p1.13.m13.2.3.cmml" xref="S3.SS1.p1.13.m13.2.3"><equivalent id="S3.SS1.p1.13.m13.2.3.1.cmml" xref="S3.SS1.p1.13.m13.2.3.1"></equivalent><apply id="S3.SS1.p1.13.m13.2.3.2.cmml" xref="S3.SS1.p1.13.m13.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.p1.13.m13.2.3.2.1.cmml" xref="S3.SS1.p1.13.m13.2.3.2">subscript</csymbol><ci id="S3.SS1.p1.13.m13.2.3.2.2.cmml" xref="S3.SS1.p1.13.m13.2.3.2.2">𝒦</ci><list id="S3.SS1.p1.13.m13.2.2.2.3.cmml" xref="S3.SS1.p1.13.m13.2.2.2.4"><cn type="integer" id="S3.SS1.p1.13.m13.1.1.1.1.cmml" xref="S3.SS1.p1.13.m13.1.1.1.1">0</cn><ci id="S3.SS1.p1.13.m13.2.2.2.2.cmml" xref="S3.SS1.p1.13.m13.2.2.2.2">𝑡</ci></list></apply><cn type="integer" id="S3.SS1.p1.13.m13.2.3.3.cmml" xref="S3.SS1.p1.13.m13.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.13.m13.2c">\mathcal{K}_{0,t}\equiv 0</annotation></semantics></math>.</p>
</div>
<div id="S3.SS1.p2" class="ltx_para">
<p id="S3.SS1.p2.1" class="ltx_p"><a href="#A4.SS5.SSS0.Px1" title="Corner case of sampling noise ‣ D.5 Implementation Details of Sampling with Guidance ‣ Appendix D Additional Method Details ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">D.5</span></a> discusses corner cases of this scheme; the hyperparameters <math id="S3.SS1.p2.1.m1.3" class="ltx_Math" alttext="(\alpha_{k},\bar{\alpha}_{k},\sigma_{k})" display="inline"><semantics id="S3.SS1.p2.1.m1.3a"><mrow id="S3.SS1.p2.1.m1.3.3.3" xref="S3.SS1.p2.1.m1.3.3.4.cmml"><mo stretchy="false" id="S3.SS1.p2.1.m1.3.3.3.4" xref="S3.SS1.p2.1.m1.3.3.4.cmml">(</mo><msub id="S3.SS1.p2.1.m1.1.1.1.1" xref="S3.SS1.p2.1.m1.1.1.1.1.cmml"><mi id="S3.SS1.p2.1.m1.1.1.1.1.2" xref="S3.SS1.p2.1.m1.1.1.1.1.2.cmml">α</mi><mi id="S3.SS1.p2.1.m1.1.1.1.1.3" xref="S3.SS1.p2.1.m1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS1.p2.1.m1.3.3.3.5" xref="S3.SS1.p2.1.m1.3.3.4.cmml">,</mo><msub id="S3.SS1.p2.1.m1.2.2.2.2" xref="S3.SS1.p2.1.m1.2.2.2.2.cmml"><mover accent="true" id="S3.SS1.p2.1.m1.2.2.2.2.2" xref="S3.SS1.p2.1.m1.2.2.2.2.2.cmml"><mi id="S3.SS1.p2.1.m1.2.2.2.2.2.2" xref="S3.SS1.p2.1.m1.2.2.2.2.2.2.cmml">α</mi><mo id="S3.SS1.p2.1.m1.2.2.2.2.2.1" xref="S3.SS1.p2.1.m1.2.2.2.2.2.1.cmml">¯</mo></mover><mi id="S3.SS1.p2.1.m1.2.2.2.2.3" xref="S3.SS1.p2.1.m1.2.2.2.2.3.cmml">k</mi></msub><mo id="S3.SS1.p2.1.m1.3.3.3.6" xref="S3.SS1.p2.1.m1.3.3.4.cmml">,</mo><msub id="S3.SS1.p2.1.m1.3.3.3.3" xref="S3.SS1.p2.1.m1.3.3.3.3.cmml"><mi id="S3.SS1.p2.1.m1.3.3.3.3.2" xref="S3.SS1.p2.1.m1.3.3.3.3.2.cmml">σ</mi><mi id="S3.SS1.p2.1.m1.3.3.3.3.3" xref="S3.SS1.p2.1.m1.3.3.3.3.3.cmml">k</mi></msub><mo stretchy="false" id="S3.SS1.p2.1.m1.3.3.3.7" xref="S3.SS1.p2.1.m1.3.3.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.3b"><vector id="S3.SS1.p2.1.m1.3.3.4.cmml" xref="S3.SS1.p2.1.m1.3.3.3"><apply id="S3.SS1.p2.1.m1.1.1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.1.1.1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p2.1.m1.1.1.1.1.2.cmml" xref="S3.SS1.p2.1.m1.1.1.1.1.2">𝛼</ci><ci id="S3.SS1.p2.1.m1.1.1.1.1.3.cmml" xref="S3.SS1.p2.1.m1.1.1.1.1.3">𝑘</ci></apply><apply id="S3.SS1.p2.1.m1.2.2.2.2.cmml" xref="S3.SS1.p2.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.2.2.2.2.1.cmml" xref="S3.SS1.p2.1.m1.2.2.2.2">subscript</csymbol><apply id="S3.SS1.p2.1.m1.2.2.2.2.2.cmml" xref="S3.SS1.p2.1.m1.2.2.2.2.2"><ci id="S3.SS1.p2.1.m1.2.2.2.2.2.1.cmml" xref="S3.SS1.p2.1.m1.2.2.2.2.2.1">¯</ci><ci id="S3.SS1.p2.1.m1.2.2.2.2.2.2.cmml" xref="S3.SS1.p2.1.m1.2.2.2.2.2.2">𝛼</ci></apply><ci id="S3.SS1.p2.1.m1.2.2.2.2.3.cmml" xref="S3.SS1.p2.1.m1.2.2.2.2.3">𝑘</ci></apply><apply id="S3.SS1.p2.1.m1.3.3.3.3.cmml" xref="S3.SS1.p2.1.m1.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.p2.1.m1.3.3.3.3.1.cmml" xref="S3.SS1.p2.1.m1.3.3.3.3">subscript</csymbol><ci id="S3.SS1.p2.1.m1.3.3.3.3.2.cmml" xref="S3.SS1.p2.1.m1.3.3.3.3.2">𝜎</ci><ci id="S3.SS1.p2.1.m1.3.3.3.3.3.cmml" xref="S3.SS1.p2.1.m1.3.3.3.3.3">𝑘</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.3c">(\alpha_{k},\bar{\alpha}_{k},\sigma_{k})</annotation></semantics></math> are set to their standard values <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib29" title="" class="ltx_ref">29</a>]</cite>. We now explain new capabilities this sampling paradigm has to offer.</p>
</div>
<figure id="S3.SS1.1" class="ltx_figure"><img src="/html/2407.01392/assets/x3.png" id="S3.SS1.1.g1" class="ltx_graphics ltx_img_landscape" width="461" height="66" alt="[Uncaptioned image]">
</figure>
<section id="S3.SS1.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Stabilizing Auto-Regressive Generation.</h5>

<div id="S3.SS1.SSS0.Px1.p1" class="ltx_para">
<p id="S3.SS1.SSS0.Px1.p1.1" class="ltx_p">For high-dimensional, continuous sequences such as video, auto-regressive architectures are known to diverge, especially when sampling past the training horizon.
In contrast, Diffusion Forcing can stably roll out long sequences even beyond the training sequence length by updating the latents using the previous latent associated with slightly “noisy tokens” for some small noise level <math id="S3.SS1.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="0&lt;k\ll K" display="inline"><semantics id="S3.SS1.SSS0.Px1.p1.1.m1.1a"><mrow id="S3.SS1.SSS0.Px1.p1.1.m1.1.1" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.cmml"><mn id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.cmml">0</mn><mo id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml">&lt;</mo><mi id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.4" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.4.cmml">k</mi><mo id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.5" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.5.cmml">≪</mo><mi id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.6" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.6.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p1.1.m1.1b"><apply id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1"><and id="S3.SS1.SSS0.Px1.p1.1.m1.1.1a.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1"></and><apply id="S3.SS1.SSS0.Px1.p1.1.m1.1.1b.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1"><lt id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.3"></lt><cn type="integer" id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.2">0</cn><ci id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.4.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.4">𝑘</ci></apply><apply id="S3.SS1.SSS0.Px1.p1.1.m1.1.1c.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="latexml" id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.5.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.5">much-less-than</csymbol><share href="#S3.SS1.SSS0.Px1.p1.1.m1.1.1.4.cmml" id="S3.SS1.SSS0.Px1.p1.1.m1.1.1d.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1"></share><ci id="S3.SS1.SSS0.Px1.p1.1.m1.1.1.6.cmml" xref="S3.SS1.SSS0.Px1.p1.1.m1.1.1.6">𝐾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p1.1.m1.1c">0&lt;k\ll K</annotation></semantics></math>. Our experiments (Sec. <a href="#S4.SS1" title="4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4.1</span></a>) illustrates the resulting marked improvements in long-horizon generation capabilities; App. <a href="#A2.SS2" title="B.2 Noising and stabilizing long-horizon generations ‣ Appendix B Additional Intuitions and Explainations ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">B.2</span></a> provides further intuition.</p>
</div>
</section>
<section id="S3.SS1.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Keeping the future uncertain.</h5>

<div id="S3.SS1.SSS0.Px2.p1" class="ltx_para">
<p id="S3.SS1.SSS0.Px2.p1.4" class="ltx_p">Beginning from a sequence of white noise tokens <math id="S3.SS1.SSS0.Px2.p1.1.m1.3" class="ltx_Math" alttext="[\mathbf{x}^{K}_{1},\mathbf{x}^{K}_{2},\mathbf{x}^{K}_{3}]^{\top}" display="inline"><semantics id="S3.SS1.SSS0.Px2.p1.1.m1.3a"><msup id="S3.SS1.SSS0.Px2.p1.1.m1.3.3" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.cmml"><mrow id="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.4.cmml"><mo stretchy="false" id="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.4" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.4.cmml">[</mo><msubsup id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.1.1.1" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.2" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.1.1.1.3" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.1.1.1.3.cmml">1</mn><mi id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.3" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.1.1.1.2.3.cmml">K</mi></msubsup><mo 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xref="S3.SS1.SSS0.Px2.p1.1.m1.2.2.2.2.2.2.2">𝐱</ci><ci id="S3.SS1.SSS0.Px2.p1.1.m1.2.2.2.2.2.2.3.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.2.2.2.2.2.2.3">𝐾</ci></apply><cn type="integer" id="S3.SS1.SSS0.Px2.p1.1.m1.2.2.2.2.2.3.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.2.2.2.2.2.3">2</cn></apply><apply id="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3.1.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3">subscript</csymbol><apply id="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3.2.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3.2.1.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3">superscript</csymbol><ci id="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3.2.2.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3.2.2">𝐱</ci><ci id="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3.2.3.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3.2.3">𝐾</ci></apply><cn type="integer" id="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3.3.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.3.3.3.3">3</cn></apply></list><csymbol cd="latexml" id="S3.SS1.SSS0.Px2.p1.1.m1.3.3.5.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.1.m1.3c">[\mathbf{x}^{K}_{1},\mathbf{x}^{K}_{2},\mathbf{x}^{K}_{3}]^{\top}</annotation></semantics></math>, we may denoise the first token fully and the second token partially, yielding <math id="S3.SS1.SSS0.Px2.p1.2.m2.3" class="ltx_Math" alttext="[\mathbf{x}^{0}_{1},\mathbf{x}^{K/2}_{2},\mathbf{x}^{K}_{3}]^{\top}" display="inline"><semantics id="S3.SS1.SSS0.Px2.p1.2.m2.3a"><msup id="S3.SS1.SSS0.Px2.p1.2.m2.3.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.cmml"><mrow id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.4.cmml"><mo stretchy="false" id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.4" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.4.cmml">[</mo><msubsup id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.2.2" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.2.2.cmml">𝐱</mi><mn id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.3.cmml">1</mn><mn id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.2.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.2.3.cmml">0</mn></msubsup><mo id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.5" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.4.cmml">,</mo><msubsup id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.cmml"><mi id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.2" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.2.cmml">𝐱</mi><mn id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.3.cmml">2</mn><mrow id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.cmml"><mi id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.2" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.2.cmml">K</mi><mo id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.1" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.1.cmml">/</mo><mn id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.3.cmml">2</mn></mrow></msubsup><mo id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.6" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.4.cmml">,</mo><msubsup id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.cmml"><mi id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.2.2" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.2.2.cmml">𝐱</mi><mn id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.3.cmml">3</mn><mi id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.2.3" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.2.3.cmml">K</mi></msubsup><mo stretchy="false" id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.7" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.4.cmml">]</mo></mrow><mo id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.5" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px2.p1.2.m2.3b"><apply id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.4.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3">superscript</csymbol><list id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.4.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3"><apply id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1">subscript</csymbol><apply id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.2.1.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.2.2.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.2.3.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.2.3">0</cn></apply><cn type="integer" id="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.1.1.1.1.1.3">1</cn></apply><apply id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.1.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2">subscript</csymbol><apply id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.1.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2">superscript</csymbol><ci id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.2.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.2">𝐱</ci><apply id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3"><divide id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.1.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.1"></divide><ci id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.2.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.2">𝐾</ci><cn type="integer" id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.3.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.2.3.3">2</cn></apply></apply><cn type="integer" id="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.3.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.2.2.2.2.2.3">2</cn></apply><apply id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.1.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3">subscript</csymbol><apply id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.2.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.2.1.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3">superscript</csymbol><ci id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.2.2.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.2.2">𝐱</ci><ci id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.2.3.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.2.3">𝐾</ci></apply><cn type="integer" id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.3.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.3.3.3.3">3</cn></apply></list><csymbol cd="latexml" id="S3.SS1.SSS0.Px2.p1.2.m2.3.3.5.cmml" xref="S3.SS1.SSS0.Px2.p1.2.m2.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.2.m2.3c">[\mathbf{x}^{0}_{1},\mathbf{x}^{K/2}_{2},\mathbf{x}^{K}_{3}]^{\top}</annotation></semantics></math>, then <math id="S3.SS1.SSS0.Px2.p1.3.m3.3" class="ltx_Math" alttext="[\mathbf{x}^{0}_{1},\mathbf{x}^{0}_{2},\mathbf{x}^{K/2_{3}}]^{\top}" display="inline"><semantics id="S3.SS1.SSS0.Px2.p1.3.m3.3a"><msup id="S3.SS1.SSS0.Px2.p1.3.m3.3.3" xref="S3.SS1.SSS0.Px2.p1.3.m3.3.3.cmml"><mrow id="S3.SS1.SSS0.Px2.p1.3.m3.3.3.3.3" xref="S3.SS1.SSS0.Px2.p1.3.m3.3.3.3.4.cmml"><mo stretchy="false" id="S3.SS1.SSS0.Px2.p1.3.m3.3.3.3.3.4" xref="S3.SS1.SSS0.Px2.p1.3.m3.3.3.3.4.cmml">[</mo><msubsup 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xref="S3.SS1.SSS0.Px2.p1.3.m3.3.3.3.3.3.3.3.2">2</cn><cn type="integer" id="S3.SS1.SSS0.Px2.p1.3.m3.3.3.3.3.3.3.3.3.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.3.3.3.3.3.3.3.3">3</cn></apply></apply></apply></list><csymbol cd="latexml" id="S3.SS1.SSS0.Px2.p1.3.m3.3.3.5.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.3.m3.3c">[\mathbf{x}^{0}_{1},\mathbf{x}^{0}_{2},\mathbf{x}^{K/2_{3}}]^{\top}</annotation></semantics></math>, and finally denoising all tokens fully to <math id="S3.SS1.SSS0.Px2.p1.4.m4.3" class="ltx_Math" alttext="[\mathbf{x}^{0}_{1},\mathbf{x}^{0}_{2},\mathbf{x}^{0}_{3}]^{\top}" display="inline"><semantics id="S3.SS1.SSS0.Px2.p1.4.m4.3a"><msup id="S3.SS1.SSS0.Px2.p1.4.m4.3.3" xref="S3.SS1.SSS0.Px2.p1.4.m4.3.3.cmml"><mrow id="S3.SS1.SSS0.Px2.p1.4.m4.3.3.3.3" xref="S3.SS1.SSS0.Px2.p1.4.m4.3.3.3.4.cmml"><mo stretchy="false" id="S3.SS1.SSS0.Px2.p1.4.m4.3.3.3.3.4" 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Interpreting the noise level as uncertainty, this “zig-zag” sampling scheme intuitively encodes the immediate future as more certain than the far future. Sec. <a href="#S3.SS2" title="3.2 Diffusion Forcing for Flexible Sequential Decision Making ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3.2</span></a> describes how this leads to more effective sequence guidance.</p>
</div>
</section>
<section id="S3.SS1.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Long-horizon Guidance.</h5>

<div id="S3.SS1.SSS0.Px3.p1" class="ltx_para">
<p id="S3.SS1.SSS0.Px3.p1.1" class="ltx_p">In Line 10 of Algorithm <a href="#alg2" title="Algorithm 2 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>, one may add guidance to the partially diffused trajectory <math id="S3.SS1.SSS0.Px3.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}" display="inline"><semantics id="S3.SS1.SSS0.Px3.p1.1.m1.1a"><msub id="S3.SS1.SSS0.Px3.p1.1.m1.1.1" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.cmml"><mi id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.2" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.2.cmml">𝐱</mi><mrow id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.cmml"><mn id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.2" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.1" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.1.cmml">:</mo><mi id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.3" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px3.p1.1.m1.1b"><apply id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.1.cmml" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.2.cmml" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.2">𝐱</ci><apply id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.cmml" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3"><ci id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.1.cmml" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.1">:</ci><cn type="integer" id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.2.cmml" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.2">1</cn><ci id="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.3.cmml" xref="S3.SS1.SSS0.Px3.p1.1.m1.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px3.p1.1.m1.1c">\mathbf{x}_{1:T}</annotation></semantics></math> as in Sec. <a href="#S2" title="2 Related Work and Preliminaries ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>. Due to the dependency of future tokens on the past, guidance gradients from future tokens can propagate backwards in time. The unique advantage of Diffusion Forcing is that, because we can diffuse future tokens without fully diffusing the past, the gradient guides the sampling of <em id="S3.SS1.SSS0.Px3.p1.1.1" class="ltx_emph ltx_font_italic">past</em> tokens, thereby achieving long-horizon guidance while respecting causality. We elaborate on implementation details in Appendix  <a href="#A2.SS1" title="B.1 Guidance as planning ‣ Appendix B Additional Intuitions and Explainations ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">B.1</span></a>. As we show in <a href="#S4.SS2" title="4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.2</span></a>, planning in this manner significantly outperforms guided full-sequence diffusion models.</p>
</div>
</section>
</section>
<section id="S3.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">3.2 </span>Diffusion Forcing for Flexible Sequential Decision Making</h3>

<div id="S3.SS2.p1" class="ltx_para">
<p id="S3.SS2.p1.18" class="ltx_p">The capabilities offered by Diffusion Forcing motivate our novel framework for sequential decision making (SDM), with key applications to robotics and autonomous agents.
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id="S3.SS2.p1.3.m3.1c">p(\mathbf{r}_{t}|\mathbf{s}_{t},\mathbf{a}_{t})</annotation></semantics></math>. The goal is to train a policy <math id="S3.SS2.p1.4.m4.1" class="ltx_Math" alttext="\pi(\mathbf{a}_{t}|\mathbf{o}_{1:t})" display="inline"><semantics id="S3.SS2.p1.4.m4.1a"><mrow id="S3.SS2.p1.4.m4.1.1" xref="S3.SS2.p1.4.m4.1.1.cmml"><mi id="S3.SS2.p1.4.m4.1.1.3" xref="S3.SS2.p1.4.m4.1.1.3.cmml">π</mi><mo lspace="0em" rspace="0em" id="S3.SS2.p1.4.m4.1.1.2" xref="S3.SS2.p1.4.m4.1.1.2.cmml">​</mo><mrow id="S3.SS2.p1.4.m4.1.1.1.1" xref="S3.SS2.p1.4.m4.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.SS2.p1.4.m4.1.1.1.1.2" xref="S3.SS2.p1.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p1.4.m4.1.1.1.1.1" xref="S3.SS2.p1.4.m4.1.1.1.1.1.cmml"><msub id="S3.SS2.p1.4.m4.1.1.1.1.1.2" xref="S3.SS2.p1.4.m4.1.1.1.1.1.2.cmml"><mi id="S3.SS2.p1.4.m4.1.1.1.1.1.2.2" xref="S3.SS2.p1.4.m4.1.1.1.1.1.2.2.cmml">𝐚</mi><mi id="S3.SS2.p1.4.m4.1.1.1.1.1.2.3" xref="S3.SS2.p1.4.m4.1.1.1.1.1.2.3.cmml">t</mi></msub><mo fence="false" id="S3.SS2.p1.4.m4.1.1.1.1.1.1" xref="S3.SS2.p1.4.m4.1.1.1.1.1.1.cmml">|</mo><msub 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xref="S3.SS2.p1.5.m5.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.5.m5.1b"><apply id="S3.SS2.p1.5.m5.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1"><ci id="S3.SS2.p1.5.m5.1.1.2.cmml" xref="S3.SS2.p1.5.m5.1.1.2">𝔼</ci><apply id="S3.SS2.p1.5.m5.1.1.1.2.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1"><csymbol cd="latexml" id="S3.SS2.p1.5.m5.1.1.1.2.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.SS2.p1.5.m5.1.1.1.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1"><apply id="S3.SS2.p1.5.m5.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.1">superscript</csymbol><apply id="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.1">subscript</csymbol><sum id="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.2.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.2"></sum><apply id="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.3.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.3"><eq id="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.3.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.3.1"></eq><ci id="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.3.2.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.3.2">𝑡</ci><cn type="integer" id="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.3.3.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="S3.SS2.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.1.3">𝑇</ci></apply><apply id="S3.SS2.p1.5.m5.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p1.5.m5.1.1.1.1.1.2.1.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS2.p1.5.m5.1.1.1.1.1.2.2.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.2.2">𝐫</ci><ci id="S3.SS2.p1.5.m5.1.1.1.1.1.2.3.cmml" xref="S3.SS2.p1.5.m5.1.1.1.1.1.2.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.5.m5.1c">\mathop{{\mathbb{E}}}[\sum_{t=1}^{T}\mathbf{r}_{t}]</annotation></semantics></math> is maximized.
We assign tokens <math id="S3.SS2.p1.6.m6.3" class="ltx_Math" alttext="\mathbf{x}_{t}=[\mathbf{a}_{t},\mathbf{r}_{t},\mathbf{o}_{t+1}]" display="inline"><semantics id="S3.SS2.p1.6.m6.3a"><mrow id="S3.SS2.p1.6.m6.3.3" xref="S3.SS2.p1.6.m6.3.3.cmml"><msub id="S3.SS2.p1.6.m6.3.3.5" xref="S3.SS2.p1.6.m6.3.3.5.cmml"><mi id="S3.SS2.p1.6.m6.3.3.5.2" xref="S3.SS2.p1.6.m6.3.3.5.2.cmml">𝐱</mi><mi id="S3.SS2.p1.6.m6.3.3.5.3" xref="S3.SS2.p1.6.m6.3.3.5.3.cmml">t</mi></msub><mo id="S3.SS2.p1.6.m6.3.3.4" xref="S3.SS2.p1.6.m6.3.3.4.cmml">=</mo><mrow id="S3.SS2.p1.6.m6.3.3.3.3" xref="S3.SS2.p1.6.m6.3.3.3.4.cmml"><mo stretchy="false" id="S3.SS2.p1.6.m6.3.3.3.3.4" xref="S3.SS2.p1.6.m6.3.3.3.4.cmml">[</mo><msub id="S3.SS2.p1.6.m6.1.1.1.1.1" xref="S3.SS2.p1.6.m6.1.1.1.1.1.cmml"><mi id="S3.SS2.p1.6.m6.1.1.1.1.1.2" xref="S3.SS2.p1.6.m6.1.1.1.1.1.2.cmml">𝐚</mi><mi id="S3.SS2.p1.6.m6.1.1.1.1.1.3" xref="S3.SS2.p1.6.m6.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S3.SS2.p1.6.m6.3.3.3.3.5" xref="S3.SS2.p1.6.m6.3.3.3.4.cmml">,</mo><msub id="S3.SS2.p1.6.m6.2.2.2.2.2" xref="S3.SS2.p1.6.m6.2.2.2.2.2.cmml"><mi id="S3.SS2.p1.6.m6.2.2.2.2.2.2" xref="S3.SS2.p1.6.m6.2.2.2.2.2.2.cmml">𝐫</mi><mi id="S3.SS2.p1.6.m6.2.2.2.2.2.3" xref="S3.SS2.p1.6.m6.2.2.2.2.2.3.cmml">t</mi></msub><mo id="S3.SS2.p1.6.m6.3.3.3.3.6" xref="S3.SS2.p1.6.m6.3.3.3.4.cmml">,</mo><msub id="S3.SS2.p1.6.m6.3.3.3.3.3" xref="S3.SS2.p1.6.m6.3.3.3.3.3.cmml"><mi id="S3.SS2.p1.6.m6.3.3.3.3.3.2" xref="S3.SS2.p1.6.m6.3.3.3.3.3.2.cmml">𝐨</mi><mrow id="S3.SS2.p1.6.m6.3.3.3.3.3.3" xref="S3.SS2.p1.6.m6.3.3.3.3.3.3.cmml"><mi id="S3.SS2.p1.6.m6.3.3.3.3.3.3.2" xref="S3.SS2.p1.6.m6.3.3.3.3.3.3.2.cmml">t</mi><mo id="S3.SS2.p1.6.m6.3.3.3.3.3.3.1" xref="S3.SS2.p1.6.m6.3.3.3.3.3.3.1.cmml">+</mo><mn id="S3.SS2.p1.6.m6.3.3.3.3.3.3.3" xref="S3.SS2.p1.6.m6.3.3.3.3.3.3.3.cmml">1</mn></mrow></msub><mo stretchy="false" id="S3.SS2.p1.6.m6.3.3.3.3.7" xref="S3.SS2.p1.6.m6.3.3.3.4.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.6.m6.3b"><apply id="S3.SS2.p1.6.m6.3.3.cmml" xref="S3.SS2.p1.6.m6.3.3"><eq id="S3.SS2.p1.6.m6.3.3.4.cmml" xref="S3.SS2.p1.6.m6.3.3.4"></eq><apply id="S3.SS2.p1.6.m6.3.3.5.cmml" xref="S3.SS2.p1.6.m6.3.3.5"><csymbol cd="ambiguous" id="S3.SS2.p1.6.m6.3.3.5.1.cmml" xref="S3.SS2.p1.6.m6.3.3.5">subscript</csymbol><ci id="S3.SS2.p1.6.m6.3.3.5.2.cmml" xref="S3.SS2.p1.6.m6.3.3.5.2">𝐱</ci><ci id="S3.SS2.p1.6.m6.3.3.5.3.cmml" xref="S3.SS2.p1.6.m6.3.3.5.3">𝑡</ci></apply><list id="S3.SS2.p1.6.m6.3.3.3.4.cmml" xref="S3.SS2.p1.6.m6.3.3.3.3"><apply id="S3.SS2.p1.6.m6.1.1.1.1.1.cmml" xref="S3.SS2.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.6.m6.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.6.m6.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.6.m6.1.1.1.1.1.2">𝐚</ci><ci id="S3.SS2.p1.6.m6.1.1.1.1.1.3.cmml" xref="S3.SS2.p1.6.m6.1.1.1.1.1.3">𝑡</ci></apply><apply id="S3.SS2.p1.6.m6.2.2.2.2.2.cmml" xref="S3.SS2.p1.6.m6.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p1.6.m6.2.2.2.2.2.1.cmml" xref="S3.SS2.p1.6.m6.2.2.2.2.2">subscript</csymbol><ci id="S3.SS2.p1.6.m6.2.2.2.2.2.2.cmml" xref="S3.SS2.p1.6.m6.2.2.2.2.2.2">𝐫</ci><ci id="S3.SS2.p1.6.m6.2.2.2.2.2.3.cmml" xref="S3.SS2.p1.6.m6.2.2.2.2.2.3">𝑡</ci></apply><apply id="S3.SS2.p1.6.m6.3.3.3.3.3.cmml" xref="S3.SS2.p1.6.m6.3.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.p1.6.m6.3.3.3.3.3.1.cmml" xref="S3.SS2.p1.6.m6.3.3.3.3.3">subscript</csymbol><ci id="S3.SS2.p1.6.m6.3.3.3.3.3.2.cmml" xref="S3.SS2.p1.6.m6.3.3.3.3.3.2">𝐨</ci><apply id="S3.SS2.p1.6.m6.3.3.3.3.3.3.cmml" xref="S3.SS2.p1.6.m6.3.3.3.3.3.3"><plus id="S3.SS2.p1.6.m6.3.3.3.3.3.3.1.cmml" xref="S3.SS2.p1.6.m6.3.3.3.3.3.3.1"></plus><ci id="S3.SS2.p1.6.m6.3.3.3.3.3.3.2.cmml" xref="S3.SS2.p1.6.m6.3.3.3.3.3.3.2">𝑡</ci><cn type="integer" id="S3.SS2.p1.6.m6.3.3.3.3.3.3.3.cmml" xref="S3.SS2.p1.6.m6.3.3.3.3.3.3.3">1</cn></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.6.m6.3c">\mathbf{x}_{t}=[\mathbf{a}_{t},\mathbf{r}_{t},\mathbf{o}_{t+1}]</annotation></semantics></math>. A trajectory is a sequence <math id="S3.SS2.p1.7.m7.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}" display="inline"><semantics id="S3.SS2.p1.7.m7.1a"><msub id="S3.SS2.p1.7.m7.1.1" xref="S3.SS2.p1.7.m7.1.1.cmml"><mi id="S3.SS2.p1.7.m7.1.1.2" xref="S3.SS2.p1.7.m7.1.1.2.cmml">𝐱</mi><mrow id="S3.SS2.p1.7.m7.1.1.3" xref="S3.SS2.p1.7.m7.1.1.3.cmml"><mn id="S3.SS2.p1.7.m7.1.1.3.2" xref="S3.SS2.p1.7.m7.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S3.SS2.p1.7.m7.1.1.3.1" xref="S3.SS2.p1.7.m7.1.1.3.1.cmml">:</mo><mi id="S3.SS2.p1.7.m7.1.1.3.3" xref="S3.SS2.p1.7.m7.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.7.m7.1b"><apply id="S3.SS2.p1.7.m7.1.1.cmml" xref="S3.SS2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.7.m7.1.1.1.cmml" xref="S3.SS2.p1.7.m7.1.1">subscript</csymbol><ci id="S3.SS2.p1.7.m7.1.1.2.cmml" xref="S3.SS2.p1.7.m7.1.1.2">𝐱</ci><apply id="S3.SS2.p1.7.m7.1.1.3.cmml" xref="S3.SS2.p1.7.m7.1.1.3"><ci id="S3.SS2.p1.7.m7.1.1.3.1.cmml" xref="S3.SS2.p1.7.m7.1.1.3.1">:</ci><cn type="integer" id="S3.SS2.p1.7.m7.1.1.3.2.cmml" xref="S3.SS2.p1.7.m7.1.1.3.2">1</cn><ci id="S3.SS2.p1.7.m7.1.1.3.3.cmml" xref="S3.SS2.p1.7.m7.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.7.m7.1c">\mathbf{x}_{1:T}</annotation></semantics></math>, possibly of variable length; training is conducted as in Algorithm <a href="#alg1" title="Algorithm 1 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>.
At each step <math id="S3.SS2.p1.8.m8.1" class="ltx_Math" alttext="t" display="inline"><semantics id="S3.SS2.p1.8.m8.1a"><mi id="S3.SS2.p1.8.m8.1.1" xref="S3.SS2.p1.8.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.8.m8.1b"><ci id="S3.SS2.p1.8.m8.1.1.cmml" xref="S3.SS2.p1.8.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.8.m8.1c">t</annotation></semantics></math> of execution, past (noise-free) tokens <math id="S3.SS2.p1.9.m9.1" class="ltx_Math" alttext="\mathbf{x}_{1:t-1}" display="inline"><semantics id="S3.SS2.p1.9.m9.1a"><msub id="S3.SS2.p1.9.m9.1.1" xref="S3.SS2.p1.9.m9.1.1.cmml"><mi id="S3.SS2.p1.9.m9.1.1.2" xref="S3.SS2.p1.9.m9.1.1.2.cmml">𝐱</mi><mrow id="S3.SS2.p1.9.m9.1.1.3" xref="S3.SS2.p1.9.m9.1.1.3.cmml"><mn id="S3.SS2.p1.9.m9.1.1.3.2" xref="S3.SS2.p1.9.m9.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="S3.SS2.p1.9.m9.1.1.3.1" xref="S3.SS2.p1.9.m9.1.1.3.1.cmml">:</mo><mrow id="S3.SS2.p1.9.m9.1.1.3.3" xref="S3.SS2.p1.9.m9.1.1.3.3.cmml"><mi id="S3.SS2.p1.9.m9.1.1.3.3.2" xref="S3.SS2.p1.9.m9.1.1.3.3.2.cmml">t</mi><mo id="S3.SS2.p1.9.m9.1.1.3.3.1" xref="S3.SS2.p1.9.m9.1.1.3.3.1.cmml">−</mo><mn id="S3.SS2.p1.9.m9.1.1.3.3.3" xref="S3.SS2.p1.9.m9.1.1.3.3.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.9.m9.1b"><apply id="S3.SS2.p1.9.m9.1.1.cmml" xref="S3.SS2.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.9.m9.1.1.1.cmml" xref="S3.SS2.p1.9.m9.1.1">subscript</csymbol><ci id="S3.SS2.p1.9.m9.1.1.2.cmml" xref="S3.SS2.p1.9.m9.1.1.2">𝐱</ci><apply id="S3.SS2.p1.9.m9.1.1.3.cmml" xref="S3.SS2.p1.9.m9.1.1.3"><ci id="S3.SS2.p1.9.m9.1.1.3.1.cmml" xref="S3.SS2.p1.9.m9.1.1.3.1">:</ci><cn type="integer" id="S3.SS2.p1.9.m9.1.1.3.2.cmml" xref="S3.SS2.p1.9.m9.1.1.3.2">1</cn><apply id="S3.SS2.p1.9.m9.1.1.3.3.cmml" xref="S3.SS2.p1.9.m9.1.1.3.3"><minus id="S3.SS2.p1.9.m9.1.1.3.3.1.cmml" xref="S3.SS2.p1.9.m9.1.1.3.3.1"></minus><ci id="S3.SS2.p1.9.m9.1.1.3.3.2.cmml" xref="S3.SS2.p1.9.m9.1.1.3.3.2">𝑡</ci><cn type="integer" id="S3.SS2.p1.9.m9.1.1.3.3.3.cmml" xref="S3.SS2.p1.9.m9.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.9.m9.1c">\mathbf{x}_{1:t-1}</annotation></semantics></math> are summarized by a latent <math id="S3.SS2.p1.10.m10.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}" display="inline"><semantics id="S3.SS2.p1.10.m10.1a"><msub id="S3.SS2.p1.10.m10.1.1" xref="S3.SS2.p1.10.m10.1.1.cmml"><mi id="S3.SS2.p1.10.m10.1.1.2" xref="S3.SS2.p1.10.m10.1.1.2.cmml">𝐳</mi><mrow id="S3.SS2.p1.10.m10.1.1.3" xref="S3.SS2.p1.10.m10.1.1.3.cmml"><mi id="S3.SS2.p1.10.m10.1.1.3.2" xref="S3.SS2.p1.10.m10.1.1.3.2.cmml">t</mi><mo id="S3.SS2.p1.10.m10.1.1.3.1" xref="S3.SS2.p1.10.m10.1.1.3.1.cmml">−</mo><mn id="S3.SS2.p1.10.m10.1.1.3.3" xref="S3.SS2.p1.10.m10.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.10.m10.1b"><apply id="S3.SS2.p1.10.m10.1.1.cmml" xref="S3.SS2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.10.m10.1.1.1.cmml" xref="S3.SS2.p1.10.m10.1.1">subscript</csymbol><ci id="S3.SS2.p1.10.m10.1.1.2.cmml" xref="S3.SS2.p1.10.m10.1.1.2">𝐳</ci><apply id="S3.SS2.p1.10.m10.1.1.3.cmml" xref="S3.SS2.p1.10.m10.1.1.3"><minus id="S3.SS2.p1.10.m10.1.1.3.1.cmml" xref="S3.SS2.p1.10.m10.1.1.3.1"></minus><ci id="S3.SS2.p1.10.m10.1.1.3.2.cmml" xref="S3.SS2.p1.10.m10.1.1.3.2">𝑡</ci><cn type="integer" id="S3.SS2.p1.10.m10.1.1.3.3.cmml" xref="S3.SS2.p1.10.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.10.m10.1c">\mathbf{z}_{t-1}</annotation></semantics></math>. Conditioned on this latent, we sample, via Algorithm <a href="#alg2" title="Algorithm 2 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>, a plan <math id="S3.SS2.p1.11.m11.1" class="ltx_Math" alttext="\hat{\mathbf{x}}_{t:t+H}" display="inline"><semantics id="S3.SS2.p1.11.m11.1a"><msub id="S3.SS2.p1.11.m11.1.1" xref="S3.SS2.p1.11.m11.1.1.cmml"><mover accent="true" id="S3.SS2.p1.11.m11.1.1.2" xref="S3.SS2.p1.11.m11.1.1.2.cmml"><mi id="S3.SS2.p1.11.m11.1.1.2.2" xref="S3.SS2.p1.11.m11.1.1.2.2.cmml">𝐱</mi><mo id="S3.SS2.p1.11.m11.1.1.2.1" xref="S3.SS2.p1.11.m11.1.1.2.1.cmml">^</mo></mover><mrow id="S3.SS2.p1.11.m11.1.1.3" xref="S3.SS2.p1.11.m11.1.1.3.cmml"><mi id="S3.SS2.p1.11.m11.1.1.3.2" xref="S3.SS2.p1.11.m11.1.1.3.2.cmml">t</mi><mo lspace="0.278em" rspace="0.278em" id="S3.SS2.p1.11.m11.1.1.3.1" 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id="S3.SS2.p1.12.m12.3c">\hat{\mathbf{x}}_{t}=[\hat{\mathbf{a}}_{t},\hat{\mathbf{r}}_{t},\hat{\mathbf{o}}_{t+1}]^{\top}</annotation></semantics></math> containing predicted actions, rewards and observations. <math id="S3.SS2.p1.13.m13.1" class="ltx_Math" alttext="H" display="inline"><semantics id="S3.SS2.p1.13.m13.1a"><mi id="S3.SS2.p1.13.m13.1.1" xref="S3.SS2.p1.13.m13.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.13.m13.1b"><ci id="S3.SS2.p1.13.m13.1.1.cmml" xref="S3.SS2.p1.13.m13.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.13.m13.1c">H</annotation></semantics></math> is a lookahead window, analogous to future predictions in model predictive control <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib20" title="" class="ltx_ref">20</a>]</cite>. After taking planned action <math id="S3.SS2.p1.14.m14.1" class="ltx_Math" alttext="\hat{\mathbf{a}}_{t}" display="inline"><semantics id="S3.SS2.p1.14.m14.1a"><msub id="S3.SS2.p1.14.m14.1.1" xref="S3.SS2.p1.14.m14.1.1.cmml"><mover accent="true" id="S3.SS2.p1.14.m14.1.1.2" xref="S3.SS2.p1.14.m14.1.1.2.cmml"><mi id="S3.SS2.p1.14.m14.1.1.2.2" xref="S3.SS2.p1.14.m14.1.1.2.2.cmml">𝐚</mi><mo id="S3.SS2.p1.14.m14.1.1.2.1" xref="S3.SS2.p1.14.m14.1.1.2.1.cmml">^</mo></mover><mi id="S3.SS2.p1.14.m14.1.1.3" xref="S3.SS2.p1.14.m14.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.14.m14.1b"><apply id="S3.SS2.p1.14.m14.1.1.cmml" xref="S3.SS2.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.14.m14.1.1.1.cmml" xref="S3.SS2.p1.14.m14.1.1">subscript</csymbol><apply id="S3.SS2.p1.14.m14.1.1.2.cmml" xref="S3.SS2.p1.14.m14.1.1.2"><ci id="S3.SS2.p1.14.m14.1.1.2.1.cmml" xref="S3.SS2.p1.14.m14.1.1.2.1">^</ci><ci id="S3.SS2.p1.14.m14.1.1.2.2.cmml" xref="S3.SS2.p1.14.m14.1.1.2.2">𝐚</ci></apply><ci id="S3.SS2.p1.14.m14.1.1.3.cmml" xref="S3.SS2.p1.14.m14.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.14.m14.1c">\hat{\mathbf{a}}_{t}</annotation></semantics></math>, the environment produces a reward <math id="S3.SS2.p1.15.m15.1" class="ltx_Math" alttext="\mathbf{r}_{t}" display="inline"><semantics id="S3.SS2.p1.15.m15.1a"><msub id="S3.SS2.p1.15.m15.1.1" xref="S3.SS2.p1.15.m15.1.1.cmml"><mi id="S3.SS2.p1.15.m15.1.1.2" xref="S3.SS2.p1.15.m15.1.1.2.cmml">𝐫</mi><mi id="S3.SS2.p1.15.m15.1.1.3" xref="S3.SS2.p1.15.m15.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.15.m15.1b"><apply id="S3.SS2.p1.15.m15.1.1.cmml" xref="S3.SS2.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.15.m15.1.1.1.cmml" xref="S3.SS2.p1.15.m15.1.1">subscript</csymbol><ci id="S3.SS2.p1.15.m15.1.1.2.cmml" xref="S3.SS2.p1.15.m15.1.1.2">𝐫</ci><ci id="S3.SS2.p1.15.m15.1.1.3.cmml" xref="S3.SS2.p1.15.m15.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.15.m15.1c">\mathbf{r}_{t}</annotation></semantics></math> and next observation <math id="S3.SS2.p1.16.m16.1" class="ltx_Math" alttext="\mathbf{o}_{t+1}" display="inline"><semantics id="S3.SS2.p1.16.m16.1a"><msub id="S3.SS2.p1.16.m16.1.1" xref="S3.SS2.p1.16.m16.1.1.cmml"><mi id="S3.SS2.p1.16.m16.1.1.2" xref="S3.SS2.p1.16.m16.1.1.2.cmml">𝐨</mi><mrow id="S3.SS2.p1.16.m16.1.1.3" xref="S3.SS2.p1.16.m16.1.1.3.cmml"><mi id="S3.SS2.p1.16.m16.1.1.3.2" xref="S3.SS2.p1.16.m16.1.1.3.2.cmml">t</mi><mo id="S3.SS2.p1.16.m16.1.1.3.1" xref="S3.SS2.p1.16.m16.1.1.3.1.cmml">+</mo><mn id="S3.SS2.p1.16.m16.1.1.3.3" xref="S3.SS2.p1.16.m16.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.16.m16.1b"><apply id="S3.SS2.p1.16.m16.1.1.cmml" xref="S3.SS2.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.16.m16.1.1.1.cmml" xref="S3.SS2.p1.16.m16.1.1">subscript</csymbol><ci id="S3.SS2.p1.16.m16.1.1.2.cmml" xref="S3.SS2.p1.16.m16.1.1.2">𝐨</ci><apply id="S3.SS2.p1.16.m16.1.1.3.cmml" xref="S3.SS2.p1.16.m16.1.1.3"><plus id="S3.SS2.p1.16.m16.1.1.3.1.cmml" xref="S3.SS2.p1.16.m16.1.1.3.1"></plus><ci id="S3.SS2.p1.16.m16.1.1.3.2.cmml" xref="S3.SS2.p1.16.m16.1.1.3.2">𝑡</ci><cn type="integer" id="S3.SS2.p1.16.m16.1.1.3.3.cmml" xref="S3.SS2.p1.16.m16.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.16.m16.1c">\mathbf{o}_{t+1}</annotation></semantics></math>, yielding next token <math id="S3.SS2.p1.17.m17.3" class="ltx_Math" alttext="\mathbf{x}_{t}=[\hat{\mathbf{a}}_{t},\mathbf{r}_{t},\mathbf{o}_{t+1}]^{\top}" display="inline"><semantics id="S3.SS2.p1.17.m17.3a"><mrow id="S3.SS2.p1.17.m17.3.3" xref="S3.SS2.p1.17.m17.3.3.cmml"><msub id="S3.SS2.p1.17.m17.3.3.5" xref="S3.SS2.p1.17.m17.3.3.5.cmml"><mi id="S3.SS2.p1.17.m17.3.3.5.2" xref="S3.SS2.p1.17.m17.3.3.5.2.cmml">𝐱</mi><mi id="S3.SS2.p1.17.m17.3.3.5.3" xref="S3.SS2.p1.17.m17.3.3.5.3.cmml">t</mi></msub><mo id="S3.SS2.p1.17.m17.3.3.4" xref="S3.SS2.p1.17.m17.3.3.4.cmml">=</mo><msup id="S3.SS2.p1.17.m17.3.3.3" xref="S3.SS2.p1.17.m17.3.3.3.cmml"><mrow id="S3.SS2.p1.17.m17.3.3.3.3.3" xref="S3.SS2.p1.17.m17.3.3.3.3.4.cmml"><mo stretchy="false" id="S3.SS2.p1.17.m17.3.3.3.3.3.4" xref="S3.SS2.p1.17.m17.3.3.3.3.4.cmml">[</mo><msub id="S3.SS2.p1.17.m17.1.1.1.1.1.1" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1.cmml"><mover accent="true" id="S3.SS2.p1.17.m17.1.1.1.1.1.1.2" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1.2.cmml"><mi id="S3.SS2.p1.17.m17.1.1.1.1.1.1.2.2" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1.2.2.cmml">𝐚</mi><mo id="S3.SS2.p1.17.m17.1.1.1.1.1.1.2.1" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1.2.1.cmml">^</mo></mover><mi id="S3.SS2.p1.17.m17.1.1.1.1.1.1.3" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="S3.SS2.p1.17.m17.3.3.3.3.3.5" xref="S3.SS2.p1.17.m17.3.3.3.3.4.cmml">,</mo><msub id="S3.SS2.p1.17.m17.2.2.2.2.2.2" xref="S3.SS2.p1.17.m17.2.2.2.2.2.2.cmml"><mi id="S3.SS2.p1.17.m17.2.2.2.2.2.2.2" xref="S3.SS2.p1.17.m17.2.2.2.2.2.2.2.cmml">𝐫</mi><mi id="S3.SS2.p1.17.m17.2.2.2.2.2.2.3" xref="S3.SS2.p1.17.m17.2.2.2.2.2.2.3.cmml">t</mi></msub><mo id="S3.SS2.p1.17.m17.3.3.3.3.3.6" xref="S3.SS2.p1.17.m17.3.3.3.3.4.cmml">,</mo><msub id="S3.SS2.p1.17.m17.3.3.3.3.3.3" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.cmml"><mi id="S3.SS2.p1.17.m17.3.3.3.3.3.3.2" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.2.cmml">𝐨</mi><mrow id="S3.SS2.p1.17.m17.3.3.3.3.3.3.3" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.cmml"><mi id="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.2" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.2.cmml">t</mi><mo id="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.1" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.1.cmml">+</mo><mn id="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.3" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.3.cmml">1</mn></mrow></msub><mo stretchy="false" id="S3.SS2.p1.17.m17.3.3.3.3.3.7" xref="S3.SS2.p1.17.m17.3.3.3.3.4.cmml">]</mo></mrow><mo id="S3.SS2.p1.17.m17.3.3.3.5" xref="S3.SS2.p1.17.m17.3.3.3.5.cmml">⊤</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.17.m17.3b"><apply id="S3.SS2.p1.17.m17.3.3.cmml" xref="S3.SS2.p1.17.m17.3.3"><eq id="S3.SS2.p1.17.m17.3.3.4.cmml" xref="S3.SS2.p1.17.m17.3.3.4"></eq><apply id="S3.SS2.p1.17.m17.3.3.5.cmml" xref="S3.SS2.p1.17.m17.3.3.5"><csymbol cd="ambiguous" id="S3.SS2.p1.17.m17.3.3.5.1.cmml" xref="S3.SS2.p1.17.m17.3.3.5">subscript</csymbol><ci id="S3.SS2.p1.17.m17.3.3.5.2.cmml" xref="S3.SS2.p1.17.m17.3.3.5.2">𝐱</ci><ci id="S3.SS2.p1.17.m17.3.3.5.3.cmml" xref="S3.SS2.p1.17.m17.3.3.5.3">𝑡</ci></apply><apply id="S3.SS2.p1.17.m17.3.3.3.cmml" xref="S3.SS2.p1.17.m17.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.p1.17.m17.3.3.3.4.cmml" xref="S3.SS2.p1.17.m17.3.3.3">superscript</csymbol><list id="S3.SS2.p1.17.m17.3.3.3.3.4.cmml" xref="S3.SS2.p1.17.m17.3.3.3.3.3"><apply id="S3.SS2.p1.17.m17.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.17.m17.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1">subscript</csymbol><apply id="S3.SS2.p1.17.m17.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1.2"><ci id="S3.SS2.p1.17.m17.1.1.1.1.1.1.2.1.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1.2.1">^</ci><ci id="S3.SS2.p1.17.m17.1.1.1.1.1.1.2.2.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1.2.2">𝐚</ci></apply><ci id="S3.SS2.p1.17.m17.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p1.17.m17.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="S3.SS2.p1.17.m17.2.2.2.2.2.2.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p1.17.m17.2.2.2.2.2.2.1.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2.2.2">subscript</csymbol><ci id="S3.SS2.p1.17.m17.2.2.2.2.2.2.2.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2.2.2.2">𝐫</ci><ci id="S3.SS2.p1.17.m17.2.2.2.2.2.2.3.cmml" xref="S3.SS2.p1.17.m17.2.2.2.2.2.2.3">𝑡</ci></apply><apply id="S3.SS2.p1.17.m17.3.3.3.3.3.3.cmml" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3"><csymbol cd="ambiguous" id="S3.SS2.p1.17.m17.3.3.3.3.3.3.1.cmml" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3">subscript</csymbol><ci id="S3.SS2.p1.17.m17.3.3.3.3.3.3.2.cmml" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.2">𝐨</ci><apply id="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.cmml" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.3"><plus id="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.1.cmml" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.1"></plus><ci id="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.2.cmml" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.2">𝑡</ci><cn type="integer" id="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.3.cmml" xref="S3.SS2.p1.17.m17.3.3.3.3.3.3.3.3">1</cn></apply></apply></list><csymbol cd="latexml" id="S3.SS2.p1.17.m17.3.3.3.5.cmml" xref="S3.SS2.p1.17.m17.3.3.3.5">top</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.17.m17.3c">\mathbf{x}_{t}=[\hat{\mathbf{a}}_{t},\mathbf{r}_{t},\mathbf{o}_{t+1}]^{\top}</annotation></semantics></math>. The latent is updated according to the posterior <math id="S3.SS2.p1.18.m18.2" class="ltx_Math" alttext="p_{\theta}(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t},0)" display="inline"><semantics id="S3.SS2.p1.18.m18.2a"><mrow id="S3.SS2.p1.18.m18.2.2" xref="S3.SS2.p1.18.m18.2.2.cmml"><msub id="S3.SS2.p1.18.m18.2.2.3" xref="S3.SS2.p1.18.m18.2.2.3.cmml"><mi id="S3.SS2.p1.18.m18.2.2.3.2" xref="S3.SS2.p1.18.m18.2.2.3.2.cmml">p</mi><mi id="S3.SS2.p1.18.m18.2.2.3.3" xref="S3.SS2.p1.18.m18.2.2.3.3.cmml">θ</mi></msub><mo lspace="0em" rspace="0em" id="S3.SS2.p1.18.m18.2.2.2" xref="S3.SS2.p1.18.m18.2.2.2.cmml">​</mo><mrow id="S3.SS2.p1.18.m18.2.2.1.1" xref="S3.SS2.p1.18.m18.2.2.1.1.1.cmml"><mo stretchy="false" id="S3.SS2.p1.18.m18.2.2.1.1.2" xref="S3.SS2.p1.18.m18.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS2.p1.18.m18.2.2.1.1.1" xref="S3.SS2.p1.18.m18.2.2.1.1.1.cmml"><msub id="S3.SS2.p1.18.m18.2.2.1.1.1.4" xref="S3.SS2.p1.18.m18.2.2.1.1.1.4.cmml"><mi id="S3.SS2.p1.18.m18.2.2.1.1.1.4.2" xref="S3.SS2.p1.18.m18.2.2.1.1.1.4.2.cmml">𝐳</mi><mi id="S3.SS2.p1.18.m18.2.2.1.1.1.4.3" xref="S3.SS2.p1.18.m18.2.2.1.1.1.4.3.cmml">t</mi></msub><mo fence="false" id="S3.SS2.p1.18.m18.2.2.1.1.1.3" xref="S3.SS2.p1.18.m18.2.2.1.1.1.3.cmml">|</mo><mrow id="S3.SS2.p1.18.m18.2.2.1.1.1.2.2" xref="S3.SS2.p1.18.m18.2.2.1.1.1.2.3.cmml"><msub id="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1" xref="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.2" xref="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.3" xref="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.3.cmml"><mi id="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.3.2" xref="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.3.2.cmml">t</mi><mo id="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.3.1" xref="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.3.3" xref="S3.SS2.p1.18.m18.2.2.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.SS2.p1.18.m18.2.2.1.1.1.2.2.3" 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Our framework enables functionality as both <em id="S3.SS2.p1.18.1" class="ltx_emph ltx_font_italic">policy</em> and <em id="S3.SS2.p1.18.2" class="ltx_emph ltx_font_italic">planner</em>:</p>
</div>
<section id="S3.SS2.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Flexible planning horizon.</h5>

<div id="S3.SS2.SSS0.Px1.p1" class="ltx_para">
<p id="S3.SS2.SSS0.Px1.p1.1" class="ltx_p">Diffusion Forcing (a) can be deployed on <em id="S3.SS2.SSS0.Px1.p1.1.1" class="ltx_emph ltx_font_italic">tasks of variable horizon</em>, because each new action is selected sequentially, and
(b) its lookahead window <math id="S3.SS2.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="H" display="inline"><semantics id="S3.SS2.SSS0.Px1.p1.1.m1.1a"><mi id="S3.SS2.SSS0.Px1.p1.1.m1.1.1" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px1.p1.1.m1.1b"><ci id="S3.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS0.Px1.p1.1.m1.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px1.p1.1.m1.1c">H</annotation></semantics></math> can be shortened to lower latency (using Diffusion Forcing as a <em id="S3.SS2.SSS0.Px1.p1.1.2" class="ltx_emph ltx_font_italic">policy</em>), or lengthened to perform long-horizon <em id="S3.SS2.SSS0.Px1.p1.1.3" class="ltx_emph ltx_font_italic">planning</em> (via guidance described below), without re-training or modifications of the architecture. Note that (a) is not possible for full-sequence diffusion models like Diffuser <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib36" title="" class="ltx_ref">36</a>]</cite> with full-trajectory generation horizons, whereas diffusion policies <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib10" title="" class="ltx_ref">10</a>]</cite> need fixed, small lookahead sizes, precluding (b).</p>
</div>
</section>
<section id="S3.SS2.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Flexible reward guidance.</h5>

<div id="S3.SS2.SSS0.Px2.p1" class="ltx_para">
<p id="S3.SS2.SSS0.Px2.p1.4" class="ltx_p">As detailed in Appendix <a href="#A2.SS1" title="B.1 Guidance as planning ‣ Appendix B Additional Intuitions and Explainations ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">B.1</span></a>,
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xref="S3.SS2.SSS0.Px2.p1.3.m3.1.1.2.2">𝐫</ci><ci id="S3.SS2.SSS0.Px2.p1.3.m3.1.1.2.3.cmml" xref="S3.SS2.SSS0.Px2.p1.3.m3.1.1.2.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p1.3.m3.1c">\sum_{t^{\prime}=t}^{t+H}\mathbf{r}_{t}</annotation></semantics></math>, and sparse rewards indicating goal completion <math id="S3.SS2.SSS0.Px2.p1.4.m4.1" class="ltx_Math" alttext="-\|\mathbf{o}_{T}-\mathbf{g}\|^{2}" display="inline"><semantics id="S3.SS2.SSS0.Px2.p1.4.m4.1a"><mrow id="S3.SS2.SSS0.Px2.p1.4.m4.1.1" xref="S3.SS2.SSS0.Px2.p1.4.m4.1.1.cmml"><mo id="S3.SS2.SSS0.Px2.p1.4.m4.1.1a" xref="S3.SS2.SSS0.Px2.p1.4.m4.1.1.cmml">−</mo><msup id="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1" xref="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.cmml"><mrow id="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.1.1" xref="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.1.2.cmml"><mo stretchy="false" id="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.1.1.2" xref="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.1.2.1.cmml">‖</mo><mrow 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id="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.1.1.1.2.2.cmml" xref="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.1.1.1.2.2">𝐨</ci><ci id="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.1.1.1.2.3.cmml" xref="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.1.1.1.2.3">𝑇</ci></apply><ci id="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.1.1.1.3">𝐠</ci></apply></apply><cn type="integer" id="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.3.cmml" xref="S3.SS2.SSS0.Px2.p1.4.m4.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px2.p1.4.m4.1c">-\|\mathbf{o}_{T}-\mathbf{g}\|^{2}</annotation></semantics></math>. Per-time step policies cannot take advantage of this latter, longer horizon guidance.</p>
</div>
</section>
<section id="S3.SS2.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Monte Carlo Tree Guidance (MCTG), future uncertainty.</h5>

<div id="S3.SS2.SSS0.Px3.p1" class="ltx_para">
<p id="S3.SS2.SSS0.Px3.p1.4" class="ltx_p">Causal Diffusion Forcing allows us to influence the generation of a token <math id="S3.SS2.SSS0.Px3.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k}" display="inline"><semantics id="S3.SS2.SSS0.Px3.p1.1.m1.1a"><msubsup id="S3.SS2.SSS0.Px3.p1.1.m1.1.1" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1.cmml"><mi id="S3.SS2.SSS0.Px3.p1.1.m1.1.1.2.2" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS2.SSS0.Px3.p1.1.m1.1.1.2.3" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1.2.3.cmml">t</mi><mi id="S3.SS2.SSS0.Px3.p1.1.m1.1.1.3" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px3.p1.1.m1.1b"><apply id="S3.SS2.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px3.p1.1.m1.1.1.1.cmml" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1">superscript</csymbol><apply id="S3.SS2.SSS0.Px3.p1.1.m1.1.1.2.cmml" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px3.p1.1.m1.1.1.2.1.cmml" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.SSS0.Px3.p1.1.m1.1.1.2.2.cmml" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1.2.2">𝐱</ci><ci id="S3.SS2.SSS0.Px3.p1.1.m1.1.1.2.3.cmml" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1.2.3">𝑡</ci></apply><ci id="S3.SS2.SSS0.Px3.p1.1.m1.1.1.3.cmml" xref="S3.SS2.SSS0.Px3.p1.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px3.p1.1.m1.1c">\mathbf{x}_{t}^{k}</annotation></semantics></math> by guidance on the whole distribution of future <math id="S3.SS2.SSS0.Px3.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{t+1:T}" display="inline"><semantics id="S3.SS2.SSS0.Px3.p1.2.m2.1a"><msub id="S3.SS2.SSS0.Px3.p1.2.m2.1.1" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.cmml"><mi id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.2" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.2.cmml">𝐱</mi><mrow id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.cmml"><mrow id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.cmml"><mi id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.2" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.2.cmml">t</mi><mo id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.1" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.1.cmml">+</mo><mn id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.3" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.3.cmml">1</mn></mrow><mo lspace="0.278em" rspace="0.278em" id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.1" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.1.cmml">:</mo><mi id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.3" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px3.p1.2.m2.1b"><apply id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.1.cmml" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.2.cmml" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.2">𝐱</ci><apply id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.cmml" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3"><ci id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.1.cmml" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.1">:</ci><apply id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.cmml" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2"><plus id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.1.cmml" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.1"></plus><ci id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.2.cmml" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.2">𝑡</ci><cn type="integer" id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.3.cmml" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.2.3">1</cn></apply><ci id="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.3.cmml" xref="S3.SS2.SSS0.Px3.p1.2.m2.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px3.p1.2.m2.1c">\mathbf{x}_{t+1:T}</annotation></semantics></math>. Instead of drawing a single trajectory sample to calculate this guidance gradient, we can draw multiple samples and average their guidance gradients. We call this Monte Carlo Tree Guidance, where “tree” comes from the fact that the the denoising of the current <math id="S3.SS2.SSS0.Px3.p1.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k}" display="inline"><semantics id="S3.SS2.SSS0.Px3.p1.3.m3.1a"><msubsup id="S3.SS2.SSS0.Px3.p1.3.m3.1.1" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1.cmml"><mi id="S3.SS2.SSS0.Px3.p1.3.m3.1.1.2.2" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS2.SSS0.Px3.p1.3.m3.1.1.2.3" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1.2.3.cmml">t</mi><mi id="S3.SS2.SSS0.Px3.p1.3.m3.1.1.3" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px3.p1.3.m3.1b"><apply id="S3.SS2.SSS0.Px3.p1.3.m3.1.1.cmml" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px3.p1.3.m3.1.1.1.cmml" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1">superscript</csymbol><apply id="S3.SS2.SSS0.Px3.p1.3.m3.1.1.2.cmml" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px3.p1.3.m3.1.1.2.1.cmml" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS2.SSS0.Px3.p1.3.m3.1.1.2.2.cmml" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1.2.2">𝐱</ci><ci id="S3.SS2.SSS0.Px3.p1.3.m3.1.1.2.3.cmml" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1.2.3">𝑡</ci></apply><ci id="S3.SS2.SSS0.Px3.p1.3.m3.1.1.3.cmml" xref="S3.SS2.SSS0.Px3.p1.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px3.p1.3.m3.1c">\mathbf{x}_{t}^{k}</annotation></semantics></math> is influenced by gradients through many paths through the future. In the spirit of so-called shooting methods like MPPI <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib61" title="" class="ltx_ref">61</a>]</cite>, <math id="S3.SS2.SSS0.Px3.p1.4.m4.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k}" display="inline"><semantics id="S3.SS2.SSS0.Px3.p1.4.m4.1a"><msubsup id="S3.SS2.SSS0.Px3.p1.4.m4.1.1" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1.cmml"><mi id="S3.SS2.SSS0.Px3.p1.4.m4.1.1.2.2" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS2.SSS0.Px3.p1.4.m4.1.1.2.3" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1.2.3.cmml">t</mi><mi id="S3.SS2.SSS0.Px3.p1.4.m4.1.1.3" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.SSS0.Px3.p1.4.m4.1b"><apply id="S3.SS2.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px3.p1.4.m4.1.1.1.cmml" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1">superscript</csymbol><apply id="S3.SS2.SSS0.Px3.p1.4.m4.1.1.2.cmml" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.SSS0.Px3.p1.4.m4.1.1.2.1.cmml" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1">subscript</csymbol><ci id="S3.SS2.SSS0.Px3.p1.4.m4.1.1.2.2.cmml" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1.2.2">𝐱</ci><ci id="S3.SS2.SSS0.Px3.p1.4.m4.1.1.2.3.cmml" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1.2.3">𝑡</ci></apply><ci id="S3.SS2.SSS0.Px3.p1.4.m4.1.1.3.cmml" xref="S3.SS2.SSS0.Px3.p1.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.SSS0.Px3.p1.4.m4.1c">\mathbf{x}_{t}^{k}</annotation></semantics></math> is then guided by the expected reward over the distribution of all future outcomes instead of one particular outcome. The effect of MCTG is enhanced when combined with sampling schedules that keep the noise level of future tokens high when denoising immediate next tokens (e.g. the zig-zag schedule described in Sec. <a href="#S3.SS1.SSS0.Px2" title="Keeping the future uncertain. ‣ 3.1 Diffusion Forcing Sampling and Resulting Capabilities ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3.1</span></a>), accounting for greater uncertainty farther into the future. Appendix <a href="#A2.SS3" title="B.3 Why Monte Carlo Tree Guidance relies on Diffusion Forcing ‣ Appendix B Additional Intuitions and Explainations ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">B.3</span></a> further justifies the significance of MCTG, and why Diffusion Forcing uniquely takes advantage of it.</p>
</div>
<figure id="S3.F3" class="ltx_figure"><img src="/html/2407.01392/assets/x4.png" id="S3.F3.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="461" height="224" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="S3.F3.3.1.1" class="ltx_text" style="font-size:90%;">Figure 3</span>: </span><span id="S3.F3.4.2" class="ltx_text ltx_font_bold" style="font-size:90%;">Video Generation.<span id="S3.F3.4.2.1" class="ltx_text ltx_font_medium"> Among tested methods, Diffusion Forcing generations are uniquely temporally consistent and do not diverge even when rolling out well past the training horizon. Please see the <a target="_blank" href="https://boyuan.space/diffusion-forcing" title="" class="ltx_ref ltx_href" style="color:#0000FF;">project website</a> for video results.</span></span></figcaption>
</figure>
</section>
</section>
</section>
<section id="S4" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">4 </span>Experiments</h2>

<div id="S4.p1" class="ltx_para">
<p id="S4.p1.1" class="ltx_p">We extensively evaluate Diffusion Forcing’s merits as a generative sequence model across diverse applications in video and time series prediction, planning, and imitation learning. Please find dataset and reproducibility details in the Appendix, as well as video results on the <a target="_blank" href="https://boyuan.space/diffusion-forcing" title="" class="ltx_ref ltx_href" style="color:#0000FF;">project website</a>.</p>
</div>
<section id="S4.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.1 </span>Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout.</h3>

<div id="S4.SS1.p1" class="ltx_para">
<p id="S4.SS1.p1.1" class="ltx_p">We train a convolutional RNN implementation of Causal Diffusion Forcing for video generative modeling on videos of Minecraft gameplay <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib65" title="" class="ltx_ref">65</a>]</cite> and DMLab navigation <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib65" title="" class="ltx_ref">65</a>]</cite>. At sampling time, we perform auto-regressive rollout with stabilization proposed in Sec. <a href="#S3.SS1.SSS0.Px1" title="Stabilizing Auto-Regressive Generation. ‣ 3.1 Diffusion Forcing Sampling and Resulting Capabilities ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3.1</span></a>.
We consider two baselines, both leveraging the same exact RNN architecture: a next-frame diffusion baseline trained with teacher forcing <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib62" title="" class="ltx_ref">62</a>]</cite> as well as a causal full-sequence diffusion model. <a href="#S3.F3" title="In Monte Carlo Tree Guidance (MCTG), future uncertainty. ‣ 3.2 Diffusion Forcing for Flexible Sequential Decision Making ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">3</span></a> displays qualitative results of roll-outs generated by Diffusion Forcing and baselines starting from unseen frames for both datasets. While Diffusion Forcing succeeds at stably rolling out even far beyond its training horizon (e.g. <math id="S4.SS1.p1.1.m1.1" class="ltx_Math" alttext="1000" display="inline"><semantics id="S4.SS1.p1.1.m1.1a"><mn id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml">1000</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><cn type="integer" id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1">1000</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">1000</annotation></semantics></math> frames), teacher forcing and full-sequence diffusion baselines diverge quickly. Further, within the training horizon, we observe that full-sequence diffusion suffers from frame-to-frame discontinuity where video sequences jump dramatically, while Diffusion Forcing roll-outs show ego-motion through a consistent 3D environment. This highlights the ability of Diffusion Forcing to stabilize rollouts of high-dimensional sequences without compounding errors.</p>
</div>
<figure id="S4.SS1.1" class="ltx_table">
<div class="ltx_flex_figure ltx_flex_table">
<div class="ltx_flex_cell ltx_flex_size_1"><svg version="1.1" fill="none" height="0.8302200083022" stroke="none" width="0" class="ltx_figure_panel" overflow="visible"><g transform="translate(0,0.8302200083022) scale(1,-1)"><g transform="translate(0,135.6)"><g transform="translate(0,140) scale(1, -1)"><foreignObject width="461" height="140" overflow="visible"><img src="/html/2407.01392/assets/x5.png" id="S4.SS1.1.pic1.1.g1" class="ltx_graphics ltx_img_landscape" width="461" height="140" alt="[Uncaptioned image]"></foreignObject></g></g><g transform="translate(0,59.5)"><g transform="translate(0,3.4592500345925) scale(1, -1)"><foreignObject width="599.972325999723" height="3.3208800332088" overflow="visible">
<div id="S4.SS1.1.pic1.25.24" class="ltx_inline-block ltx_transformed_outer" style="width:433.6pt;height:2.5pt;vertical-align:-0.1pt;"><span class="ltx_transformed_inner" style="transform:translate(-1421.4pt,7.8pt) scale(0.132346102467378,0.132346102467378) ;">
<table id="S4.SS1.1.pic1.25.24.24" class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr id="S4.SS1.1.pic1.25.24.24.24" class="ltx_tr">
<td id="S4.SS1.1.pic1.25.24.24.24.25" class="ltx_td ltx_align_left">
<span id="S4.SS1.1.pic1.25.24.24.24.25.1" class="ltx_ERROR undefined">\FL</span>      <span id="S4.SS1.1.pic1.25.24.24.24.25.2" class="ltx_text ltx_font_bold">Environment</span>
</td>
<td id="S4.SS1.1.pic1.25.24.24.24.26" class="ltx_td ltx_align_center"><span id="S4.SS1.1.pic1.25.24.24.24.26.1" class="ltx_text ltx_font_bold">MPPI</span></td>
<td id="S4.SS1.1.pic1.25.24.24.24.27" class="ltx_td ltx_align_center"><span id="S4.SS1.1.pic1.25.24.24.24.27.1" class="ltx_text ltx_font_bold">CQL</span></td>
<td id="S4.SS1.1.pic1.25.24.24.24.28" class="ltx_td ltx_align_center"><span id="S4.SS1.1.pic1.25.24.24.24.28.1" class="ltx_text ltx_font_bold">IQL</span></td>
<td id="S4.SS1.1.pic1.25.24.24.24.29" class="ltx_td ltx_align_center"><span id="S4.SS1.1.pic1.25.24.24.24.29.1" class="ltx_text ltx_font_bold">Diffuser*</span></td>
<td id="S4.SS1.1.pic1.25.24.24.24.30" class="ltx_td ltx_align_center"><span id="S4.SS1.1.pic1.25.24.24.24.30.1" class="ltx_text ltx_font_bold">Diffuser w/ diffused action</span></td>
<td id="S4.SS1.1.pic1.25.24.24.24.31" class="ltx_td ltx_align_center"><span id="S4.SS1.1.pic1.25.24.24.24.31.1" class="ltx_text ltx_font_bold">Ours wo/ MCTG</span></td>
<td id="S4.SS1.1.pic1.25.24.24.24.32" class="ltx_td ltx_nopad_r ltx_align_center">
<span id="S4.SS1.1.pic1.25.24.24.24.32.1" class="ltx_text ltx_font_bold">Ours<span id="S4.SS1.1.pic1.25.24.24.24.32.1.1" class="ltx_ERROR undefined">\ML</span></span>Maze2D</td>
<td id="S4.SS1.1.pic1.25.24.24.24.33" class="ltx_td ltx_nopad_r ltx_align_center">U-Maze</td>
<td id="S4.SS1.1.pic1.25.24.24.24.34" class="ltx_td ltx_nopad_r ltx_align_center">33.2</td>
<td id="S4.SS1.1.pic1.25.24.24.24.35" class="ltx_td ltx_nopad_r ltx_align_center">5.7</td>
<td id="S4.SS1.1.pic1.25.24.24.24.36" class="ltx_td ltx_nopad_r ltx_align_center">47.4</td>
<td id="S4.SS1.1.pic1.2.1.1.1.1" class="ltx_td ltx_nopad_r ltx_align_center">113.9 <math id="S4.SS1.1.pic1.2.1.1.1.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.2.1.1.1.1.m1.1a"><mo id="S4.SS1.1.pic1.2.1.1.1.1.m1.1.1" xref="S4.SS1.1.pic1.2.1.1.1.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.2.1.1.1.1.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.2.1.1.1.1.m1.1.1.cmml" xref="S4.SS1.1.pic1.2.1.1.1.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.2.1.1.1.1.m1.1c">\pm</annotation></semantics></math> 3.1</td>
<td id="S4.SS1.1.pic1.3.2.2.2.2" class="ltx_td ltx_nopad_r ltx_align_center">6.3 <math id="S4.SS1.1.pic1.3.2.2.2.2.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.3.2.2.2.2.m1.1a"><mo id="S4.SS1.1.pic1.3.2.2.2.2.m1.1.1" xref="S4.SS1.1.pic1.3.2.2.2.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.3.2.2.2.2.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.3.2.2.2.2.m1.1.1.cmml" xref="S4.SS1.1.pic1.3.2.2.2.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.3.2.2.2.2.m1.1c">\pm</annotation></semantics></math> 2.1</td>
<td id="S4.SS1.1.pic1.4.3.3.3.3" class="ltx_td ltx_nopad_r ltx_align_center">110.1 <math id="S4.SS1.1.pic1.4.3.3.3.3.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.4.3.3.3.3.m1.1a"><mo id="S4.SS1.1.pic1.4.3.3.3.3.m1.1.1" xref="S4.SS1.1.pic1.4.3.3.3.3.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.4.3.3.3.3.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.4.3.3.3.3.m1.1.1.cmml" xref="S4.SS1.1.pic1.4.3.3.3.3.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.4.3.3.3.3.m1.1c">\pm</annotation></semantics></math> 3.9</td>
<td id="S4.SS1.1.pic1.5.4.4.4.4" class="ltx_td ltx_nopad_r ltx_align_center">
<span id="S4.SS1.1.pic1.5.4.4.4.4.1" class="ltx_text ltx_font_bold">116.7 <math id="S4.SS1.1.pic1.5.4.4.4.4.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.5.4.4.4.4.1.m1.1a"><mo id="S4.SS1.1.pic1.5.4.4.4.4.1.m1.1.1" xref="S4.SS1.1.pic1.5.4.4.4.4.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.5.4.4.4.4.1.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.5.4.4.4.4.1.m1.1.1.cmml" xref="S4.SS1.1.pic1.5.4.4.4.4.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.5.4.4.4.4.1.m1.1c">\pm</annotation></semantics></math> 2.0</span> <span id="S4.SS1.1.pic1.5.4.4.4.4.2" class="ltx_ERROR undefined">\NN</span>Maze2D</td>
<td id="S4.SS1.1.pic1.25.24.24.24.37" class="ltx_td ltx_nopad_r ltx_align_center">Medium</td>
<td id="S4.SS1.1.pic1.25.24.24.24.38" class="ltx_td ltx_nopad_r ltx_align_center">10.2</td>
<td id="S4.SS1.1.pic1.25.24.24.24.39" class="ltx_td ltx_nopad_r ltx_align_center">5.0</td>
<td id="S4.SS1.1.pic1.25.24.24.24.40" class="ltx_td ltx_nopad_r ltx_align_center">34.9</td>
<td id="S4.SS1.1.pic1.6.5.5.5.5" class="ltx_td ltx_nopad_r ltx_align_center">121.5 <math id="S4.SS1.1.pic1.6.5.5.5.5.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.6.5.5.5.5.m1.1a"><mo id="S4.SS1.1.pic1.6.5.5.5.5.m1.1.1" xref="S4.SS1.1.pic1.6.5.5.5.5.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.6.5.5.5.5.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.6.5.5.5.5.m1.1.1.cmml" xref="S4.SS1.1.pic1.6.5.5.5.5.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.6.5.5.5.5.m1.1c">\pm</annotation></semantics></math> 2.7</td>
<td id="S4.SS1.1.pic1.7.6.6.6.6" class="ltx_td ltx_nopad_r ltx_align_center">13.5<math id="S4.SS1.1.pic1.7.6.6.6.6.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.7.6.6.6.6.m1.1a"><mo id="S4.SS1.1.pic1.7.6.6.6.6.m1.1.1" xref="S4.SS1.1.pic1.7.6.6.6.6.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.7.6.6.6.6.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.7.6.6.6.6.m1.1.1.cmml" xref="S4.SS1.1.pic1.7.6.6.6.6.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.7.6.6.6.6.m1.1c">\pm</annotation></semantics></math>2.3</td>
<td id="S4.SS1.1.pic1.8.7.7.7.7" class="ltx_td ltx_nopad_r ltx_align_center">136.1 <math id="S4.SS1.1.pic1.8.7.7.7.7.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.8.7.7.7.7.m1.1a"><mo id="S4.SS1.1.pic1.8.7.7.7.7.m1.1.1" xref="S4.SS1.1.pic1.8.7.7.7.7.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.8.7.7.7.7.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.8.7.7.7.7.m1.1.1.cmml" xref="S4.SS1.1.pic1.8.7.7.7.7.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.8.7.7.7.7.m1.1c">\pm</annotation></semantics></math> 10.2</td>
<td id="S4.SS1.1.pic1.9.8.8.8.8" class="ltx_td ltx_nopad_r ltx_align_center">
<span id="S4.SS1.1.pic1.9.8.8.8.8.1" class="ltx_text ltx_font_bold">149.4 <math id="S4.SS1.1.pic1.9.8.8.8.8.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.9.8.8.8.8.1.m1.1a"><mo id="S4.SS1.1.pic1.9.8.8.8.8.1.m1.1.1" xref="S4.SS1.1.pic1.9.8.8.8.8.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.9.8.8.8.8.1.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.9.8.8.8.8.1.m1.1.1.cmml" xref="S4.SS1.1.pic1.9.8.8.8.8.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.9.8.8.8.8.1.m1.1c">\pm</annotation></semantics></math> 7.5</span> <span id="S4.SS1.1.pic1.9.8.8.8.8.2" class="ltx_ERROR undefined">\NN</span>Maze2D</td>
<td id="S4.SS1.1.pic1.25.24.24.24.41" class="ltx_td ltx_nopad_r ltx_align_center">Large</td>
<td id="S4.SS1.1.pic1.25.24.24.24.42" class="ltx_td ltx_nopad_r ltx_align_center">5.1</td>
<td id="S4.SS1.1.pic1.25.24.24.24.43" class="ltx_td ltx_nopad_r ltx_align_center">12.5</td>
<td id="S4.SS1.1.pic1.25.24.24.24.44" class="ltx_td ltx_nopad_r ltx_align_center">58.6</td>
<td id="S4.SS1.1.pic1.10.9.9.9.9" class="ltx_td ltx_nopad_r ltx_align_center">123.0 <math id="S4.SS1.1.pic1.10.9.9.9.9.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.10.9.9.9.9.m1.1a"><mo id="S4.SS1.1.pic1.10.9.9.9.9.m1.1.1" xref="S4.SS1.1.pic1.10.9.9.9.9.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.10.9.9.9.9.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.10.9.9.9.9.m1.1.1.cmml" xref="S4.SS1.1.pic1.10.9.9.9.9.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.10.9.9.9.9.m1.1c">\pm</annotation></semantics></math> 6.4</td>
<td id="S4.SS1.1.pic1.11.10.10.10.10" class="ltx_td ltx_nopad_r ltx_align_center">6.3 <math id="S4.SS1.1.pic1.11.10.10.10.10.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.11.10.10.10.10.m1.1a"><mo id="S4.SS1.1.pic1.11.10.10.10.10.m1.1.1" xref="S4.SS1.1.pic1.11.10.10.10.10.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.11.10.10.10.10.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.11.10.10.10.10.m1.1.1.cmml" xref="S4.SS1.1.pic1.11.10.10.10.10.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.11.10.10.10.10.m1.1c">\pm</annotation></semantics></math>2.1</td>
<td id="S4.SS1.1.pic1.12.11.11.11.11" class="ltx_td ltx_nopad_r ltx_align_center">142.8 <math id="S4.SS1.1.pic1.12.11.11.11.11.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.12.11.11.11.11.m1.1a"><mo id="S4.SS1.1.pic1.12.11.11.11.11.m1.1.1" xref="S4.SS1.1.pic1.12.11.11.11.11.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.12.11.11.11.11.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.12.11.11.11.11.m1.1.1.cmml" xref="S4.SS1.1.pic1.12.11.11.11.11.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.12.11.11.11.11.m1.1c">\pm</annotation></semantics></math> 5.6</td>
<td id="S4.SS1.1.pic1.13.12.12.12.12" class="ltx_td ltx_align_center">
<span id="S4.SS1.1.pic1.13.12.12.12.12.1" class="ltx_text ltx_font_bold">159.0 <math id="S4.SS1.1.pic1.13.12.12.12.12.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.13.12.12.12.12.1.m1.1a"><mo id="S4.SS1.1.pic1.13.12.12.12.12.1.m1.1.1" xref="S4.SS1.1.pic1.13.12.12.12.12.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.13.12.12.12.12.1.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.13.12.12.12.12.1.m1.1.1.cmml" xref="S4.SS1.1.pic1.13.12.12.12.12.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.13.12.12.12.12.1.m1.1c">\pm</annotation></semantics></math> 2.7</span> <span id="S4.SS1.1.pic1.13.12.12.12.12.2" class="ltx_ERROR undefined">\ML</span>     <span id="S4.SS1.1.pic1.13.12.12.12.12.3" class="ltx_text ltx_font_bold">Single-task Average</span>
</td>
<td id="S4.SS1.1.pic1.25.24.24.24.45" class="ltx_td ltx_nopad_r ltx_align_center">16.2</td>
<td id="S4.SS1.1.pic1.25.24.24.24.46" class="ltx_td ltx_nopad_r ltx_align_center">7.7</td>
<td id="S4.SS1.1.pic1.25.24.24.24.47" class="ltx_td ltx_nopad_r ltx_align_center">47.0</td>
<td id="S4.SS1.1.pic1.25.24.24.24.48" class="ltx_td ltx_nopad_r ltx_align_center">119.5</td>
<td id="S4.SS1.1.pic1.25.24.24.24.49" class="ltx_td ltx_nopad_r ltx_align_center">8.7</td>
<td id="S4.SS1.1.pic1.25.24.24.24.50" class="ltx_td ltx_nopad_r ltx_align_center">129.67</td>
<td id="S4.SS1.1.pic1.25.24.24.24.51" class="ltx_td ltx_nopad_r ltx_align_center">
<span id="S4.SS1.1.pic1.25.24.24.24.51.1" class="ltx_text ltx_font_bold">141.7</span> <span id="S4.SS1.1.pic1.25.24.24.24.51.2" class="ltx_ERROR undefined">\ML</span>[0.11em]
Multi2D</td>
<td id="S4.SS1.1.pic1.25.24.24.24.52" class="ltx_td ltx_nopad_r ltx_align_center">U-Maze</td>
<td id="S4.SS1.1.pic1.25.24.24.24.53" class="ltx_td ltx_nopad_r ltx_align_center">41.2</td>
<td id="S4.SS1.1.pic1.25.24.24.24.54" class="ltx_td ltx_nopad_r ltx_align_center">-</td>
<td id="S4.SS1.1.pic1.25.24.24.24.55" class="ltx_td ltx_nopad_r ltx_align_center">24.8</td>
<td id="S4.SS1.1.pic1.14.13.13.13.13" class="ltx_td ltx_nopad_r ltx_align_center">128.9 <math id="S4.SS1.1.pic1.14.13.13.13.13.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.14.13.13.13.13.m1.1a"><mo id="S4.SS1.1.pic1.14.13.13.13.13.m1.1.1" xref="S4.SS1.1.pic1.14.13.13.13.13.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.14.13.13.13.13.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.14.13.13.13.13.m1.1.1.cmml" xref="S4.SS1.1.pic1.14.13.13.13.13.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.14.13.13.13.13.m1.1c">\pm</annotation></semantics></math> 1.8</td>
<td id="S4.SS1.1.pic1.15.14.14.14.14" class="ltx_td ltx_nopad_r ltx_align_center">32.8<math id="S4.SS1.1.pic1.15.14.14.14.14.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.15.14.14.14.14.m1.1a"><mo id="S4.SS1.1.pic1.15.14.14.14.14.m1.1.1" xref="S4.SS1.1.pic1.15.14.14.14.14.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.15.14.14.14.14.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.15.14.14.14.14.m1.1.1.cmml" xref="S4.SS1.1.pic1.15.14.14.14.14.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.15.14.14.14.14.m1.1c">\pm</annotation></semantics></math>1.7</td>
<td id="S4.SS1.1.pic1.16.15.15.15.15" class="ltx_td ltx_nopad_r ltx_align_center">107.7 <math id="S4.SS1.1.pic1.16.15.15.15.15.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.16.15.15.15.15.m1.1a"><mo id="S4.SS1.1.pic1.16.15.15.15.15.m1.1.1" xref="S4.SS1.1.pic1.16.15.15.15.15.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.16.15.15.15.15.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.16.15.15.15.15.m1.1.1.cmml" xref="S4.SS1.1.pic1.16.15.15.15.15.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.16.15.15.15.15.m1.1c">\pm</annotation></semantics></math> 4.9</td>
<td id="S4.SS1.1.pic1.17.16.16.16.16" class="ltx_td ltx_nopad_r ltx_align_center">
<span id="S4.SS1.1.pic1.17.16.16.16.16.1" class="ltx_text ltx_font_bold">119.1 <math id="S4.SS1.1.pic1.17.16.16.16.16.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.17.16.16.16.16.1.m1.1a"><mo id="S4.SS1.1.pic1.17.16.16.16.16.1.m1.1.1" xref="S4.SS1.1.pic1.17.16.16.16.16.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.17.16.16.16.16.1.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.17.16.16.16.16.1.m1.1.1.cmml" xref="S4.SS1.1.pic1.17.16.16.16.16.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.17.16.16.16.16.1.m1.1c">\pm</annotation></semantics></math> 4.0</span> <span id="S4.SS1.1.pic1.17.16.16.16.16.2" class="ltx_ERROR undefined">\NN</span>Multi2D</td>
<td id="S4.SS1.1.pic1.25.24.24.24.56" class="ltx_td ltx_nopad_r ltx_align_center">Medium</td>
<td id="S4.SS1.1.pic1.25.24.24.24.57" class="ltx_td ltx_nopad_r ltx_align_center">15.4</td>
<td id="S4.SS1.1.pic1.25.24.24.24.58" class="ltx_td ltx_nopad_r ltx_align_center">-</td>
<td id="S4.SS1.1.pic1.25.24.24.24.59" class="ltx_td ltx_nopad_r ltx_align_center">12.1</td>
<td id="S4.SS1.1.pic1.18.17.17.17.17" class="ltx_td ltx_nopad_r ltx_align_center">127.2 <math id="S4.SS1.1.pic1.18.17.17.17.17.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.18.17.17.17.17.m1.1a"><mo id="S4.SS1.1.pic1.18.17.17.17.17.m1.1.1" xref="S4.SS1.1.pic1.18.17.17.17.17.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.18.17.17.17.17.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.18.17.17.17.17.m1.1.1.cmml" xref="S4.SS1.1.pic1.18.17.17.17.17.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.18.17.17.17.17.m1.1c">\pm</annotation></semantics></math> 3.4</td>
<td id="S4.SS1.1.pic1.19.18.18.18.18" class="ltx_td ltx_nopad_r ltx_align_center">22.0<math id="S4.SS1.1.pic1.19.18.18.18.18.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.19.18.18.18.18.m1.1a"><mo id="S4.SS1.1.pic1.19.18.18.18.18.m1.1.1" xref="S4.SS1.1.pic1.19.18.18.18.18.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.19.18.18.18.18.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.19.18.18.18.18.m1.1.1.cmml" xref="S4.SS1.1.pic1.19.18.18.18.18.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.19.18.18.18.18.m1.1c">\pm</annotation></semantics></math>2.7</td>
<td id="S4.SS1.1.pic1.20.19.19.19.19" class="ltx_td ltx_nopad_r ltx_align_center">145.6 <math id="S4.SS1.1.pic1.20.19.19.19.19.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.20.19.19.19.19.m1.1a"><mo id="S4.SS1.1.pic1.20.19.19.19.19.m1.1.1" xref="S4.SS1.1.pic1.20.19.19.19.19.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.20.19.19.19.19.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.20.19.19.19.19.m1.1.1.cmml" xref="S4.SS1.1.pic1.20.19.19.19.19.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.20.19.19.19.19.m1.1c">\pm</annotation></semantics></math> 6.5</td>
<td id="S4.SS1.1.pic1.21.20.20.20.20" class="ltx_td ltx_nopad_r ltx_align_center">
<span id="S4.SS1.1.pic1.21.20.20.20.20.1" class="ltx_text ltx_font_bold">152.3 <math id="S4.SS1.1.pic1.21.20.20.20.20.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.21.20.20.20.20.1.m1.1a"><mo id="S4.SS1.1.pic1.21.20.20.20.20.1.m1.1.1" xref="S4.SS1.1.pic1.21.20.20.20.20.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.21.20.20.20.20.1.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.21.20.20.20.20.1.m1.1.1.cmml" xref="S4.SS1.1.pic1.21.20.20.20.20.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.21.20.20.20.20.1.m1.1c">\pm</annotation></semantics></math> 9.9</span> <span id="S4.SS1.1.pic1.21.20.20.20.20.2" class="ltx_ERROR undefined">\NN</span>Multi2D</td>
<td id="S4.SS1.1.pic1.25.24.24.24.60" class="ltx_td ltx_nopad_r ltx_align_center">Large</td>
<td id="S4.SS1.1.pic1.25.24.24.24.61" class="ltx_td ltx_nopad_r ltx_align_center">8.0</td>
<td id="S4.SS1.1.pic1.25.24.24.24.62" class="ltx_td ltx_nopad_r ltx_align_center">-</td>
<td id="S4.SS1.1.pic1.25.24.24.24.63" class="ltx_td ltx_nopad_r ltx_align_center">13.9</td>
<td id="S4.SS1.1.pic1.22.21.21.21.21" class="ltx_td ltx_nopad_r ltx_align_center">132.1 <math id="S4.SS1.1.pic1.22.21.21.21.21.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.22.21.21.21.21.m1.1a"><mo id="S4.SS1.1.pic1.22.21.21.21.21.m1.1.1" xref="S4.SS1.1.pic1.22.21.21.21.21.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.22.21.21.21.21.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.22.21.21.21.21.m1.1.1.cmml" xref="S4.SS1.1.pic1.22.21.21.21.21.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.22.21.21.21.21.m1.1c">\pm</annotation></semantics></math> 5.8</td>
<td id="S4.SS1.1.pic1.23.22.22.22.22" class="ltx_td ltx_nopad_r ltx_align_center">6.9 <math id="S4.SS1.1.pic1.23.22.22.22.22.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.23.22.22.22.22.m1.1a"><mo id="S4.SS1.1.pic1.23.22.22.22.22.m1.1.1" xref="S4.SS1.1.pic1.23.22.22.22.22.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.23.22.22.22.22.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.23.22.22.22.22.m1.1.1.cmml" xref="S4.SS1.1.pic1.23.22.22.22.22.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.23.22.22.22.22.m1.1c">\pm</annotation></semantics></math>1.7</td>
<td id="S4.SS1.1.pic1.24.23.23.23.23" class="ltx_td ltx_nopad_r ltx_align_center">129.8 <math id="S4.SS1.1.pic1.24.23.23.23.23.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.24.23.23.23.23.m1.1a"><mo id="S4.SS1.1.pic1.24.23.23.23.23.m1.1.1" xref="S4.SS1.1.pic1.24.23.23.23.23.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.24.23.23.23.23.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.24.23.23.23.23.m1.1.1.cmml" xref="S4.SS1.1.pic1.24.23.23.23.23.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.24.23.23.23.23.m1.1c">\pm</annotation></semantics></math> 1.5</td>
<td id="S4.SS1.1.pic1.25.24.24.24.24" class="ltx_td ltx_align_center">
<span id="S4.SS1.1.pic1.25.24.24.24.24.1" class="ltx_text ltx_font_bold">167.1 <math id="S4.SS1.1.pic1.25.24.24.24.24.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S4.SS1.1.pic1.25.24.24.24.24.1.m1.1a"><mo id="S4.SS1.1.pic1.25.24.24.24.24.1.m1.1.1" xref="S4.SS1.1.pic1.25.24.24.24.24.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.1.pic1.25.24.24.24.24.1.m1.1b"><csymbol cd="latexml" id="S4.SS1.1.pic1.25.24.24.24.24.1.m1.1.1.cmml" xref="S4.SS1.1.pic1.25.24.24.24.24.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.pic1.25.24.24.24.24.1.m1.1c">\pm</annotation></semantics></math>2.7</span> <span id="S4.SS1.1.pic1.25.24.24.24.24.2" class="ltx_ERROR undefined">\ML</span>     <span id="S4.SS1.1.pic1.25.24.24.24.24.3" class="ltx_text ltx_font_bold">Multi-task Average</span>
</td>
<td id="S4.SS1.1.pic1.25.24.24.24.64" class="ltx_td ltx_nopad_r ltx_align_center">21.5</td>
<td id="S4.SS1.1.pic1.25.24.24.24.65" class="ltx_td ltx_nopad_r ltx_align_center">-</td>
<td id="S4.SS1.1.pic1.25.24.24.24.66" class="ltx_td ltx_nopad_r ltx_align_center">16.9</td>
<td id="S4.SS1.1.pic1.25.24.24.24.67" class="ltx_td ltx_nopad_r ltx_align_center">129.4</td>
<td id="S4.SS1.1.pic1.25.24.24.24.68" class="ltx_td ltx_nopad_r ltx_align_center">20.57</td>
<td id="S4.SS1.1.pic1.25.24.24.24.69" class="ltx_td ltx_nopad_r ltx_align_center">127.7</td>
<td id="S4.SS1.1.pic1.25.24.24.24.70" class="ltx_td ltx_nopad_r ltx_align_center">
<span id="S4.SS1.1.pic1.25.24.24.24.70.1" class="ltx_text ltx_font_bold">146.17</span> <span id="S4.SS1.1.pic1.25.24.24.24.70.2" class="ltx_ERROR undefined">\LL</span>
</td>
</tr>
</tbody>
</table>
</span></div></foreignObject></g></g></g></svg></div>
</div>
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span id="S4.SS1.1.4.1.1" class="ltx_text" style="font-size:90%;">Table 1</span>: </span><span id="S4.SS1.1.5.2" class="ltx_text" style="font-size:90%;"> <span id="S4.SS1.1.5.2.1" class="ltx_text ltx_font_bold">Diffusion Forcing for Planning.</span> (<span id="S4.SS1.1.5.2.2" class="ltx_text ltx_font_bold">top</span>) During sampling, Diffusion Forcingallows each time step to be denoised on different noise schedules, enabling us to account for causal uncertainty during guided planning. Diffusion Forcingkeeps the far future more uncertain than the near future while Diffuser <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib36" title="" class="ltx_ref">36</a>]</cite> puts them at the same noise level during sampling. (<span id="S4.SS1.1.5.2.3" class="ltx_text ltx_font_bold">bottom)</span> Quantitatively, Diffusion Forcingachieves the highest average reward across runs. Diffuser fails dramatically when executing the actually generated actions, requiring a hand-crafted PD controller (indicated by the asterisk) and ignoring generated actions.</span></figcaption><div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_1">
<section id="S4.SS2" class="ltx_subsection ltx_figure_panel">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.2 </span>Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control.</h3>

<div id="S4.SS2.p1" class="ltx_para">
<p id="S4.SS2.p1.1" class="ltx_p">Decision-making uniquely benefits from Diffusion Forcing’s capabilities. We evaluate our proposed decision-making framework in a standard offline RL benchmark, D4RL <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib18" title="" class="ltx_ref">18</a>]</cite>. Specifically, we benchmark Diffusion Forcing on a set of 2D maze environments with sparse reward. An agent is tasked with reaching a designated goal position starting from a random starting position. In Appendix <a href="#A5.F16" title="Figure 16 ‣ E.5 Real robot experiment setup ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">16</span></a> we provide a detailed description of the environment. The benchmark provides a dataset of <em id="S4.SS2.p1.1.1" class="ltx_emph ltx_font_italic">random walks</em> through mazes (thus stochastic). We train one model per maze.</p>
</div>
<div id="S4.SS2.p2" class="ltx_para">
<p id="S4.SS2.p2.1" class="ltx_p">We benchmark the proposed decision-making framework <a href="#S3.SS2" title="3.2 Diffusion Forcing for Flexible Sequential Decision Making ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3.2</span></a> with state-of-the-art offline RL methods and the recently introduced Diffuser <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib36" title="" class="ltx_ref">36</a>]</cite>, a diffusion planning framework. See Fig. <a href="#S4.SS1" title="4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4.1</span></a> for qualitative and quantitative reuslts: DF outperforms Diffuser and all baselines across all <math id="S4.SS2.p2.1.m1.1" class="ltx_Math" alttext="6" display="inline"><semantics id="S4.SS2.p2.1.m1.1a"><mn id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><cn type="integer" id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">6</annotation></semantics></math> environments.</p>
</div>
<section id="S4.SS2.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Benefit of Monte Carlo Tree Guidance.</h5>

<div id="S4.SS2.SSS0.Px1.p1" class="ltx_para">
<p id="S4.SS2.SSS0.Px1.p1.1" class="ltx_p">The typical goal for an RL problem is to find actions that maximize the <em id="S4.SS2.SSS0.Px1.p1.1.1" class="ltx_emph ltx_font_italic">expected</em> future rewards, which we achieve through MCTG. Full-sequence diffusion models such as Diffuser do not support sampling to maximize expected reward, as we formally derive in <a href="#A2.SS3" title="B.3 Why Monte Carlo Tree Guidance relies on Diffusion Forcing ‣ Appendix B Additional Intuitions and Explainations ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">B.3</span></a>. To understand MCTG’s importance, we ablate it in <a href="#S4.SS1" title="4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4.1</span></a>. Removing MCTG guidance degrades our performance, though Diffusion Forcing remains competitive even then.</p>
</div>
</section>
<section id="S4.SS2.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Benefit of Modeling Causality.</h5>

<div id="S4.SS2.SSS0.Px2.p1" class="ltx_para">
<p id="S4.SS2.SSS0.Px2.p1.1" class="ltx_p">Unlike pure generative modeling, sequential decision-making takes actions and receives feedback. Due to compounding uncertainty, the immediate next actions are more important than those in the far future.
Though Diffuser and subsequent models are trained to generate sequences of action-reward-state tuples <math id="S4.SS2.SSS0.Px2.p1.1.m1.3" class="ltx_Math" alttext="[\mathbf{a}_{t},\mathbf{r}_{t},\mathbf{o}_{t}]" display="inline"><semantics id="S4.SS2.SSS0.Px2.p1.1.m1.3a"><mrow id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.4.cmml"><mo stretchy="false" id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.4" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.4.cmml">[</mo><msub id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.cmml"><mi id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.2" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.2.cmml">𝐚</mi><mi id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.3" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.3.cmml">t</mi></msub><mo id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.5" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.4.cmml">,</mo><msub id="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2" xref="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.cmml"><mi id="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.2" xref="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.2.cmml">𝐫</mi><mi id="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.3" xref="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.3.cmml">t</mi></msub><mo id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.6" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.4.cmml">,</mo><msub id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.cmml"><mi id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.2" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.2.cmml">𝐨</mi><mi id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.3" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.3.cmml">t</mi></msub><mo stretchy="false" id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.7" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.4.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS0.Px2.p1.1.m1.3b"><list id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.4.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3"><apply id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.1.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.2.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.2">𝐚</ci><ci id="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.3.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.1.1.1.1.3">𝑡</ci></apply><apply id="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.1.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.2.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.2">𝐫</ci><ci id="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.3.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.2.2.2.2.3">𝑡</ci></apply><apply id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.1.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3">subscript</csymbol><ci id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.2.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.2">𝐨</ci><ci id="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.3.cmml" xref="S4.SS2.SSS0.Px2.p1.1.m1.3.3.3.3.3">𝑡</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS0.Px2.p1.1.m1.3c">[\mathbf{a}_{t},\mathbf{r}_{t},\mathbf{o}_{t}]</annotation></semantics></math>, directly executing the actions will lead to a trajectory that deviates significantly from the generated states. In other words, the generated states and actions are not causally consistent with each other. To address this shortcoming, Diffuser’s implementation ignores the generated actions and instead relies on a hand-crafted PD controller to infer actions from generated states. In Table <a href="#S4.SS1" title="4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4.1</span></a>, we see that Diffuser’s performance drops dramatically when directly executing generated actions. In contrast, Diffusion Forcing’s raw action generations are self-consistent, outperforming even actions selected by combining Diffuser’s state predictions with a handcrafted PD controller.</p>
</div>
</section>
<section id="S4.SS2.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Benefit of Flexible Horizon.</h5>

<div id="S4.SS2.SSS0.Px3.p1" class="ltx_para">
<p id="S4.SS2.SSS0.Px3.p1.1" class="ltx_p">Many RL tasks have a fixed horizon, requiring the planning horizon to shrink as an agent makes progress in the task. Diffusion Forcing accomplishes this by design, while full-sequence models like Diffuser perform poorly even with tweaks, as we explain in Appendix <a href="#A2.SS4" title="B.4 Does the replacement technique lead to flexible horizons in full-sequence diffusion? ‣ Appendix B Additional Intuitions and Explainations ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">B.4</span></a>.</p>
</div>
<section id="S4.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.3 </span>Controllable Sequential Compositional Generation</h3>

<div id="S4.SS3.p1" class="ltx_para">
<p id="S4.SS3.p1.1" class="ltx_p">We demonstrate that by only modifying the sampling scheme, we can flexibly compose sub-sequences of sequences observed at training time.
We consider a dataset of trajectories on a 2D, square plane, where all trajectories start from one corner and end up in the opposite corner, forming a cross shape. As shown in Fig. <a href="#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>, when no compositional behavior is desired, one can let DF keep full memory, replicating the cross-shaped distribution. When one desires compositionality, one can let the model generate shorter plans without memory using MPC, leading to stitching of the cross’s sub-trajectories, forming a V-shaped trajectory. Due to limited space, we defer the result to Appendix <a href="#A5.SS2" title="E.2 Additional results in compositional generation ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">E.2</span></a>.</p>
</div>
<figure id="S4.F4" class="ltx_figure"><img src="/html/2407.01392/assets/x6.png" id="S4.F4.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="461" height="98" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="S4.F4.2.1.1" class="ltx_text" style="font-size:90%;">Figure 4</span>: </span><span id="S4.F4.3.2" class="ltx_text" style="font-size:90%;">In our real robot task, a robot arm is asked to swap the slots of two fruits using a third slot. Since the fruits are input in random slots at the beginning, one cannot determine the next steps from a single observation without knowledge of the initial placement of the fruits. As illustrated in (a) and (b), the upper observation is the same but the desired outcome illustrated below can vary—the task thus requires remembering the initial configuration. In addition, as shown in (c), the same model that generates actions also synthesizes realistic video from just a single frame.</span></figcaption>
</figure>
<section id="S4.SS4" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.4 </span>Robotics: Long horizon imitation learning and robust visuomotor control</h3>

<div id="S4.SS4.p1" class="ltx_para">
<p id="S4.SS4.p1.1" class="ltx_p">Finally, we illustrate that Diffusion Forcing (DF) opens up new opportunities in visuomotor control of real-world robots. Imitation learning <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib10" title="" class="ltx_ref">10</a>]</cite> is a popular technique in robotic manipulation where one learns an observation-to-action mapping from expert demonstrations. However, the lack of memory often prevents imitation learning from accomplishing long-horizon tasks. DF not only alleviates this shortcoming but also provides a way to make imitation learning robust.</p>
</div>
<section id="S4.SS4.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Imitation Learning with Memory.</h5>

<div id="S4.SS4.SSS0.Px1.p1" class="ltx_para">
<p id="S4.SS4.SSS0.Px1.p1.1" class="ltx_p">We collect a dataset of videos and actions by teleoperating a Franka robot. In the chosen task, one needs to swap the position of an apple and an orange, using a third slot. See Fig. <a href="#S4.F4" title="Figure 4 ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4</span></a> for an illustration. The initial positions of the fruits are randomized such that there are two possible goal states. As illustrated in Fig. <a href="#S4.F4" title="Figure 4 ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4</span></a>, when one fruit is in the third slot, the desired outcome cannot be inferred from the current observation—a policy must remember the initial configuration to determine which fruit to move. In contrast to common behavior cloning methods, DF naturally incorporates memory in its latent state. We found that DF achieves <math id="S4.SS4.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="80\%" display="inline"><semantics id="S4.SS4.SSS0.Px1.p1.1.m1.1a"><mrow id="S4.SS4.SSS0.Px1.p1.1.m1.1.1" xref="S4.SS4.SSS0.Px1.p1.1.m1.1.1.cmml"><mn id="S4.SS4.SSS0.Px1.p1.1.m1.1.1.2" xref="S4.SS4.SSS0.Px1.p1.1.m1.1.1.2.cmml">80</mn><mo id="S4.SS4.SSS0.Px1.p1.1.m1.1.1.1" xref="S4.SS4.SSS0.Px1.p1.1.m1.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.SSS0.Px1.p1.1.m1.1b"><apply id="S4.SS4.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S4.SS4.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="latexml" id="S4.SS4.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S4.SS4.SSS0.Px1.p1.1.m1.1.1.1">percent</csymbol><cn type="integer" id="S4.SS4.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S4.SS4.SSS0.Px1.p1.1.m1.1.1.2">80</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.SSS0.Px1.p1.1.m1.1c">80\%</annotation></semantics></math> success rate while diffusion policy <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib10" title="" class="ltx_ref">10</a>]</cite>, a state-of-the-art imitation learning algorithm without memory, fails.</p>
</div>
</section>
<section id="S4.SS4.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Robustness to missing or noisy observations.</h5>

<div id="S4.SS4.SSS0.Px2.p1" class="ltx_para">
<p id="S4.SS4.SSS0.Px2.p1.4" class="ltx_p">Because it incorporates principles from Bayes filtering,
Diffusion Forcing can perform imitation learning while being robust to noisy or missing observations. We demonstrate this by adding visual distractions and even fully occluding the camera during execution.
DF allows us to easily indicate these observations as “noisy” by using <math id="S4.SS4.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="k&gt;0" display="inline"><semantics id="S4.SS4.SSS0.Px2.p1.1.m1.1a"><mrow id="S4.SS4.SSS0.Px2.p1.1.m1.1.1" xref="S4.SS4.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="S4.SS4.SSS0.Px2.p1.1.m1.1.1.2" xref="S4.SS4.SSS0.Px2.p1.1.m1.1.1.2.cmml">k</mi><mo id="S4.SS4.SSS0.Px2.p1.1.m1.1.1.1" xref="S4.SS4.SSS0.Px2.p1.1.m1.1.1.1.cmml">&gt;</mo><mn id="S4.SS4.SSS0.Px2.p1.1.m1.1.1.3" xref="S4.SS4.SSS0.Px2.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.SSS0.Px2.p1.1.m1.1b"><apply id="S4.SS4.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S4.SS4.SSS0.Px2.p1.1.m1.1.1"><gt id="S4.SS4.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S4.SS4.SSS0.Px2.p1.1.m1.1.1.1"></gt><ci id="S4.SS4.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S4.SS4.SSS0.Px2.p1.1.m1.1.1.2">𝑘</ci><cn type="integer" id="S4.SS4.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="S4.SS4.SSS0.Px2.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.SSS0.Px2.p1.1.m1.1c">k&gt;0</annotation></semantics></math>, in which case DF relies heavily on its prior model to predict actions. Consequently, the succes rate is only lowered by <math id="S4.SS4.SSS0.Px2.p1.2.m2.1" class="ltx_Math" alttext="4\%" display="inline"><semantics id="S4.SS4.SSS0.Px2.p1.2.m2.1a"><mrow id="S4.SS4.SSS0.Px2.p1.2.m2.1.1" xref="S4.SS4.SSS0.Px2.p1.2.m2.1.1.cmml"><mn id="S4.SS4.SSS0.Px2.p1.2.m2.1.1.2" xref="S4.SS4.SSS0.Px2.p1.2.m2.1.1.2.cmml">4</mn><mo id="S4.SS4.SSS0.Px2.p1.2.m2.1.1.1" xref="S4.SS4.SSS0.Px2.p1.2.m2.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.SSS0.Px2.p1.2.m2.1b"><apply id="S4.SS4.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S4.SS4.SSS0.Px2.p1.2.m2.1.1"><csymbol cd="latexml" id="S4.SS4.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S4.SS4.SSS0.Px2.p1.2.m2.1.1.1">percent</csymbol><cn type="integer" id="S4.SS4.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S4.SS4.SSS0.Px2.p1.2.m2.1.1.2">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.SSS0.Px2.p1.2.m2.1c">4\%</annotation></semantics></math> to <math id="S4.SS4.SSS0.Px2.p1.3.m3.1" class="ltx_Math" alttext="76\%" display="inline"><semantics id="S4.SS4.SSS0.Px2.p1.3.m3.1a"><mrow id="S4.SS4.SSS0.Px2.p1.3.m3.1.1" xref="S4.SS4.SSS0.Px2.p1.3.m3.1.1.cmml"><mn id="S4.SS4.SSS0.Px2.p1.3.m3.1.1.2" xref="S4.SS4.SSS0.Px2.p1.3.m3.1.1.2.cmml">76</mn><mo id="S4.SS4.SSS0.Px2.p1.3.m3.1.1.1" xref="S4.SS4.SSS0.Px2.p1.3.m3.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.SSS0.Px2.p1.3.m3.1b"><apply id="S4.SS4.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S4.SS4.SSS0.Px2.p1.3.m3.1.1"><csymbol cd="latexml" id="S4.SS4.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="S4.SS4.SSS0.Px2.p1.3.m3.1.1.1">percent</csymbol><cn type="integer" id="S4.SS4.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="S4.SS4.SSS0.Px2.p1.3.m3.1.1.2">76</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.SSS0.Px2.p1.3.m3.1c">76\%</annotation></semantics></math>.
In contrast, a next-frame diffusion model baseline attains a success rate of <math id="S4.SS4.SSS0.Px2.p1.4.m4.1" class="ltx_Math" alttext="48\%" display="inline"><semantics id="S4.SS4.SSS0.Px2.p1.4.m4.1a"><mrow id="S4.SS4.SSS0.Px2.p1.4.m4.1.1" xref="S4.SS4.SSS0.Px2.p1.4.m4.1.1.cmml"><mn id="S4.SS4.SSS0.Px2.p1.4.m4.1.1.2" xref="S4.SS4.SSS0.Px2.p1.4.m4.1.1.2.cmml">48</mn><mo id="S4.SS4.SSS0.Px2.p1.4.m4.1.1.1" xref="S4.SS4.SSS0.Px2.p1.4.m4.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.SSS0.Px2.p1.4.m4.1b"><apply id="S4.SS4.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S4.SS4.SSS0.Px2.p1.4.m4.1.1"><csymbol cd="latexml" id="S4.SS4.SSS0.Px2.p1.4.m4.1.1.1.cmml" xref="S4.SS4.SSS0.Px2.p1.4.m4.1.1.1">percent</csymbol><cn type="integer" id="S4.SS4.SSS0.Px2.p1.4.m4.1.1.2.cmml" xref="S4.SS4.SSS0.Px2.p1.4.m4.1.1.2">48</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.SSS0.Px2.p1.4.m4.1c">48\%</annotation></semantics></math>: it must treat perturbed observations as ground truth and suffers out-of-distribution error.</p>
</div>
</section>
<section id="S4.SS4.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Potential for pre-training with video.</h5>

<div id="S4.SS4.SSS0.Px3.p1" class="ltx_para">
<p id="S4.SS4.SSS0.Px3.p1.1" class="ltx_p">Finally in parallel to generating actions, Fig. <a href="#S4.F4" title="Figure 4 ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4</span></a> illustrates that Diffusion Forcing is capable of generating a video of the robot performing the task given only an initial frame, unifying diffusion policy / imitation learning and video generative modeling and paving the way to pre-training on unlabeled video.</p>
</div>
<section id="S4.SS5" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.5 </span>Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model</h3>

<div id="S4.SS5.p1" class="ltx_para">
<p id="S4.SS5.p1.1" class="ltx_p">In Appendix <a href="#A5" title="Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">E</span></a>, we show that DF is competitive with prior diffusion <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib48" title="" class="ltx_ref">48</a>]</cite> and transformer-based <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib49" title="" class="ltx_ref">49</a>]</cite> work on multivariate time series forecasting, following the experimental setup of  <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib51" title="" class="ltx_ref">51</a>]</cite>.</p>
</div>
<section id="S5" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">5 </span>Discussion</h2>

<section id="S5.SS0.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Limitations.</h5>

<div id="S5.SS0.SSS0.Px1.p1" class="ltx_para">
<p id="S5.SS0.SSS0.Px1.p1.1" class="ltx_p">Our current causal implementation is based on a small RNN, and applications to higher-resolution video or more complex distributions likely require large transformer models. We do not investigate the scaling behavior of Diffusion Forcing to internet-scale datasets and tasks.</p>
</div>
</section>
<section id="S5.SS0.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">Conclusion.</h5>

<div id="S5.SS0.SSS0.Px2.p1" class="ltx_para">
<p id="S5.SS0.SSS0.Px2.p1.1" class="ltx_p">In this paper, we introduced Diffusion Forcing, a new training paradigm where a model is trained to denoise sets of tokens with independent, per-token noise levels.
Applied to time series data, we show how a next-token prediction model trained with Diffusion Forcing combines benefits of both next-token models and full-sequence diffusion models.
We introduced new sampling and guidance schemes that lead to dramatic performance gains when applied to tasks in sequential decision making.
Future work may investigate the application of Diffusion Forcing to domains other than time series generative modeling, and scale up Diffusion Forcing to larger datasets.</p>
</div>
</section>
<section id="S5.SS0.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Acknowledgements.</h5>

<div id="S5.SS0.SSS0.Px3.p1" class="ltx_para">
<p id="S5.SS0.SSS0.Px3.p1.1" class="ltx_p"><span id="S5.SS0.SSS0.Px3.p1.1.1" class="ltx_text" style="font-size:90%;">This work was supported by the National Science Foundation under Grant No. 2211259, by the Singapore DSTA under DST00OECI20300823 (3D Self-Supervised Learning for Label-Efficient Vision), by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/ Interior Business Center (DOI/IBC) under 140D0423C0075, and by the Amazon Science Hub. Special thanks to Kiwhan Song (MIT) for 3D-Unet and transformer reimplementation of the paper after initial release.</span></p>
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<div class="ltx_pagination ltx_role_newpage"></div>
<section id="A1" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix" style="font-size:90%;">
<span class="ltx_tag ltx_tag_appendix">Appendix A </span>Theoretical Justification</h2>

<div id="A1.p1" class="ltx_para">
<p id="A1.p1.1" class="ltx_p"><span id="A1.p1.1.1" class="ltx_text" style="font-size:90%;">In this section, we provide theoretical justification for the train of Diffusion Forcing. The main contributions can be summarized as follows:</span></p>
<ul id="A1.I1" class="ltx_itemize">
<li id="A1.I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A1.I1.i1.p1" class="ltx_para">
<p id="A1.I1.i1.p1.1" class="ltx_p"><span id="A1.I1.i1.p1.1.1" class="ltx_text" style="font-size:90%;">We show that our training methods optimizes a reweighting of Evidence Lower Bound (ELBO) on the average log-likelihood of our data. We first establish this in full abstraction (</span><a href="#A1.Thmtheorem1" title="Theorem A.1. ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">A.1</span></a><span id="A1.I1.i1.p1.1.2" class="ltx_text" style="font-size:90%;">), and then specialize to the form of Gaussian diffusion (</span><a href="#A1.Thmtheorem2" title="Corollary A.2. ‣ A.1.1 Specializing to Gaussian diffusion ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">A.2</span></a><span id="A1.I1.i1.p1.1.3" class="ltx_text" style="font-size:90%;">). We show that the resulting terms decouple in such a fashion that, in the limit of a fully expressive latent and model, makes the reweighting terms immaterial.</span></p>
</div>
</li>
<li id="A1.I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A1.I1.i2.p1" class="ltx_para">
<p id="A1.I1.i2.p1.1" class="ltx_p"><span id="A1.I1.i2.p1.1.1" class="ltx_text" style="font-size:90%;">We show that the expected likelihood over </span><em id="A1.I1.i2.p1.1.2" class="ltx_emph ltx_font_italic" style="font-size:90%;">any</em><span id="A1.I1.i2.p1.1.3" class="ltx_text" style="font-size:90%;"> distribution over sequences of noise levels can be lower bounded by a sum over nonnegative terms which, when reweighted, correspond to the terms optimized in the Diffusion Forcing training objective maximizes. Thus, for a fully expressive network that can drive all terms to their minimal value, Diffusion Forcing optimizes a valid surrogate of the likelihood for </span><em id="A1.I1.i2.p1.1.4" class="ltx_emph ltx_font_italic" style="font-size:90%;">likelihood of all sequences of noise levels simultaneously.</em><span id="A1.I1.i2.p1.1.5" class="ltx_text" style="font-size:90%;"></span></p>
</div>
</li>
</ul>
</div>
<div id="A1.p2" class="ltx_para">
<p id="A1.p2.3" class="ltx_p"><span id="A1.p2.3.1" class="ltx_text" style="font-size:90%;">We begin by stating an ELBO for general Markov forward processes </span><math id="A1.p2.1.m1.1" class="ltx_Math" alttext="q(\cdot)" display="inline"><semantics id="A1.p2.1.m1.1a"><mrow id="A1.p2.1.m1.1.2" xref="A1.p2.1.m1.1.2.cmml"><mi mathsize="90%" id="A1.p2.1.m1.1.2.2" xref="A1.p2.1.m1.1.2.2.cmml">q</mi><mo lspace="0em" rspace="0em" id="A1.p2.1.m1.1.2.1" xref="A1.p2.1.m1.1.2.1.cmml">​</mo><mrow id="A1.p2.1.m1.1.2.3.2" xref="A1.p2.1.m1.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A1.p2.1.m1.1.2.3.2.1" xref="A1.p2.1.m1.1.2.cmml">(</mo><mo lspace="0em" mathsize="90%" rspace="0em" id="A1.p2.1.m1.1.1" xref="A1.p2.1.m1.1.1.cmml">⋅</mo><mo maxsize="90%" minsize="90%" id="A1.p2.1.m1.1.2.3.2.2" xref="A1.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.1.m1.1b"><apply id="A1.p2.1.m1.1.2.cmml" xref="A1.p2.1.m1.1.2"><times id="A1.p2.1.m1.1.2.1.cmml" xref="A1.p2.1.m1.1.2.1"></times><ci id="A1.p2.1.m1.1.2.2.cmml" xref="A1.p2.1.m1.1.2.2">𝑞</ci><ci id="A1.p2.1.m1.1.1.cmml" xref="A1.p2.1.m1.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.1.m1.1c">q(\cdot)</annotation></semantics></math><span id="A1.p2.3.2" class="ltx_text" style="font-size:90%;">, and generative models </span><math id="A1.p2.2.m2.1" class="ltx_Math" alttext="p_{\bm{\theta}}(\cdot)" display="inline"><semantics id="A1.p2.2.m2.1a"><mrow id="A1.p2.2.m2.1.2" xref="A1.p2.2.m2.1.2.cmml"><msub id="A1.p2.2.m2.1.2.2" xref="A1.p2.2.m2.1.2.2.cmml"><mi mathsize="90%" id="A1.p2.2.m2.1.2.2.2" xref="A1.p2.2.m2.1.2.2.2.cmml">p</mi><mi mathsize="90%" id="A1.p2.2.m2.1.2.2.3" xref="A1.p2.2.m2.1.2.2.3.cmml">𝜽</mi></msub><mo lspace="0em" rspace="0em" id="A1.p2.2.m2.1.2.1" xref="A1.p2.2.m2.1.2.1.cmml">​</mo><mrow id="A1.p2.2.m2.1.2.3.2" xref="A1.p2.2.m2.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A1.p2.2.m2.1.2.3.2.1" xref="A1.p2.2.m2.1.2.cmml">(</mo><mo lspace="0em" mathsize="90%" rspace="0em" id="A1.p2.2.m2.1.1" xref="A1.p2.2.m2.1.1.cmml">⋅</mo><mo maxsize="90%" minsize="90%" id="A1.p2.2.m2.1.2.3.2.2" xref="A1.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.2.m2.1b"><apply id="A1.p2.2.m2.1.2.cmml" xref="A1.p2.2.m2.1.2"><times id="A1.p2.2.m2.1.2.1.cmml" xref="A1.p2.2.m2.1.2.1"></times><apply id="A1.p2.2.m2.1.2.2.cmml" xref="A1.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="A1.p2.2.m2.1.2.2.1.cmml" xref="A1.p2.2.m2.1.2.2">subscript</csymbol><ci id="A1.p2.2.m2.1.2.2.2.cmml" xref="A1.p2.2.m2.1.2.2.2">𝑝</ci><ci id="A1.p2.2.m2.1.2.2.3.cmml" xref="A1.p2.2.m2.1.2.2.3">𝜽</ci></apply><ci id="A1.p2.2.m2.1.1.cmml" xref="A1.p2.2.m2.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.2.m2.1c">p_{\bm{\theta}}(\cdot)</annotation></semantics></math><span id="A1.p2.3.3" class="ltx_text" style="font-size:90%;">, and then specialize to Gaussian diffusion, thereby recovering our loss. We denote our Markov forward process </span><math id="A1.p2.3.m3.1" class="ltx_Math" alttext="q(\cdot)" display="inline"><semantics id="A1.p2.3.m3.1a"><mrow id="A1.p2.3.m3.1.2" xref="A1.p2.3.m3.1.2.cmml"><mi mathsize="90%" id="A1.p2.3.m3.1.2.2" xref="A1.p2.3.m3.1.2.2.cmml">q</mi><mo lspace="0em" rspace="0em" id="A1.p2.3.m3.1.2.1" xref="A1.p2.3.m3.1.2.1.cmml">​</mo><mrow id="A1.p2.3.m3.1.2.3.2" xref="A1.p2.3.m3.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A1.p2.3.m3.1.2.3.2.1" xref="A1.p2.3.m3.1.2.cmml">(</mo><mo lspace="0em" mathsize="90%" rspace="0em" id="A1.p2.3.m3.1.1" xref="A1.p2.3.m3.1.1.cmml">⋅</mo><mo maxsize="90%" minsize="90%" id="A1.p2.3.m3.1.2.3.2.2" xref="A1.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.p2.3.m3.1b"><apply id="A1.p2.3.m3.1.2.cmml" xref="A1.p2.3.m3.1.2"><times id="A1.p2.3.m3.1.2.1.cmml" xref="A1.p2.3.m3.1.2.1"></times><ci id="A1.p2.3.m3.1.2.2.cmml" xref="A1.p2.3.m3.1.2.2">𝑞</ci><ci id="A1.p2.3.m3.1.1.cmml" xref="A1.p2.3.m3.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.3.m3.1c">q(\cdot)</annotation></semantics></math><span id="A1.p2.3.4" class="ltx_text" style="font-size:90%;"> as</span></p>
<table id="A6.EGx1" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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mathsize="90%" id="A1.E1.m1.1.1.1.1.3" xref="A1.E1.m1.1.1.1.1.3.cmml">=</mo><mrow id="A1.E1.m1.1.1.1.1.2" xref="A1.E1.m1.1.1.1.1.2.cmml"><mstyle displaystyle="true" id="A1.E1.m1.1.1.1.1.2.2" xref="A1.E1.m1.1.1.1.1.2.2.cmml"><munderover id="A1.E1.m1.1.1.1.1.2.2a" xref="A1.E1.m1.1.1.1.1.2.2.cmml"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.E1.m1.1.1.1.1.2.2.2.2" xref="A1.E1.m1.1.1.1.1.2.2.2.2.cmml">∏</mo><mrow id="A1.E1.m1.1.1.1.1.2.2.2.3" xref="A1.E1.m1.1.1.1.1.2.2.2.3.cmml"><mi mathsize="90%" id="A1.E1.m1.1.1.1.1.2.2.2.3.2" xref="A1.E1.m1.1.1.1.1.2.2.2.3.2.cmml">k</mi><mo mathsize="90%" id="A1.E1.m1.1.1.1.1.2.2.2.3.1" xref="A1.E1.m1.1.1.1.1.2.2.2.3.1.cmml">=</mo><mn mathsize="90%" id="A1.E1.m1.1.1.1.1.2.2.2.3.3" xref="A1.E1.m1.1.1.1.1.2.2.2.3.3.cmml">1</mn></mrow><mi mathsize="90%" id="A1.E1.m1.1.1.1.1.2.2.3" xref="A1.E1.m1.1.1.1.1.2.2.3.cmml">K</mi></munderover></mstyle><mrow id="A1.E1.m1.1.1.1.1.2.1" xref="A1.E1.m1.1.1.1.1.2.1.cmml"><mi mathsize="90%" 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xref="A1.E1.m1.1.1.1.1.1.2"></times><ci id="A1.E1.m1.1.1.1.1.1.3.cmml" xref="A1.E1.m1.1.1.1.1.1.3">𝑞</ci><apply id="A1.E1.m1.1.1.1.1.1.1.1.1.cmml" xref="A1.E1.m1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="A1.E1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.E1.m1.1.1.1.1.1.1.1.1.1">conditional</csymbol><apply id="A1.E1.m1.1.1.1.1.1.1.1.1.2.cmml" xref="A1.E1.m1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.E1.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A1.E1.m1.1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="A1.E1.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A1.E1.m1.1.1.1.1.1.1.1.1.2.2">𝐱</ci><apply id="A1.E1.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="A1.E1.m1.1.1.1.1.1.1.1.1.2.3"><ci id="A1.E1.m1.1.1.1.1.1.1.1.1.2.3.1.cmml" xref="A1.E1.m1.1.1.1.1.1.1.1.1.2.3.1">:</ci><cn type="integer" id="A1.E1.m1.1.1.1.1.1.1.1.1.2.3.2.cmml" xref="A1.E1.m1.1.1.1.1.1.1.1.1.2.3.2">1</cn><ci id="A1.E1.m1.1.1.1.1.1.1.1.1.2.3.3.cmml" xref="A1.E1.m1.1.1.1.1.1.1.1.1.2.3.3">𝐾</ci></apply></apply><apply id="A1.E1.m1.1.1.1.1.1.1.1.1.3.cmml" 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xref="A1.E1.m1.1.1.1.1.2.2.2.3.1"></eq><ci id="A1.E1.m1.1.1.1.1.2.2.2.3.2.cmml" xref="A1.E1.m1.1.1.1.1.2.2.2.3.2">𝑘</ci><cn type="integer" id="A1.E1.m1.1.1.1.1.2.2.2.3.3.cmml" xref="A1.E1.m1.1.1.1.1.2.2.2.3.3">1</cn></apply></apply><ci id="A1.E1.m1.1.1.1.1.2.2.3.cmml" xref="A1.E1.m1.1.1.1.1.2.2.3">𝐾</ci></apply><apply id="A1.E1.m1.1.1.1.1.2.1.cmml" xref="A1.E1.m1.1.1.1.1.2.1"><times id="A1.E1.m1.1.1.1.1.2.1.2.cmml" xref="A1.E1.m1.1.1.1.1.2.1.2"></times><ci id="A1.E1.m1.1.1.1.1.2.1.3.cmml" xref="A1.E1.m1.1.1.1.1.2.1.3">𝑞</ci><apply id="A1.E1.m1.1.1.1.1.2.1.1.1.1.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1"><csymbol cd="latexml" id="A1.E1.m1.1.1.1.1.2.1.1.1.1.1.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.1">conditional</csymbol><apply id="A1.E1.m1.1.1.1.1.2.1.1.1.1.2.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.E1.m1.1.1.1.1.2.1.1.1.1.2.1.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.2">superscript</csymbol><ci id="A1.E1.m1.1.1.1.1.2.1.1.1.1.2.2.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.2.2">𝐱</ci><ci id="A1.E1.m1.1.1.1.1.2.1.1.1.1.2.3.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.2.3">𝑘</ci></apply><apply id="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.1.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.3">superscript</csymbol><ci id="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.2.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.2">𝐱</ci><apply id="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.3.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.3"><minus id="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.3.1.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.3.1"></minus><ci id="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.3.2.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.3.2">𝑘</ci><cn type="integer" id="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.3.3.cmml" xref="A1.E1.m1.1.1.1.1.2.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E1.m1.1c">\displaystyle q(\mathbf{x}^{1:K}\mid\mathbf{x}^{0})=\prod_{k=1}^{K}q(\mathbf{x}^{k}\mid\mathbf{x}^{k-1}),</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.1)</span></td>
</tr></tbody>
</table>
<p id="A1.p2.9" class="ltx_p"><span id="A1.p2.9.1" class="ltx_text" style="font-size:90%;">and a parmeterized probability model</span></p>
<table id="A6.EGx2" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E2.m1.1" class="ltx_Math" alttext="\displaystyle p_{\bm{\theta}}(((\mathbf{x}^{k}_{t})_{1\leq k\leq K},\mathbf{z}_{t})_{t\geq 1})" display="inline"><semantics id="A1.E2.m1.1a"><mrow id="A1.E2.m1.1.1" xref="A1.E2.m1.1.1.cmml"><msub id="A1.E2.m1.1.1.3" xref="A1.E2.m1.1.1.3.cmml"><mi mathsize="90%" id="A1.E2.m1.1.1.3.2" xref="A1.E2.m1.1.1.3.2.cmml">p</mi><mi mathsize="90%" id="A1.E2.m1.1.1.3.3" xref="A1.E2.m1.1.1.3.3.cmml">𝜽</mi></msub><mo lspace="0em" rspace="0em" id="A1.E2.m1.1.1.2" xref="A1.E2.m1.1.1.2.cmml">​</mo><mrow id="A1.E2.m1.1.1.1.1" xref="A1.E2.m1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.E2.m1.1.1.1.1.2" xref="A1.E2.m1.1.1.1.1.1.cmml">(</mo><msub id="A1.E2.m1.1.1.1.1.1" xref="A1.E2.m1.1.1.1.1.1.cmml"><mrow id="A1.E2.m1.1.1.1.1.1.2.2" xref="A1.E2.m1.1.1.1.1.1.2.3.cmml"><mo maxsize="90%" minsize="90%" id="A1.E2.m1.1.1.1.1.1.2.2.3" xref="A1.E2.m1.1.1.1.1.1.2.3.cmml">(</mo><msub id="A1.E2.m1.1.1.1.1.1.1.1.1" xref="A1.E2.m1.1.1.1.1.1.1.1.1.cmml"><mrow id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.2" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2.2" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.3" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">t</mi><mi mathsize="90%" id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">k</mi></msubsup><mo maxsize="90%" minsize="90%" id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.3" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.E2.m1.1.1.1.1.1.1.1.1.3" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.cmml"><mn mathsize="90%" id="A1.E2.m1.1.1.1.1.1.1.1.1.3.2" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.2.cmml">1</mn><mo mathsize="90%" id="A1.E2.m1.1.1.1.1.1.1.1.1.3.3" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.3.cmml">≤</mo><mi mathsize="90%" id="A1.E2.m1.1.1.1.1.1.1.1.1.3.4" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.4.cmml">k</mi><mo mathsize="90%" id="A1.E2.m1.1.1.1.1.1.1.1.1.3.5" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.5.cmml">≤</mo><mi mathsize="90%" id="A1.E2.m1.1.1.1.1.1.1.1.1.3.6" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.6.cmml">K</mi></mrow></msub><mo mathsize="90%" id="A1.E2.m1.1.1.1.1.1.2.2.4" xref="A1.E2.m1.1.1.1.1.1.2.3.cmml">,</mo><msub id="A1.E2.m1.1.1.1.1.1.2.2.2" xref="A1.E2.m1.1.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A1.E2.m1.1.1.1.1.1.2.2.2.2" xref="A1.E2.m1.1.1.1.1.1.2.2.2.2.cmml">𝐳</mi><mi mathsize="90%" id="A1.E2.m1.1.1.1.1.1.2.2.2.3" xref="A1.E2.m1.1.1.1.1.1.2.2.2.3.cmml">t</mi></msub><mo maxsize="90%" minsize="90%" id="A1.E2.m1.1.1.1.1.1.2.2.5" xref="A1.E2.m1.1.1.1.1.1.2.3.cmml">)</mo></mrow><mrow id="A1.E2.m1.1.1.1.1.1.4" xref="A1.E2.m1.1.1.1.1.1.4.cmml"><mi mathsize="90%" id="A1.E2.m1.1.1.1.1.1.4.2" xref="A1.E2.m1.1.1.1.1.1.4.2.cmml">t</mi><mo mathsize="90%" id="A1.E2.m1.1.1.1.1.1.4.1" xref="A1.E2.m1.1.1.1.1.1.4.1.cmml">≥</mo><mn mathsize="90%" id="A1.E2.m1.1.1.1.1.1.4.3" xref="A1.E2.m1.1.1.1.1.1.4.3.cmml">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.E2.m1.1.1.1.1.3" xref="A1.E2.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E2.m1.1b"><apply id="A1.E2.m1.1.1.cmml" xref="A1.E2.m1.1.1"><times id="A1.E2.m1.1.1.2.cmml" xref="A1.E2.m1.1.1.2"></times><apply id="A1.E2.m1.1.1.3.cmml" xref="A1.E2.m1.1.1.3"><csymbol cd="ambiguous" id="A1.E2.m1.1.1.3.1.cmml" xref="A1.E2.m1.1.1.3">subscript</csymbol><ci id="A1.E2.m1.1.1.3.2.cmml" xref="A1.E2.m1.1.1.3.2">𝑝</ci><ci id="A1.E2.m1.1.1.3.3.cmml" xref="A1.E2.m1.1.1.3.3">𝜽</ci></apply><apply id="A1.E2.m1.1.1.1.1.1.cmml" xref="A1.E2.m1.1.1.1.1"><csymbol cd="ambiguous" id="A1.E2.m1.1.1.1.1.1.3.cmml" xref="A1.E2.m1.1.1.1.1">subscript</csymbol><interval closure="open" id="A1.E2.m1.1.1.1.1.1.2.3.cmml" xref="A1.E2.m1.1.1.1.1.1.2.2"><apply id="A1.E2.m1.1.1.1.1.1.1.1.1.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.E2.m1.1.1.1.1.1.1.1.1.2.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1">subscript</csymbol><apply id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><apply id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2.2">𝐱</ci><ci id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.2.3">𝑘</ci></apply><ci id="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.1.1.1.3">𝑡</ci></apply><apply id="A1.E2.m1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3"><and id="A1.E2.m1.1.1.1.1.1.1.1.1.3a.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3"></and><apply id="A1.E2.m1.1.1.1.1.1.1.1.1.3b.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3"><leq id="A1.E2.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.3"></leq><cn type="integer" id="A1.E2.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.2">1</cn><ci id="A1.E2.m1.1.1.1.1.1.1.1.1.3.4.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.4">𝑘</ci></apply><apply id="A1.E2.m1.1.1.1.1.1.1.1.1.3c.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3"><leq id="A1.E2.m1.1.1.1.1.1.1.1.1.3.5.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.5"></leq><share href="#A1.E2.m1.1.1.1.1.1.1.1.1.3.4.cmml" id="A1.E2.m1.1.1.1.1.1.1.1.1.3d.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3"></share><ci id="A1.E2.m1.1.1.1.1.1.1.1.1.3.6.cmml" xref="A1.E2.m1.1.1.1.1.1.1.1.1.3.6">𝐾</ci></apply></apply></apply><apply id="A1.E2.m1.1.1.1.1.1.2.2.2.cmml" xref="A1.E2.m1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.E2.m1.1.1.1.1.1.2.2.2.1.cmml" xref="A1.E2.m1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A1.E2.m1.1.1.1.1.1.2.2.2.2.cmml" xref="A1.E2.m1.1.1.1.1.1.2.2.2.2">𝐳</ci><ci id="A1.E2.m1.1.1.1.1.1.2.2.2.3.cmml" xref="A1.E2.m1.1.1.1.1.1.2.2.2.3">𝑡</ci></apply></interval><apply id="A1.E2.m1.1.1.1.1.1.4.cmml" xref="A1.E2.m1.1.1.1.1.1.4"><geq id="A1.E2.m1.1.1.1.1.1.4.1.cmml" xref="A1.E2.m1.1.1.1.1.1.4.1"></geq><ci id="A1.E2.m1.1.1.1.1.1.4.2.cmml" xref="A1.E2.m1.1.1.1.1.1.4.2">𝑡</ci><cn type="integer" id="A1.E2.m1.1.1.1.1.1.4.3.cmml" xref="A1.E2.m1.1.1.1.1.1.4.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E2.m1.1c">\displaystyle p_{\bm{\theta}}(((\mathbf{x}^{k}_{t})_{1\leq k\leq K},\mathbf{z}_{t})_{t\geq 1})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.2)</span></td>
</tr></tbody>
</table>
<p id="A1.p2.4" class="ltx_p"><span id="A1.p2.4.1" class="ltx_text" style="font-size:90%;">We assume that </span><math id="A1.p2.4.m1.1" class="ltx_Math" alttext="p_{\bm{\theta}}" display="inline"><semantics id="A1.p2.4.m1.1a"><msub id="A1.p2.4.m1.1.1" xref="A1.p2.4.m1.1.1.cmml"><mi mathsize="90%" id="A1.p2.4.m1.1.1.2" xref="A1.p2.4.m1.1.1.2.cmml">p</mi><mi mathsize="90%" id="A1.p2.4.m1.1.1.3" xref="A1.p2.4.m1.1.1.3.cmml">𝜽</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p2.4.m1.1b"><apply id="A1.p2.4.m1.1.1.cmml" xref="A1.p2.4.m1.1.1"><csymbol cd="ambiguous" id="A1.p2.4.m1.1.1.1.cmml" xref="A1.p2.4.m1.1.1">subscript</csymbol><ci id="A1.p2.4.m1.1.1.2.cmml" xref="A1.p2.4.m1.1.1.2">𝑝</ci><ci id="A1.p2.4.m1.1.1.3.cmml" xref="A1.p2.4.m1.1.1.3">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.4.m1.1c">p_{\bm{\theta}}</annotation></semantics></math><span id="A1.p2.4.2" class="ltx_text" style="font-size:90%;"> satisfies the </span><em id="A1.p2.4.3" class="ltx_emph ltx_font_italic" style="font-size:90%;">Markov property</em><span id="A1.p2.4.4" class="ltx_text" style="font-size:90%;"> that</span></p>
<table id="A6.EGx3" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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p_{\bm{\theta}}(\mathbf{z}_{t},\mathbf{x}_{t}^{k_{t}}\mid\mathbf{z}_{1:t-1},(\mathbf{x}_{s}^{k_{2}})_{1\leq s&lt;t})=p_{\bm{\theta}}(\mathbf{z}_{t},\mathbf{x}^{k_{t}}\mid\mathbf{z}_{t-1})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.3)</span></td>
</tr></tbody>
</table>
<p id="A1.p2.8" class="ltx_p"><span id="A1.p2.8.1" class="ltx_text" style="font-size:90%;">that is, the latent codes </span><math id="A1.p2.5.m1.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}" display="inline"><semantics id="A1.p2.5.m1.1a"><msub id="A1.p2.5.m1.1.1" xref="A1.p2.5.m1.1.1.cmml"><mi mathsize="90%" id="A1.p2.5.m1.1.1.2" xref="A1.p2.5.m1.1.1.2.cmml">𝐳</mi><mrow id="A1.p2.5.m1.1.1.3" xref="A1.p2.5.m1.1.1.3.cmml"><mi mathsize="90%" id="A1.p2.5.m1.1.1.3.2" xref="A1.p2.5.m1.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A1.p2.5.m1.1.1.3.1" xref="A1.p2.5.m1.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A1.p2.5.m1.1.1.3.3" xref="A1.p2.5.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.p2.5.m1.1b"><apply id="A1.p2.5.m1.1.1.cmml" xref="A1.p2.5.m1.1.1"><csymbol cd="ambiguous" id="A1.p2.5.m1.1.1.1.cmml" xref="A1.p2.5.m1.1.1">subscript</csymbol><ci id="A1.p2.5.m1.1.1.2.cmml" xref="A1.p2.5.m1.1.1.2">𝐳</ci><apply id="A1.p2.5.m1.1.1.3.cmml" xref="A1.p2.5.m1.1.1.3"><minus id="A1.p2.5.m1.1.1.3.1.cmml" xref="A1.p2.5.m1.1.1.3.1"></minus><ci id="A1.p2.5.m1.1.1.3.2.cmml" xref="A1.p2.5.m1.1.1.3.2">𝑡</ci><cn type="integer" id="A1.p2.5.m1.1.1.3.3.cmml" xref="A1.p2.5.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.5.m1.1c">\mathbf{z}_{t-1}</annotation></semantics></math><span id="A1.p2.8.2" class="ltx_text" style="font-size:90%;"> is a sufficient statistic for </span><math id="A1.p2.6.m2.1" class="ltx_Math" alttext="\mathbf{x}^{k_{t}}" display="inline"><semantics id="A1.p2.6.m2.1a"><msup id="A1.p2.6.m2.1.1" xref="A1.p2.6.m2.1.1.cmml"><mi mathsize="90%" id="A1.p2.6.m2.1.1.2" xref="A1.p2.6.m2.1.1.2.cmml">𝐱</mi><msub id="A1.p2.6.m2.1.1.3" xref="A1.p2.6.m2.1.1.3.cmml"><mi mathsize="90%" id="A1.p2.6.m2.1.1.3.2" xref="A1.p2.6.m2.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A1.p2.6.m2.1.1.3.3" xref="A1.p2.6.m2.1.1.3.3.cmml">t</mi></msub></msup><annotation-xml encoding="MathML-Content" id="A1.p2.6.m2.1b"><apply id="A1.p2.6.m2.1.1.cmml" xref="A1.p2.6.m2.1.1"><csymbol cd="ambiguous" id="A1.p2.6.m2.1.1.1.cmml" xref="A1.p2.6.m2.1.1">superscript</csymbol><ci id="A1.p2.6.m2.1.1.2.cmml" xref="A1.p2.6.m2.1.1.2">𝐱</ci><apply id="A1.p2.6.m2.1.1.3.cmml" xref="A1.p2.6.m2.1.1.3"><csymbol cd="ambiguous" id="A1.p2.6.m2.1.1.3.1.cmml" xref="A1.p2.6.m2.1.1.3">subscript</csymbol><ci id="A1.p2.6.m2.1.1.3.2.cmml" xref="A1.p2.6.m2.1.1.3.2">𝑘</ci><ci id="A1.p2.6.m2.1.1.3.3.cmml" xref="A1.p2.6.m2.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.6.m2.1c">\mathbf{x}^{k_{t}}</annotation></semantics></math><span id="A1.p2.8.3" class="ltx_text" style="font-size:90%;"> given the history. We say that </span><math id="A1.p2.7.m3.1" class="ltx_Math" alttext="p_{\bm{\theta}}" display="inline"><semantics id="A1.p2.7.m3.1a"><msub id="A1.p2.7.m3.1.1" xref="A1.p2.7.m3.1.1.cmml"><mi mathsize="90%" id="A1.p2.7.m3.1.1.2" xref="A1.p2.7.m3.1.1.2.cmml">p</mi><mi mathsize="90%" id="A1.p2.7.m3.1.1.3" xref="A1.p2.7.m3.1.1.3.cmml">𝜽</mi></msub><annotation-xml encoding="MathML-Content" id="A1.p2.7.m3.1b"><apply id="A1.p2.7.m3.1.1.cmml" xref="A1.p2.7.m3.1.1"><csymbol cd="ambiguous" id="A1.p2.7.m3.1.1.1.cmml" xref="A1.p2.7.m3.1.1">subscript</csymbol><ci id="A1.p2.7.m3.1.1.2.cmml" xref="A1.p2.7.m3.1.1.2">𝑝</ci><ci id="A1.p2.7.m3.1.1.3.cmml" xref="A1.p2.7.m3.1.1.3">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.7.m3.1c">p_{\bm{\theta}}</annotation></semantics></math><span id="A1.p2.8.4" class="ltx_text" style="font-size:90%;"> has </span><em id="A1.p2.8.5" class="ltx_emph ltx_font_italic" style="font-size:90%;">deterministic latents</em><span id="A1.p2.8.6" 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href="#A1.p2.8.m4.1.1.1.1.1.2.2.2.3.4.cmml" id="A1.p2.8.m4.1.1.1.1.1.2.2.2.3d.cmml" xref="A1.p2.8.m4.1.1.1.1.1.2.2.2.3"></share><ci id="A1.p2.8.m4.1.1.1.1.1.2.2.2.3.6.cmml" xref="A1.p2.8.m4.1.1.1.1.1.2.2.2.3.6">𝑡</ci></apply></apply></apply><apply id="A1.p2.8.m4.1.1.1.1.1.3.3.3.cmml" xref="A1.p2.8.m4.1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A1.p2.8.m4.1.1.1.1.1.3.3.3.1.cmml" xref="A1.p2.8.m4.1.1.1.1.1.3.3.3">superscript</csymbol><apply id="A1.p2.8.m4.1.1.1.1.1.3.3.3.2.cmml" xref="A1.p2.8.m4.1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A1.p2.8.m4.1.1.1.1.1.3.3.3.2.1.cmml" xref="A1.p2.8.m4.1.1.1.1.1.3.3.3">subscript</csymbol><ci id="A1.p2.8.m4.1.1.1.1.1.3.3.3.2.2.cmml" xref="A1.p2.8.m4.1.1.1.1.1.3.3.3.2.2">𝐱</ci><ci id="A1.p2.8.m4.1.1.1.1.1.3.3.3.2.3.cmml" xref="A1.p2.8.m4.1.1.1.1.1.3.3.3.2.3">𝑡</ci></apply><apply id="A1.p2.8.m4.1.1.1.1.1.3.3.3.3.cmml" xref="A1.p2.8.m4.1.1.1.1.1.3.3.3.3"><csymbol cd="ambiguous" id="A1.p2.8.m4.1.1.1.1.1.3.3.3.3.1.cmml" xref="A1.p2.8.m4.1.1.1.1.1.3.3.3.3">subscript</csymbol><ci id="A1.p2.8.m4.1.1.1.1.1.3.3.3.3.2.cmml" xref="A1.p2.8.m4.1.1.1.1.1.3.3.3.3.2">𝑘</ci><ci id="A1.p2.8.m4.1.1.1.1.1.3.3.3.3.3.cmml" xref="A1.p2.8.m4.1.1.1.1.1.3.3.3.3.3">𝑡</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.8.m4.1c">p_{\bm{\theta}}(\mathbf{z}_{t}\mid\mathbf{z}_{1:t-1},(\mathbf{x}_{s}^{k_{s}})_{1\leq s&lt;t},\mathbf{x}_{t}^{k_{t}})</annotation></semantics></math><span id="A1.p2.8.7" class="ltx_text" style="font-size:90%;"> is a Dirac delta.</span></p>
</div>
<div id="Thmremark1" class="ltx_theorem ltx_theorem_remark">
<h6 class="ltx_title ltx_runin ltx_title_theorem" style="font-size:90%;">
<span class="ltx_tag ltx_tag_theorem"><span id="Thmremark1.1.1.1" class="ltx_text ltx_font_italic">Remark 1</span></span><span id="Thmremark1.2.2" class="ltx_text ltx_font_italic">.</span>
</h6>
<div id="Thmremark1.p1" class="ltx_para">
<p id="Thmremark1.p1.15" class="ltx_p"><span id="Thmremark1.p1.15.1" class="ltx_text" style="font-size:90%;">In order for </span><math id="Thmremark1.p1.1.m1.1" class="ltx_Math" alttext="p_{\bm{\theta}}" display="inline"><semantics id="Thmremark1.p1.1.m1.1a"><msub id="Thmremark1.p1.1.m1.1.1" xref="Thmremark1.p1.1.m1.1.1.cmml"><mi mathsize="90%" id="Thmremark1.p1.1.m1.1.1.2" xref="Thmremark1.p1.1.m1.1.1.2.cmml">p</mi><mi mathsize="90%" id="Thmremark1.p1.1.m1.1.1.3" xref="Thmremark1.p1.1.m1.1.1.3.cmml">𝜽</mi></msub><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.1.m1.1b"><apply id="Thmremark1.p1.1.m1.1.1.cmml" xref="Thmremark1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmremark1.p1.1.m1.1.1.1.cmml" xref="Thmremark1.p1.1.m1.1.1">subscript</csymbol><ci id="Thmremark1.p1.1.m1.1.1.2.cmml" xref="Thmremark1.p1.1.m1.1.1.2">𝑝</ci><ci id="Thmremark1.p1.1.m1.1.1.3.cmml" xref="Thmremark1.p1.1.m1.1.1.3">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.1.m1.1c">p_{\bm{\theta}}</annotation></semantics></math><span id="Thmremark1.p1.15.2" class="ltx_text" style="font-size:90%;"> to have deterministic latents and correspond to a valid probability distribution, we need to view the latents </span><math id="Thmremark1.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{z}_{t}" display="inline"><semantics id="Thmremark1.p1.2.m2.1a"><msub id="Thmremark1.p1.2.m2.1.1" xref="Thmremark1.p1.2.m2.1.1.cmml"><mi mathsize="90%" id="Thmremark1.p1.2.m2.1.1.2" xref="Thmremark1.p1.2.m2.1.1.2.cmml">𝐳</mi><mi mathsize="90%" id="Thmremark1.p1.2.m2.1.1.3" xref="Thmremark1.p1.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.2.m2.1b"><apply id="Thmremark1.p1.2.m2.1.1.cmml" xref="Thmremark1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="Thmremark1.p1.2.m2.1.1.1.cmml" xref="Thmremark1.p1.2.m2.1.1">subscript</csymbol><ci id="Thmremark1.p1.2.m2.1.1.2.cmml" xref="Thmremark1.p1.2.m2.1.1.2">𝐳</ci><ci 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id="Thmremark1.p1.3.m3.1.1.cmml" xref="Thmremark1.p1.3.m3.1.1"><times id="Thmremark1.p1.3.m3.1.1.2.cmml" xref="Thmremark1.p1.3.m3.1.1.2"></times><apply id="Thmremark1.p1.3.m3.1.1.3.cmml" xref="Thmremark1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="Thmremark1.p1.3.m3.1.1.3.1.cmml" xref="Thmremark1.p1.3.m3.1.1.3">subscript</csymbol><ci id="Thmremark1.p1.3.m3.1.1.3.2.cmml" xref="Thmremark1.p1.3.m3.1.1.3.2">𝐳</ci><ci id="Thmremark1.p1.3.m3.1.1.3.3.cmml" xref="Thmremark1.p1.3.m3.1.1.3.3">𝑡</ci></apply><apply id="Thmremark1.p1.3.m3.1.1.1.1.1.cmml" xref="Thmremark1.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="Thmremark1.p1.3.m3.1.1.1.1.1.1.cmml" xref="Thmremark1.p1.3.m3.1.1.1.1">subscript</csymbol><ci id="Thmremark1.p1.3.m3.1.1.1.1.1.2.cmml" xref="Thmremark1.p1.3.m3.1.1.1.1.1.2">𝑘</ci><apply id="Thmremark1.p1.3.m3.1.1.1.1.1.3.cmml" xref="Thmremark1.p1.3.m3.1.1.1.1.1.3"><ci id="Thmremark1.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="Thmremark1.p1.3.m3.1.1.1.1.1.3.1">:</ci><cn type="integer" id="Thmremark1.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="Thmremark1.p1.3.m3.1.1.1.1.1.3.2">1</cn><ci id="Thmremark1.p1.3.m3.1.1.1.1.1.3.3.cmml" xref="Thmremark1.p1.3.m3.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.3.m3.1c">\mathbf{z}_{t}(k_{1:t})</annotation></semantics></math><span id="Thmremark1.p1.15.4" class="ltx_text" style="font-size:90%;"> indexed by </span><math id="Thmremark1.p1.4.m4.1" class="ltx_Math" alttext="t\in[T]" display="inline"><semantics id="Thmremark1.p1.4.m4.1a"><mrow id="Thmremark1.p1.4.m4.1.2" xref="Thmremark1.p1.4.m4.1.2.cmml"><mi mathsize="90%" id="Thmremark1.p1.4.m4.1.2.2" xref="Thmremark1.p1.4.m4.1.2.2.cmml">t</mi><mo mathsize="90%" id="Thmremark1.p1.4.m4.1.2.1" xref="Thmremark1.p1.4.m4.1.2.1.cmml">∈</mo><mrow id="Thmremark1.p1.4.m4.1.2.3.2" xref="Thmremark1.p1.4.m4.1.2.3.1.cmml"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.4.m4.1.2.3.2.1" xref="Thmremark1.p1.4.m4.1.2.3.1.1.cmml">[</mo><mi mathsize="90%" id="Thmremark1.p1.4.m4.1.1" xref="Thmremark1.p1.4.m4.1.1.cmml">T</mi><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.4.m4.1.2.3.2.2" xref="Thmremark1.p1.4.m4.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.4.m4.1b"><apply id="Thmremark1.p1.4.m4.1.2.cmml" xref="Thmremark1.p1.4.m4.1.2"><in id="Thmremark1.p1.4.m4.1.2.1.cmml" xref="Thmremark1.p1.4.m4.1.2.1"></in><ci id="Thmremark1.p1.4.m4.1.2.2.cmml" xref="Thmremark1.p1.4.m4.1.2.2">𝑡</ci><apply id="Thmremark1.p1.4.m4.1.2.3.1.cmml" xref="Thmremark1.p1.4.m4.1.2.3.2"><csymbol cd="latexml" id="Thmremark1.p1.4.m4.1.2.3.1.1.cmml" xref="Thmremark1.p1.4.m4.1.2.3.2.1">delimited-[]</csymbol><ci id="Thmremark1.p1.4.m4.1.1.cmml" xref="Thmremark1.p1.4.m4.1.1">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.4.m4.1c">t\in[T]</annotation></semantics></math><span id="Thmremark1.p1.15.5" class="ltx_text" style="font-size:90%;"> and and the </span><em id="Thmremark1.p1.15.6" class="ltx_emph ltx_font_italic" style="font-size:90%;">history</em><span id="Thmremark1.p1.15.7" class="ltx_text" style="font-size:90%;"> of noise levels </span><math id="Thmremark1.p1.5.m5.4" class="ltx_Math" alttext="k_{1:t}\in\{0,1,\dots,K\}^{t}" display="inline"><semantics id="Thmremark1.p1.5.m5.4a"><mrow id="Thmremark1.p1.5.m5.4.5" xref="Thmremark1.p1.5.m5.4.5.cmml"><msub id="Thmremark1.p1.5.m5.4.5.2" xref="Thmremark1.p1.5.m5.4.5.2.cmml"><mi mathsize="90%" id="Thmremark1.p1.5.m5.4.5.2.2" xref="Thmremark1.p1.5.m5.4.5.2.2.cmml">k</mi><mrow id="Thmremark1.p1.5.m5.4.5.2.3" xref="Thmremark1.p1.5.m5.4.5.2.3.cmml"><mn mathsize="90%" id="Thmremark1.p1.5.m5.4.5.2.3.2" xref="Thmremark1.p1.5.m5.4.5.2.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="Thmremark1.p1.5.m5.4.5.2.3.1" xref="Thmremark1.p1.5.m5.4.5.2.3.1.cmml">:</mo><mi mathsize="90%" id="Thmremark1.p1.5.m5.4.5.2.3.3" xref="Thmremark1.p1.5.m5.4.5.2.3.3.cmml">t</mi></mrow></msub><mo mathsize="90%" id="Thmremark1.p1.5.m5.4.5.1" xref="Thmremark1.p1.5.m5.4.5.1.cmml">∈</mo><msup id="Thmremark1.p1.5.m5.4.5.3" xref="Thmremark1.p1.5.m5.4.5.3.cmml"><mrow id="Thmremark1.p1.5.m5.4.5.3.2.2" xref="Thmremark1.p1.5.m5.4.5.3.2.1.cmml"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.5.m5.4.5.3.2.2.1" xref="Thmremark1.p1.5.m5.4.5.3.2.1.cmml">{</mo><mn mathsize="90%" id="Thmremark1.p1.5.m5.1.1" xref="Thmremark1.p1.5.m5.1.1.cmml">0</mn><mo mathsize="90%" id="Thmremark1.p1.5.m5.4.5.3.2.2.2" xref="Thmremark1.p1.5.m5.4.5.3.2.1.cmml">,</mo><mn mathsize="90%" id="Thmremark1.p1.5.m5.2.2" xref="Thmremark1.p1.5.m5.2.2.cmml">1</mn><mo mathsize="90%" id="Thmremark1.p1.5.m5.4.5.3.2.2.3" xref="Thmremark1.p1.5.m5.4.5.3.2.1.cmml">,</mo><mi mathsize="90%" mathvariant="normal" id="Thmremark1.p1.5.m5.3.3" xref="Thmremark1.p1.5.m5.3.3.cmml">…</mi><mo mathsize="90%" id="Thmremark1.p1.5.m5.4.5.3.2.2.4" xref="Thmremark1.p1.5.m5.4.5.3.2.1.cmml">,</mo><mi mathsize="90%" id="Thmremark1.p1.5.m5.4.4" xref="Thmremark1.p1.5.m5.4.4.cmml">K</mi><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.5.m5.4.5.3.2.2.5" xref="Thmremark1.p1.5.m5.4.5.3.2.1.cmml">}</mo></mrow><mi mathsize="90%" id="Thmremark1.p1.5.m5.4.5.3.3" xref="Thmremark1.p1.5.m5.4.5.3.3.cmml">t</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.5.m5.4b"><apply id="Thmremark1.p1.5.m5.4.5.cmml" xref="Thmremark1.p1.5.m5.4.5"><in id="Thmremark1.p1.5.m5.4.5.1.cmml" xref="Thmremark1.p1.5.m5.4.5.1"></in><apply id="Thmremark1.p1.5.m5.4.5.2.cmml" xref="Thmremark1.p1.5.m5.4.5.2"><csymbol cd="ambiguous" id="Thmremark1.p1.5.m5.4.5.2.1.cmml" xref="Thmremark1.p1.5.m5.4.5.2">subscript</csymbol><ci id="Thmremark1.p1.5.m5.4.5.2.2.cmml" xref="Thmremark1.p1.5.m5.4.5.2.2">𝑘</ci><apply id="Thmremark1.p1.5.m5.4.5.2.3.cmml" xref="Thmremark1.p1.5.m5.4.5.2.3"><ci id="Thmremark1.p1.5.m5.4.5.2.3.1.cmml" xref="Thmremark1.p1.5.m5.4.5.2.3.1">:</ci><cn type="integer" id="Thmremark1.p1.5.m5.4.5.2.3.2.cmml" xref="Thmremark1.p1.5.m5.4.5.2.3.2">1</cn><ci id="Thmremark1.p1.5.m5.4.5.2.3.3.cmml" xref="Thmremark1.p1.5.m5.4.5.2.3.3">𝑡</ci></apply></apply><apply id="Thmremark1.p1.5.m5.4.5.3.cmml" xref="Thmremark1.p1.5.m5.4.5.3"><csymbol cd="ambiguous" id="Thmremark1.p1.5.m5.4.5.3.1.cmml" xref="Thmremark1.p1.5.m5.4.5.3">superscript</csymbol><set id="Thmremark1.p1.5.m5.4.5.3.2.1.cmml" xref="Thmremark1.p1.5.m5.4.5.3.2.2"><cn type="integer" id="Thmremark1.p1.5.m5.1.1.cmml" xref="Thmremark1.p1.5.m5.1.1">0</cn><cn type="integer" id="Thmremark1.p1.5.m5.2.2.cmml" xref="Thmremark1.p1.5.m5.2.2">1</cn><ci id="Thmremark1.p1.5.m5.3.3.cmml" xref="Thmremark1.p1.5.m5.3.3">…</ci><ci id="Thmremark1.p1.5.m5.4.4.cmml" xref="Thmremark1.p1.5.m5.4.4">𝐾</ci></set><ci id="Thmremark1.p1.5.m5.4.5.3.3.cmml" xref="Thmremark1.p1.5.m5.4.5.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.5.m5.4c">k_{1:t}\in\{0,1,\dots,K\}^{t}</annotation></semantics></math><span id="Thmremark1.p1.15.8" class="ltx_text" style="font-size:90%;">.. In this case, simply setting </span><math id="Thmremark1.p1.6.m6.1" class="ltx_math_unparsed" alttext="\mathbf{z}_{t}(k_{1:t})=(k_{1:t},(\mathbf{x}_{s}^{k_{s}})_{1\leq s\leq t}" display="inline"><semantics id="Thmremark1.p1.6.m6.1a"><mrow id="Thmremark1.p1.6.m6.1b"><msub id="Thmremark1.p1.6.m6.1.1"><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.1.2">𝐳</mi><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.1.3">t</mi></msub><mrow id="Thmremark1.p1.6.m6.1.2"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.6.m6.1.2.1">(</mo><msub id="Thmremark1.p1.6.m6.1.2.2"><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.2.2.2">k</mi><mrow id="Thmremark1.p1.6.m6.1.2.2.3"><mn mathsize="90%" id="Thmremark1.p1.6.m6.1.2.2.3.2">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="Thmremark1.p1.6.m6.1.2.2.3.1">:</mo><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.2.2.3.3">t</mi></mrow></msub><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.6.m6.1.2.3">)</mo></mrow><mo mathsize="90%" id="Thmremark1.p1.6.m6.1.3">=</mo><mrow id="Thmremark1.p1.6.m6.1.4"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.6.m6.1.4.1">(</mo><msub id="Thmremark1.p1.6.m6.1.4.2"><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.4.2.2">k</mi><mrow id="Thmremark1.p1.6.m6.1.4.2.3"><mn mathsize="90%" id="Thmremark1.p1.6.m6.1.4.2.3.2">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="Thmremark1.p1.6.m6.1.4.2.3.1">:</mo><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.4.2.3.3">t</mi></mrow></msub><mo mathsize="90%" id="Thmremark1.p1.6.m6.1.4.3">,</mo><msub id="Thmremark1.p1.6.m6.1.4.4"><mrow id="Thmremark1.p1.6.m6.1.4.4.2"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.6.m6.1.4.4.2.1">(</mo><msubsup id="Thmremark1.p1.6.m6.1.4.4.2.2"><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.4.4.2.2.2.2">𝐱</mi><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.4.4.2.2.2.3">s</mi><msub id="Thmremark1.p1.6.m6.1.4.4.2.2.3"><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.4.4.2.2.3.2">k</mi><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.4.4.2.2.3.3">s</mi></msub></msubsup><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.6.m6.1.4.4.2.3">)</mo></mrow><mrow id="Thmremark1.p1.6.m6.1.4.4.3"><mn mathsize="90%" id="Thmremark1.p1.6.m6.1.4.4.3.2">1</mn><mo mathsize="90%" id="Thmremark1.p1.6.m6.1.4.4.3.3">≤</mo><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.4.4.3.4">s</mi><mo mathsize="90%" id="Thmremark1.p1.6.m6.1.4.4.3.5">≤</mo><mi mathsize="90%" id="Thmremark1.p1.6.m6.1.4.4.3.6">t</mi></mrow></msub></mrow></mrow><annotation encoding="application/x-tex" id="Thmremark1.p1.6.m6.1c">\mathbf{z}_{t}(k_{1:t})=(k_{1:t},(\mathbf{x}_{s}^{k_{s}})_{1\leq s\leq t}</annotation></semantics></math><span id="Thmremark1.p1.15.9" class="ltx_text" style="font-size:90%;"> tautologically produces deterministic latents. The reason for indexing </span><math id="Thmremark1.p1.7.m7.1" class="ltx_Math" alttext="\mathbf{z}_{t}(k_{1:t})" display="inline"><semantics id="Thmremark1.p1.7.m7.1a"><mrow id="Thmremark1.p1.7.m7.1.1" xref="Thmremark1.p1.7.m7.1.1.cmml"><msub id="Thmremark1.p1.7.m7.1.1.3" xref="Thmremark1.p1.7.m7.1.1.3.cmml"><mi mathsize="90%" id="Thmremark1.p1.7.m7.1.1.3.2" xref="Thmremark1.p1.7.m7.1.1.3.2.cmml">𝐳</mi><mi mathsize="90%" id="Thmremark1.p1.7.m7.1.1.3.3" xref="Thmremark1.p1.7.m7.1.1.3.3.cmml">t</mi></msub><mo lspace="0em" rspace="0em" id="Thmremark1.p1.7.m7.1.1.2" xref="Thmremark1.p1.7.m7.1.1.2.cmml">​</mo><mrow id="Thmremark1.p1.7.m7.1.1.1.1" xref="Thmremark1.p1.7.m7.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.7.m7.1.1.1.1.2" xref="Thmremark1.p1.7.m7.1.1.1.1.1.cmml">(</mo><msub id="Thmremark1.p1.7.m7.1.1.1.1.1" xref="Thmremark1.p1.7.m7.1.1.1.1.1.cmml"><mi mathsize="90%" id="Thmremark1.p1.7.m7.1.1.1.1.1.2" xref="Thmremark1.p1.7.m7.1.1.1.1.1.2.cmml">k</mi><mrow id="Thmremark1.p1.7.m7.1.1.1.1.1.3" xref="Thmremark1.p1.7.m7.1.1.1.1.1.3.cmml"><mn mathsize="90%" id="Thmremark1.p1.7.m7.1.1.1.1.1.3.2" xref="Thmremark1.p1.7.m7.1.1.1.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="Thmremark1.p1.7.m7.1.1.1.1.1.3.1" xref="Thmremark1.p1.7.m7.1.1.1.1.1.3.1.cmml">:</mo><mi mathsize="90%" id="Thmremark1.p1.7.m7.1.1.1.1.1.3.3" xref="Thmremark1.p1.7.m7.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.7.m7.1.1.1.1.3" xref="Thmremark1.p1.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.7.m7.1b"><apply id="Thmremark1.p1.7.m7.1.1.cmml" xref="Thmremark1.p1.7.m7.1.1"><times id="Thmremark1.p1.7.m7.1.1.2.cmml" xref="Thmremark1.p1.7.m7.1.1.2"></times><apply id="Thmremark1.p1.7.m7.1.1.3.cmml" xref="Thmremark1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="Thmremark1.p1.7.m7.1.1.3.1.cmml" xref="Thmremark1.p1.7.m7.1.1.3">subscript</csymbol><ci id="Thmremark1.p1.7.m7.1.1.3.2.cmml" xref="Thmremark1.p1.7.m7.1.1.3.2">𝐳</ci><ci id="Thmremark1.p1.7.m7.1.1.3.3.cmml" xref="Thmremark1.p1.7.m7.1.1.3.3">𝑡</ci></apply><apply id="Thmremark1.p1.7.m7.1.1.1.1.1.cmml" xref="Thmremark1.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="Thmremark1.p1.7.m7.1.1.1.1.1.1.cmml" xref="Thmremark1.p1.7.m7.1.1.1.1">subscript</csymbol><ci id="Thmremark1.p1.7.m7.1.1.1.1.1.2.cmml" xref="Thmremark1.p1.7.m7.1.1.1.1.1.2">𝑘</ci><apply id="Thmremark1.p1.7.m7.1.1.1.1.1.3.cmml" xref="Thmremark1.p1.7.m7.1.1.1.1.1.3"><ci id="Thmremark1.p1.7.m7.1.1.1.1.1.3.1.cmml" xref="Thmremark1.p1.7.m7.1.1.1.1.1.3.1">:</ci><cn type="integer" id="Thmremark1.p1.7.m7.1.1.1.1.1.3.2.cmml" xref="Thmremark1.p1.7.m7.1.1.1.1.1.3.2">1</cn><ci id="Thmremark1.p1.7.m7.1.1.1.1.1.3.3.cmml" xref="Thmremark1.p1.7.m7.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.7.m7.1c">\mathbf{z}_{t}(k_{1:t})</annotation></semantics></math><span id="Thmremark1.p1.15.10" class="ltx_text" style="font-size:90%;"> with </span><math id="Thmremark1.p1.8.m8.1" class="ltx_Math" alttext="k_{1:t}" display="inline"><semantics id="Thmremark1.p1.8.m8.1a"><msub id="Thmremark1.p1.8.m8.1.1" xref="Thmremark1.p1.8.m8.1.1.cmml"><mi mathsize="90%" id="Thmremark1.p1.8.m8.1.1.2" xref="Thmremark1.p1.8.m8.1.1.2.cmml">k</mi><mrow id="Thmremark1.p1.8.m8.1.1.3" xref="Thmremark1.p1.8.m8.1.1.3.cmml"><mn mathsize="90%" id="Thmremark1.p1.8.m8.1.1.3.2" xref="Thmremark1.p1.8.m8.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="Thmremark1.p1.8.m8.1.1.3.1" xref="Thmremark1.p1.8.m8.1.1.3.1.cmml">:</mo><mi mathsize="90%" id="Thmremark1.p1.8.m8.1.1.3.3" xref="Thmremark1.p1.8.m8.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.8.m8.1b"><apply id="Thmremark1.p1.8.m8.1.1.cmml" xref="Thmremark1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="Thmremark1.p1.8.m8.1.1.1.cmml" xref="Thmremark1.p1.8.m8.1.1">subscript</csymbol><ci id="Thmremark1.p1.8.m8.1.1.2.cmml" xref="Thmremark1.p1.8.m8.1.1.2">𝑘</ci><apply id="Thmremark1.p1.8.m8.1.1.3.cmml" xref="Thmremark1.p1.8.m8.1.1.3"><ci id="Thmremark1.p1.8.m8.1.1.3.1.cmml" xref="Thmremark1.p1.8.m8.1.1.3.1">:</ci><cn type="integer" id="Thmremark1.p1.8.m8.1.1.3.2.cmml" xref="Thmremark1.p1.8.m8.1.1.3.2">1</cn><ci id="Thmremark1.p1.8.m8.1.1.3.3.cmml" xref="Thmremark1.p1.8.m8.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.8.m8.1c">k_{1:t}</annotation></semantics></math><span id="Thmremark1.p1.15.11" class="ltx_text" style="font-size:90%;"> then arises because, otherwise, </span><math id="Thmremark1.p1.9.m9.1" class="ltx_math_unparsed" alttext="p_{\bm{\theta}}(\mathbf{z}_{t}\mid((\mathbf{x}_{s}^{k_{s}})_{1\leq s\leq t},(\mathbf{x}_{s}^{k_{s}^{\prime}})_{1\leq s\leq t})" display="inline"><semantics id="Thmremark1.p1.9.m9.1a"><mrow id="Thmremark1.p1.9.m9.1b"><msub id="Thmremark1.p1.9.m9.1.1"><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.1.2">p</mi><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.1.3">𝜽</mi></msub><mrow id="Thmremark1.p1.9.m9.1.2"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.9.m9.1.2.1">(</mo><msub id="Thmremark1.p1.9.m9.1.2.2"><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.2.2">𝐳</mi><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.2.3">t</mi></msub><mo lspace="0em" mathsize="90%" rspace="0.167em" id="Thmremark1.p1.9.m9.1.2.3">∣</mo><mrow id="Thmremark1.p1.9.m9.1.2.4"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.9.m9.1.2.4.1">(</mo><msub id="Thmremark1.p1.9.m9.1.2.4.2"><mrow id="Thmremark1.p1.9.m9.1.2.4.2.2"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.2.1">(</mo><msubsup id="Thmremark1.p1.9.m9.1.2.4.2.2.2"><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.2.2.2.2">𝐱</mi><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.2.2.2.3">s</mi><msub id="Thmremark1.p1.9.m9.1.2.4.2.2.2.3"><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.2.2.3.2">k</mi><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.2.2.3.3">s</mi></msub></msubsup><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.2.3">)</mo></mrow><mrow id="Thmremark1.p1.9.m9.1.2.4.2.3"><mn mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.3.2">1</mn><mo mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.3.3">≤</mo><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.3.4">s</mi><mo mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.3.5">≤</mo><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.2.3.6">t</mi></mrow></msub><mo mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.3">,</mo><msub id="Thmremark1.p1.9.m9.1.2.4.4"><mrow id="Thmremark1.p1.9.m9.1.2.4.4.2"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.2.1">(</mo><msubsup id="Thmremark1.p1.9.m9.1.2.4.4.2.2"><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.2.2.2.2">𝐱</mi><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.2.2.2.3">s</mi><msubsup id="Thmremark1.p1.9.m9.1.2.4.4.2.2.3"><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.2.2.3.2.2">k</mi><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.2.2.3.2.3">s</mi><mo mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.2.2.3.3">′</mo></msubsup></msubsup><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.2.3">)</mo></mrow><mrow id="Thmremark1.p1.9.m9.1.2.4.4.3"><mn mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.3.2">1</mn><mo mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.3.3">≤</mo><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.3.4">s</mi><mo mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.3.5">≤</mo><mi mathsize="90%" id="Thmremark1.p1.9.m9.1.2.4.4.3.6">t</mi></mrow></msub><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.9.m9.1.2.4.5">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="Thmremark1.p1.9.m9.1c">p_{\bm{\theta}}(\mathbf{z}_{t}\mid((\mathbf{x}_{s}^{k_{s}})_{1\leq s\leq t},(\mathbf{x}_{s}^{k_{s}^{\prime}})_{1\leq s\leq t})</annotation></semantics></math><span id="Thmremark1.p1.15.12" class="ltx_text" style="font-size:90%;"> would be ill-defined unless </span><math id="Thmremark1.p1.10.m10.1" class="ltx_Math" alttext="k_{s}=k_{s}^{\prime}" display="inline"><semantics id="Thmremark1.p1.10.m10.1a"><mrow id="Thmremark1.p1.10.m10.1.1" xref="Thmremark1.p1.10.m10.1.1.cmml"><msub id="Thmremark1.p1.10.m10.1.1.2" xref="Thmremark1.p1.10.m10.1.1.2.cmml"><mi mathsize="90%" id="Thmremark1.p1.10.m10.1.1.2.2" xref="Thmremark1.p1.10.m10.1.1.2.2.cmml">k</mi><mi mathsize="90%" id="Thmremark1.p1.10.m10.1.1.2.3" xref="Thmremark1.p1.10.m10.1.1.2.3.cmml">s</mi></msub><mo mathsize="90%" id="Thmremark1.p1.10.m10.1.1.1" xref="Thmremark1.p1.10.m10.1.1.1.cmml">=</mo><msubsup id="Thmremark1.p1.10.m10.1.1.3" xref="Thmremark1.p1.10.m10.1.1.3.cmml"><mi mathsize="90%" id="Thmremark1.p1.10.m10.1.1.3.2.2" xref="Thmremark1.p1.10.m10.1.1.3.2.2.cmml">k</mi><mi mathsize="90%" id="Thmremark1.p1.10.m10.1.1.3.2.3" xref="Thmremark1.p1.10.m10.1.1.3.2.3.cmml">s</mi><mo mathsize="90%" id="Thmremark1.p1.10.m10.1.1.3.3" xref="Thmremark1.p1.10.m10.1.1.3.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.10.m10.1b"><apply id="Thmremark1.p1.10.m10.1.1.cmml" xref="Thmremark1.p1.10.m10.1.1"><eq id="Thmremark1.p1.10.m10.1.1.1.cmml" xref="Thmremark1.p1.10.m10.1.1.1"></eq><apply id="Thmremark1.p1.10.m10.1.1.2.cmml" xref="Thmremark1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="Thmremark1.p1.10.m10.1.1.2.1.cmml" xref="Thmremark1.p1.10.m10.1.1.2">subscript</csymbol><ci id="Thmremark1.p1.10.m10.1.1.2.2.cmml" xref="Thmremark1.p1.10.m10.1.1.2.2">𝑘</ci><ci id="Thmremark1.p1.10.m10.1.1.2.3.cmml" xref="Thmremark1.p1.10.m10.1.1.2.3">𝑠</ci></apply><apply id="Thmremark1.p1.10.m10.1.1.3.cmml" xref="Thmremark1.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="Thmremark1.p1.10.m10.1.1.3.1.cmml" xref="Thmremark1.p1.10.m10.1.1.3">superscript</csymbol><apply id="Thmremark1.p1.10.m10.1.1.3.2.cmml" xref="Thmremark1.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="Thmremark1.p1.10.m10.1.1.3.2.1.cmml" xref="Thmremark1.p1.10.m10.1.1.3">subscript</csymbol><ci id="Thmremark1.p1.10.m10.1.1.3.2.2.cmml" xref="Thmremark1.p1.10.m10.1.1.3.2.2">𝑘</ci><ci id="Thmremark1.p1.10.m10.1.1.3.2.3.cmml" xref="Thmremark1.p1.10.m10.1.1.3.2.3">𝑠</ci></apply><ci id="Thmremark1.p1.10.m10.1.1.3.3.cmml" xref="Thmremark1.p1.10.m10.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.10.m10.1c">k_{s}=k_{s}^{\prime}</annotation></semantics></math><span id="Thmremark1.p1.15.13" class="ltx_text" style="font-size:90%;"> for all </span><math id="Thmremark1.p1.11.m11.1" class="ltx_Math" alttext="1\leq s\leq t" display="inline"><semantics id="Thmremark1.p1.11.m11.1a"><mrow id="Thmremark1.p1.11.m11.1.1" xref="Thmremark1.p1.11.m11.1.1.cmml"><mn mathsize="90%" id="Thmremark1.p1.11.m11.1.1.2" xref="Thmremark1.p1.11.m11.1.1.2.cmml">1</mn><mo mathsize="90%" id="Thmremark1.p1.11.m11.1.1.3" xref="Thmremark1.p1.11.m11.1.1.3.cmml">≤</mo><mi mathsize="90%" id="Thmremark1.p1.11.m11.1.1.4" xref="Thmremark1.p1.11.m11.1.1.4.cmml">s</mi><mo mathsize="90%" id="Thmremark1.p1.11.m11.1.1.5" xref="Thmremark1.p1.11.m11.1.1.5.cmml">≤</mo><mi mathsize="90%" id="Thmremark1.p1.11.m11.1.1.6" xref="Thmremark1.p1.11.m11.1.1.6.cmml">t</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.11.m11.1b"><apply id="Thmremark1.p1.11.m11.1.1.cmml" xref="Thmremark1.p1.11.m11.1.1"><and id="Thmremark1.p1.11.m11.1.1a.cmml" xref="Thmremark1.p1.11.m11.1.1"></and><apply id="Thmremark1.p1.11.m11.1.1b.cmml" xref="Thmremark1.p1.11.m11.1.1"><leq id="Thmremark1.p1.11.m11.1.1.3.cmml" xref="Thmremark1.p1.11.m11.1.1.3"></leq><cn type="integer" id="Thmremark1.p1.11.m11.1.1.2.cmml" xref="Thmremark1.p1.11.m11.1.1.2">1</cn><ci id="Thmremark1.p1.11.m11.1.1.4.cmml" xref="Thmremark1.p1.11.m11.1.1.4">𝑠</ci></apply><apply id="Thmremark1.p1.11.m11.1.1c.cmml" xref="Thmremark1.p1.11.m11.1.1"><leq id="Thmremark1.p1.11.m11.1.1.5.cmml" xref="Thmremark1.p1.11.m11.1.1.5"></leq><share href="#Thmremark1.p1.11.m11.1.1.4.cmml" id="Thmremark1.p1.11.m11.1.1d.cmml" xref="Thmremark1.p1.11.m11.1.1"></share><ci id="Thmremark1.p1.11.m11.1.1.6.cmml" xref="Thmremark1.p1.11.m11.1.1.6">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.11.m11.1c">1\leq s\leq t</annotation></semantics></math><span id="Thmremark1.p1.15.14" class="ltx_text" style="font-size:90%;">, and thus, </span><math id="Thmremark1.p1.12.m12.1" class="ltx_Math" alttext="p_{\bm{\theta}}" display="inline"><semantics id="Thmremark1.p1.12.m12.1a"><msub id="Thmremark1.p1.12.m12.1.1" xref="Thmremark1.p1.12.m12.1.1.cmml"><mi mathsize="90%" id="Thmremark1.p1.12.m12.1.1.2" xref="Thmremark1.p1.12.m12.1.1.2.cmml">p</mi><mi mathsize="90%" id="Thmremark1.p1.12.m12.1.1.3" xref="Thmremark1.p1.12.m12.1.1.3.cmml">𝜽</mi></msub><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.12.m12.1b"><apply id="Thmremark1.p1.12.m12.1.1.cmml" xref="Thmremark1.p1.12.m12.1.1"><csymbol cd="ambiguous" id="Thmremark1.p1.12.m12.1.1.1.cmml" xref="Thmremark1.p1.12.m12.1.1">subscript</csymbol><ci id="Thmremark1.p1.12.m12.1.1.2.cmml" xref="Thmremark1.p1.12.m12.1.1.2">𝑝</ci><ci id="Thmremark1.p1.12.m12.1.1.3.cmml" xref="Thmremark1.p1.12.m12.1.1.3">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.12.m12.1c">p_{\bm{\theta}}</annotation></semantics></math><span id="Thmremark1.p1.15.15" class="ltx_text" style="font-size:90%;"> would not correspond to a joint probability measure. The exposition and theorem that follows allows </span><math id="Thmremark1.p1.13.m13.1" class="ltx_Math" alttext="\mathbf{z}_{t}(k_{1:t})" display="inline"><semantics id="Thmremark1.p1.13.m13.1a"><mrow id="Thmremark1.p1.13.m13.1.1" xref="Thmremark1.p1.13.m13.1.1.cmml"><msub id="Thmremark1.p1.13.m13.1.1.3" xref="Thmremark1.p1.13.m13.1.1.3.cmml"><mi mathsize="90%" id="Thmremark1.p1.13.m13.1.1.3.2" xref="Thmremark1.p1.13.m13.1.1.3.2.cmml">𝐳</mi><mi mathsize="90%" id="Thmremark1.p1.13.m13.1.1.3.3" xref="Thmremark1.p1.13.m13.1.1.3.3.cmml">t</mi></msub><mo lspace="0em" rspace="0em" id="Thmremark1.p1.13.m13.1.1.2" xref="Thmremark1.p1.13.m13.1.1.2.cmml">​</mo><mrow id="Thmremark1.p1.13.m13.1.1.1.1" xref="Thmremark1.p1.13.m13.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.13.m13.1.1.1.1.2" xref="Thmremark1.p1.13.m13.1.1.1.1.1.cmml">(</mo><msub id="Thmremark1.p1.13.m13.1.1.1.1.1" xref="Thmremark1.p1.13.m13.1.1.1.1.1.cmml"><mi mathsize="90%" id="Thmremark1.p1.13.m13.1.1.1.1.1.2" xref="Thmremark1.p1.13.m13.1.1.1.1.1.2.cmml">k</mi><mrow id="Thmremark1.p1.13.m13.1.1.1.1.1.3" xref="Thmremark1.p1.13.m13.1.1.1.1.1.3.cmml"><mn mathsize="90%" id="Thmremark1.p1.13.m13.1.1.1.1.1.3.2" xref="Thmremark1.p1.13.m13.1.1.1.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="Thmremark1.p1.13.m13.1.1.1.1.1.3.1" xref="Thmremark1.p1.13.m13.1.1.1.1.1.3.1.cmml">:</mo><mi mathsize="90%" id="Thmremark1.p1.13.m13.1.1.1.1.1.3.3" xref="Thmremark1.p1.13.m13.1.1.1.1.1.3.3.cmml">t</mi></mrow></msub><mo maxsize="90%" minsize="90%" id="Thmremark1.p1.13.m13.1.1.1.1.3" xref="Thmremark1.p1.13.m13.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.13.m13.1b"><apply id="Thmremark1.p1.13.m13.1.1.cmml" xref="Thmremark1.p1.13.m13.1.1"><times id="Thmremark1.p1.13.m13.1.1.2.cmml" xref="Thmremark1.p1.13.m13.1.1.2"></times><apply id="Thmremark1.p1.13.m13.1.1.3.cmml" xref="Thmremark1.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="Thmremark1.p1.13.m13.1.1.3.1.cmml" xref="Thmremark1.p1.13.m13.1.1.3">subscript</csymbol><ci id="Thmremark1.p1.13.m13.1.1.3.2.cmml" xref="Thmremark1.p1.13.m13.1.1.3.2">𝐳</ci><ci id="Thmremark1.p1.13.m13.1.1.3.3.cmml" xref="Thmremark1.p1.13.m13.1.1.3.3">𝑡</ci></apply><apply id="Thmremark1.p1.13.m13.1.1.1.1.1.cmml" xref="Thmremark1.p1.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="Thmremark1.p1.13.m13.1.1.1.1.1.1.cmml" xref="Thmremark1.p1.13.m13.1.1.1.1">subscript</csymbol><ci id="Thmremark1.p1.13.m13.1.1.1.1.1.2.cmml" xref="Thmremark1.p1.13.m13.1.1.1.1.1.2">𝑘</ci><apply id="Thmremark1.p1.13.m13.1.1.1.1.1.3.cmml" xref="Thmremark1.p1.13.m13.1.1.1.1.1.3"><ci id="Thmremark1.p1.13.m13.1.1.1.1.1.3.1.cmml" xref="Thmremark1.p1.13.m13.1.1.1.1.1.3.1">:</ci><cn type="integer" id="Thmremark1.p1.13.m13.1.1.1.1.1.3.2.cmml" xref="Thmremark1.p1.13.m13.1.1.1.1.1.3.2">1</cn><ci id="Thmremark1.p1.13.m13.1.1.1.1.1.3.3.cmml" xref="Thmremark1.p1.13.m13.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.13.m13.1c">\mathbf{z}_{t}(k_{1:t})</annotation></semantics></math><span id="Thmremark1.p1.15.16" class="ltx_text" style="font-size:90%;"> to be indexed on past noise levels </span><math id="Thmremark1.p1.14.m14.1" class="ltx_Math" alttext="k_{1:t}" display="inline"><semantics id="Thmremark1.p1.14.m14.1a"><msub id="Thmremark1.p1.14.m14.1.1" xref="Thmremark1.p1.14.m14.1.1.cmml"><mi mathsize="90%" id="Thmremark1.p1.14.m14.1.1.2" xref="Thmremark1.p1.14.m14.1.1.2.cmml">k</mi><mrow id="Thmremark1.p1.14.m14.1.1.3" xref="Thmremark1.p1.14.m14.1.1.3.cmml"><mn mathsize="90%" id="Thmremark1.p1.14.m14.1.1.3.2" xref="Thmremark1.p1.14.m14.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="Thmremark1.p1.14.m14.1.1.3.1" xref="Thmremark1.p1.14.m14.1.1.3.1.cmml">:</mo><mi mathsize="90%" id="Thmremark1.p1.14.m14.1.1.3.3" xref="Thmremark1.p1.14.m14.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.14.m14.1b"><apply id="Thmremark1.p1.14.m14.1.1.cmml" xref="Thmremark1.p1.14.m14.1.1"><csymbol cd="ambiguous" id="Thmremark1.p1.14.m14.1.1.1.cmml" xref="Thmremark1.p1.14.m14.1.1">subscript</csymbol><ci id="Thmremark1.p1.14.m14.1.1.2.cmml" xref="Thmremark1.p1.14.m14.1.1.2">𝑘</ci><apply id="Thmremark1.p1.14.m14.1.1.3.cmml" xref="Thmremark1.p1.14.m14.1.1.3"><ci id="Thmremark1.p1.14.m14.1.1.3.1.cmml" xref="Thmremark1.p1.14.m14.1.1.3.1">:</ci><cn type="integer" id="Thmremark1.p1.14.m14.1.1.3.2.cmml" xref="Thmremark1.p1.14.m14.1.1.3.2">1</cn><ci id="Thmremark1.p1.14.m14.1.1.3.3.cmml" xref="Thmremark1.p1.14.m14.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.14.m14.1c">k_{1:t}</annotation></semantics></math><span id="Thmremark1.p1.15.17" class="ltx_text" style="font-size:90%;">, but suppresses dependence on </span><math id="Thmremark1.p1.15.m15.1" class="ltx_Math" alttext="k_{1:t}" display="inline"><semantics id="Thmremark1.p1.15.m15.1a"><msub id="Thmremark1.p1.15.m15.1.1" xref="Thmremark1.p1.15.m15.1.1.cmml"><mi mathsize="90%" id="Thmremark1.p1.15.m15.1.1.2" xref="Thmremark1.p1.15.m15.1.1.2.cmml">k</mi><mrow id="Thmremark1.p1.15.m15.1.1.3" xref="Thmremark1.p1.15.m15.1.1.3.cmml"><mn mathsize="90%" id="Thmremark1.p1.15.m15.1.1.3.2" xref="Thmremark1.p1.15.m15.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="Thmremark1.p1.15.m15.1.1.3.1" xref="Thmremark1.p1.15.m15.1.1.3.1.cmml">:</mo><mi mathsize="90%" id="Thmremark1.p1.15.m15.1.1.3.3" xref="Thmremark1.p1.15.m15.1.1.3.3.cmml">t</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="Thmremark1.p1.15.m15.1b"><apply id="Thmremark1.p1.15.m15.1.1.cmml" xref="Thmremark1.p1.15.m15.1.1"><csymbol cd="ambiguous" id="Thmremark1.p1.15.m15.1.1.1.cmml" xref="Thmremark1.p1.15.m15.1.1">subscript</csymbol><ci id="Thmremark1.p1.15.m15.1.1.2.cmml" xref="Thmremark1.p1.15.m15.1.1.2">𝑘</ci><apply id="Thmremark1.p1.15.m15.1.1.3.cmml" xref="Thmremark1.p1.15.m15.1.1.3"><ci id="Thmremark1.p1.15.m15.1.1.3.1.cmml" xref="Thmremark1.p1.15.m15.1.1.3.1">:</ci><cn type="integer" id="Thmremark1.p1.15.m15.1.1.3.2.cmml" xref="Thmremark1.p1.15.m15.1.1.3.2">1</cn><ci id="Thmremark1.p1.15.m15.1.1.3.3.cmml" xref="Thmremark1.p1.15.m15.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmremark1.p1.15.m15.1c">k_{1:t}</annotation></semantics></math><span id="Thmremark1.p1.15.18" class="ltx_text" style="font-size:90%;"> to avoid notational confusion.</span></p>
</div>
</div>
<section id="A1.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">A.1 </span>Main Results</h3>

<div id="A1.SS1.p1" class="ltx_para">
<p id="A1.SS1.p1.6" class="ltx_p"><span id="A1.SS1.p1.6.1" class="ltx_text" style="font-size:90%;">We can now state our main theorem, which provides an evidence lower bound (ELBO) on the expected log likelihood of partially-noised sequences </span><math id="A1.SS1.p1.1.m1.1" class="ltx_math_unparsed" alttext="(\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})" display="inline"><semantics id="A1.SS1.p1.1.m1.1a"><mrow id="A1.SS1.p1.1.m1.1b"><msub id="A1.SS1.p1.1.m1.1.1"><mrow id="A1.SS1.p1.1.m1.1.1.2"><mo maxsize="90%" minsize="90%" id="A1.SS1.p1.1.m1.1.1.2.1">(</mo><msubsup id="A1.SS1.p1.1.m1.1.1.2.2"><mi mathsize="90%" id="A1.SS1.p1.1.m1.1.1.2.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.SS1.p1.1.m1.1.1.2.2.2.3">t</mi><msub id="A1.SS1.p1.1.m1.1.1.2.2.3"><mi mathsize="90%" id="A1.SS1.p1.1.m1.1.1.2.2.3.2">k</mi><mi mathsize="90%" id="A1.SS1.p1.1.m1.1.1.2.2.3.3">t</mi></msub></msubsup><mo maxsize="90%" minsize="90%" id="A1.SS1.p1.1.m1.1.1.2.3">)</mo></mrow><mrow id="A1.SS1.p1.1.m1.1.1.3"><mn mathsize="90%" id="A1.SS1.p1.1.m1.1.1.3.2">1</mn><mo mathsize="90%" id="A1.SS1.p1.1.m1.1.1.3.3">≤</mo><mi mathsize="90%" id="A1.SS1.p1.1.m1.1.1.3.4">t</mi><mo mathsize="90%" id="A1.SS1.p1.1.m1.1.1.3.5">≤</mo><mi mathsize="90%" id="A1.SS1.p1.1.m1.1.1.3.6">T</mi></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.SS1.p1.1.m1.1.2">)</mo></mrow><annotation encoding="application/x-tex" id="A1.SS1.p1.1.m1.1c">(\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})</annotation></semantics></math><span id="A1.SS1.p1.6.2" class="ltx_text" style="font-size:90%;">, under uniform levels </span><math id="A1.SS1.p1.2.m2.1" class="ltx_Math" alttext="k_{t}" display="inline"><semantics id="A1.SS1.p1.2.m2.1a"><msub id="A1.SS1.p1.2.m2.1.1" xref="A1.SS1.p1.2.m2.1.1.cmml"><mi mathsize="90%" id="A1.SS1.p1.2.m2.1.1.2" xref="A1.SS1.p1.2.m2.1.1.2.cmml">k</mi><mi mathsize="90%" id="A1.SS1.p1.2.m2.1.1.3" xref="A1.SS1.p1.2.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.2.m2.1b"><apply id="A1.SS1.p1.2.m2.1.1.cmml" xref="A1.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.2.m2.1.1.1.cmml" xref="A1.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="A1.SS1.p1.2.m2.1.1.2.cmml" xref="A1.SS1.p1.2.m2.1.1.2">𝑘</ci><ci id="A1.SS1.p1.2.m2.1.1.3.cmml" xref="A1.SS1.p1.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.2.m2.1c">k_{t}</annotation></semantics></math><span id="A1.SS1.p1.6.3" class="ltx_text" style="font-size:90%;"> and </span><math id="A1.SS1.p1.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="A1.SS1.p1.3.m3.1a"><msubsup id="A1.SS1.p1.3.m3.1.1" xref="A1.SS1.p1.3.m3.1.1.cmml"><mi mathsize="90%" id="A1.SS1.p1.3.m3.1.1.2.2" xref="A1.SS1.p1.3.m3.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.SS1.p1.3.m3.1.1.2.3" xref="A1.SS1.p1.3.m3.1.1.2.3.cmml">t</mi><msub id="A1.SS1.p1.3.m3.1.1.3" xref="A1.SS1.p1.3.m3.1.1.3.cmml"><mi mathsize="90%" id="A1.SS1.p1.3.m3.1.1.3.2" xref="A1.SS1.p1.3.m3.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A1.SS1.p1.3.m3.1.1.3.3" xref="A1.SS1.p1.3.m3.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.3.m3.1b"><apply id="A1.SS1.p1.3.m3.1.1.cmml" xref="A1.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.3.m3.1.1.1.cmml" xref="A1.SS1.p1.3.m3.1.1">superscript</csymbol><apply id="A1.SS1.p1.3.m3.1.1.2.cmml" xref="A1.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.3.m3.1.1.2.1.cmml" xref="A1.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="A1.SS1.p1.3.m3.1.1.2.2.cmml" xref="A1.SS1.p1.3.m3.1.1.2.2">𝐱</ci><ci id="A1.SS1.p1.3.m3.1.1.2.3.cmml" xref="A1.SS1.p1.3.m3.1.1.2.3">𝑡</ci></apply><apply id="A1.SS1.p1.3.m3.1.1.3.cmml" xref="A1.SS1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.p1.3.m3.1.1.3.1.cmml" xref="A1.SS1.p1.3.m3.1.1.3">subscript</csymbol><ci id="A1.SS1.p1.3.m3.1.1.3.2.cmml" xref="A1.SS1.p1.3.m3.1.1.3.2">𝑘</ci><ci id="A1.SS1.p1.3.m3.1.1.3.3.cmml" xref="A1.SS1.p1.3.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.3.m3.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math><span id="A1.SS1.p1.6.4" class="ltx_text" style="font-size:90%;"> obtained by noising according to </span><math id="A1.SS1.p1.4.m4.1" class="ltx_Math" alttext="q(\cdot)" display="inline"><semantics id="A1.SS1.p1.4.m4.1a"><mrow id="A1.SS1.p1.4.m4.1.2" xref="A1.SS1.p1.4.m4.1.2.cmml"><mi mathsize="90%" id="A1.SS1.p1.4.m4.1.2.2" xref="A1.SS1.p1.4.m4.1.2.2.cmml">q</mi><mo lspace="0em" rspace="0em" id="A1.SS1.p1.4.m4.1.2.1" xref="A1.SS1.p1.4.m4.1.2.1.cmml">​</mo><mrow id="A1.SS1.p1.4.m4.1.2.3.2" xref="A1.SS1.p1.4.m4.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS1.p1.4.m4.1.2.3.2.1" xref="A1.SS1.p1.4.m4.1.2.cmml">(</mo><mo lspace="0em" mathsize="90%" rspace="0em" id="A1.SS1.p1.4.m4.1.1" xref="A1.SS1.p1.4.m4.1.1.cmml">⋅</mo><mo maxsize="90%" minsize="90%" id="A1.SS1.p1.4.m4.1.2.3.2.2" xref="A1.SS1.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.4.m4.1b"><apply id="A1.SS1.p1.4.m4.1.2.cmml" xref="A1.SS1.p1.4.m4.1.2"><times id="A1.SS1.p1.4.m4.1.2.1.cmml" xref="A1.SS1.p1.4.m4.1.2.1"></times><ci id="A1.SS1.p1.4.m4.1.2.2.cmml" xref="A1.SS1.p1.4.m4.1.2.2">𝑞</ci><ci id="A1.SS1.p1.4.m4.1.1.cmml" xref="A1.SS1.p1.4.m4.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.4.m4.1c">q(\cdot)</annotation></semantics></math><span id="A1.SS1.p1.6.5" class="ltx_text" style="font-size:90%;"> as in (</span><a href="#A1.E1" title="Equation A.1 ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">A.1</span></a><span id="A1.SS1.p1.6.6" class="ltx_text" style="font-size:90%;">). Notice that this formulation does not require an explicit for of </span><math id="A1.SS1.p1.5.m5.1" class="ltx_Math" alttext="q(\cdot)" display="inline"><semantics id="A1.SS1.p1.5.m5.1a"><mrow id="A1.SS1.p1.5.m5.1.2" xref="A1.SS1.p1.5.m5.1.2.cmml"><mi mathsize="90%" id="A1.SS1.p1.5.m5.1.2.2" xref="A1.SS1.p1.5.m5.1.2.2.cmml">q</mi><mo lspace="0em" rspace="0em" id="A1.SS1.p1.5.m5.1.2.1" xref="A1.SS1.p1.5.m5.1.2.1.cmml">​</mo><mrow id="A1.SS1.p1.5.m5.1.2.3.2" xref="A1.SS1.p1.5.m5.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS1.p1.5.m5.1.2.3.2.1" xref="A1.SS1.p1.5.m5.1.2.cmml">(</mo><mo lspace="0em" mathsize="90%" rspace="0em" id="A1.SS1.p1.5.m5.1.1" xref="A1.SS1.p1.5.m5.1.1.cmml">⋅</mo><mo maxsize="90%" minsize="90%" id="A1.SS1.p1.5.m5.1.2.3.2.2" xref="A1.SS1.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.5.m5.1b"><apply id="A1.SS1.p1.5.m5.1.2.cmml" xref="A1.SS1.p1.5.m5.1.2"><times id="A1.SS1.p1.5.m5.1.2.1.cmml" xref="A1.SS1.p1.5.m5.1.2.1"></times><ci id="A1.SS1.p1.5.m5.1.2.2.cmml" xref="A1.SS1.p1.5.m5.1.2.2">𝑞</ci><ci id="A1.SS1.p1.5.m5.1.1.cmml" xref="A1.SS1.p1.5.m5.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.5.m5.1c">q(\cdot)</annotation></semantics></math><span id="A1.SS1.p1.6.7" class="ltx_text" style="font-size:90%;"> or </span><math id="A1.SS1.p1.6.m6.1" class="ltx_Math" alttext="p_{\bm{\theta}}" display="inline"><semantics id="A1.SS1.p1.6.m6.1a"><msub id="A1.SS1.p1.6.m6.1.1" xref="A1.SS1.p1.6.m6.1.1.cmml"><mi mathsize="90%" id="A1.SS1.p1.6.m6.1.1.2" xref="A1.SS1.p1.6.m6.1.1.2.cmml">p</mi><mi mathsize="90%" id="A1.SS1.p1.6.m6.1.1.3" xref="A1.SS1.p1.6.m6.1.1.3.cmml">𝜽</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS1.p1.6.m6.1b"><apply id="A1.SS1.p1.6.m6.1.1.cmml" xref="A1.SS1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A1.SS1.p1.6.m6.1.1.1.cmml" xref="A1.SS1.p1.6.m6.1.1">subscript</csymbol><ci id="A1.SS1.p1.6.m6.1.1.2.cmml" xref="A1.SS1.p1.6.m6.1.1.2">𝑝</ci><ci id="A1.SS1.p1.6.m6.1.1.3.cmml" xref="A1.SS1.p1.6.m6.1.1.3">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p1.6.m6.1c">p_{\bm{\theta}}</annotation></semantics></math><span id="A1.SS1.p1.6.8" class="ltx_text" style="font-size:90%;">, but we will specialize to Gaussian diffusion in the sequel.</span></p>
</div>
<div id="A1.Thmtheorem1" class="ltx_theorem ltx_theorem_theorem">
<h6 class="ltx_title ltx_runin ltx_title_theorem" style="font-size:90%;">
<span class="ltx_tag ltx_tag_theorem"><span id="A1.Thmtheorem1.1.1.1" class="ltx_text ltx_font_bold">Theorem A.1</span></span><span id="A1.Thmtheorem1.2.2" class="ltx_text ltx_font_bold">.</span>
</h6>
<div id="A1.Thmtheorem1.p1" class="ltx_para">
<p id="A1.Thmtheorem1.p1.2" class="ltx_p"><span id="A1.Thmtheorem1.p1.2.2" class="ltx_text ltx_font_italic" style="font-size:90%;">Fix <math id="A1.Thmtheorem1.p1.1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}^{0}" display="inline"><semantics id="A1.Thmtheorem1.p1.1.1.m1.1a"><msubsup id="A1.Thmtheorem1.p1.1.1.m1.1.1" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.cmml"><mi id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.2" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.2.2.cmml">𝐱</mi><mrow id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.cmml"><mn id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.2" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.1" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.1.cmml">:</mo><mi id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.3" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.3.cmml">T</mi></mrow><mn id="A1.Thmtheorem1.p1.1.1.m1.1.1.3" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.3.cmml">0</mn></msubsup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.1.1.m1.1b"><apply id="A1.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.1.1.m1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1">superscript</csymbol><apply id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1">subscript</csymbol><ci id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.2.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.2.2">𝐱</ci><apply id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3"><ci id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.1.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.1">:</ci><cn type="integer" id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.2.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.2">1</cn><ci id="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.3.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.2.3.3">𝑇</ci></apply></apply><cn type="integer" id="A1.Thmtheorem1.p1.1.1.m1.1.1.3.cmml" xref="A1.Thmtheorem1.p1.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.1.1.m1.1c">\mathbf{x}_{1:T}^{0}</annotation></semantics></math>. Define the expectation over the forward process with random noise level <math id="A1.Thmtheorem1.p1.2.2.m2.1" class="ltx_Math" alttext="k_{1:T}" display="inline"><semantics id="A1.Thmtheorem1.p1.2.2.m2.1a"><msub id="A1.Thmtheorem1.p1.2.2.m2.1.1" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="A1.Thmtheorem1.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">k</mi><mrow id="A1.Thmtheorem1.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.cmml"><mn id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.2" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.1" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.1.cmml">:</mo><mi id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.3" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.2.2.m2.1b"><apply id="A1.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1">subscript</csymbol><ci id="A1.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.2">𝑘</ci><apply id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3"><ci id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.1.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.1">:</ci><cn type="integer" id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.2">1</cn><ci id="A1.Thmtheorem1.p1.2.2.m2.1.1.3.3.cmml" xref="A1.Thmtheorem1.p1.2.2.m2.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.2.2.m2.1c">k_{1:T}</annotation></semantics></math> as</span></p>
<table id="A6.EGx4" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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s\leq T}[\cdot],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.4)</span></td>
</tr></tbody>
</table>
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<table id="A6.EGx5" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.4.2">𝐳</mi><mi mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.4.3">s</mi></msub><mo mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.3">∣</mo><mrow id="A1.E5.m1.3.3.1.1.1.1.1.1.1.2.2"><msub id="A1.E5.m1.3.3.1.1.1.1.1.1.1.1.1.1"><mi mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.1.1.1.2">𝐳</mi><mrow id="A1.E5.m1.3.3.1.1.1.1.1.1.1.1.1.1.3"><mi mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.1.1.1.3.2">s</mi><mo mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.1.1.1.3.1">−</mo><mn mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.1.1.1.3.3">1</mn></mrow></msub><mo mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.2.2.3">,</mo><msubsup id="A1.E5.m1.3.3.1.1.1.1.1.1.1.2.2.2"><mi mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.2.2.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.2.2.2.2.3">s</mi><msub id="A1.E5.m1.3.3.1.1.1.1.1.1.1.2.2.2.3"><mi mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.2.2.2.3.2">k</mi><mi mathsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.1.2.2.2.3.3">s</mi></msub></msubsup></mrow></mrow><mo maxsize="90%" minsize="90%" id="A1.E5.m1.3.3.1.1.1.1.1.1.3">)</mo></mrow></mrow></mrow><mo mathsize="90%" id="A1.E5.m1.4.4.2.2.3">,</mo><mrow id="A1.E5.m1.4.4.2.2.2"><mi mathsize="90%" id="A1.E5.m1.4.4.2.2.2.2">s</mi><mo mathsize="90%" id="A1.E5.m1.4.4.2.2.2.1">≤</mo><mi mathsize="90%" id="A1.E5.m1.4.4.2.2.2.3">T</mi></mrow></mrow></munder><mrow id="A1.E5.m1.5.10"><mo id="A1.E5.m1.5.10.1">[</mo><mo lspace="0em" mathsize="90%" rspace="0em" id="A1.E5.m1.5.10.2">⋅</mo><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.E5.m1.5.10.3">∣</mo><msub id="A1.E5.m1.5.10.4"><mi mathsize="90%" id="A1.E5.m1.5.10.4.2">k</mi><mrow id="A1.E5.m1.5.10.4.3"><mn mathsize="90%" id="A1.E5.m1.5.10.4.3.2">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.E5.m1.5.10.4.3.1">:</mo><mi mathsize="90%" id="A1.E5.m1.5.10.4.3.3">T</mi></mrow></msub><mo mathsize="90%" id="A1.E5.m1.5.10.5">,</mo><msub id="A1.E5.m1.5.10.6"><mrow id="A1.E5.m1.5.10.6.2"><mo maxsize="90%" minsize="90%" id="A1.E5.m1.5.10.6.2.1">(</mo><msubsup id="A1.E5.m1.5.10.6.2.2"><mi mathsize="90%" id="A1.E5.m1.5.10.6.2.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.E5.m1.5.10.6.2.2.2.3">t</mi><msub id="A1.E5.m1.5.10.6.2.2.3"><mi mathsize="90%" id="A1.E5.m1.5.10.6.2.2.3.2">k</mi><mi mathsize="90%" id="A1.E5.m1.5.10.6.2.2.3.3">t</mi></msub></msubsup><mo maxsize="90%" minsize="90%" id="A1.E5.m1.5.10.6.2.3">)</mo></mrow><mrow id="A1.E5.m1.5.10.6.3"><mn mathsize="90%" id="A1.E5.m1.5.10.6.3.2">1</mn><mo mathsize="90%" id="A1.E5.m1.5.10.6.3.3">≤</mo><mi mathsize="90%" id="A1.E5.m1.5.10.6.3.4">t</mi><mo mathsize="90%" id="A1.E5.m1.5.10.6.3.5">≤</mo><mi mathsize="90%" id="A1.E5.m1.5.10.6.3.6">T</mi></mrow></msub><mo id="A1.E5.m1.5.10.7">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="A1.E5.m1.5c">\displaystyle\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}[\cdot]:=\mathop{{\mathbb{E}}}_{\mathbf{z}_{s}\sim p(\mathbf{z}_{s}\mid\mathbf{z}_{s-1},\mathbf{x}_{s}^{k_{s}}),s\leq T}\left[\cdot\mid k_{1:T},(\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T}\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.5)</span></td>
</tr></tbody>
</table>
<p id="A1.Thmtheorem1.p1.5" class="ltx_p"><span id="A1.Thmtheorem1.p1.5.1" class="ltx_text ltx_font_italic" style="font-size:90%;">Then, as long as <math id="A1.Thmtheorem1.p1.5.1.m1.1" class="ltx_Math" alttext="p_{\bm{\theta}}" display="inline"><semantics id="A1.Thmtheorem1.p1.5.1.m1.1a"><msub id="A1.Thmtheorem1.p1.5.1.m1.1.1" xref="A1.Thmtheorem1.p1.5.1.m1.1.1.cmml"><mi id="A1.Thmtheorem1.p1.5.1.m1.1.1.2" xref="A1.Thmtheorem1.p1.5.1.m1.1.1.2.cmml">p</mi><mi id="A1.Thmtheorem1.p1.5.1.m1.1.1.3" xref="A1.Thmtheorem1.p1.5.1.m1.1.1.3.cmml">𝛉</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.5.1.m1.1b"><apply id="A1.Thmtheorem1.p1.5.1.m1.1.1.cmml" xref="A1.Thmtheorem1.p1.5.1.m1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.5.1.m1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.5.1.m1.1.1">subscript</csymbol><ci id="A1.Thmtheorem1.p1.5.1.m1.1.1.2.cmml" xref="A1.Thmtheorem1.p1.5.1.m1.1.1.2">𝑝</ci><ci id="A1.Thmtheorem1.p1.5.1.m1.1.1.3.cmml" xref="A1.Thmtheorem1.p1.5.1.m1.1.1.3">𝛉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.5.1.m1.1c">p_{\bm{\theta}}</annotation></semantics></math> satisfies the Markov property,</span></p>
<table id="A6.EGx6" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.Ex1.m1.2c">\displaystyle\mathop{{\mathbb{E}}}_{\mathrm{forward}}[\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})]\geq C(\mathbf{x}_{1:T}^{0})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.Ex2.m1.2.6.3.10.3.2"><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.2.2.3">t</mi><mrow id="A1.Ex2.m1.2.6.3.10.3.2.3"><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.2.3.2">j</mi><mo mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.2.3.1">−</mo><mn mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.2.3.3">1</mn></mrow></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex2.m1.2.6.3.10.3.3">∣</mo><msubsup id="A1.Ex2.m1.2.6.3.10.3.4"><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.4.2.3">t</mi><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.4.3">j</mi></msubsup><mo mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.5">,</mo><msubsup id="A1.Ex2.m1.2.6.3.10.3.6"><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex2.m1.2.6.3.10.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex2.m1.2.6.3.10.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex2.m1.2.6.3.10.4">∥</mo><msub id="A1.Ex2.m1.2.6.3.10.5"><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.5.2">p</mi><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.5.3">𝜽</mi></msub><mrow id="A1.Ex2.m1.2.6.3.10.6"><mo maxsize="90%" minsize="90%" id="A1.Ex2.m1.2.6.3.10.6.1">(</mo><msubsup id="A1.Ex2.m1.2.6.3.10.6.2"><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex2.m1.2.6.3.10.6.3">∣</mo><msubsup id="A1.Ex2.m1.2.6.3.10.6.4"><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.4.2.3">t</mi><mrow id="A1.Ex2.m1.2.6.3.10.6.4.3"><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.4.3.1">−</mo><mn mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.5">,</mo><msub id="A1.Ex2.m1.2.6.3.10.6.6"><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.6.2">𝐳</mi><mrow id="A1.Ex2.m1.2.6.3.10.6.6.3"><mi mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex2.m1.2.6.3.10.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex2.m1.2.6.3.10.6.7">)</mo></mrow><mo id="A1.Ex2.m1.2.6.3.10.7">)</mo></mrow><mo id="A1.Ex2.m1.2.6.3.11">)</mo></mrow><mo id="A1.Ex2.m1.2.6.4">]</mo></mrow><mo mathsize="90%" id="A1.Ex2.m1.2.7">,</mo></mrow><annotation encoding="application/x-tex" id="A1.Ex2.m1.2c">\displaystyle+\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\left(\frac{1}{K+1}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=2}^{K}\frac{j}{K+1}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{j},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j-1},\mathbf{z}_{t-1})\right)\right)\right],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.Thmtheorem1.p1.9" class="ltx_p"><span id="A1.Thmtheorem1.p1.9.4" class="ltx_text ltx_font_italic" style="font-size:90%;">where <math id="A1.Thmtheorem1.p1.6.1.m1.1" class="ltx_Math" alttext="C(\mathbf{x}_{1:T}^{0})" display="inline"><semantics id="A1.Thmtheorem1.p1.6.1.m1.1a"><mrow id="A1.Thmtheorem1.p1.6.1.m1.1.1" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.cmml"><mi id="A1.Thmtheorem1.p1.6.1.m1.1.1.3" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.3.cmml">C</mi><mo lspace="0em" rspace="0em" id="A1.Thmtheorem1.p1.6.1.m1.1.1.2" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.2.cmml">​</mo><mrow id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.cmml"><mo stretchy="false" id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.2" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.cmml">(</mo><msubsup id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.cmml"><mi id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.2" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mrow id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.cmml"><mn id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.2" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.1" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.1.cmml">:</mo><mi id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.3" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.3.cmml">T</mi></mrow><mn id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.3" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.3.cmml">0</mn></msubsup><mo stretchy="false" id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.3" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.6.1.m1.1b"><apply id="A1.Thmtheorem1.p1.6.1.m1.1.1.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1"><times id="A1.Thmtheorem1.p1.6.1.m1.1.1.2.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.2"></times><ci id="A1.Thmtheorem1.p1.6.1.m1.1.1.3.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.3">𝐶</ci><apply id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1">superscript</csymbol><apply id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.1.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1">subscript</csymbol><ci id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.2.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.2">𝐱</ci><apply id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3"><ci id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.1.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.1">:</ci><cn type="integer" id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.2.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.2">1</cn><ci id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.3.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.2.3.3">𝑇</ci></apply></apply><cn type="integer" id="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem1.p1.6.1.m1.1.1.1.1.1.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.6.1.m1.1c">C(\mathbf{x}_{1:T}^{0})</annotation></semantics></math> is a constant depending only on <math id="A1.Thmtheorem1.p1.7.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}^{0}" display="inline"><semantics id="A1.Thmtheorem1.p1.7.2.m2.1a"><msubsup id="A1.Thmtheorem1.p1.7.2.m2.1.1" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.cmml"><mi id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.2" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.2.2.cmml">𝐱</mi><mrow id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.cmml"><mn id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.2" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.1" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.1.cmml">:</mo><mi id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.3" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.3.cmml">T</mi></mrow><mn id="A1.Thmtheorem1.p1.7.2.m2.1.1.3" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.3.cmml">0</mn></msubsup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem1.p1.7.2.m2.1b"><apply id="A1.Thmtheorem1.p1.7.2.m2.1.1.cmml" xref="A1.Thmtheorem1.p1.7.2.m2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.7.2.m2.1.1.1.cmml" xref="A1.Thmtheorem1.p1.7.2.m2.1.1">superscript</csymbol><apply id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.cmml" xref="A1.Thmtheorem1.p1.7.2.m2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.1.cmml" xref="A1.Thmtheorem1.p1.7.2.m2.1.1">subscript</csymbol><ci id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.2.cmml" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.2.2">𝐱</ci><apply id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.cmml" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3"><ci id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.1.cmml" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.1">:</ci><cn type="integer" id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.2.cmml" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.2">1</cn><ci id="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.3.cmml" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.2.3.3">𝑇</ci></apply></apply><cn type="integer" id="A1.Thmtheorem1.p1.7.2.m2.1.1.3.cmml" xref="A1.Thmtheorem1.p1.7.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem1.p1.7.2.m2.1c">\mathbf{x}_{1:T}^{0}</annotation></semantics></math> (the unnoised data). Moreover, if the latents are deterministic (i.e. <math id="A1.Thmtheorem1.p1.8.3.m3.1" class="ltx_Math" alttext="p_{\bm{\theta}}(\mathbf{z}_{t}\mid\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}})" display="inline"><semantics id="A1.Thmtheorem1.p1.8.3.m3.1a"><mrow id="A1.Thmtheorem1.p1.8.3.m3.1.1" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.cmml"><msub id="A1.Thmtheorem1.p1.8.3.m3.1.1.3" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.3.cmml"><mi id="A1.Thmtheorem1.p1.8.3.m3.1.1.3.2" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.3.2.cmml">p</mi><mi id="A1.Thmtheorem1.p1.8.3.m3.1.1.3.3" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.3.3.cmml">𝛉</mi></msub><mo lspace="0em" rspace="0em" id="A1.Thmtheorem1.p1.8.3.m3.1.1.2" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.2.cmml">​</mo><mrow id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.cmml"><mo stretchy="false" id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.2" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.cmml"><msub id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.4" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.4.cmml"><mi id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.4.2" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.4.2.cmml">𝐳</mi><mi id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.4.3" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.4.3.cmml">t</mi></msub><mo id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.3" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.3.cmml">∣</mo><mrow id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.2.2" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.2.3.cmml"><msub id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.1.1.1" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.1.1.1.cmml"><mi id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.1.1.1.2" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.1.1.1.3" xref="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.1.1.1.3.cmml"><mi id="A1.Thmtheorem1.p1.8.3.m3.1.1.1.1.1.1.1.1.3.2" 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</div>
</div>
<div id="A1.SS1.p2" class="ltx_para">
<p id="A1.SS1.p2.1" class="ltx_p"><span id="A1.SS1.p2.1.1" class="ltx_text" style="font-size:90%;">The proof of the amove theorem is given in </span><a href="#A1.SS2" title="A.2 Proof of Theorem A.1 ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">A.2</span></a><span id="A1.SS1.p2.1.2" class="ltx_text" style="font-size:90%;">. Remarkably, it involves </span><em id="A1.SS1.p2.1.3" class="ltx_emph ltx_font_italic" style="font-size:90%;">only two</em><span id="A1.SS1.p2.1.4" class="ltx_text" style="font-size:90%;"> inequalities! 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id="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.cmml" xref="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.1.cmml" xref="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2">subscript</csymbol><ci id="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.2.cmml" xref="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.2">𝐳</ci><apply id="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.3.cmml" xref="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.3"><minus id="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.3.1.cmml" xref="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.3.1"></minus><ci id="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.3.2.cmml" xref="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.3.2">𝑡</ci><cn type="integer" id="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.3.3.cmml" xref="A1.SS1.p2.1.m1.2.2.2.1.1.1.2.2.2.3.3">1</cn></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.p2.1.m1.2c">q(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}})\equiv p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1})</annotation></semantics></math><span id="A1.SS1.p2.1.5" class="ltx_text" style="font-size:90%;">. This tightness of the ELBO suggests that the expression in </span><a href="#A1.Thmtheorem1" title="Theorem A.1. ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">A.1</span></a><span id="A1.SS1.p2.1.6" class="ltx_text" style="font-size:90%;"> is a relatively strong surrogate objective for optimizing the likelihoods.</span></p>
</div>
<section id="A1.SS1.SSS1" class="ltx_subsubsection">
<h4 class="ltx_title ltx_title_subsubsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsubsection">A.1.1 </span>Specializing to Gaussian diffusion</h4>

<div id="A1.SS1.SSS1.p1" class="ltx_para">
<p id="A1.SS1.SSS1.p1.2" class="ltx_p"><span id="A1.SS1.SSS1.p1.2.1" class="ltx_text" style="font-size:90%;">We now special </span><a href="#A1.Thmtheorem1" title="Theorem A.1. ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">A.1</span></a><span id="A1.SS1.SSS1.p1.2.2" class="ltx_text" style="font-size:90%;"> to Gaussian diffusion. For now, we focus on the “</span><math id="A1.SS1.SSS1.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}" display="inline"><semantics id="A1.SS1.SSS1.p1.1.m1.1a"><mi mathsize="90%" id="A1.SS1.SSS1.p1.1.m1.1.1" xref="A1.SS1.SSS1.p1.1.m1.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS1.p1.1.m1.1b"><ci id="A1.SS1.SSS1.p1.1.m1.1.1.cmml" xref="A1.SS1.SSS1.p1.1.m1.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS1.p1.1.m1.1c">\mathbf{x}</annotation></semantics></math><span id="A1.SS1.SSS1.p1.2.3" class="ltx_text" style="font-size:90%;">-prediction” formulation of diffusion, which is the one used in our implementation. The “</span><math id="A1.SS1.SSS1.p1.2.m2.1" class="ltx_Math" alttext="\bm{\epsilon}" display="inline"><semantics id="A1.SS1.SSS1.p1.2.m2.1a"><mi class="ltx_mathvariant_bold-italic" mathsize="90%" mathvariant="bold-italic" id="A1.SS1.SSS1.p1.2.m2.1.1" xref="A1.SS1.SSS1.p1.2.m2.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS1.p1.2.m2.1b"><ci id="A1.SS1.SSS1.p1.2.m2.1.1.cmml" xref="A1.SS1.SSS1.p1.2.m2.1.1">bold-italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS1.p1.2.m2.1c">\bm{\epsilon}</annotation></semantics></math><span id="A1.SS1.SSS1.p1.2.4" class="ltx_text" style="font-size:90%;">-prediction” formalism, used throughout the main body of the text, can be derived similarly (see Section 2 of </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A1.SS1.SSS1.p1.2.5.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib7" title="" class="ltx_ref">7</a><span id="A1.SS1.SSS1.p1.2.6.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A1.SS1.SSS1.p1.2.7" class="ltx_text" style="font-size:90%;"> for a clean exposition). The following theorem follows directly by apply standard likelihood and KL-divergence computations for the DDPM </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A1.SS1.SSS1.p1.2.8.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib28" title="" class="ltx_ref">28</a>, <a href="#bib.bib7" title="" class="ltx_ref">7</a><span id="A1.SS1.SSS1.p1.2.9.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A1.SS1.SSS1.p1.2.10" class="ltx_text" style="font-size:90%;"> to </span><a href="#A1.Thmtheorem1" title="Theorem A.1. ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">A.1</span></a><span id="A1.SS1.SSS1.p1.2.11" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
<div id="A1.Thmtheorem2" class="ltx_theorem ltx_theorem_corollary">
<h6 class="ltx_title ltx_runin ltx_title_theorem" style="font-size:90%;">
<span class="ltx_tag ltx_tag_theorem"><span id="A1.Thmtheorem2.1.1.1" class="ltx_text ltx_font_bold">Corollary A.2</span></span><span id="A1.Thmtheorem2.2.2" class="ltx_text ltx_font_bold">.</span>
</h6>
<div id="A1.Thmtheorem2.p1" class="ltx_para">
<p id="A1.Thmtheorem2.p1.6" class="ltx_p"><span id="A1.Thmtheorem2.p1.6.1" class="ltx_text ltx_font_italic" style="font-size:90%;">Let</span></p>
<table id="A6.EGx7" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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cd="ambiguous" id="A1.E6.m1.1.1.1.1.4.3.3.3.2.1.cmml" xref="A1.E6.m1.1.1.1.1.4.3.3.3.2">subscript</csymbol><ci id="A1.E6.m1.1.1.1.1.4.3.3.3.2.2.cmml" xref="A1.E6.m1.1.1.1.1.4.3.3.3.2.2">𝛽</ci><ci id="A1.E6.m1.1.1.1.1.4.3.3.3.2.3.cmml" xref="A1.E6.m1.1.1.1.1.4.3.3.3.2.3">𝑘</ci></apply><ci id="A1.E6.m1.1.1.1.1.4.3.3.3.3.cmml" xref="A1.E6.m1.1.1.1.1.4.3.3.3.3">𝐈</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E6.m1.1c">\displaystyle q(\mathbf{x}^{k+1}\mid\mathbf{x}_{t}^{k})=\mathcal{N}(\mathbf{x}^{k};\sqrt{1-\beta_{k}}\mathbf{x}^{k-1},\beta_{k}\mathbf{I}),</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.6)</span></td>
</tr></tbody>
</table>
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xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.cmml"><mn id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.2" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.2.cmml">1</mn><mo id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.1" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.1.cmml">−</mo><msub id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.cmml"><mi id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.2" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.2.cmml">β</mi><mi id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.3" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.3.cmml">k</mi></msub></mrow><mo stretchy="false" id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.3" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.1.1.m1.1b"><apply id="A1.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1"><eq id="A1.Thmtheorem2.p1.1.1.m1.1.1.2.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.2"></eq><apply id="A1.Thmtheorem2.p1.1.1.m1.1.1.3.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.1.1.m1.1.1.3.1.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="A1.Thmtheorem2.p1.1.1.m1.1.1.3.2.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3.2">𝛼</ci><ci id="A1.Thmtheorem2.p1.1.1.m1.1.1.3.3.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.3.3">𝑘</ci></apply><apply id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1"><minus id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.1"></minus><cn type="integer" id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.2">1</cn><apply id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.1.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.2.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.2">𝛽</ci><ci id="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.3.cmml" xref="A1.Thmtheorem2.p1.1.1.m1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.1.1.m1.1c">\alpha_{k}=(1-\beta_{k})</annotation></semantics></math>, <math id="A1.Thmtheorem2.p1.2.2.m2.1" class="ltx_Math" alttext="\bar{\alpha}_{k}=\prod_{j=1}^{k}\alpha_{j}" display="inline"><semantics id="A1.Thmtheorem2.p1.2.2.m2.1a"><mrow id="A1.Thmtheorem2.p1.2.2.m2.1.1" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.cmml"><msub id="A1.Thmtheorem2.p1.2.2.m2.1.1.2" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2.cmml"><mover accent="true" id="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2.cmml"><mi id="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2.2" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2.2.cmml">α</mi><mo id="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2.1" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2.1.cmml">¯</mo></mover><mi id="A1.Thmtheorem2.p1.2.2.m2.1.1.2.3" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2.3.cmml">k</mi></msub><mo rspace="0.111em" id="A1.Thmtheorem2.p1.2.2.m2.1.1.1" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">=</mo><mrow id="A1.Thmtheorem2.p1.2.2.m2.1.1.3" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.cmml"><msubsup id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.cmml"><mo id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.2" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.2.cmml">∏</mo><mrow id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.cmml"><mi id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.2" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.2.cmml">j</mi><mo id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.1" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.1.cmml">=</mo><mn id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.3" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.3.cmml">1</mn></mrow><mi id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.3" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.3.cmml">k</mi></msubsup><msub id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml"><mi id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.2" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.2.cmml">α</mi><mi id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.3" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.3.cmml">j</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.2.2.m2.1b"><apply id="A1.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1"><eq id="A1.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.1"></eq><apply id="A1.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.2.2.m2.1.1.2.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2">subscript</csymbol><apply id="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2"><ci id="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2.1">¯</ci><ci id="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2.2.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2.2.2">𝛼</ci></apply><ci id="A1.Thmtheorem2.p1.2.2.m2.1.1.2.3.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.2.3">𝑘</ci></apply><apply id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3"><apply id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1">superscript</csymbol><apply id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1">subscript</csymbol><csymbol cd="latexml" id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.2.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.2">product</csymbol><apply id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3"><eq id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.1"></eq><ci id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.2.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.2">𝑗</ci><cn type="integer" id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.3.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.2.3.3">1</cn></apply></apply><ci id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.3.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.1.3">𝑘</ci></apply><apply id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.1.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2">subscript</csymbol><ci id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.2.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.2">𝛼</ci><ci id="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.3.cmml" xref="A1.Thmtheorem2.p1.2.2.m2.1.1.3.2.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.2.2.m2.1c">\bar{\alpha}_{k}=\prod_{j=1}^{k}\alpha_{j}</annotation></semantics></math>. Suppose that we parameterize <math id="A1.Thmtheorem2.p1.3.3.m3.4" class="ltx_Math" alttext="p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})=\mathcal{N}(\mu_{\bm{\theta}}(\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1},j),\sigma_{j}^{2})" display="inline"><semantics id="A1.Thmtheorem2.p1.3.3.m3.4a"><mrow id="A1.Thmtheorem2.p1.3.3.m3.4.4" xref="A1.Thmtheorem2.p1.3.3.m3.4.4.cmml"><mrow id="A1.Thmtheorem2.p1.3.3.m3.2.2.1" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.cmml"><msub id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.3" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.3.cmml"><mi id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.3.2" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.3.2.cmml">p</mi><mi id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.3.3" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.3.3.cmml">𝛉</mi></msub><mo lspace="0em" rspace="0em" id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.2" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.2.cmml">​</mo><mrow id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.cmml"><mo stretchy="false" id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.2" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.cmml"><msubsup id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.4" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.4.cmml"><mi id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.4.2.2" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.4.2.2.cmml">𝐱</mi><mi id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.4.2.3" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.4.2.3.cmml">t</mi><mi id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.4.3" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.4.3.cmml">j</mi></msubsup><mo id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.3" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.3.cmml">∣</mo><mrow id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.2.2" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.2.3.cmml"><msubsup id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.cmml"><mi id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.2.2" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.2.3" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.2.3.cmml">t</mi><mrow id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3.cmml"><mi id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3.2" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3.2.cmml">j</mi><mo id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3.1" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3.3" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msubsup><mo id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.2.2.3" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.2.3.cmml">,</mo><msub id="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.2.2.2" xref="A1.Thmtheorem2.p1.3.3.m3.2.2.1.1.1.1.2.2.2.cmml"><mi 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xref="A1.Thmtheorem2.p1.3.3.m3.4.4.3.4.cmml">𝒩</mi><mo lspace="0em" rspace="0em" id="A1.Thmtheorem2.p1.3.3.m3.4.4.3.3" xref="A1.Thmtheorem2.p1.3.3.m3.4.4.3.3.cmml">​</mo><mrow id="A1.Thmtheorem2.p1.3.3.m3.4.4.3.2.2" xref="A1.Thmtheorem2.p1.3.3.m3.4.4.3.2.3.cmml"><mo stretchy="false" id="A1.Thmtheorem2.p1.3.3.m3.4.4.3.2.2.3" xref="A1.Thmtheorem2.p1.3.3.m3.4.4.3.2.3.cmml">(</mo><mrow id="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1" xref="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1.cmml"><msub id="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1.4" xref="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1.4.cmml"><mi id="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1.4.2" xref="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1.4.2.cmml">μ</mi><mi id="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1.4.3" xref="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1.4.3.cmml">𝛉</mi></msub><mo lspace="0em" rspace="0em" id="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1.3" xref="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1.3.cmml">​</mo><mrow id="A1.Thmtheorem2.p1.3.3.m3.3.3.2.1.1.1.2.2" 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<table id="A6.EGx8" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.Ex3.m1.7" class="ltx_Math" alttext="\displaystyle\mu_{\bm{\theta}}(\mathbf{x}_{t}^{j},\mathbf{z}_{t-1},j)=\frac{(1-\bar{\alpha}_{j-1})\sqrt{{\alpha}_{j}}}{1-\bar{\alpha}_{j}}\mathbf{x}_{t}^{j}+\frac{(1-{\alpha}_{j})\sqrt{\bar{\alpha}_{j-1}}}{1-\bar{\alpha}_{j}}\hat{\mathbf{x}}_{\bm{\theta}}(\mathbf{x}_{t}^{j},\mathbf{z}_{t-1},j),\quad\sigma_{j}^{2}:=\frac{(1-\alpha_{j})(1-\sqrt{\bar{\alpha}_{j-1}})}{1-\bar{\alpha}_{j}}." display="inline"><semantics id="A1.Ex3.m1.7a"><mrow id="A1.Ex3.m1.7.7.1"><mrow id="A1.Ex3.m1.7.7.1.1.2" xref="A1.Ex3.m1.7.7.1.1.3.cmml"><mrow id="A1.Ex3.m1.7.7.1.1.1.1" xref="A1.Ex3.m1.7.7.1.1.1.1.cmml"><mrow id="A1.Ex3.m1.7.7.1.1.1.1.2" xref="A1.Ex3.m1.7.7.1.1.1.1.2.cmml"><msub id="A1.Ex3.m1.7.7.1.1.1.1.2.4" xref="A1.Ex3.m1.7.7.1.1.1.1.2.4.cmml"><mi mathsize="90%" id="A1.Ex3.m1.7.7.1.1.1.1.2.4.2" xref="A1.Ex3.m1.7.7.1.1.1.1.2.4.2.cmml">μ</mi><mi mathsize="90%" id="A1.Ex3.m1.7.7.1.1.1.1.2.4.3" 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id="A1.Ex3.m1.7c">\displaystyle\mu_{\bm{\theta}}(\mathbf{x}_{t}^{j},\mathbf{z}_{t-1},j)=\frac{(1-\bar{\alpha}_{j-1})\sqrt{{\alpha}_{j}}}{1-\bar{\alpha}_{j}}\mathbf{x}_{t}^{j}+\frac{(1-{\alpha}_{j})\sqrt{\bar{\alpha}_{j-1}}}{1-\bar{\alpha}_{j}}\hat{\mathbf{x}}_{\bm{\theta}}(\mathbf{x}_{t}^{j},\mathbf{z}_{t-1},j),\quad\sigma_{j}^{2}:=\frac{(1-\alpha_{j})(1-\sqrt{\bar{\alpha}_{j-1}})}{1-\bar{\alpha}_{j}}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.Thmtheorem2.p1.4" class="ltx_p"><span id="A1.Thmtheorem2.p1.4.1" class="ltx_text ltx_font_italic" style="font-size:90%;">Then, as long as <math id="A1.Thmtheorem2.p1.4.1.m1.1" class="ltx_Math" alttext="p_{\bm{\theta}}" display="inline"><semantics id="A1.Thmtheorem2.p1.4.1.m1.1a"><msub id="A1.Thmtheorem2.p1.4.1.m1.1.1" xref="A1.Thmtheorem2.p1.4.1.m1.1.1.cmml"><mi id="A1.Thmtheorem2.p1.4.1.m1.1.1.2" xref="A1.Thmtheorem2.p1.4.1.m1.1.1.2.cmml">p</mi><mi id="A1.Thmtheorem2.p1.4.1.m1.1.1.3" xref="A1.Thmtheorem2.p1.4.1.m1.1.1.3.cmml">𝛉</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.4.1.m1.1b"><apply id="A1.Thmtheorem2.p1.4.1.m1.1.1.cmml" xref="A1.Thmtheorem2.p1.4.1.m1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.4.1.m1.1.1.1.cmml" xref="A1.Thmtheorem2.p1.4.1.m1.1.1">subscript</csymbol><ci id="A1.Thmtheorem2.p1.4.1.m1.1.1.2.cmml" xref="A1.Thmtheorem2.p1.4.1.m1.1.1.2">𝑝</ci><ci id="A1.Thmtheorem2.p1.4.1.m1.1.1.3.cmml" xref="A1.Thmtheorem2.p1.4.1.m1.1.1.3">𝛉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.4.1.m1.1c">p_{\bm{\theta}}</annotation></semantics></math> satisfies the Markov property, we obtained</span></p>
<table id="A6.EGx9" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.Ex4.m1.2c">\displaystyle\mathop{{\mathbb{E}}}_{\mathrm{forward}}[\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})]+C(\mathbf{x}_{1:T}^{0})</annotation></semantics></math></td>
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xref="A1.Ex4.m2.3.3">𝑗</ci></vector></apply><apply id="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4.cmml" xref="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4.1.cmml" xref="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4">superscript</csymbol><apply id="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4.2.cmml" xref="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4.2.1.cmml" xref="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4">subscript</csymbol><ci id="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4.2.2.cmml" xref="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4.2.2">𝐱</ci><ci id="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4.2.3.cmml" xref="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4.3.cmml" xref="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.1.1.1.4.3">0</cn></apply></apply></apply><cn type="integer" id="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.Ex4.m2.4.4.1.1.1.1.1.1.1.1.1.3">2</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex4.m2.4c">\displaystyle\geq\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\frac{j}{K+1}\sum_{j=1}^{K}c_{j}\|\hat{\mathbf{x}}^{0}_{\bm{\theta}}(\mathbf{x}_{t}^{j},\mathbf{z}_{t-1},j)-\mathbf{x}_{t}^{0}\|^{2}\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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id="A1.Ex5.m1.3c">\displaystyle=\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\mathbf{1}\{k_{t}\geq 1\}\cdot k_{t}c_{k_{t}}\|\hat{\mathbf{x}}^{0}_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1},{k_{t}})-\mathbf{x}_{t}^{0}\|^{2}\right],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
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xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.2.1.cmml">¯</mo></mover><mi id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.3" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.3.cmml">j</mi></msub></mrow><mo stretchy="false" id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.3" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.cmml">)</mo></mrow><mn id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.3" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.3.cmml">2</mn></msup></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem2.p1.5.1.m1.2b"><apply id="A1.Thmtheorem2.p1.5.1.m1.2.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.3"><eq id="A1.Thmtheorem2.p1.5.1.m1.2.3.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.3.1"></eq><apply id="A1.Thmtheorem2.p1.5.1.m1.2.3.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.3.2"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.5.1.m1.2.3.2.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.3.2">subscript</csymbol><ci id="A1.Thmtheorem2.p1.5.1.m1.2.3.2.2.cmml" 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id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.1.1.1.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.1.1.1.2">1</cn><apply id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.1.1.1.3.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.1.1.1.3.2">𝛼</ci><ci id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.1.1.1.3.3">𝑗</ci></apply></apply><cn type="integer" id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.1.3">2</cn></apply><apply id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3">subscript</csymbol><apply id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.2"><ci id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.2.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.2.1">¯</ci><ci id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.2.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.2.2">𝛼</ci></apply><apply id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.3"><minus id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.3.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.3.1"></minus><ci id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.3.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.3.2">𝑗</ci><cn type="integer" id="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.3.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.1.1.1.3.3.3">1</cn></apply></apply></apply><apply id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2"><times id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.2"></times><cn type="integer" id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.3">2</cn><apply id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.4.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.4"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.4.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.4">superscript</csymbol><ci id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.4.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.4.2">𝜎</ci><cn type="integer" id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.4.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.4.3">2</cn></apply><apply id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1">superscript</csymbol><apply id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1"><minus id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.1"></minus><cn type="integer" id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.2">1</cn><apply id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3">subscript</csymbol><apply id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.2"><ci id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.2.1.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.2.1">¯</ci><ci id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.2.2.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.2.2">𝛼</ci></apply><ci id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.1.1.1.3.3">𝑗</ci></apply></apply><cn type="integer" id="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.3.cmml" xref="A1.Thmtheorem2.p1.5.1.m1.2.2.2.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem2.p1.5.1.m1.2c">c_{j}=\frac{(1-\alpha_{j})^{2}\bar{\alpha}_{j-1}}{2\sigma^{2}(1-\bar{\alpha}_{j})^{2}}</annotation></semantics></math>.</span></p>
</div>
</div>
<div id="A1.SS1.SSS1.1" class="ltx_proof">
<h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof" style="font-size:90%;">Proof.</h6>
<div id="A1.SS1.SSS1.1.p1" class="ltx_para">
<p id="A1.SS1.SSS1.1.p1.3" class="ltx_p"><span id="A1.SS1.SSS1.1.p1.3.1" class="ltx_text" style="font-size:90%;">The first inequality follows from the standard computations for the “</span><math id="A1.SS1.SSS1.1.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}" display="inline"><semantics id="A1.SS1.SSS1.1.p1.1.m1.1a"><mi mathsize="90%" id="A1.SS1.SSS1.1.p1.1.m1.1.1" xref="A1.SS1.SSS1.1.p1.1.m1.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS1.1.p1.1.m1.1b"><ci id="A1.SS1.SSS1.1.p1.1.m1.1.1.cmml" xref="A1.SS1.SSS1.1.p1.1.m1.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS1.1.p1.1.m1.1c">\mathbf{x}</annotation></semantics></math><span id="A1.SS1.SSS1.1.p1.3.2" class="ltx_text" style="font-size:90%;">-prediction” formulation of Diffusion (see Section 2.7 of </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A1.SS1.SSS1.1.p1.3.3.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib7" title="" class="ltx_ref">7</a><span id="A1.SS1.SSS1.1.p1.3.4.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A1.SS1.SSS1.1.p1.3.5" class="ltx_text" style="font-size:90%;"> and references therein). The second follows by replacing the sum over </span><math id="A1.SS1.SSS1.1.p1.2.m2.1" class="ltx_Math" alttext="j" display="inline"><semantics id="A1.SS1.SSS1.1.p1.2.m2.1a"><mi mathsize="90%" id="A1.SS1.SSS1.1.p1.2.m2.1.1" xref="A1.SS1.SSS1.1.p1.2.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS1.1.p1.2.m2.1b"><ci id="A1.SS1.SSS1.1.p1.2.m2.1.1.cmml" xref="A1.SS1.SSS1.1.p1.2.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS1.1.p1.2.m2.1c">j</annotation></semantics></math><span id="A1.SS1.SSS1.1.p1.3.6" class="ltx_text" style="font-size:90%;"> with an expectation over </span><math id="A1.SS1.SSS1.1.p1.3.m3.4" class="ltx_Math" alttext="k_{t}\overset{\mathrm{unif}}{\sim}\{0,1,\dots,K\}" display="inline"><semantics id="A1.SS1.SSS1.1.p1.3.m3.4a"><mrow id="A1.SS1.SSS1.1.p1.3.m3.4.5" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.cmml"><msub id="A1.SS1.SSS1.1.p1.3.m3.4.5.2" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.2.cmml"><mi mathsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.4.5.2.2" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.2.2.cmml">k</mi><mi mathsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.4.5.2.3" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.2.3.cmml">t</mi></msub><mo lspace="0em" rspace="0em" id="A1.SS1.SSS1.1.p1.3.m3.4.5.1" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.1.cmml">​</mo><mover accent="true" id="A1.SS1.SSS1.1.p1.3.m3.4.5.3" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.3.cmml"><mo mathsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.4.5.3.2" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.3.2.cmml">∼</mo><mo mathsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.4.5.3.1" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.3.1.cmml">unif</mo></mover><mo lspace="0em" rspace="0em" id="A1.SS1.SSS1.1.p1.3.m3.4.5.1a" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.1.cmml">​</mo><mrow id="A1.SS1.SSS1.1.p1.3.m3.4.5.4.2" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.4.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.4.5.4.2.1" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.4.1.cmml">{</mo><mn mathsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.1.1" xref="A1.SS1.SSS1.1.p1.3.m3.1.1.cmml">0</mn><mo mathsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.4.5.4.2.2" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.4.1.cmml">,</mo><mn mathsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.2.2" xref="A1.SS1.SSS1.1.p1.3.m3.2.2.cmml">1</mn><mo mathsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.4.5.4.2.3" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.4.1.cmml">,</mo><mi mathsize="90%" mathvariant="normal" id="A1.SS1.SSS1.1.p1.3.m3.3.3" xref="A1.SS1.SSS1.1.p1.3.m3.3.3.cmml">…</mi><mo mathsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.4.5.4.2.4" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.4.1.cmml">,</mo><mi mathsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.4.4" xref="A1.SS1.SSS1.1.p1.3.m3.4.4.cmml">K</mi><mo maxsize="90%" minsize="90%" id="A1.SS1.SSS1.1.p1.3.m3.4.5.4.2.5" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.4.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS1.1.p1.3.m3.4b"><apply id="A1.SS1.SSS1.1.p1.3.m3.4.5.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.5"><times id="A1.SS1.SSS1.1.p1.3.m3.4.5.1.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.1"></times><apply id="A1.SS1.SSS1.1.p1.3.m3.4.5.2.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.2"><csymbol cd="ambiguous" id="A1.SS1.SSS1.1.p1.3.m3.4.5.2.1.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.2">subscript</csymbol><ci id="A1.SS1.SSS1.1.p1.3.m3.4.5.2.2.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.2.2">𝑘</ci><ci id="A1.SS1.SSS1.1.p1.3.m3.4.5.2.3.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.2.3">𝑡</ci></apply><apply id="A1.SS1.SSS1.1.p1.3.m3.4.5.3.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.3"><ci id="A1.SS1.SSS1.1.p1.3.m3.4.5.3.1.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.3.1">unif</ci><csymbol cd="latexml" id="A1.SS1.SSS1.1.p1.3.m3.4.5.3.2.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.3.2">similar-to</csymbol></apply><set id="A1.SS1.SSS1.1.p1.3.m3.4.5.4.1.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.5.4.2"><cn type="integer" id="A1.SS1.SSS1.1.p1.3.m3.1.1.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.1.1">0</cn><cn type="integer" id="A1.SS1.SSS1.1.p1.3.m3.2.2.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.2.2">1</cn><ci id="A1.SS1.SSS1.1.p1.3.m3.3.3.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.3.3">…</ci><ci id="A1.SS1.SSS1.1.p1.3.m3.4.4.cmml" xref="A1.SS1.SSS1.1.p1.3.m3.4.4">𝐾</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS1.1.p1.3.m3.4c">k_{t}\overset{\mathrm{unif}}{\sim}\{0,1,\dots,K\}</annotation></semantics></math><span id="A1.SS1.SSS1.1.p1.3.7" class="ltx_text" style="font-size:90%;">.
∎</span></p>
</div>
</div>
<div id="A1.SS1.SSS1.p2" class="ltx_para">
<p id="A1.SS1.SSS1.p2.1" class="ltx_p"><span id="A1.SS1.SSS1.p2.1.1" class="ltx_text" style="font-size:90%;">We make a couple remarks:</span></p>
<ul id="A1.I2" class="ltx_itemize">
<li id="A1.I2.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A1.I2.i1.p1" class="ltx_para">
<p id="A1.I2.i1.p1.2" class="ltx_p"><span id="A1.I2.i1.p1.2.1" class="ltx_text" style="font-size:90%;">As noted above, </span><a href="#A1.Thmtheorem2" title="Corollary A.2. ‣ A.1.1 Specializing to Gaussian diffusion ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">A.2</span></a><span id="A1.I2.i1.p1.2.2" class="ltx_text" style="font-size:90%;"> can also be stated for </span><math id="A1.I2.i1.p1.1.m1.1" class="ltx_Math" alttext="\bm{\epsilon}" display="inline"><semantics id="A1.I2.i1.p1.1.m1.1a"><mi class="ltx_mathvariant_bold-italic" mathsize="90%" mathvariant="bold-italic" id="A1.I2.i1.p1.1.m1.1.1" xref="A1.I2.i1.p1.1.m1.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="A1.I2.i1.p1.1.m1.1b"><ci id="A1.I2.i1.p1.1.m1.1.1.cmml" xref="A1.I2.i1.p1.1.m1.1.1">bold-italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i1.p1.1.m1.1c">\bm{\epsilon}</annotation></semantics></math><span id="A1.I2.i1.p1.2.3" class="ltx_text" style="font-size:90%;">-prediction, or the so-called “</span><math id="A1.I2.i1.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{v}" display="inline"><semantics id="A1.I2.i1.p1.2.m2.1a"><mi mathsize="90%" id="A1.I2.i1.p1.2.m2.1.1" xref="A1.I2.i1.p1.2.m2.1.1.cmml">𝐯</mi><annotation-xml encoding="MathML-Content" id="A1.I2.i1.p1.2.m2.1b"><ci id="A1.I2.i1.p1.2.m2.1.1.cmml" xref="A1.I2.i1.p1.2.m2.1.1">𝐯</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i1.p1.2.m2.1c">\mathbf{v}</annotation></semantics></math><span id="A1.I2.i1.p1.2.4" class="ltx_text" style="font-size:90%;">-prediction” formalism, as all are affinely related.</span></p>
</div>
</li>
<li id="A1.I2.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A1.I2.i2.p1" class="ltx_para">
<p id="A1.I2.i2.p1.9" class="ltx_p"><span id="A1.I2.i2.p1.9.1" class="ltx_text" style="font-size:90%;">Define an idealized latent </span><math id="A1.I2.i2.p1.1.m1.1" class="ltx_Math" alttext="\tilde{\mathbf{z}}_{t-1}" display="inline"><semantics id="A1.I2.i2.p1.1.m1.1a"><msub id="A1.I2.i2.p1.1.m1.1.1" xref="A1.I2.i2.p1.1.m1.1.1.cmml"><mover accent="true" id="A1.I2.i2.p1.1.m1.1.1.2" xref="A1.I2.i2.p1.1.m1.1.1.2.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.1.m1.1.1.2.2" xref="A1.I2.i2.p1.1.m1.1.1.2.2.cmml">𝐳</mi><mo mathsize="90%" id="A1.I2.i2.p1.1.m1.1.1.2.1" xref="A1.I2.i2.p1.1.m1.1.1.2.1.cmml">~</mo></mover><mrow id="A1.I2.i2.p1.1.m1.1.1.3" xref="A1.I2.i2.p1.1.m1.1.1.3.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.1.m1.1.1.3.2" xref="A1.I2.i2.p1.1.m1.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A1.I2.i2.p1.1.m1.1.1.3.1" xref="A1.I2.i2.p1.1.m1.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A1.I2.i2.p1.1.m1.1.1.3.3" xref="A1.I2.i2.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.I2.i2.p1.1.m1.1b"><apply id="A1.I2.i2.p1.1.m1.1.1.cmml" xref="A1.I2.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.I2.i2.p1.1.m1.1.1.1.cmml" xref="A1.I2.i2.p1.1.m1.1.1">subscript</csymbol><apply id="A1.I2.i2.p1.1.m1.1.1.2.cmml" xref="A1.I2.i2.p1.1.m1.1.1.2"><ci id="A1.I2.i2.p1.1.m1.1.1.2.1.cmml" xref="A1.I2.i2.p1.1.m1.1.1.2.1">~</ci><ci id="A1.I2.i2.p1.1.m1.1.1.2.2.cmml" xref="A1.I2.i2.p1.1.m1.1.1.2.2">𝐳</ci></apply><apply id="A1.I2.i2.p1.1.m1.1.1.3.cmml" xref="A1.I2.i2.p1.1.m1.1.1.3"><minus id="A1.I2.i2.p1.1.m1.1.1.3.1.cmml" xref="A1.I2.i2.p1.1.m1.1.1.3.1"></minus><ci id="A1.I2.i2.p1.1.m1.1.1.3.2.cmml" xref="A1.I2.i2.p1.1.m1.1.1.3.2">𝑡</ci><cn type="integer" id="A1.I2.i2.p1.1.m1.1.1.3.3.cmml" xref="A1.I2.i2.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i2.p1.1.m1.1c">\tilde{\mathbf{z}}_{t-1}</annotation></semantics></math><span id="A1.I2.i2.p1.9.2" class="ltx_text" style="font-size:90%;"> consisting of all past tokens </span><math id="A1.I2.i2.p1.2.m2.1" class="ltx_Math" alttext="(\mathbf{x}_{t}^{k_{t}})" display="inline"><semantics id="A1.I2.i2.p1.2.m2.1a"><mrow id="A1.I2.i2.p1.2.m2.1.1.1" xref="A1.I2.i2.p1.2.m2.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.I2.i2.p1.2.m2.1.1.1.2" xref="A1.I2.i2.p1.2.m2.1.1.1.1.cmml">(</mo><msubsup id="A1.I2.i2.p1.2.m2.1.1.1.1" xref="A1.I2.i2.p1.2.m2.1.1.1.1.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.2.m2.1.1.1.1.2.2" xref="A1.I2.i2.p1.2.m2.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.I2.i2.p1.2.m2.1.1.1.1.2.3" xref="A1.I2.i2.p1.2.m2.1.1.1.1.2.3.cmml">t</mi><msub id="A1.I2.i2.p1.2.m2.1.1.1.1.3" xref="A1.I2.i2.p1.2.m2.1.1.1.1.3.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.2.m2.1.1.1.1.3.2" xref="A1.I2.i2.p1.2.m2.1.1.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A1.I2.i2.p1.2.m2.1.1.1.1.3.3" xref="A1.I2.i2.p1.2.m2.1.1.1.1.3.3.cmml">t</mi></msub></msubsup><mo maxsize="90%" minsize="90%" id="A1.I2.i2.p1.2.m2.1.1.1.3" xref="A1.I2.i2.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.I2.i2.p1.2.m2.1b"><apply id="A1.I2.i2.p1.2.m2.1.1.1.1.cmml" xref="A1.I2.i2.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="A1.I2.i2.p1.2.m2.1.1.1.1.1.cmml" xref="A1.I2.i2.p1.2.m2.1.1.1">superscript</csymbol><apply id="A1.I2.i2.p1.2.m2.1.1.1.1.2.cmml" xref="A1.I2.i2.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="A1.I2.i2.p1.2.m2.1.1.1.1.2.1.cmml" xref="A1.I2.i2.p1.2.m2.1.1.1">subscript</csymbol><ci id="A1.I2.i2.p1.2.m2.1.1.1.1.2.2.cmml" xref="A1.I2.i2.p1.2.m2.1.1.1.1.2.2">𝐱</ci><ci id="A1.I2.i2.p1.2.m2.1.1.1.1.2.3.cmml" xref="A1.I2.i2.p1.2.m2.1.1.1.1.2.3">𝑡</ci></apply><apply id="A1.I2.i2.p1.2.m2.1.1.1.1.3.cmml" xref="A1.I2.i2.p1.2.m2.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.I2.i2.p1.2.m2.1.1.1.1.3.1.cmml" xref="A1.I2.i2.p1.2.m2.1.1.1.1.3">subscript</csymbol><ci id="A1.I2.i2.p1.2.m2.1.1.1.1.3.2.cmml" xref="A1.I2.i2.p1.2.m2.1.1.1.1.3.2">𝑘</ci><ci id="A1.I2.i2.p1.2.m2.1.1.1.1.3.3.cmml" xref="A1.I2.i2.p1.2.m2.1.1.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i2.p1.2.m2.1c">(\mathbf{x}_{t}^{k_{t}})</annotation></semantics></math><span id="A1.I2.i2.p1.9.3" class="ltx_text" style="font-size:90%;"> as well as of their noise levels </span><math id="A1.I2.i2.p1.3.m3.1" class="ltx_Math" alttext="k_{t}" display="inline"><semantics id="A1.I2.i2.p1.3.m3.1a"><msub id="A1.I2.i2.p1.3.m3.1.1" xref="A1.I2.i2.p1.3.m3.1.1.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.3.m3.1.1.2" xref="A1.I2.i2.p1.3.m3.1.1.2.cmml">k</mi><mi mathsize="90%" id="A1.I2.i2.p1.3.m3.1.1.3" xref="A1.I2.i2.p1.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.I2.i2.p1.3.m3.1b"><apply id="A1.I2.i2.p1.3.m3.1.1.cmml" xref="A1.I2.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A1.I2.i2.p1.3.m3.1.1.1.cmml" xref="A1.I2.i2.p1.3.m3.1.1">subscript</csymbol><ci id="A1.I2.i2.p1.3.m3.1.1.2.cmml" xref="A1.I2.i2.p1.3.m3.1.1.2">𝑘</ci><ci id="A1.I2.i2.p1.3.m3.1.1.3.cmml" xref="A1.I2.i2.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i2.p1.3.m3.1c">k_{t}</annotation></semantics></math><span id="A1.I2.i2.p1.9.4" class="ltx_text" style="font-size:90%;">. This is a sufficient statistic for </span><math id="A1.I2.i2.p1.4.m4.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}" display="inline"><semantics id="A1.I2.i2.p1.4.m4.1a"><msub id="A1.I2.i2.p1.4.m4.1.1" xref="A1.I2.i2.p1.4.m4.1.1.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.4.m4.1.1.2" xref="A1.I2.i2.p1.4.m4.1.1.2.cmml">𝐳</mi><mrow id="A1.I2.i2.p1.4.m4.1.1.3" xref="A1.I2.i2.p1.4.m4.1.1.3.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.4.m4.1.1.3.2" xref="A1.I2.i2.p1.4.m4.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A1.I2.i2.p1.4.m4.1.1.3.1" xref="A1.I2.i2.p1.4.m4.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A1.I2.i2.p1.4.m4.1.1.3.3" xref="A1.I2.i2.p1.4.m4.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.I2.i2.p1.4.m4.1b"><apply id="A1.I2.i2.p1.4.m4.1.1.cmml" xref="A1.I2.i2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A1.I2.i2.p1.4.m4.1.1.1.cmml" xref="A1.I2.i2.p1.4.m4.1.1">subscript</csymbol><ci id="A1.I2.i2.p1.4.m4.1.1.2.cmml" xref="A1.I2.i2.p1.4.m4.1.1.2">𝐳</ci><apply id="A1.I2.i2.p1.4.m4.1.1.3.cmml" xref="A1.I2.i2.p1.4.m4.1.1.3"><minus id="A1.I2.i2.p1.4.m4.1.1.3.1.cmml" xref="A1.I2.i2.p1.4.m4.1.1.3.1"></minus><ci id="A1.I2.i2.p1.4.m4.1.1.3.2.cmml" xref="A1.I2.i2.p1.4.m4.1.1.3.2">𝑡</ci><cn type="integer" id="A1.I2.i2.p1.4.m4.1.1.3.3.cmml" xref="A1.I2.i2.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i2.p1.4.m4.1c">\mathbf{z}_{t-1}</annotation></semantics></math><span id="A1.I2.i2.p1.9.5" class="ltx_text" style="font-size:90%;">, and thus we can always view </span><math id="A1.I2.i2.p1.5.m5.6" class="ltx_Math" alttext="\hat{\mathbf{x}}^{0}_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1},{k_{t}})=\hat{\mathbf{x}}^{0}_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}},\bar{\mathbf{z}}_{t-1},{k_{t}})" display="inline"><semantics id="A1.I2.i2.p1.5.m5.6a"><mrow id="A1.I2.i2.p1.5.m5.6.6" xref="A1.I2.i2.p1.5.m5.6.6.cmml"><mrow id="A1.I2.i2.p1.5.m5.3.3.3" xref="A1.I2.i2.p1.5.m5.3.3.3.cmml"><msubsup 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xref="A1.I2.i2.p1.5.m5.5.5.5.2.2.2.3"><minus id="A1.I2.i2.p1.5.m5.5.5.5.2.2.2.3.1.cmml" xref="A1.I2.i2.p1.5.m5.5.5.5.2.2.2.3.1"></minus><ci id="A1.I2.i2.p1.5.m5.5.5.5.2.2.2.3.2.cmml" xref="A1.I2.i2.p1.5.m5.5.5.5.2.2.2.3.2">𝑡</ci><cn type="integer" id="A1.I2.i2.p1.5.m5.5.5.5.2.2.2.3.3.cmml" xref="A1.I2.i2.p1.5.m5.5.5.5.2.2.2.3.3">1</cn></apply></apply><apply id="A1.I2.i2.p1.5.m5.6.6.6.3.3.3.cmml" xref="A1.I2.i2.p1.5.m5.6.6.6.3.3.3"><csymbol cd="ambiguous" id="A1.I2.i2.p1.5.m5.6.6.6.3.3.3.1.cmml" xref="A1.I2.i2.p1.5.m5.6.6.6.3.3.3">subscript</csymbol><ci id="A1.I2.i2.p1.5.m5.6.6.6.3.3.3.2.cmml" xref="A1.I2.i2.p1.5.m5.6.6.6.3.3.3.2">𝑘</ci><ci id="A1.I2.i2.p1.5.m5.6.6.6.3.3.3.3.cmml" xref="A1.I2.i2.p1.5.m5.6.6.6.3.3.3.3">𝑡</ci></apply></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i2.p1.5.m5.6c">\hat{\mathbf{x}}^{0}_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1},{k_{t}})=\hat{\mathbf{x}}^{0}_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}},\bar{\mathbf{z}}_{t-1},{k_{t}})</annotation></semantics></math><span id="A1.I2.i2.p1.9.6" class="ltx_text" style="font-size:90%;">, where </span><math id="A1.I2.i2.p1.6.m6.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}" display="inline"><semantics id="A1.I2.i2.p1.6.m6.1a"><msub id="A1.I2.i2.p1.6.m6.1.1" xref="A1.I2.i2.p1.6.m6.1.1.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.6.m6.1.1.2" xref="A1.I2.i2.p1.6.m6.1.1.2.cmml">𝐳</mi><mrow id="A1.I2.i2.p1.6.m6.1.1.3" xref="A1.I2.i2.p1.6.m6.1.1.3.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.6.m6.1.1.3.2" xref="A1.I2.i2.p1.6.m6.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A1.I2.i2.p1.6.m6.1.1.3.1" xref="A1.I2.i2.p1.6.m6.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A1.I2.i2.p1.6.m6.1.1.3.3" xref="A1.I2.i2.p1.6.m6.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.I2.i2.p1.6.m6.1b"><apply id="A1.I2.i2.p1.6.m6.1.1.cmml" xref="A1.I2.i2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A1.I2.i2.p1.6.m6.1.1.1.cmml" xref="A1.I2.i2.p1.6.m6.1.1">subscript</csymbol><ci id="A1.I2.i2.p1.6.m6.1.1.2.cmml" xref="A1.I2.i2.p1.6.m6.1.1.2">𝐳</ci><apply id="A1.I2.i2.p1.6.m6.1.1.3.cmml" xref="A1.I2.i2.p1.6.m6.1.1.3"><minus id="A1.I2.i2.p1.6.m6.1.1.3.1.cmml" xref="A1.I2.i2.p1.6.m6.1.1.3.1"></minus><ci id="A1.I2.i2.p1.6.m6.1.1.3.2.cmml" xref="A1.I2.i2.p1.6.m6.1.1.3.2">𝑡</ci><cn type="integer" id="A1.I2.i2.p1.6.m6.1.1.3.3.cmml" xref="A1.I2.i2.p1.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i2.p1.6.m6.1c">\mathbf{z}_{t-1}</annotation></semantics></math><span id="A1.I2.i2.p1.9.7" class="ltx_text" style="font-size:90%;"> is just compressing </span><math id="A1.I2.i2.p1.7.m7.1" class="ltx_Math" alttext="\bar{\mathbf{z}}_{t-1}" display="inline"><semantics id="A1.I2.i2.p1.7.m7.1a"><msub id="A1.I2.i2.p1.7.m7.1.1" xref="A1.I2.i2.p1.7.m7.1.1.cmml"><mover accent="true" id="A1.I2.i2.p1.7.m7.1.1.2" xref="A1.I2.i2.p1.7.m7.1.1.2.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.7.m7.1.1.2.2" xref="A1.I2.i2.p1.7.m7.1.1.2.2.cmml">𝐳</mi><mo mathsize="90%" id="A1.I2.i2.p1.7.m7.1.1.2.1" xref="A1.I2.i2.p1.7.m7.1.1.2.1.cmml">¯</mo></mover><mrow id="A1.I2.i2.p1.7.m7.1.1.3" xref="A1.I2.i2.p1.7.m7.1.1.3.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.7.m7.1.1.3.2" xref="A1.I2.i2.p1.7.m7.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A1.I2.i2.p1.7.m7.1.1.3.1" xref="A1.I2.i2.p1.7.m7.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A1.I2.i2.p1.7.m7.1.1.3.3" xref="A1.I2.i2.p1.7.m7.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.I2.i2.p1.7.m7.1b"><apply id="A1.I2.i2.p1.7.m7.1.1.cmml" xref="A1.I2.i2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A1.I2.i2.p1.7.m7.1.1.1.cmml" xref="A1.I2.i2.p1.7.m7.1.1">subscript</csymbol><apply id="A1.I2.i2.p1.7.m7.1.1.2.cmml" xref="A1.I2.i2.p1.7.m7.1.1.2"><ci id="A1.I2.i2.p1.7.m7.1.1.2.1.cmml" xref="A1.I2.i2.p1.7.m7.1.1.2.1">¯</ci><ci id="A1.I2.i2.p1.7.m7.1.1.2.2.cmml" xref="A1.I2.i2.p1.7.m7.1.1.2.2">𝐳</ci></apply><apply id="A1.I2.i2.p1.7.m7.1.1.3.cmml" xref="A1.I2.i2.p1.7.m7.1.1.3"><minus id="A1.I2.i2.p1.7.m7.1.1.3.1.cmml" xref="A1.I2.i2.p1.7.m7.1.1.3.1"></minus><ci id="A1.I2.i2.p1.7.m7.1.1.3.2.cmml" xref="A1.I2.i2.p1.7.m7.1.1.3.2">𝑡</ci><cn type="integer" id="A1.I2.i2.p1.7.m7.1.1.3.3.cmml" xref="A1.I2.i2.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i2.p1.7.m7.1c">\bar{\mathbf{z}}_{t-1}</annotation></semantics></math><span id="A1.I2.i2.p1.9.8" class="ltx_text" style="font-size:90%;">. When applying the expectation of </span><math id="A1.I2.i2.p1.8.m8.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}\sim q" display="inline"><semantics id="A1.I2.i2.p1.8.m8.1a"><mrow id="A1.I2.i2.p1.8.m8.1.1" xref="A1.I2.i2.p1.8.m8.1.1.cmml"><msub id="A1.I2.i2.p1.8.m8.1.1.2" xref="A1.I2.i2.p1.8.m8.1.1.2.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.8.m8.1.1.2.2" xref="A1.I2.i2.p1.8.m8.1.1.2.2.cmml">𝐱</mi><mrow id="A1.I2.i2.p1.8.m8.1.1.2.3" xref="A1.I2.i2.p1.8.m8.1.1.2.3.cmml"><mn mathsize="90%" id="A1.I2.i2.p1.8.m8.1.1.2.3.2" xref="A1.I2.i2.p1.8.m8.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.I2.i2.p1.8.m8.1.1.2.3.1" xref="A1.I2.i2.p1.8.m8.1.1.2.3.1.cmml">:</mo><mi mathsize="90%" id="A1.I2.i2.p1.8.m8.1.1.2.3.3" xref="A1.I2.i2.p1.8.m8.1.1.2.3.3.cmml">T</mi></mrow></msub><mo mathsize="90%" id="A1.I2.i2.p1.8.m8.1.1.1" xref="A1.I2.i2.p1.8.m8.1.1.1.cmml">∼</mo><mi mathsize="90%" id="A1.I2.i2.p1.8.m8.1.1.3" xref="A1.I2.i2.p1.8.m8.1.1.3.cmml">q</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.I2.i2.p1.8.m8.1b"><apply id="A1.I2.i2.p1.8.m8.1.1.cmml" xref="A1.I2.i2.p1.8.m8.1.1"><csymbol cd="latexml" id="A1.I2.i2.p1.8.m8.1.1.1.cmml" xref="A1.I2.i2.p1.8.m8.1.1.1">similar-to</csymbol><apply id="A1.I2.i2.p1.8.m8.1.1.2.cmml" xref="A1.I2.i2.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="A1.I2.i2.p1.8.m8.1.1.2.1.cmml" xref="A1.I2.i2.p1.8.m8.1.1.2">subscript</csymbol><ci id="A1.I2.i2.p1.8.m8.1.1.2.2.cmml" xref="A1.I2.i2.p1.8.m8.1.1.2.2">𝐱</ci><apply id="A1.I2.i2.p1.8.m8.1.1.2.3.cmml" xref="A1.I2.i2.p1.8.m8.1.1.2.3"><ci id="A1.I2.i2.p1.8.m8.1.1.2.3.1.cmml" xref="A1.I2.i2.p1.8.m8.1.1.2.3.1">:</ci><cn type="integer" id="A1.I2.i2.p1.8.m8.1.1.2.3.2.cmml" xref="A1.I2.i2.p1.8.m8.1.1.2.3.2">1</cn><ci id="A1.I2.i2.p1.8.m8.1.1.2.3.3.cmml" xref="A1.I2.i2.p1.8.m8.1.1.2.3.3">𝑇</ci></apply></apply><ci id="A1.I2.i2.p1.8.m8.1.1.3.cmml" xref="A1.I2.i2.p1.8.m8.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i2.p1.8.m8.1c">\mathbf{x}_{1:T}\sim q</annotation></semantics></math><span id="A1.I2.i2.p1.9.9" class="ltx_text" style="font-size:90%;"> to both sides of the bound in </span><a href="#A1.Thmtheorem2" title="Corollary A.2. ‣ A.1.1 Specializing to Gaussian diffusion ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">A.2</span></a><span id="A1.I2.i2.p1.9.10" class="ltx_text" style="font-size:90%;">, and taking an infimum over possible function approximator </span><math id="A1.I2.i2.p1.9.m9.1" class="ltx_Math" alttext="\hat{\mathbf{x}}^{0}_{\theta}" display="inline"><semantics id="A1.I2.i2.p1.9.m9.1a"><msubsup id="A1.I2.i2.p1.9.m9.1.1" xref="A1.I2.i2.p1.9.m9.1.1.cmml"><mover accent="true" id="A1.I2.i2.p1.9.m9.1.1.2.2" xref="A1.I2.i2.p1.9.m9.1.1.2.2.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.9.m9.1.1.2.2.2" xref="A1.I2.i2.p1.9.m9.1.1.2.2.2.cmml">𝐱</mi><mo mathsize="90%" id="A1.I2.i2.p1.9.m9.1.1.2.2.1" xref="A1.I2.i2.p1.9.m9.1.1.2.2.1.cmml">^</mo></mover><mi mathsize="90%" id="A1.I2.i2.p1.9.m9.1.1.3" xref="A1.I2.i2.p1.9.m9.1.1.3.cmml">θ</mi><mn mathsize="90%" id="A1.I2.i2.p1.9.m9.1.1.2.3" xref="A1.I2.i2.p1.9.m9.1.1.2.3.cmml">0</mn></msubsup><annotation-xml encoding="MathML-Content" id="A1.I2.i2.p1.9.m9.1b"><apply id="A1.I2.i2.p1.9.m9.1.1.cmml" xref="A1.I2.i2.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A1.I2.i2.p1.9.m9.1.1.1.cmml" xref="A1.I2.i2.p1.9.m9.1.1">subscript</csymbol><apply id="A1.I2.i2.p1.9.m9.1.1.2.cmml" xref="A1.I2.i2.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A1.I2.i2.p1.9.m9.1.1.2.1.cmml" xref="A1.I2.i2.p1.9.m9.1.1">superscript</csymbol><apply id="A1.I2.i2.p1.9.m9.1.1.2.2.cmml" xref="A1.I2.i2.p1.9.m9.1.1.2.2"><ci id="A1.I2.i2.p1.9.m9.1.1.2.2.1.cmml" xref="A1.I2.i2.p1.9.m9.1.1.2.2.1">^</ci><ci id="A1.I2.i2.p1.9.m9.1.1.2.2.2.cmml" xref="A1.I2.i2.p1.9.m9.1.1.2.2.2">𝐱</ci></apply><cn type="integer" id="A1.I2.i2.p1.9.m9.1.1.2.3.cmml" xref="A1.I2.i2.p1.9.m9.1.1.2.3">0</cn></apply><ci id="A1.I2.i2.p1.9.m9.1.1.3.cmml" xref="A1.I2.i2.p1.9.m9.1.1.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i2.p1.9.m9.1c">\hat{\mathbf{x}}^{0}_{\theta}</annotation></semantics></math><span id="A1.I2.i2.p1.9.11" class="ltx_text" style="font-size:90%;">, we obtain</span></p>
<table id="A6.EGx10" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.Ex6.m1.3" class="ltx_Math" alttext="\displaystyle\inf_{p_{\theta}}\mathop{{\mathbb{E}}}_{q}\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\|\hat{\mathbf{x}}^{0}_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1},{k_{t}})-\mathbf{x}_{t}^{0}\|^{2}" display="inline"><semantics id="A1.Ex6.m1.3a"><mrow id="A1.Ex6.m1.3.3" xref="A1.Ex6.m1.3.3.cmml"><munder id="A1.Ex6.m1.3.3.2" xref="A1.Ex6.m1.3.3.2.cmml"><mo mathsize="90%" movablelimits="false" id="A1.Ex6.m1.3.3.2.2" xref="A1.Ex6.m1.3.3.2.2.cmml">inf</mo><msub id="A1.Ex6.m1.3.3.2.3" xref="A1.Ex6.m1.3.3.2.3.cmml"><mi mathsize="90%" id="A1.Ex6.m1.3.3.2.3.2" xref="A1.Ex6.m1.3.3.2.3.2.cmml">p</mi><mi mathsize="90%" id="A1.Ex6.m1.3.3.2.3.3" xref="A1.Ex6.m1.3.3.2.3.3.cmml">θ</mi></msub></munder><mrow id="A1.Ex6.m1.3.3.1" xref="A1.Ex6.m1.3.3.1.cmml"><munder id="A1.Ex6.m1.3.3.1.2" xref="A1.Ex6.m1.3.3.1.2.cmml"><mo lspace="0.167em" mathsize="108%" 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xref="A1.Ex6.m2.3.3.1.1.1.1.1.1.1.1.4.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex6.m2.3.3.1.1.1.1.1.1.1.1.4.3.cmml" xref="A1.Ex6.m2.3.3.1.1.1.1.1.1.1.1.4.3">0</cn></apply></apply></apply><cn type="integer" id="A1.Ex6.m2.3.3.1.1.1.1.1.3.cmml" xref="A1.Ex6.m2.3.3.1.1.1.1.1.3">2</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex6.m2.3c">\displaystyle=\inf_{p_{\theta}}\mathop{{\mathbb{E}}}_{q}\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\|\hat{\mathbf{x}}^{0}_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}},\bar{\mathbf{z}}_{t-1})-\mathbf{x}_{t}^{0}\|^{2}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.I2.i2.p1.10" class="ltx_p"><span id="A1.I2.i2.p1.10.1" class="ltx_text" style="font-size:90%;">This leads to a striking finding: with expressive enough latents and </span><math id="A1.I2.i2.p1.10.m1.1" class="ltx_Math" alttext="p_{\theta}" display="inline"><semantics id="A1.I2.i2.p1.10.m1.1a"><msub id="A1.I2.i2.p1.10.m1.1.1" xref="A1.I2.i2.p1.10.m1.1.1.cmml"><mi mathsize="90%" id="A1.I2.i2.p1.10.m1.1.1.2" xref="A1.I2.i2.p1.10.m1.1.1.2.cmml">p</mi><mi mathsize="90%" id="A1.I2.i2.p1.10.m1.1.1.3" xref="A1.I2.i2.p1.10.m1.1.1.3.cmml">θ</mi></msub><annotation-xml encoding="MathML-Content" id="A1.I2.i2.p1.10.m1.1b"><apply id="A1.I2.i2.p1.10.m1.1.1.cmml" xref="A1.I2.i2.p1.10.m1.1.1"><csymbol cd="ambiguous" id="A1.I2.i2.p1.10.m1.1.1.1.cmml" xref="A1.I2.i2.p1.10.m1.1.1">subscript</csymbol><ci id="A1.I2.i2.p1.10.m1.1.1.2.cmml" xref="A1.I2.i2.p1.10.m1.1.1.2">𝑝</ci><ci id="A1.I2.i2.p1.10.m1.1.1.3.cmml" xref="A1.I2.i2.p1.10.m1.1.1.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i2.p1.10.m1.1c">p_{\theta}</annotation></semantics></math><span id="A1.I2.i2.p1.10.2" class="ltx_text" style="font-size:90%;">, we can view the maximization of each term in </span><a href="#A1.Thmtheorem2" title="Corollary A.2. ‣ A.1.1 Specializing to Gaussian diffusion ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">A.2</span></a><span id="A1.I2.i2.p1.10.3" class="ltx_text" style="font-size:90%;"> separately across time steps. The absence of this coupling means that the weighting terms are immaterial to the optimization, and thus can be ignored.</span></p>
</div>
</li>
<li id="A1.I2.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="A1.I2.i3.p1" class="ltx_para">
<p id="A1.I2.i3.p1.1" class="ltx_p"><span id="A1.I2.i3.p1.1.1" class="ltx_text" style="font-size:90%;">Given the above remarks, we can optimize the ELBO by taking gradients through the objective specified by </span><a href="#A1.Thmtheorem2" title="Corollary A.2. ‣ A.1.1 Specializing to Gaussian diffusion ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">A.2</span></a><span id="A1.I2.i3.p1.1.2" class="ltx_text" style="font-size:90%;">, and are free to drop any weighting terms (or rescale them) as desired. Backpropogating through </span><math id="A1.I2.i3.p1.1.m1.2" class="ltx_Math" alttext="\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}" display="inline"><semantics id="A1.I2.i3.p1.1.m1.2a"><msub id="A1.I2.i3.p1.1.m1.2.3" xref="A1.I2.i3.p1.1.m1.2.3.cmml"><mo mathsize="108%" id="A1.I2.i3.p1.1.m1.2.3.2" xref="A1.I2.i3.p1.1.m1.2.3.2.cmml">𝔼</mo><mrow id="A1.I2.i3.p1.1.m1.2.2.2.2" xref="A1.I2.i3.p1.1.m1.2.2.2.3.cmml"><mi mathsize="90%" id="A1.I2.i3.p1.1.m1.1.1.1.1" xref="A1.I2.i3.p1.1.m1.1.1.1.1.cmml">p</mi><mo mathsize="90%" id="A1.I2.i3.p1.1.m1.2.2.2.2.2" xref="A1.I2.i3.p1.1.m1.2.2.2.3.cmml">,</mo><msub id="A1.I2.i3.p1.1.m1.2.2.2.2.1" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.cmml"><mi mathsize="90%" id="A1.I2.i3.p1.1.m1.2.2.2.2.1.2" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.2.cmml">𝐳</mi><mrow id="A1.I2.i3.p1.1.m1.2.2.2.2.1.3" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.cmml"><mn mathsize="90%" id="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.2" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.1" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.1.cmml">:</mo><mi mathsize="90%" id="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.3" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.3.cmml">T</mi></mrow></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.I2.i3.p1.1.m1.2b"><apply id="A1.I2.i3.p1.1.m1.2.3.cmml" xref="A1.I2.i3.p1.1.m1.2.3"><csymbol cd="ambiguous" id="A1.I2.i3.p1.1.m1.2.3.1.cmml" xref="A1.I2.i3.p1.1.m1.2.3">subscript</csymbol><ci id="A1.I2.i3.p1.1.m1.2.3.2.cmml" xref="A1.I2.i3.p1.1.m1.2.3.2">𝔼</ci><list id="A1.I2.i3.p1.1.m1.2.2.2.3.cmml" xref="A1.I2.i3.p1.1.m1.2.2.2.2"><ci id="A1.I2.i3.p1.1.m1.1.1.1.1.cmml" xref="A1.I2.i3.p1.1.m1.1.1.1.1">𝑝</ci><apply id="A1.I2.i3.p1.1.m1.2.2.2.2.1.cmml" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="A1.I2.i3.p1.1.m1.2.2.2.2.1.1.cmml" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1">subscript</csymbol><ci id="A1.I2.i3.p1.1.m1.2.2.2.2.1.2.cmml" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.2">𝐳</ci><apply id="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.cmml" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.3"><ci id="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.1.cmml" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.1">:</ci><cn type="integer" id="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.2.cmml" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.2">1</cn><ci id="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.3.cmml" xref="A1.I2.i3.p1.1.m1.2.2.2.2.1.3.3">𝑇</ci></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.I2.i3.p1.1.m1.2c">\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}</annotation></semantics></math><span id="A1.I2.i3.p1.1.3" class="ltx_text" style="font-size:90%;"> is straightforward due to deterministic latents. This justifies the correctness of our training objective (</span><a href="#S3.E1" title="Equation 3.1 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">3.1</span></a><span id="A1.I2.i3.p1.1.4" class="ltx_text" style="font-size:90%;">) and protocol </span><a href="#alg1" title="In Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">1</span></a><span id="A1.I2.i3.p1.1.5" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
</li>
</ul>
</div>
</section>
<section id="A1.SS1.SSS2" class="ltx_subsubsection">
<h4 class="ltx_title ltx_title_subsubsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsubsection">A.1.2 </span>Capturing all subsequences</h4>

<div id="A1.SS1.SSS2.p1" class="ltx_para">
<p id="A1.SS1.SSS2.p1.1" class="ltx_p"><a href="#A1.Thmtheorem1" title="Theorem A.1. ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">A.1</span></a><span id="A1.SS1.SSS2.p1.1.1" class="ltx_text" style="font-size:90%;"> stipulates that, up to reweighting, the Diffusion Forcing objective optimizes a valid ELBO on the expected log-likelihoods over uniformly sampled noise levels. The following theorem can be obtained by a straightforward modification of the proof of </span><a href="#A1.Thmtheorem1" title="Theorem A.1. ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">A.1</span></a><span id="A1.SS1.SSS2.p1.1.2" class="ltx_text" style="font-size:90%;"> generalizes this to arbitrary (possible temporally correlated) sequences of noise.</span></p>
</div>
<div id="A1.Thmtheorem3" class="ltx_theorem ltx_theorem_theorem">
<h6 class="ltx_title ltx_runin ltx_title_theorem" style="font-size:90%;">
<span class="ltx_tag ltx_tag_theorem"><span id="A1.Thmtheorem3.1.1.1" class="ltx_text ltx_font_bold">Theorem A.3</span></span><span id="A1.Thmtheorem3.2.2" class="ltx_text ltx_font_bold">.</span>
</h6>
<div id="A1.Thmtheorem3.p1" class="ltx_para">
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xref="A1.Thmtheorem3.p1.2.2.m2.1.2.2.1.cmml"><mo stretchy="false" id="A1.Thmtheorem3.p1.2.2.m2.1.2.2.2.1" xref="A1.Thmtheorem3.p1.2.2.m2.1.2.2.1.1.cmml">[</mo><mi id="A1.Thmtheorem3.p1.2.2.m2.1.1" xref="A1.Thmtheorem3.p1.2.2.m2.1.1.cmml">K</mi><mo stretchy="false" id="A1.Thmtheorem3.p1.2.2.m2.1.2.2.2.2" xref="A1.Thmtheorem3.p1.2.2.m2.1.2.2.1.1.cmml">]</mo></mrow><mi id="A1.Thmtheorem3.p1.2.2.m2.1.2.3" xref="A1.Thmtheorem3.p1.2.2.m2.1.2.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.2.2.m2.1b"><apply id="A1.Thmtheorem3.p1.2.2.m2.1.2.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.2"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.2.2.m2.1.2.1.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.2">superscript</csymbol><apply id="A1.Thmtheorem3.p1.2.2.m2.1.2.2.1.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.2.2.2"><csymbol cd="latexml" id="A1.Thmtheorem3.p1.2.2.m2.1.2.2.1.1.cmml" xref="A1.Thmtheorem3.p1.2.2.m2.1.2.2.2.1">delimited-[]</csymbol><ci 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xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.3.cmml"><msub id="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1" xref="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.cmml"><mi id="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.2" xref="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.2.cmml">Pr</mi><mi class="ltx_font_mathcaligraphic" id="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.3" xref="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.3.cmml">𝒟</mi></msub><mo id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2a" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.3.cmml">⁡</mo><mrow id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.3.cmml"><mo stretchy="false" id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.2" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.3.cmml">[</mo><mrow id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.cmml"><msub id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.cmml"><mi id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.2" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.2.cmml">k</mi><mi id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.3" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.3.cmml">t</mi></msub><mo id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.1" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.1.cmml">=</mo><mrow id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.cmml"><mi id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.2" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.2.cmml">j</mi><mo id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.1" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.1.cmml">∣</mo><msub id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.cmml"><mi id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.2" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.2.cmml">k</mi><mrow id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.cmml"><mn id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.2" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.1" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.1.cmml">:</mo><mrow id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.cmml"><mi id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.2" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.2.cmml">t</mi><mo id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.1" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.1.cmml">−</mo><mn id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.3" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.3.cmml">1</mn></mrow></mrow></msub></mrow></mrow><mo stretchy="false" id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.3" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.3.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.3.3.m3.3b"><apply id="A1.Thmtheorem3.p1.3.3.m3.3.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3"><csymbol cd="latexml" id="A1.Thmtheorem3.p1.3.3.m3.3.3.4.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.4">assign</csymbol><apply id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1"><times id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.2"></times><apply id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.3.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.3">subscript</csymbol><ci id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.3.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.3.2">𝑃</ci><ci id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.3.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.3.3">𝑡</ci></apply><apply id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1"><csymbol cd="latexml" id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.1">conditional</csymbol><ci id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.2">𝑗</ci><apply id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.2">𝑘</ci><apply id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3"><ci id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.1">:</ci><cn type="integer" id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.2">1</cn><apply id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.3"><minus id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.3.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.3.1"></minus><ci id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.3.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.3.2">𝑡</ci><cn type="integer" id="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.3.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.1.1.1.1.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></apply><apply id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2"><apply id="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1">subscript</csymbol><csymbol cd="latexml" id="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.2">probability</csymbol><ci id="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.2.2.2.1.1.3">𝒟</ci></apply><apply id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1"><eq id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.1"></eq><apply id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2">subscript</csymbol><ci id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.2">𝑘</ci><ci id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.2.3">𝑡</ci></apply><apply id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3"><csymbol cd="latexml" id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.1">conditional</csymbol><ci id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.2">𝑗</ci><apply id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3">subscript</csymbol><ci id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.2">𝑘</ci><apply id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3"><ci id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.1">:</ci><cn type="integer" id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.2">1</cn><apply id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3"><minus id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.1.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.1"></minus><ci id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.2.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.2">𝑡</ci><cn type="integer" id="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.3.cmml" xref="A1.Thmtheorem3.p1.3.3.m3.3.3.3.2.2.1.3.3.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.3.3.m3.3c">P_{t}(j\mid k_{1:t-1}):=\Pr_{\mathcal{D}}[k_{t}=j\mid k_{1:t-1}]</annotation></semantics></math>.
Fix <math id="A1.Thmtheorem3.p1.4.4.m4.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}^{0}" display="inline"><semantics id="A1.Thmtheorem3.p1.4.4.m4.1a"><msubsup id="A1.Thmtheorem3.p1.4.4.m4.1.1" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.cmml"><mi id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.2" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.2.2.cmml">𝐱</mi><mrow id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.cmml"><mn id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.2" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.1" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.1.cmml">:</mo><mi id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.3" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.3.cmml">T</mi></mrow><mn id="A1.Thmtheorem3.p1.4.4.m4.1.1.3" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.3.cmml">0</mn></msubsup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.4.4.m4.1b"><apply id="A1.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.4.4.m4.1.1.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1">superscript</csymbol><apply id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1">subscript</csymbol><ci id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.2.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.2.2">𝐱</ci><apply id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3"><ci id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.1.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.1">:</ci><cn type="integer" id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.2.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.2">1</cn><ci id="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.3.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.2.3.3">𝑇</ci></apply></apply><cn type="integer" id="A1.Thmtheorem3.p1.4.4.m4.1.1.3.cmml" xref="A1.Thmtheorem3.p1.4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.4.4.m4.1c">\mathbf{x}_{1:T}^{0}</annotation></semantics></math>. Define the expectation over the forward process with random noise level <math id="A1.Thmtheorem3.p1.5.5.m5.1" class="ltx_Math" alttext="k_{1:T}" display="inline"><semantics id="A1.Thmtheorem3.p1.5.5.m5.1a"><msub id="A1.Thmtheorem3.p1.5.5.m5.1.1" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.cmml"><mi id="A1.Thmtheorem3.p1.5.5.m5.1.1.2" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.2.cmml">k</mi><mrow id="A1.Thmtheorem3.p1.5.5.m5.1.1.3" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.3.cmml"><mn id="A1.Thmtheorem3.p1.5.5.m5.1.1.3.2" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="A1.Thmtheorem3.p1.5.5.m5.1.1.3.1" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.3.1.cmml">:</mo><mi id="A1.Thmtheorem3.p1.5.5.m5.1.1.3.3" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.5.5.m5.1b"><apply id="A1.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.5.5.m5.1.1.1.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.1">subscript</csymbol><ci id="A1.Thmtheorem3.p1.5.5.m5.1.1.2.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.2">𝑘</ci><apply id="A1.Thmtheorem3.p1.5.5.m5.1.1.3.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.3"><ci id="A1.Thmtheorem3.p1.5.5.m5.1.1.3.1.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.3.1">:</ci><cn type="integer" id="A1.Thmtheorem3.p1.5.5.m5.1.1.3.2.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.3.2">1</cn><ci id="A1.Thmtheorem3.p1.5.5.m5.1.1.3.3.cmml" xref="A1.Thmtheorem3.p1.5.5.m5.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.5.5.m5.1c">k_{1:T}</annotation></semantics></math> as</span></p>
<table id="A6.EGx11" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.E7.m1.10.10.1.1.3.2.2.2"><csymbol cd="latexml" id="A1.E7.m1.10.10.1.1.3.2.2.1.1.cmml" xref="A1.E7.m1.10.10.1.1.3.2.2.2.1">delimited-[]</csymbol><ci id="A1.E7.m1.9.9.cmml" xref="A1.E7.m1.9.9">⋅</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E7.m1.10c">\displaystyle\mathop{{\mathbb{E}}}_{\mathrm{forward},\mathcal{D}}[\cdot]:=\mathop{{\mathbb{E}}}_{k_{1},\dots,k_{T}\sim\mathcal{D}}\mathop{{\mathbb{E}}}_{\mathbf{x}_{s}^{k_{s}}\sim q(\mathbf{x}_{s}^{k_{s}}\mid\mathbf{x}_{s}^{0}),1\leq s\leq T}[\cdot],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.7)</span></td>
</tr></tbody>
</table>
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<table id="A6.EGx12" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.4.2">𝐳</mi><mi mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.4.3">s</mi></msub><mo mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.3">∣</mo><mrow id="A1.E8.m1.3.3.1.1.1.1.1.1.1.2.2"><msub id="A1.E8.m1.3.3.1.1.1.1.1.1.1.1.1.1"><mi mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.1.1.1.2">𝐳</mi><mrow id="A1.E8.m1.3.3.1.1.1.1.1.1.1.1.1.1.3"><mi mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.1.1.1.3.2">s</mi><mo mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.1.1.1.3.1">−</mo><mn mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.1.1.1.3.3">1</mn></mrow></msub><mo mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.2.2.3">,</mo><msubsup id="A1.E8.m1.3.3.1.1.1.1.1.1.1.2.2.2"><mi mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.2.2.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.2.2.2.2.3">s</mi><msub id="A1.E8.m1.3.3.1.1.1.1.1.1.1.2.2.2.3"><mi mathsize="90%" id="A1.E8.m1.3.3.1.1.1.1.1.1.1.2.2.2.3.2">k</mi><mi mathsize="90%" 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id="A1.E8.m1.5.10.6"><mrow id="A1.E8.m1.5.10.6.2"><mo maxsize="90%" minsize="90%" id="A1.E8.m1.5.10.6.2.1">(</mo><msubsup id="A1.E8.m1.5.10.6.2.2"><mi mathsize="90%" id="A1.E8.m1.5.10.6.2.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.E8.m1.5.10.6.2.2.2.3">t</mi><msub id="A1.E8.m1.5.10.6.2.2.3"><mi mathsize="90%" id="A1.E8.m1.5.10.6.2.2.3.2">k</mi><mi mathsize="90%" id="A1.E8.m1.5.10.6.2.2.3.3">t</mi></msub></msubsup><mo maxsize="90%" minsize="90%" id="A1.E8.m1.5.10.6.2.3">)</mo></mrow><mrow id="A1.E8.m1.5.10.6.3"><mn mathsize="90%" id="A1.E8.m1.5.10.6.3.2">1</mn><mo mathsize="90%" id="A1.E8.m1.5.10.6.3.3">≤</mo><mi mathsize="90%" id="A1.E8.m1.5.10.6.3.4">t</mi><mo mathsize="90%" id="A1.E8.m1.5.10.6.3.5">≤</mo><mi mathsize="90%" id="A1.E8.m1.5.10.6.3.6">T</mi></mrow></msub><mo id="A1.E8.m1.5.10.7">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="A1.E8.m1.5c">\displaystyle\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}[\cdot]:=\mathop{{\mathbb{E}}}_{\mathbf{z}_{s}\sim p(\mathbf{z}_{s}\mid\mathbf{z}_{s-1},\mathbf{x}_{s}^{k_{s}}),s\leq T}\left[\cdot\mid k_{1:T},(\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T}\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.8)</span></td>
</tr></tbody>
</table>
<p id="A1.Thmtheorem3.p1.8" class="ltx_p"><span id="A1.Thmtheorem3.p1.8.1" class="ltx_text ltx_font_italic" style="font-size:90%;">Then, as long as <math id="A1.Thmtheorem3.p1.8.1.m1.1" class="ltx_Math" alttext="p_{\bm{\theta}}" display="inline"><semantics id="A1.Thmtheorem3.p1.8.1.m1.1a"><msub id="A1.Thmtheorem3.p1.8.1.m1.1.1" xref="A1.Thmtheorem3.p1.8.1.m1.1.1.cmml"><mi id="A1.Thmtheorem3.p1.8.1.m1.1.1.2" xref="A1.Thmtheorem3.p1.8.1.m1.1.1.2.cmml">p</mi><mi id="A1.Thmtheorem3.p1.8.1.m1.1.1.3" xref="A1.Thmtheorem3.p1.8.1.m1.1.1.3.cmml">𝛉</mi></msub><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.8.1.m1.1b"><apply id="A1.Thmtheorem3.p1.8.1.m1.1.1.cmml" xref="A1.Thmtheorem3.p1.8.1.m1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.8.1.m1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.8.1.m1.1.1">subscript</csymbol><ci id="A1.Thmtheorem3.p1.8.1.m1.1.1.2.cmml" xref="A1.Thmtheorem3.p1.8.1.m1.1.1.2">𝑝</ci><ci id="A1.Thmtheorem3.p1.8.1.m1.1.1.3.cmml" xref="A1.Thmtheorem3.p1.8.1.m1.1.1.3">𝛉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.8.1.m1.1c">p_{\bm{\theta}}</annotation></semantics></math> satisfies the Markov property,</span></p>
<table id="A6.EGx13" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex8.m1.9" class="ltx_Math" alttext="\displaystyle\mathop{{\mathbb{E}}}_{\mathrm{forward},\mathcal{D}}[\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})]\geq C(\mathbf{x}_{1:T}^{0})+\mathop{{\mathbb{E}}}_{\mathrm{forward},\mathcal{D}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\Xi_{t}\right],\text{where }" display="inline"><semantics id="A1.Ex8.m1.9a"><mrow id="A1.Ex8.m1.9.9" xref="A1.Ex8.m1.9.9.cmml"><mrow id="A1.Ex8.m1.8.8.1" xref="A1.Ex8.m1.8.8.1.cmml"><munder id="A1.Ex8.m1.8.8.1.2" xref="A1.Ex8.m1.8.8.1.2.cmml"><mo mathsize="108%" movablelimits="false" id="A1.Ex8.m1.8.8.1.2.2" xref="A1.Ex8.m1.8.8.1.2.2.cmml">𝔼</mo><mrow id="A1.Ex8.m1.2.2.2.4" xref="A1.Ex8.m1.2.2.2.3.cmml"><mi mathsize="90%" id="A1.Ex8.m1.1.1.1.1" xref="A1.Ex8.m1.1.1.1.1.cmml">forward</mi><mo mathsize="90%" id="A1.Ex8.m1.2.2.2.4.1" xref="A1.Ex8.m1.2.2.2.3.cmml">,</mo><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A1.Ex8.m1.2.2.2.2" xref="A1.Ex8.m1.2.2.2.2.cmml">𝒟</mi></mrow></munder><mrow id="A1.Ex8.m1.8.8.1.1.1" xref="A1.Ex8.m1.8.8.1.1.2.cmml"><mo lspace="0em" maxsize="90%" minsize="90%" id="A1.Ex8.m1.8.8.1.1.1.2" xref="A1.Ex8.m1.8.8.1.1.2.1.cmml">[</mo><mrow id="A1.Ex8.m1.8.8.1.1.1.1" xref="A1.Ex8.m1.8.8.1.1.1.1.cmml"><mrow id="A1.Ex8.m1.8.8.1.1.1.1.3" xref="A1.Ex8.m1.8.8.1.1.1.1.3.cmml"><mi mathsize="90%" id="A1.Ex8.m1.8.8.1.1.1.1.3.1" xref="A1.Ex8.m1.8.8.1.1.1.1.3.1.cmml">ln</mi><mo lspace="0.167em" id="A1.Ex8.m1.8.8.1.1.1.1.3a" xref="A1.Ex8.m1.8.8.1.1.1.1.3.cmml">⁡</mo><msub id="A1.Ex8.m1.8.8.1.1.1.1.3.2" xref="A1.Ex8.m1.8.8.1.1.1.1.3.2.cmml"><mi mathsize="90%" id="A1.Ex8.m1.8.8.1.1.1.1.3.2.2" xref="A1.Ex8.m1.8.8.1.1.1.1.3.2.2.cmml">p</mi><mi mathsize="90%" id="A1.Ex8.m1.8.8.1.1.1.1.3.2.3" xref="A1.Ex8.m1.8.8.1.1.1.1.3.2.3.cmml">𝜽</mi></msub></mrow><mo lspace="0em" rspace="0em" id="A1.Ex8.m1.8.8.1.1.1.1.2" xref="A1.Ex8.m1.8.8.1.1.1.1.2.cmml">​</mo><mrow id="A1.Ex8.m1.8.8.1.1.1.1.1.1" xref="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.Ex8.m1.8.8.1.1.1.1.1.1.2" xref="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.cmml">(</mo><msub id="A1.Ex8.m1.8.8.1.1.1.1.1.1.1" xref="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.cmml"><mrow id="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1" xref="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.2" xref="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1" xref="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.2.2" xref="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.2.3" xref="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.2.3.cmml">t</mi><msub id="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.3" xref="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.3.2" xref="A1.Ex8.m1.8.8.1.1.1.1.1.1.1.1.1.1.3.2.cmml">k</mi><mi 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xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1">superscript</csymbol><apply id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.1.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1">subscript</csymbol><sum id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.2.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.2"></sum><apply id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.3.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.3"><eq id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.3.1.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.3.1"></eq><ci id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.3.2.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.3.2">𝑡</ci><cn type="integer" id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.3.3.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.3.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.1.3">𝑇</ci></apply><apply id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.2.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.2.1.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.2">subscript</csymbol><ci id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.2.2.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.2.2">Ξ</ci><ci id="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.2.3.cmml" xref="A1.Ex8.m1.9.9.2.1.1.2.1.1.1.1.2.3">𝑡</ci></apply></apply></apply></apply></apply></apply><ci id="A1.Ex8.m1.7.7a.cmml" xref="A1.Ex8.m1.7.7"><mtext class="ltx_mathvariant_italic" mathsize="90%" id="A1.Ex8.m1.7.7.cmml" xref="A1.Ex8.m1.7.7">where</mtext></ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex8.m1.9c">\displaystyle\mathop{{\mathbb{E}}}_{\mathrm{forward},\mathcal{D}}[\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})]\geq C(\mathbf{x}_{1:T}^{0})+\mathop{{\mathbb{E}}}_{\mathrm{forward},\mathcal{D}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\Xi_{t}\right],\text{where }</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex9.m1.1" class="ltx_math_unparsed" alttext="\displaystyle\Xi_{t}:=\left(P_{t}(1\mid k_{1:t-1})\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=2}^{K}jP_{t}(j\mid k_{1:t-1})\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{j},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j-1},\mathbf{z}_{t-1})\right)\right)," display="inline"><semantics id="A1.Ex9.m1.1a"><mrow id="A1.Ex9.m1.1b"><msub id="A1.Ex9.m1.1.1"><mi mathsize="90%" mathvariant="normal" id="A1.Ex9.m1.1.1.2">Ξ</mi><mi mathsize="90%" id="A1.Ex9.m1.1.1.3">t</mi></msub><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.Ex9.m1.1.2">:=</mo><mrow id="A1.Ex9.m1.1.3"><mo id="A1.Ex9.m1.1.3.1">(</mo><msub id="A1.Ex9.m1.1.3.2"><mi mathsize="90%" id="A1.Ex9.m1.1.3.2.2">P</mi><mi mathsize="90%" id="A1.Ex9.m1.1.3.2.3">t</mi></msub><mrow id="A1.Ex9.m1.1.3.3"><mo maxsize="90%" 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maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.Ex9.m1.1.3.8.2.2">∑</mo><mrow id="A1.Ex9.m1.1.3.8.2.3"><mi mathsize="90%" id="A1.Ex9.m1.1.3.8.2.3.2">j</mi><mo mathsize="90%" id="A1.Ex9.m1.1.3.8.2.3.1">=</mo><mn mathsize="90%" id="A1.Ex9.m1.1.3.8.2.3.3">2</mn></mrow><mi mathsize="90%" id="A1.Ex9.m1.1.3.8.3">K</mi></munderover></mstyle><mi mathsize="90%" id="A1.Ex9.m1.1.3.9">j</mi><msub id="A1.Ex9.m1.1.3.10"><mi mathsize="90%" id="A1.Ex9.m1.1.3.10.2">P</mi><mi mathsize="90%" id="A1.Ex9.m1.1.3.10.3">t</mi></msub><mrow id="A1.Ex9.m1.1.3.11"><mo maxsize="90%" minsize="90%" id="A1.Ex9.m1.1.3.11.1">(</mo><mi mathsize="90%" id="A1.Ex9.m1.1.3.11.2">j</mi><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex9.m1.1.3.11.3">∣</mo><msub id="A1.Ex9.m1.1.3.11.4"><mi mathsize="90%" id="A1.Ex9.m1.1.3.11.4.2">k</mi><mrow id="A1.Ex9.m1.1.3.11.4.3"><mn mathsize="90%" id="A1.Ex9.m1.1.3.11.4.3.2">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.Ex9.m1.1.3.11.4.3.1">:</mo><mrow id="A1.Ex9.m1.1.3.11.4.3.3"><mi mathsize="90%" id="A1.Ex9.m1.1.3.11.4.3.3.2">t</mi><mo mathsize="90%" id="A1.Ex9.m1.1.3.11.4.3.3.1">−</mo><mn mathsize="90%" id="A1.Ex9.m1.1.3.11.4.3.3.3">1</mn></mrow></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex9.m1.1.3.11.5">)</mo></mrow><msub id="A1.Ex9.m1.1.3.12"><mi mathsize="90%" mathvariant="normal" id="A1.Ex9.m1.1.3.12.2">D</mi><mrow id="A1.Ex9.m1.1.3.12.3"><mi mathsize="90%" id="A1.Ex9.m1.1.3.12.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex9.m1.1.3.12.3.1">​</mo><mi mathsize="90%" id="A1.Ex9.m1.1.3.12.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex9.m1.1.3.13"><mo id="A1.Ex9.m1.1.3.13.1">(</mo><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.2">q</mi><mrow id="A1.Ex9.m1.1.3.13.3"><mo maxsize="90%" minsize="90%" id="A1.Ex9.m1.1.3.13.3.1">(</mo><msubsup id="A1.Ex9.m1.1.3.13.3.2"><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.3.2.2.3">t</mi><mrow id="A1.Ex9.m1.1.3.13.3.2.3"><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.3.2.3.2">j</mi><mo mathsize="90%" id="A1.Ex9.m1.1.3.13.3.2.3.1">−</mo><mn mathsize="90%" id="A1.Ex9.m1.1.3.13.3.2.3.3">1</mn></mrow></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex9.m1.1.3.13.3.3">∣</mo><msubsup id="A1.Ex9.m1.1.3.13.3.4"><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.3.4.2.3">t</mi><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.3.4.3">j</mi></msubsup><mo mathsize="90%" id="A1.Ex9.m1.1.3.13.3.5">,</mo><msubsup id="A1.Ex9.m1.1.3.13.3.6"><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex9.m1.1.3.13.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex9.m1.1.3.13.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex9.m1.1.3.13.4">∥</mo><msub id="A1.Ex9.m1.1.3.13.5"><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.5.2">p</mi><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.5.3">𝜽</mi></msub><mrow id="A1.Ex9.m1.1.3.13.6"><mo maxsize="90%" minsize="90%" id="A1.Ex9.m1.1.3.13.6.1">(</mo><msubsup id="A1.Ex9.m1.1.3.13.6.2"><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex9.m1.1.3.13.6.3">∣</mo><msubsup id="A1.Ex9.m1.1.3.13.6.4"><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.6.4.2.3">t</mi><mrow id="A1.Ex9.m1.1.3.13.6.4.3"><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex9.m1.1.3.13.6.4.3.1">−</mo><mn mathsize="90%" id="A1.Ex9.m1.1.3.13.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex9.m1.1.3.13.6.5">,</mo><msub id="A1.Ex9.m1.1.3.13.6.6"><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.6.6.2">𝐳</mi><mrow id="A1.Ex9.m1.1.3.13.6.6.3"><mi mathsize="90%" id="A1.Ex9.m1.1.3.13.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex9.m1.1.3.13.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex9.m1.1.3.13.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex9.m1.1.3.13.6.7">)</mo></mrow><mo id="A1.Ex9.m1.1.3.13.7">)</mo></mrow><mo id="A1.Ex9.m1.1.3.14">)</mo></mrow><mo mathsize="90%" id="A1.Ex9.m1.1.4">,</mo></mrow><annotation encoding="application/x-tex" id="A1.Ex9.m1.1c">\displaystyle\Xi_{t}:=\left(P_{t}(1\mid k_{1:t-1})\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=2}^{K}jP_{t}(j\mid k_{1:t-1})\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{j},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j-1},\mathbf{z}_{t-1})\right)\right),</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.Thmtheorem3.p1.12" class="ltx_p"><span id="A1.Thmtheorem3.p1.12.4" class="ltx_text ltx_font_italic" style="font-size:90%;">where <math id="A1.Thmtheorem3.p1.9.1.m1.1" class="ltx_Math" alttext="C(\mathbf{x}_{1:T}^{0})" display="inline"><semantics id="A1.Thmtheorem3.p1.9.1.m1.1a"><mrow id="A1.Thmtheorem3.p1.9.1.m1.1.1" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.cmml"><mi id="A1.Thmtheorem3.p1.9.1.m1.1.1.3" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.3.cmml">C</mi><mo lspace="0em" rspace="0em" id="A1.Thmtheorem3.p1.9.1.m1.1.1.2" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.2.cmml">​</mo><mrow id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.cmml"><mo stretchy="false" id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.2" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.cmml">(</mo><msubsup id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.cmml"><mi id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.2" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mrow id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.cmml"><mn id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.2" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.1" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.1.cmml">:</mo><mi id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.3" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.3.cmml">T</mi></mrow><mn id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.3" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.3.cmml">0</mn></msubsup><mo stretchy="false" id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.3" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.9.1.m1.1b"><apply id="A1.Thmtheorem3.p1.9.1.m1.1.1.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1"><times id="A1.Thmtheorem3.p1.9.1.m1.1.1.2.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.2"></times><ci id="A1.Thmtheorem3.p1.9.1.m1.1.1.3.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.3">𝐶</ci><apply id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1">superscript</csymbol><apply id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.1.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1">subscript</csymbol><ci id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.2.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.2">𝐱</ci><apply id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3"><ci id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.1.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.1">:</ci><cn type="integer" id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.2.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.2">1</cn><ci id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.3.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.2.3.3">𝑇</ci></apply></apply><cn type="integer" id="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem3.p1.9.1.m1.1.1.1.1.1.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.9.1.m1.1c">C(\mathbf{x}_{1:T}^{0})</annotation></semantics></math> is a constant depending only on <math id="A1.Thmtheorem3.p1.10.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}^{0}" display="inline"><semantics id="A1.Thmtheorem3.p1.10.2.m2.1a"><msubsup id="A1.Thmtheorem3.p1.10.2.m2.1.1" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.cmml"><mi id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.2" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.2.2.cmml">𝐱</mi><mrow id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.cmml"><mn id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.2" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.2.cmml">1</mn><mo lspace="0.278em" rspace="0.278em" id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.1" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.1.cmml">:</mo><mi id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.3" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.3.cmml">T</mi></mrow><mn id="A1.Thmtheorem3.p1.10.2.m2.1.1.3" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.3.cmml">0</mn></msubsup><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.10.2.m2.1b"><apply id="A1.Thmtheorem3.p1.10.2.m2.1.1.cmml" xref="A1.Thmtheorem3.p1.10.2.m2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.10.2.m2.1.1.1.cmml" xref="A1.Thmtheorem3.p1.10.2.m2.1.1">superscript</csymbol><apply id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.cmml" xref="A1.Thmtheorem3.p1.10.2.m2.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.1.cmml" xref="A1.Thmtheorem3.p1.10.2.m2.1.1">subscript</csymbol><ci id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.2.cmml" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.2.2">𝐱</ci><apply id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.cmml" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3"><ci id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.1.cmml" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.1">:</ci><cn type="integer" id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.2.cmml" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.2">1</cn><ci id="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.3.cmml" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.2.3.3">𝑇</ci></apply></apply><cn type="integer" id="A1.Thmtheorem3.p1.10.2.m2.1.1.3.cmml" xref="A1.Thmtheorem3.p1.10.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.10.2.m2.1c">\mathbf{x}_{1:T}^{0}</annotation></semantics></math> (the unnoised data), and where the inequality is an <em id="A1.Thmtheorem3.p1.12.4.1" class="ltx_emph ltx_font_upright">equality</em> under the conditions that (a) <math id="A1.Thmtheorem3.p1.11.3.m3.1" class="ltx_Math" alttext="p_{\bm{\theta}}(\mathbf{z}_{t}\mid\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}})" display="inline"><semantics id="A1.Thmtheorem3.p1.11.3.m3.1a"><mrow id="A1.Thmtheorem3.p1.11.3.m3.1.1" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.cmml"><msub id="A1.Thmtheorem3.p1.11.3.m3.1.1.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.3.cmml"><mi id="A1.Thmtheorem3.p1.11.3.m3.1.1.3.2" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.3.2.cmml">p</mi><mi id="A1.Thmtheorem3.p1.11.3.m3.1.1.3.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.3.3.cmml">𝛉</mi></msub><mo lspace="0em" rspace="0em" id="A1.Thmtheorem3.p1.11.3.m3.1.1.2" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.2.cmml">​</mo><mrow id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.cmml"><mo stretchy="false" id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.2" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.cmml"><msub id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.cmml"><mi id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.2" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.2.cmml">𝐳</mi><mi id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.3.cmml">t</mi></msub><mo id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.3.cmml">∣</mo><mrow id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.3.cmml"><msub id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.cmml"><mi id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.2" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.cmml"><mi id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.2" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.2.cmml">t</mi><mo id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.1" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.3.cmml">,</mo><msubsup id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.cmml"><mi id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.2.2" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mi id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.2.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.2.3.cmml">t</mi><msub id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.cmml"><mi id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.2" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.2.cmml">k</mi><mi id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.3.cmml">t</mi></msub></msubsup></mrow></mrow><mo stretchy="false" id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.3" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Thmtheorem3.p1.11.3.m3.1b"><apply id="A1.Thmtheorem3.p1.11.3.m3.1.1.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1"><times id="A1.Thmtheorem3.p1.11.3.m3.1.1.2.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.2"></times><apply id="A1.Thmtheorem3.p1.11.3.m3.1.1.3.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.3"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.11.3.m3.1.1.3.1.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.3">subscript</csymbol><ci id="A1.Thmtheorem3.p1.11.3.m3.1.1.3.2.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.3.2">𝑝</ci><ci id="A1.Thmtheorem3.p1.11.3.m3.1.1.3.3.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.3.3">𝛉</ci></apply><apply id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1"><csymbol cd="latexml" id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.3">conditional</csymbol><apply id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.1.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4">subscript</csymbol><ci id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.2.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.2">𝐳</ci><ci id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.3.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.4.3">𝑡</ci></apply><list id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.3.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2"><apply id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.1.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.2.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.2">𝐳</ci><apply id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3"><minus id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.1.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.1"></minus><ci id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.2">𝑡</ci><cn type="integer" id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.1.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2">superscript</csymbol><apply id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.2.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.2.1.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.2.2.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.2.2">𝐱</ci><ci id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.2.3.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.2.3">𝑡</ci></apply><apply id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3"><csymbol cd="ambiguous" id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.1.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3">subscript</csymbol><ci id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.2.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.2">𝑘</ci><ci id="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.3.cmml" xref="A1.Thmtheorem3.p1.11.3.m3.1.1.1.1.1.2.2.2.3.3">𝑡</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.11.3.m3.1c">p_{\bm{\theta}}(\mathbf{z}_{t}\mid\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}})</annotation></semantics></math> is a dirac distribution (determinstic latents), and (b) <math id="A1.Thmtheorem3.p1.12.4.m4.2" class="ltx_Math" 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id="A1.Thmtheorem3.p1.12.4.m4.2.2.2.1.1.1.2.2.2.3.2.cmml" xref="A1.Thmtheorem3.p1.12.4.m4.2.2.2.1.1.1.2.2.2.3.2">𝑡</ci><cn type="integer" id="A1.Thmtheorem3.p1.12.4.m4.2.2.2.1.1.1.2.2.2.3.3.cmml" xref="A1.Thmtheorem3.p1.12.4.m4.2.2.2.1.1.1.2.2.2.3.3">1</cn></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Thmtheorem3.p1.12.4.m4.2c">q(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}})\equiv p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1})</annotation></semantics></math>, i.e. the variational approximation is sharp.</span></p>
</div>
<div id="A1.Thmtheorem3.p2" class="ltx_para">
<p id="A1.Thmtheorem3.p2.1" class="ltx_p"><span id="A1.Thmtheorem3.p2.1.1" class="ltx_text ltx_font_italic" style="font-size:90%;">In particular, in the Gaussian case of <a href="#A1.Thmtheorem2" title="Corollary A.2. ‣ A.1.1 Specializing to Gaussian diffusion ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">A.2</span></a>, we have</span></p>
<table id="A6.EGx14" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.Ex10.m1.4.4.2.1.1.1.2.3"><ci id="A1.Ex10.m1.4.4.2.1.1.1.2.3.1.cmml" xref="A1.Ex10.m1.4.4.2.1.1.1.2.3.1">:</ci><cn type="integer" id="A1.Ex10.m1.4.4.2.1.1.1.2.3.2.cmml" xref="A1.Ex10.m1.4.4.2.1.1.1.2.3.2">1</cn><ci id="A1.Ex10.m1.4.4.2.1.1.1.2.3.3.cmml" xref="A1.Ex10.m1.4.4.2.1.1.1.2.3.3">𝑇</ci></apply></apply><cn type="integer" id="A1.Ex10.m1.4.4.2.1.1.1.3.cmml" xref="A1.Ex10.m1.4.4.2.1.1.1.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex10.m1.4c">\displaystyle\mathop{{\mathbb{E}}}_{\mathrm{forward},\mathcal{D}}[\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})]+C(\mathbf{x}_{1:T}^{0})</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex10.m2.5" class="ltx_Math" alttext="\displaystyle\geq\mathop{{\mathbb{E}}}_{\mathrm{forward},\mathcal{D}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\mathbf{1}\{k_{t}\geq 1\}k_{t}c_{k_{t}}\|\hat{\mathbf{x}}^{0}_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1},{k_{t}})-\mathbf{x}_{t}^{0}\|^{2}\right]," display="inline"><semantics id="A1.Ex10.m2.5a"><mrow id="A1.Ex10.m2.5.5.1" xref="A1.Ex10.m2.5.5.1.1.cmml"><mrow id="A1.Ex10.m2.5.5.1.1" xref="A1.Ex10.m2.5.5.1.1.cmml"><mi id="A1.Ex10.m2.5.5.1.1.3" xref="A1.Ex10.m2.5.5.1.1.3.cmml"></mi><mo mathsize="90%" rspace="0.1389em" id="A1.Ex10.m2.5.5.1.1.2" xref="A1.Ex10.m2.5.5.1.1.2.cmml">≥</mo><mrow id="A1.Ex10.m2.5.5.1.1.1" xref="A1.Ex10.m2.5.5.1.1.1.cmml"><munder id="A1.Ex10.m2.5.5.1.1.1.2" xref="A1.Ex10.m2.5.5.1.1.1.2.cmml"><mo lspace="0.1389em" mathsize="108%" movablelimits="false" rspace="0.0835em" id="A1.Ex10.m2.5.5.1.1.1.2.2" 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id="A1.Ex10.m2.5c">\displaystyle\geq\mathop{{\mathbb{E}}}_{\mathrm{forward},\mathcal{D}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\mathbf{1}\{k_{t}\geq 1\}k_{t}c_{k_{t}}\|\hat{\mathbf{x}}^{0}_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1},{k_{t}})-\mathbf{x}_{t}^{0}\|^{2}\right],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
</div>
</div>
<div id="A1.SS1.SSS2.p2" class="ltx_para">
<p id="A1.SS1.SSS2.p2.6" class="ltx_p"><span id="A1.SS1.SSS2.p2.6.1" class="ltx_text" style="font-size:90%;">The most salient case for us is the restriction of </span><math id="A1.SS1.SSS2.p2.1.m1.1" class="ltx_Math" alttext="\mathcal{D}" display="inline"><semantics id="A1.SS1.SSS2.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A1.SS1.SSS2.p2.1.m1.1.1" xref="A1.SS1.SSS2.p2.1.m1.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS2.p2.1.m1.1b"><ci id="A1.SS1.SSS2.p2.1.m1.1.1.cmml" xref="A1.SS1.SSS2.p2.1.m1.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS2.p2.1.m1.1c">\mathcal{D}</annotation></semantics></math><span id="A1.SS1.SSS2.p2.6.2" class="ltx_text" style="font-size:90%;"> to fixed sequences of noise </span><math id="A1.SS1.SSS2.p2.2.m2.3" class="ltx_Math" alttext="k_{1},\dots,k_{T}" display="inline"><semantics id="A1.SS1.SSS2.p2.2.m2.3a"><mrow id="A1.SS1.SSS2.p2.2.m2.3.3.2" xref="A1.SS1.SSS2.p2.2.m2.3.3.3.cmml"><msub id="A1.SS1.SSS2.p2.2.m2.2.2.1.1" xref="A1.SS1.SSS2.p2.2.m2.2.2.1.1.cmml"><mi mathsize="90%" id="A1.SS1.SSS2.p2.2.m2.2.2.1.1.2" xref="A1.SS1.SSS2.p2.2.m2.2.2.1.1.2.cmml">k</mi><mn mathsize="90%" id="A1.SS1.SSS2.p2.2.m2.2.2.1.1.3" xref="A1.SS1.SSS2.p2.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo mathsize="90%" id="A1.SS1.SSS2.p2.2.m2.3.3.2.3" xref="A1.SS1.SSS2.p2.2.m2.3.3.3.cmml">,</mo><mi mathsize="90%" mathvariant="normal" id="A1.SS1.SSS2.p2.2.m2.1.1" xref="A1.SS1.SSS2.p2.2.m2.1.1.cmml">…</mi><mo mathsize="90%" id="A1.SS1.SSS2.p2.2.m2.3.3.2.4" xref="A1.SS1.SSS2.p2.2.m2.3.3.3.cmml">,</mo><msub id="A1.SS1.SSS2.p2.2.m2.3.3.2.2" xref="A1.SS1.SSS2.p2.2.m2.3.3.2.2.cmml"><mi mathsize="90%" id="A1.SS1.SSS2.p2.2.m2.3.3.2.2.2" xref="A1.SS1.SSS2.p2.2.m2.3.3.2.2.2.cmml">k</mi><mi mathsize="90%" id="A1.SS1.SSS2.p2.2.m2.3.3.2.2.3" xref="A1.SS1.SSS2.p2.2.m2.3.3.2.2.3.cmml">T</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS2.p2.2.m2.3b"><list id="A1.SS1.SSS2.p2.2.m2.3.3.3.cmml" xref="A1.SS1.SSS2.p2.2.m2.3.3.2"><apply id="A1.SS1.SSS2.p2.2.m2.2.2.1.1.cmml" xref="A1.SS1.SSS2.p2.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="A1.SS1.SSS2.p2.2.m2.2.2.1.1.1.cmml" xref="A1.SS1.SSS2.p2.2.m2.2.2.1.1">subscript</csymbol><ci id="A1.SS1.SSS2.p2.2.m2.2.2.1.1.2.cmml" xref="A1.SS1.SSS2.p2.2.m2.2.2.1.1.2">𝑘</ci><cn type="integer" id="A1.SS1.SSS2.p2.2.m2.2.2.1.1.3.cmml" xref="A1.SS1.SSS2.p2.2.m2.2.2.1.1.3">1</cn></apply><ci id="A1.SS1.SSS2.p2.2.m2.1.1.cmml" xref="A1.SS1.SSS2.p2.2.m2.1.1">…</ci><apply id="A1.SS1.SSS2.p2.2.m2.3.3.2.2.cmml" xref="A1.SS1.SSS2.p2.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="A1.SS1.SSS2.p2.2.m2.3.3.2.2.1.cmml" xref="A1.SS1.SSS2.p2.2.m2.3.3.2.2">subscript</csymbol><ci id="A1.SS1.SSS2.p2.2.m2.3.3.2.2.2.cmml" xref="A1.SS1.SSS2.p2.2.m2.3.3.2.2.2">𝑘</ci><ci id="A1.SS1.SSS2.p2.2.m2.3.3.2.2.3.cmml" xref="A1.SS1.SSS2.p2.2.m2.3.3.2.2.3">𝑇</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS2.p2.2.m2.3c">k_{1},\dots,k_{T}</annotation></semantics></math><span id="A1.SS1.SSS2.p2.6.3" class="ltx_text" style="font-size:90%;"> (i.e, Dirac distributions on </span><math id="A1.SS1.SSS2.p2.3.m3.1" class="ltx_Math" alttext="[K]^{T}" display="inline"><semantics id="A1.SS1.SSS2.p2.3.m3.1a"><msup id="A1.SS1.SSS2.p2.3.m3.1.2" xref="A1.SS1.SSS2.p2.3.m3.1.2.cmml"><mrow id="A1.SS1.SSS2.p2.3.m3.1.2.2.2" xref="A1.SS1.SSS2.p2.3.m3.1.2.2.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS1.SSS2.p2.3.m3.1.2.2.2.1" xref="A1.SS1.SSS2.p2.3.m3.1.2.2.1.1.cmml">[</mo><mi mathsize="90%" id="A1.SS1.SSS2.p2.3.m3.1.1" xref="A1.SS1.SSS2.p2.3.m3.1.1.cmml">K</mi><mo maxsize="90%" minsize="90%" id="A1.SS1.SSS2.p2.3.m3.1.2.2.2.2" xref="A1.SS1.SSS2.p2.3.m3.1.2.2.1.1.cmml">]</mo></mrow><mi mathsize="90%" id="A1.SS1.SSS2.p2.3.m3.1.2.3" xref="A1.SS1.SSS2.p2.3.m3.1.2.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS2.p2.3.m3.1b"><apply id="A1.SS1.SSS2.p2.3.m3.1.2.cmml" xref="A1.SS1.SSS2.p2.3.m3.1.2"><csymbol cd="ambiguous" id="A1.SS1.SSS2.p2.3.m3.1.2.1.cmml" xref="A1.SS1.SSS2.p2.3.m3.1.2">superscript</csymbol><apply id="A1.SS1.SSS2.p2.3.m3.1.2.2.1.cmml" xref="A1.SS1.SSS2.p2.3.m3.1.2.2.2"><csymbol cd="latexml" id="A1.SS1.SSS2.p2.3.m3.1.2.2.1.1.cmml" xref="A1.SS1.SSS2.p2.3.m3.1.2.2.2.1">delimited-[]</csymbol><ci id="A1.SS1.SSS2.p2.3.m3.1.1.cmml" xref="A1.SS1.SSS2.p2.3.m3.1.1">𝐾</ci></apply><ci id="A1.SS1.SSS2.p2.3.m3.1.2.3.cmml" xref="A1.SS1.SSS2.p2.3.m3.1.2.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS2.p2.3.m3.1c">[K]^{T}</annotation></semantics></math><span id="A1.SS1.SSS2.p2.6.4" class="ltx_text" style="font-size:90%;">). In this case, </span><math id="A1.SS1.SSS2.p2.4.m4.1" class="ltx_Math" alttext="P_{t}(j\mid k_{1:t-1})=0" display="inline"><semantics id="A1.SS1.SSS2.p2.4.m4.1a"><mrow id="A1.SS1.SSS2.p2.4.m4.1.1" xref="A1.SS1.SSS2.p2.4.m4.1.1.cmml"><mrow id="A1.SS1.SSS2.p2.4.m4.1.1.1" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.cmml"><msub id="A1.SS1.SSS2.p2.4.m4.1.1.1.3" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.3.cmml"><mi mathsize="90%" id="A1.SS1.SSS2.p2.4.m4.1.1.1.3.2" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.3.2.cmml">P</mi><mi mathsize="90%" id="A1.SS1.SSS2.p2.4.m4.1.1.1.3.3" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.3.3.cmml">t</mi></msub><mo lspace="0em" rspace="0em" id="A1.SS1.SSS2.p2.4.m4.1.1.1.2" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.2.cmml">​</mo><mrow id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.2" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1" 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xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.2.cmml">t</mi><mo mathsize="90%" id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.1" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.1.cmml">−</mo><mn mathsize="90%" id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.3" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.3.cmml">1</mn></mrow></mrow></msub></mrow><mo maxsize="90%" minsize="90%" id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.3" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo mathsize="90%" id="A1.SS1.SSS2.p2.4.m4.1.1.2" xref="A1.SS1.SSS2.p2.4.m4.1.1.2.cmml">=</mo><mn mathsize="90%" id="A1.SS1.SSS2.p2.4.m4.1.1.3" xref="A1.SS1.SSS2.p2.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS2.p2.4.m4.1b"><apply id="A1.SS1.SSS2.p2.4.m4.1.1.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1"><eq id="A1.SS1.SSS2.p2.4.m4.1.1.2.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.2"></eq><apply id="A1.SS1.SSS2.p2.4.m4.1.1.1.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1"><times id="A1.SS1.SSS2.p2.4.m4.1.1.1.2.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.2"></times><apply id="A1.SS1.SSS2.p2.4.m4.1.1.1.3.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.SSS2.p2.4.m4.1.1.1.3.1.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.3">subscript</csymbol><ci id="A1.SS1.SSS2.p2.4.m4.1.1.1.3.2.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.3.2">𝑃</ci><ci id="A1.SS1.SSS2.p2.4.m4.1.1.1.3.3.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.3.3">𝑡</ci></apply><apply id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.1">conditional</csymbol><ci id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.2">𝑗</ci><apply id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.1.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.2.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.2">𝑘</ci><apply id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3"><ci id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.1.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.1">:</ci><cn type="integer" id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.2.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.2">1</cn><apply id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3"><minus id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.1.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.1"></minus><ci id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.2.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.2">𝑡</ci><cn type="integer" id="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.3.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></apply><cn type="integer" id="A1.SS1.SSS2.p2.4.m4.1.1.3.cmml" xref="A1.SS1.SSS2.p2.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS2.p2.4.m4.1c">P_{t}(j\mid k_{1:t-1})=0</annotation></semantics></math><span id="A1.SS1.SSS2.p2.6.5" class="ltx_text" style="font-size:90%;"> for all but </span><math id="A1.SS1.SSS2.p2.5.m5.1" class="ltx_Math" alttext="j=k_{t}" display="inline"><semantics id="A1.SS1.SSS2.p2.5.m5.1a"><mrow id="A1.SS1.SSS2.p2.5.m5.1.1" xref="A1.SS1.SSS2.p2.5.m5.1.1.cmml"><mi mathsize="90%" id="A1.SS1.SSS2.p2.5.m5.1.1.2" xref="A1.SS1.SSS2.p2.5.m5.1.1.2.cmml">j</mi><mo mathsize="90%" id="A1.SS1.SSS2.p2.5.m5.1.1.1" xref="A1.SS1.SSS2.p2.5.m5.1.1.1.cmml">=</mo><msub id="A1.SS1.SSS2.p2.5.m5.1.1.3" xref="A1.SS1.SSS2.p2.5.m5.1.1.3.cmml"><mi mathsize="90%" id="A1.SS1.SSS2.p2.5.m5.1.1.3.2" xref="A1.SS1.SSS2.p2.5.m5.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A1.SS1.SSS2.p2.5.m5.1.1.3.3" xref="A1.SS1.SSS2.p2.5.m5.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS2.p2.5.m5.1b"><apply id="A1.SS1.SSS2.p2.5.m5.1.1.cmml" xref="A1.SS1.SSS2.p2.5.m5.1.1"><eq id="A1.SS1.SSS2.p2.5.m5.1.1.1.cmml" xref="A1.SS1.SSS2.p2.5.m5.1.1.1"></eq><ci id="A1.SS1.SSS2.p2.5.m5.1.1.2.cmml" xref="A1.SS1.SSS2.p2.5.m5.1.1.2">𝑗</ci><apply id="A1.SS1.SSS2.p2.5.m5.1.1.3.cmml" xref="A1.SS1.SSS2.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.SSS2.p2.5.m5.1.1.3.1.cmml" xref="A1.SS1.SSS2.p2.5.m5.1.1.3">subscript</csymbol><ci id="A1.SS1.SSS2.p2.5.m5.1.1.3.2.cmml" xref="A1.SS1.SSS2.p2.5.m5.1.1.3.2">𝑘</ci><ci id="A1.SS1.SSS2.p2.5.m5.1.1.3.3.cmml" xref="A1.SS1.SSS2.p2.5.m5.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS2.p2.5.m5.1c">j=k_{t}</annotation></semantics></math><span id="A1.SS1.SSS2.p2.6.6" class="ltx_text" style="font-size:90%;">, and thus our training objective need not be a lower bound on </span><math id="A1.SS1.SSS2.p2.6.m6.3" class="ltx_Math" 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xref="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3"></and><apply id="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3b.cmml" xref="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3"><leq id="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.3.cmml" xref="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.3"></leq><cn type="integer" id="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.2.cmml" xref="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.2">1</cn><ci id="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.4.cmml" xref="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.4">𝑡</ci></apply><apply id="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3c.cmml" xref="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3"><leq id="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.5.cmml" xref="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.5"></leq><share href="#A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.4.cmml" id="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3d.cmml" xref="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3"></share><ci id="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.6.cmml" xref="A1.SS1.SSS2.p2.6.m6.3.3.1.1.1.1.1.1.3.6">𝑇</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS2.p2.6.m6.3c">\mathop{{\mathbb{E}}}_{\mathrm{forward},\mathcal{D}}[\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})]</annotation></semantics></math><span id="A1.SS1.SSS2.p2.6.7" class="ltx_text" style="font-size:90%;">. However, the terms in the lower bound are, up to reweighting, an </span><em id="A1.SS1.SSS2.p2.6.8" class="ltx_emph ltx_font_italic" style="font-size:90%;">subset</em><span id="A1.SS1.SSS2.p2.6.9" class="ltx_text" style="font-size:90%;"> of those terms optimized in the training objective. Thus, in light of the remarks following </span><a href="#A1.Thmtheorem2" title="Corollary A.2. ‣ A.1.1 Specializing to Gaussian diffusion ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">A.2</span></a><span id="A1.SS1.SSS2.p2.6.10" class="ltx_text" style="font-size:90%;">, a fully expressive network can optimize all the terms in the loss simultaneously. We conclude that, for a fully expressive neural network, optimizing the training objective (</span><a href="#S3.E1" title="Equation 3.1 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">3.1</span></a><span id="A1.SS1.SSS2.p2.6.11" class="ltx_text" style="font-size:90%;">) is a valid surrogate for maximizing the likelihood of all possible noise sequences.</span></p>
</div>
</section>
</section>
<section id="A1.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">A.2 </span>Proof of <a href="#A1.Thmtheorem1" title="Theorem A.1. ‣ A.1 Main Results ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">A.1</span></a>
</h3>

<div id="A1.SS2.p1" class="ltx_para">
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id="A1.SS2.p1.2.m2.6.7.2.2.2.1.cmml" xref="A1.SS2.p1.2.m2.6.7.2.2.2.2"><csymbol cd="latexml" id="A1.SS2.p1.2.m2.6.7.2.2.2.1.1.cmml" xref="A1.SS2.p1.2.m2.6.7.2.2.2.2.1">delimited-[]</csymbol><ci id="A1.SS2.p1.2.m2.6.6.cmml" xref="A1.SS2.p1.2.m2.6.6">⋅</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.p1.2.m2.6c">\mathop{{\mathbb{E}}}_{k_{1:s}\overset{\mathrm{unif}}{\sim}[K]}\mathop{{\mathbb{E}}}_{\mathbf{x}_{s}^{k_{s}}\sim q(\mathbf{x}_{s}^{k_{s}}\mid\mathbf{x}_{s}^{0}),1\leq s\leq t-1}\mathop{{\mathbb{E}}}_{\mathbf{z}_{s}\sim p(\mathbf{z}_{s}\mid\mathbf{z}_{s-1},\mathbf{x}_{s}^{k_{s}}),s\leq t}[\cdot]</annotation></semantics></math><span id="A1.SS2.p1.2.3" class="ltx_text" style="font-size:90%;">. We begin with the following claim</span></p>
</div>
<div id="Thmclaim1" class="ltx_theorem ltx_theorem_claim">
<h6 class="ltx_title ltx_runin ltx_title_theorem" style="font-size:90%;">
<span class="ltx_tag ltx_tag_theorem"><span id="Thmclaim1.1.1.1" class="ltx_text ltx_font_bold">Claim 1</span></span><span id="Thmclaim1.2.2" class="ltx_text ltx_font_bold"> </span>(Expanding the latents)<span id="Thmclaim1.3.3" class="ltx_text ltx_font_bold">.</span>
</h6>
<div id="Thmclaim1.p1" class="ltx_para">
<p id="Thmclaim1.p1.2" class="ltx_p"><span id="Thmclaim1.p1.2.1" class="ltx_text ltx_font_italic" style="font-size:90%;">The following lower bound holds:</span></p>
<table id="A6.EGx15" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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mathsize="90%" id="A1.E9.m1.1.1.1.1.1.1.1.1.3.5" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3.5.cmml">≤</mo><mi mathsize="90%" id="A1.E9.m1.1.1.1.1.1.1.1.1.3.6" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3.6.cmml">T</mi></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.E9.m1.1.1.1.1.1.1.1.3" xref="A1.E9.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo maxsize="90%" minsize="90%" id="A1.E9.m1.1.1.1.1.3" xref="A1.E9.m1.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E9.m1.1b"><apply id="A1.E9.m1.1.1.cmml" xref="A1.E9.m1.1.1"><apply id="A1.E9.m1.1.1.2.cmml" xref="A1.E9.m1.1.1.2"><csymbol cd="ambiguous" id="A1.E9.m1.1.1.2.1.cmml" xref="A1.E9.m1.1.1.2">subscript</csymbol><ci id="A1.E9.m1.1.1.2.2.cmml" xref="A1.E9.m1.1.1.2.2">𝔼</ci><ci id="A1.E9.m1.1.1.2.3.cmml" xref="A1.E9.m1.1.1.2.3">forward</ci></apply><apply id="A1.E9.m1.1.1.1.2.cmml" xref="A1.E9.m1.1.1.1.1"><csymbol cd="latexml" id="A1.E9.m1.1.1.1.2.1.cmml" xref="A1.E9.m1.1.1.1.1.2">delimited-[]</csymbol><apply 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id="A1.E9.m1.1.1.1.1.1.1.1.1.3a.cmml" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3"></and><apply id="A1.E9.m1.1.1.1.1.1.1.1.1.3b.cmml" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3"><leq id="A1.E9.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3.3"></leq><cn type="integer" id="A1.E9.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3.2">1</cn><ci id="A1.E9.m1.1.1.1.1.1.1.1.1.3.4.cmml" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3.4">𝑡</ci></apply><apply id="A1.E9.m1.1.1.1.1.1.1.1.1.3c.cmml" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3"><leq id="A1.E9.m1.1.1.1.1.1.1.1.1.3.5.cmml" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3.5"></leq><share href="#A1.E9.m1.1.1.1.1.1.1.1.1.3.4.cmml" id="A1.E9.m1.1.1.1.1.1.1.1.1.3d.cmml" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3"></share><ci id="A1.E9.m1.1.1.1.1.1.1.1.1.3.6.cmml" xref="A1.E9.m1.1.1.1.1.1.1.1.1.3.6">𝑇</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E9.m1.1c">\displaystyle\mathop{{\mathbb{E}}}_{\mathrm{forward}}[\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})]</annotation></semantics></math></td>
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id="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3.cmml" xref="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3">𝑘</ci></apply></apply><apply id="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.2">𝐳</ci><apply id="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.3"><minus id="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.3.1"></minus><ci id="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.3.2">𝑡</ci><cn type="integer" id="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.3.3.cmml" xref="A1.E9.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E9.m2.6c">\displaystyle\geq\sum_{t=1}^{T}\mathop{{\mathbb{E}}}_{&lt;t}\mathop{{\mathbb{E}}}_{k_{t}\overset{\mathrm{unif}}{\sim}\{0,1,\dots,K\}}\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{t_{k}}\mid\mathbf{z}_{t-1})\right],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.9)</span></td>
</tr></tbody>
</table>
<p id="Thmclaim1.p1.1" class="ltx_p"><span id="Thmclaim1.p1.1.1" class="ltx_text ltx_font_italic" style="font-size:90%;">Moreover, this lower bound holds with equality if <math id="Thmclaim1.p1.1.1.m1.1" class="ltx_Math" alttext="\mathbf{z}_{s}\sim p(\mathbf{z}_{s}\mid\mathbf{z}_{s-1},\mathbf{x}_{s}^{k_{s}})" display="inline"><semantics id="Thmclaim1.p1.1.1.m1.1a"><mrow id="Thmclaim1.p1.1.1.m1.1.1" xref="Thmclaim1.p1.1.1.m1.1.1.cmml"><msub id="Thmclaim1.p1.1.1.m1.1.1.3" xref="Thmclaim1.p1.1.1.m1.1.1.3.cmml"><mi id="Thmclaim1.p1.1.1.m1.1.1.3.2" xref="Thmclaim1.p1.1.1.m1.1.1.3.2.cmml">𝐳</mi><mi id="Thmclaim1.p1.1.1.m1.1.1.3.3" xref="Thmclaim1.p1.1.1.m1.1.1.3.3.cmml">s</mi></msub><mo id="Thmclaim1.p1.1.1.m1.1.1.2" xref="Thmclaim1.p1.1.1.m1.1.1.2.cmml">∼</mo><mrow id="Thmclaim1.p1.1.1.m1.1.1.1" xref="Thmclaim1.p1.1.1.m1.1.1.1.cmml"><mi id="Thmclaim1.p1.1.1.m1.1.1.1.3" xref="Thmclaim1.p1.1.1.m1.1.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="Thmclaim1.p1.1.1.m1.1.1.1.2" xref="Thmclaim1.p1.1.1.m1.1.1.1.2.cmml">​</mo><mrow id="Thmclaim1.p1.1.1.m1.1.1.1.1.1" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.2" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.cmml"><msub id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.cmml"><mi id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.2" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.2.cmml">𝐳</mi><mi id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.3" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.3.cmml">s</mi></msub><mo id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.3" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.3.cmml">∣</mo><mrow id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.3.cmml"><msub id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.3" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.3.cmml">,</mo><msubsup id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.cmml"><mi id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.2.2" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mi id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.2.3" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.2.3.cmml">s</mi><msub id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.cmml"><mi id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.2" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.2.cmml">k</mi><mi id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.3" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.3.cmml">s</mi></msub></msubsup></mrow></mrow><mo stretchy="false" id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.3" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmclaim1.p1.1.1.m1.1b"><apply id="Thmclaim1.p1.1.1.m1.1.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1"><csymbol cd="latexml" id="Thmclaim1.p1.1.1.m1.1.1.2.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.2">similar-to</csymbol><apply id="Thmclaim1.p1.1.1.m1.1.1.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmclaim1.p1.1.1.m1.1.1.3.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="Thmclaim1.p1.1.1.m1.1.1.3.2.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.3.2">𝐳</ci><ci id="Thmclaim1.p1.1.1.m1.1.1.3.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.3.3">𝑠</ci></apply><apply id="Thmclaim1.p1.1.1.m1.1.1.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1"><times id="Thmclaim1.p1.1.1.m1.1.1.1.2.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.2"></times><ci id="Thmclaim1.p1.1.1.m1.1.1.1.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.3">𝑝</ci><apply id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1"><csymbol cd="latexml" id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.3">conditional</csymbol><apply id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4">subscript</csymbol><ci id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.2.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.2">𝐳</ci><ci id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.4.3">𝑠</ci></apply><list id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2"><apply id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.2">𝐳</ci><apply id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3"><minus id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.1"></minus><ci id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.2">𝑠</ci><cn type="integer" id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2">superscript</csymbol><apply id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.2.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.2.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.2.2.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.2.2">𝐱</ci><ci id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.2.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.2.3">𝑠</ci></apply><apply id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3"><csymbol cd="ambiguous" id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.1.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3">subscript</csymbol><ci id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.2.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.2">𝑘</ci><ci id="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.3.cmml" xref="Thmclaim1.p1.1.1.m1.1.1.1.1.1.1.2.2.2.3.3">𝑠</ci></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmclaim1.p1.1.1.m1.1c">\mathbf{z}_{s}\sim p(\mathbf{z}_{s}\mid\mathbf{z}_{s-1},\mathbf{x}_{s}^{k_{s}})</annotation></semantics></math> is a Dirac distribution (i.e., deterministic latents).</span></p>
</div>
</div>
<div id="A1.SS2.1" class="ltx_proof">
<h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof" style="font-size:90%;">Proof.</h6>
<div id="A1.SS2.1.p1" class="ltx_para">
<p id="A1.SS2.1.p1.1" class="ltx_p"><span id="A1.SS2.1.p1.1.1" class="ltx_text" style="font-size:90%;">Let’s fix a sequence </span><math id="A1.SS2.1.p1.1.m1.1" class="ltx_Math" alttext="k_{1:T}" display="inline"><semantics id="A1.SS2.1.p1.1.m1.1a"><msub id="A1.SS2.1.p1.1.m1.1.1" xref="A1.SS2.1.p1.1.m1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.1.p1.1.m1.1.1.2" xref="A1.SS2.1.p1.1.m1.1.1.2.cmml">k</mi><mrow id="A1.SS2.1.p1.1.m1.1.1.3" xref="A1.SS2.1.p1.1.m1.1.1.3.cmml"><mn mathsize="90%" id="A1.SS2.1.p1.1.m1.1.1.3.2" xref="A1.SS2.1.p1.1.m1.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.SS2.1.p1.1.m1.1.1.3.1" xref="A1.SS2.1.p1.1.m1.1.1.3.1.cmml">:</mo><mi mathsize="90%" id="A1.SS2.1.p1.1.m1.1.1.3.3" xref="A1.SS2.1.p1.1.m1.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.1.m1.1b"><apply id="A1.SS2.1.p1.1.m1.1.1.cmml" xref="A1.SS2.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS2.1.p1.1.m1.1.1.1.cmml" xref="A1.SS2.1.p1.1.m1.1.1">subscript</csymbol><ci id="A1.SS2.1.p1.1.m1.1.1.2.cmml" xref="A1.SS2.1.p1.1.m1.1.1.2">𝑘</ci><apply id="A1.SS2.1.p1.1.m1.1.1.3.cmml" xref="A1.SS2.1.p1.1.m1.1.1.3"><ci id="A1.SS2.1.p1.1.m1.1.1.3.1.cmml" xref="A1.SS2.1.p1.1.m1.1.1.3.1">:</ci><cn type="integer" id="A1.SS2.1.p1.1.m1.1.1.3.2.cmml" xref="A1.SS2.1.p1.1.m1.1.1.3.2">1</cn><ci id="A1.SS2.1.p1.1.m1.1.1.3.3.cmml" xref="A1.SS2.1.p1.1.m1.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.1.m1.1c">k_{1:T}</annotation></semantics></math><span id="A1.SS2.1.p1.1.2" class="ltx_text" style="font-size:90%;">. It holds that</span></p>
<table id="A6.EGx16" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.Ex11.m1.1" class="ltx_Math" alttext="\displaystyle p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})" display="inline"><semantics id="A1.Ex11.m1.1a"><mrow id="A1.Ex11.m1.1.1" xref="A1.Ex11.m1.1.1.cmml"><msub id="A1.Ex11.m1.1.1.3" xref="A1.Ex11.m1.1.1.3.cmml"><mi mathsize="90%" id="A1.Ex11.m1.1.1.3.2" xref="A1.Ex11.m1.1.1.3.2.cmml">p</mi><mi mathsize="90%" id="A1.Ex11.m1.1.1.3.3" xref="A1.Ex11.m1.1.1.3.3.cmml">𝜽</mi></msub><mo lspace="0em" rspace="0em" id="A1.Ex11.m1.1.1.2" xref="A1.Ex11.m1.1.1.2.cmml">​</mo><mrow id="A1.Ex11.m1.1.1.1.1" xref="A1.Ex11.m1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.Ex11.m1.1.1.1.1.2" xref="A1.Ex11.m1.1.1.1.1.1.cmml">(</mo><msub id="A1.Ex11.m1.1.1.1.1.1" xref="A1.Ex11.m1.1.1.1.1.1.cmml"><mrow id="A1.Ex11.m1.1.1.1.1.1.1.1" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.Ex11.m1.1.1.1.1.1.1.1.2" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="A1.Ex11.m1.1.1.1.1.1.1.1.1" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.Ex11.m1.1.1.1.1.1.1.1.1.2.2" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.Ex11.m1.1.1.1.1.1.1.1.1.2.3" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.2.3.cmml">t</mi><msub id="A1.Ex11.m1.1.1.1.1.1.1.1.1.3" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.2" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.3" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></msub></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex11.m1.1.1.1.1.1.1.1.3" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.Ex11.m1.1.1.1.1.1.3" xref="A1.Ex11.m1.1.1.1.1.1.3.cmml"><mn mathsize="90%" id="A1.Ex11.m1.1.1.1.1.1.3.2" xref="A1.Ex11.m1.1.1.1.1.1.3.2.cmml">1</mn><mo mathsize="90%" id="A1.Ex11.m1.1.1.1.1.1.3.3" xref="A1.Ex11.m1.1.1.1.1.1.3.3.cmml">≤</mo><mi mathsize="90%" id="A1.Ex11.m1.1.1.1.1.1.3.4" xref="A1.Ex11.m1.1.1.1.1.1.3.4.cmml">t</mi><mo mathsize="90%" id="A1.Ex11.m1.1.1.1.1.1.3.5" xref="A1.Ex11.m1.1.1.1.1.1.3.5.cmml">≤</mo><mi mathsize="90%" id="A1.Ex11.m1.1.1.1.1.1.3.6" xref="A1.Ex11.m1.1.1.1.1.1.3.6.cmml">T</mi></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex11.m1.1.1.1.1.3" xref="A1.Ex11.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex11.m1.1b"><apply id="A1.Ex11.m1.1.1.cmml" xref="A1.Ex11.m1.1.1"><times id="A1.Ex11.m1.1.1.2.cmml" xref="A1.Ex11.m1.1.1.2"></times><apply id="A1.Ex11.m1.1.1.3.cmml" xref="A1.Ex11.m1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex11.m1.1.1.3.1.cmml" xref="A1.Ex11.m1.1.1.3">subscript</csymbol><ci id="A1.Ex11.m1.1.1.3.2.cmml" xref="A1.Ex11.m1.1.1.3.2">𝑝</ci><ci id="A1.Ex11.m1.1.1.3.3.cmml" xref="A1.Ex11.m1.1.1.3.3">𝜽</ci></apply><apply id="A1.Ex11.m1.1.1.1.1.1.cmml" xref="A1.Ex11.m1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex11.m1.1.1.1.1.1.2.cmml" xref="A1.Ex11.m1.1.1.1.1">subscript</csymbol><apply id="A1.Ex11.m1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex11.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex11.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex11.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="A1.Ex11.m1.1.1.1.1.1.1.1.1.2.cmml" xref="A1.Ex11.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex11.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A1.Ex11.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A1.Ex11.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.2.2">𝐱</ci><ci id="A1.Ex11.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.2">𝑘</ci><ci id="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex11.m1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply><apply id="A1.Ex11.m1.1.1.1.1.1.3.cmml" xref="A1.Ex11.m1.1.1.1.1.1.3"><and id="A1.Ex11.m1.1.1.1.1.1.3a.cmml" xref="A1.Ex11.m1.1.1.1.1.1.3"></and><apply id="A1.Ex11.m1.1.1.1.1.1.3b.cmml" xref="A1.Ex11.m1.1.1.1.1.1.3"><leq id="A1.Ex11.m1.1.1.1.1.1.3.3.cmml" xref="A1.Ex11.m1.1.1.1.1.1.3.3"></leq><cn type="integer" id="A1.Ex11.m1.1.1.1.1.1.3.2.cmml" xref="A1.Ex11.m1.1.1.1.1.1.3.2">1</cn><ci id="A1.Ex11.m1.1.1.1.1.1.3.4.cmml" xref="A1.Ex11.m1.1.1.1.1.1.3.4">𝑡</ci></apply><apply id="A1.Ex11.m1.1.1.1.1.1.3c.cmml" xref="A1.Ex11.m1.1.1.1.1.1.3"><leq id="A1.Ex11.m1.1.1.1.1.1.3.5.cmml" xref="A1.Ex11.m1.1.1.1.1.1.3.5"></leq><share href="#A1.Ex11.m1.1.1.1.1.1.3.4.cmml" id="A1.Ex11.m1.1.1.1.1.1.3d.cmml" xref="A1.Ex11.m1.1.1.1.1.1.3"></share><ci id="A1.Ex11.m1.1.1.1.1.1.3.6.cmml" xref="A1.Ex11.m1.1.1.1.1.1.3.6">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex11.m1.1c">\displaystyle p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})</annotation></semantics></math></td>
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xref="A1.Ex11.m2.2.2.2.2.2.2.2.2.2.2.2.2.3">𝑠</ci></apply></interval><apply id="A1.Ex11.m2.2.2.2.2.2.2.2.2.2.4.cmml" xref="A1.Ex11.m2.2.2.2.2.2.2.2.2.2.4"><lt id="A1.Ex11.m2.2.2.2.2.2.2.2.2.2.4.1.cmml" xref="A1.Ex11.m2.2.2.2.2.2.2.2.2.2.4.1"></lt><ci id="A1.Ex11.m2.2.2.2.2.2.2.2.2.2.4.2.cmml" xref="A1.Ex11.m2.2.2.2.2.2.2.2.2.2.4.2">𝑠</ci><ci id="A1.Ex11.m2.2.2.2.2.2.2.2.2.2.4.3.cmml" xref="A1.Ex11.m2.2.2.2.2.2.2.2.2.2.4.3">𝑡</ci></apply></apply></apply></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex11.m2.2c">\displaystyle=\int_{\mathbf{z}_{1:T}}\prod_{t=1}^{T}p(\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t}\mid(\mathbf{x}_{s}^{k_{s}},\mathbf{z}_{s})_{s&lt;t})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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xref="A1.Ex12.m1.2.2.2.2.2.2.2.2.3.3"><minus id="A1.Ex12.m1.2.2.2.2.2.2.2.2.3.3.1.cmml" xref="A1.Ex12.m1.2.2.2.2.2.2.2.2.3.3.1"></minus><ci id="A1.Ex12.m1.2.2.2.2.2.2.2.2.3.3.2.cmml" xref="A1.Ex12.m1.2.2.2.2.2.2.2.2.3.3.2">𝑡</ci><cn type="integer" id="A1.Ex12.m1.2.2.2.2.2.2.2.2.3.3.3.cmml" xref="A1.Ex12.m1.2.2.2.2.2.2.2.2.3.3.3">1</cn></apply></apply></apply></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex12.m1.2c">\displaystyle=\int_{\mathbf{z}_{1:T}}\prod_{t=1}^{T}p(\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t}\mid\mathbf{z}_{t-1})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Markov Property)</span></td>
</tr></tbody>
<tbody id="A1.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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xref="A1.Ex13.m1.3.3.2.2.2.2.1.1.2.3">subscript</csymbol><ci id="A1.Ex13.m1.3.3.2.2.2.2.1.1.2.3.2.cmml" xref="A1.Ex13.m1.3.3.2.2.2.2.1.1.2.3.2">𝑘</ci><ci id="A1.Ex13.m1.3.3.2.2.2.2.1.1.2.3.3.cmml" xref="A1.Ex13.m1.3.3.2.2.2.2.1.1.2.3.3">𝑡</ci></apply></apply><apply id="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.cmml" xref="A1.Ex13.m1.3.3.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.1.cmml" xref="A1.Ex13.m1.3.3.2.2.2.2.1.1.3">subscript</csymbol><ci id="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.2.cmml" xref="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.2">𝐳</ci><apply id="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.3.cmml" xref="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.3"><minus id="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.3.1.cmml" xref="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.3.1"></minus><ci id="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.3.2.cmml" xref="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.3.2">𝑡</ci><cn type="integer" id="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.3.3.cmml" xref="A1.Ex13.m1.3.3.2.2.2.2.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex13.m1.3c">\displaystyle=\int_{\mathbf{z}_{1:T}(\mathbf{k})}\prod_{t=1}^{T}p(\mathbf{z}_{t}\mid\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}})p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{z}_{t-1})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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id="A1.Ex14.m1.3.3.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="A1.Ex14.m1.3.3.1.1.1.1.1.1.1.1.3.3.1"></minus><ci id="A1.Ex14.m1.3.3.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="A1.Ex14.m1.3.3.1.1.1.1.1.1.1.1.3.3.2">𝑡</ci><cn type="integer" id="A1.Ex14.m1.3.3.1.1.1.1.1.1.1.1.3.3.3.cmml" xref="A1.Ex14.m1.3.3.1.1.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex14.m1.3c">\displaystyle=\mathop{{\mathbb{E}}}_{\mathbf{z}_{s}\sim p(\mathbf{z}_{s}\mid\mathbf{z}_{s-1},\mathbf{x}_{s}^{k_{s}}),s\leq T}\prod_{t=1}^{T}p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{z}_{t-1}).</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Importance Sampling)</span></td>
</tr></tbody>
</table>
<p id="A1.SS2.1.p1.4" class="ltx_p"><span id="A1.SS2.1.p1.4.1" class="ltx_text" style="font-size:90%;">Thus, by Jensen’s inequality,</span></p>
<table id="A6.EGx17" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.Ex15.m1.1" class="ltx_Math" alttext="\displaystyle\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})" display="inline"><semantics id="A1.Ex15.m1.1a"><mrow id="A1.Ex15.m1.1.1" xref="A1.Ex15.m1.1.1.cmml"><mrow id="A1.Ex15.m1.1.1.3" xref="A1.Ex15.m1.1.1.3.cmml"><mi mathsize="90%" id="A1.Ex15.m1.1.1.3.1" xref="A1.Ex15.m1.1.1.3.1.cmml">ln</mi><mo lspace="0.167em" id="A1.Ex15.m1.1.1.3a" xref="A1.Ex15.m1.1.1.3.cmml">⁡</mo><msub id="A1.Ex15.m1.1.1.3.2" xref="A1.Ex15.m1.1.1.3.2.cmml"><mi mathsize="90%" id="A1.Ex15.m1.1.1.3.2.2" xref="A1.Ex15.m1.1.1.3.2.2.cmml">p</mi><mi mathsize="90%" id="A1.Ex15.m1.1.1.3.2.3" xref="A1.Ex15.m1.1.1.3.2.3.cmml">𝜽</mi></msub></mrow><mo lspace="0em" rspace="0em" id="A1.Ex15.m1.1.1.2" xref="A1.Ex15.m1.1.1.2.cmml">​</mo><mrow id="A1.Ex15.m1.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.Ex15.m1.1.1.1.1.2" xref="A1.Ex15.m1.1.1.1.1.1.cmml">(</mo><msub id="A1.Ex15.m1.1.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.1.cmml"><mrow id="A1.Ex15.m1.1.1.1.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.Ex15.m1.1.1.1.1.1.1.1.2" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="A1.Ex15.m1.1.1.1.1.1.1.1.1" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.Ex15.m1.1.1.1.1.1.1.1.1.2.2" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.Ex15.m1.1.1.1.1.1.1.1.1.2.3" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.2.3.cmml">t</mi><msub id="A1.Ex15.m1.1.1.1.1.1.1.1.1.3" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.2" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.3" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></msub></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex15.m1.1.1.1.1.1.1.1.3" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A1.Ex15.m1.1.1.1.1.1.3" xref="A1.Ex15.m1.1.1.1.1.1.3.cmml"><mn mathsize="90%" id="A1.Ex15.m1.1.1.1.1.1.3.2" xref="A1.Ex15.m1.1.1.1.1.1.3.2.cmml">1</mn><mo mathsize="90%" id="A1.Ex15.m1.1.1.1.1.1.3.3" xref="A1.Ex15.m1.1.1.1.1.1.3.3.cmml">≤</mo><mi mathsize="90%" id="A1.Ex15.m1.1.1.1.1.1.3.4" xref="A1.Ex15.m1.1.1.1.1.1.3.4.cmml">t</mi><mo mathsize="90%" id="A1.Ex15.m1.1.1.1.1.1.3.5" xref="A1.Ex15.m1.1.1.1.1.1.3.5.cmml">≤</mo><mi mathsize="90%" id="A1.Ex15.m1.1.1.1.1.1.3.6" xref="A1.Ex15.m1.1.1.1.1.1.3.6.cmml">T</mi></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex15.m1.1.1.1.1.3" xref="A1.Ex15.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex15.m1.1b"><apply id="A1.Ex15.m1.1.1.cmml" xref="A1.Ex15.m1.1.1"><times id="A1.Ex15.m1.1.1.2.cmml" xref="A1.Ex15.m1.1.1.2"></times><apply id="A1.Ex15.m1.1.1.3.cmml" xref="A1.Ex15.m1.1.1.3"><ln id="A1.Ex15.m1.1.1.3.1.cmml" xref="A1.Ex15.m1.1.1.3.1"></ln><apply id="A1.Ex15.m1.1.1.3.2.cmml" xref="A1.Ex15.m1.1.1.3.2"><csymbol cd="ambiguous" id="A1.Ex15.m1.1.1.3.2.1.cmml" xref="A1.Ex15.m1.1.1.3.2">subscript</csymbol><ci id="A1.Ex15.m1.1.1.3.2.2.cmml" xref="A1.Ex15.m1.1.1.3.2.2">𝑝</ci><ci id="A1.Ex15.m1.1.1.3.2.3.cmml" xref="A1.Ex15.m1.1.1.3.2.3">𝜽</ci></apply></apply><apply id="A1.Ex15.m1.1.1.1.1.1.cmml" xref="A1.Ex15.m1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex15.m1.1.1.1.1.1.2.cmml" xref="A1.Ex15.m1.1.1.1.1">subscript</csymbol><apply id="A1.Ex15.m1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex15.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="A1.Ex15.m1.1.1.1.1.1.1.1.1.2.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex15.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A1.Ex15.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.2.2">𝐱</ci><ci id="A1.Ex15.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.2">𝑘</ci><ci id="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex15.m1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply><apply id="A1.Ex15.m1.1.1.1.1.1.3.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3"><and id="A1.Ex15.m1.1.1.1.1.1.3a.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3"></and><apply id="A1.Ex15.m1.1.1.1.1.1.3b.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3"><leq id="A1.Ex15.m1.1.1.1.1.1.3.3.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3.3"></leq><cn type="integer" id="A1.Ex15.m1.1.1.1.1.1.3.2.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3.2">1</cn><ci id="A1.Ex15.m1.1.1.1.1.1.3.4.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3.4">𝑡</ci></apply><apply id="A1.Ex15.m1.1.1.1.1.1.3c.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3"><leq id="A1.Ex15.m1.1.1.1.1.1.3.5.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3.5"></leq><share href="#A1.Ex15.m1.1.1.1.1.1.3.4.cmml" id="A1.Ex15.m1.1.1.1.1.1.3d.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3"></share><ci id="A1.Ex15.m1.1.1.1.1.1.3.6.cmml" xref="A1.Ex15.m1.1.1.1.1.1.3.6">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex15.m1.1c">\displaystyle\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex15.m2.5" class="ltx_Math" alttext="\displaystyle\geq\mathop{{\mathbb{E}}}_{\mathbf{z}_{s}\sim p(\mathbf{z}_{s}\mid\mathbf{z}_{s-1},\mathbf{x}_{s}^{k_{s}}),s\leq T}\sum_{t=1}^{T}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{t_{k}}\mid\mathbf{z}_{t-1})=\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{t_{k}}\mid\mathbf{z}_{t-1})\right]," display="inline"><semantics id="A1.Ex15.m2.5a"><mrow id="A1.Ex15.m2.5.5.1" xref="A1.Ex15.m2.5.5.1.1.cmml"><mrow id="A1.Ex15.m2.5.5.1.1" xref="A1.Ex15.m2.5.5.1.1.cmml"><mi id="A1.Ex15.m2.5.5.1.1.4" xref="A1.Ex15.m2.5.5.1.1.4.cmml"></mi><mo mathsize="90%" rspace="0.1389em" id="A1.Ex15.m2.5.5.1.1.5" xref="A1.Ex15.m2.5.5.1.1.5.cmml">≥</mo><mrow id="A1.Ex15.m2.5.5.1.1.1" xref="A1.Ex15.m2.5.5.1.1.1.cmml"><munder id="A1.Ex15.m2.5.5.1.1.1.2" xref="A1.Ex15.m2.5.5.1.1.1.2.cmml"><mo lspace="0.1389em" mathsize="108%" movablelimits="false" rspace="0.167em" 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xref="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.2.3.2.cmml" xref="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.2.3.2">𝑡</ci><ci id="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.2.3.3.cmml" xref="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.2.3.3">𝑘</ci></apply></apply><apply id="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.cmml" xref="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.2">𝐳</ci><apply id="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.3"><minus id="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.3.1.cmml" xref="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.3.1"></minus><ci id="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.3.2.cmml" xref="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.3.2">𝑡</ci><cn type="integer" id="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.3.3.cmml" xref="A1.Ex15.m2.5.5.1.1.2.1.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex15.m2.5c">\displaystyle\geq\mathop{{\mathbb{E}}}_{\mathbf{z}_{s}\sim p(\mathbf{z}_{s}\mid\mathbf{z}_{s-1},\mathbf{x}_{s}^{k_{s}}),s\leq T}\sum_{t=1}^{T}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{t_{k}}\mid\mathbf{z}_{t-1})=\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{t_{k}}\mid\mathbf{z}_{t-1})\right],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.1.p1.3" class="ltx_p"><span id="A1.SS2.1.p1.3.1" class="ltx_text" style="font-size:90%;">where the inequality is and equality when </span><math id="A1.SS2.1.p1.2.m1.1" class="ltx_Math" alttext="p_{\bm{\theta}}(\mathbf{z}_{s}\mid\mathbf{z}_{s-1},\mathbf{x}_{s}^{k_{s}})" display="inline"><semantics id="A1.SS2.1.p1.2.m1.1a"><mrow id="A1.SS2.1.p1.2.m1.1.1" xref="A1.SS2.1.p1.2.m1.1.1.cmml"><msub id="A1.SS2.1.p1.2.m1.1.1.3" xref="A1.SS2.1.p1.2.m1.1.1.3.cmml"><mi mathsize="90%" id="A1.SS2.1.p1.2.m1.1.1.3.2" xref="A1.SS2.1.p1.2.m1.1.1.3.2.cmml">p</mi><mi mathsize="90%" id="A1.SS2.1.p1.2.m1.1.1.3.3" xref="A1.SS2.1.p1.2.m1.1.1.3.3.cmml">𝜽</mi></msub><mo lspace="0em" rspace="0em" id="A1.SS2.1.p1.2.m1.1.1.2" xref="A1.SS2.1.p1.2.m1.1.1.2.cmml">​</mo><mrow id="A1.SS2.1.p1.2.m1.1.1.1.1" xref="A1.SS2.1.p1.2.m1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS2.1.p1.2.m1.1.1.1.1.2" xref="A1.SS2.1.p1.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS2.1.p1.2.m1.1.1.1.1.1" 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id="A1.SS2.1.p1.2.m1.1.1.1.1.1.2.2.2.3.2.cmml" xref="A1.SS2.1.p1.2.m1.1.1.1.1.1.2.2.2.3.2">𝑘</ci><ci id="A1.SS2.1.p1.2.m1.1.1.1.1.1.2.2.2.3.3.cmml" xref="A1.SS2.1.p1.2.m1.1.1.1.1.1.2.2.2.3.3">𝑠</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.2.m1.1c">p_{\bm{\theta}}(\mathbf{z}_{s}\mid\mathbf{z}_{s-1},\mathbf{x}_{s}^{k_{s}})</annotation></semantics></math><span id="A1.SS2.1.p1.3.2" class="ltx_text" style="font-size:90%;"> is a Dirac distribution. By applying </span><math id="A1.SS2.1.p1.3.m2.1" class="ltx_Math" alttext="\mathop{{\mathbb{E}}}_{\mathrm{forward}}" display="inline"><semantics id="A1.SS2.1.p1.3.m2.1a"><msub id="A1.SS2.1.p1.3.m2.1.1" xref="A1.SS2.1.p1.3.m2.1.1.cmml"><mo mathsize="108%" id="A1.SS2.1.p1.3.m2.1.1.2" xref="A1.SS2.1.p1.3.m2.1.1.2.cmml">𝔼</mo><mi mathsize="90%" id="A1.SS2.1.p1.3.m2.1.1.3" xref="A1.SS2.1.p1.3.m2.1.1.3.cmml">forward</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.1.p1.3.m2.1b"><apply id="A1.SS2.1.p1.3.m2.1.1.cmml" xref="A1.SS2.1.p1.3.m2.1.1"><csymbol cd="ambiguous" id="A1.SS2.1.p1.3.m2.1.1.1.cmml" xref="A1.SS2.1.p1.3.m2.1.1">subscript</csymbol><ci id="A1.SS2.1.p1.3.m2.1.1.2.cmml" xref="A1.SS2.1.p1.3.m2.1.1.2">𝔼</ci><ci id="A1.SS2.1.p1.3.m2.1.1.3.cmml" xref="A1.SS2.1.p1.3.m2.1.1.3">forward</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.1.p1.3.m2.1c">\mathop{{\mathbb{E}}}_{\mathrm{forward}}</annotation></semantics></math><span id="A1.SS2.1.p1.3.3" class="ltx_text" style="font-size:90%;"> to both sides of the above display, and invoking the Markov property of the latents, we conclude that</span></p>
<table id="A6.EGx18" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.Ex16.m1.1" class="ltx_Math" alttext="\displaystyle\mathop{{\mathbb{E}}}_{\mathrm{forward}}[\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})]" display="inline"><semantics id="A1.Ex16.m1.1a"><mrow id="A1.Ex16.m1.1.1" xref="A1.Ex16.m1.1.1.cmml"><munder id="A1.Ex16.m1.1.1.2" xref="A1.Ex16.m1.1.1.2.cmml"><mo mathsize="108%" movablelimits="false" id="A1.Ex16.m1.1.1.2.2" xref="A1.Ex16.m1.1.1.2.2.cmml">𝔼</mo><mi mathsize="90%" id="A1.Ex16.m1.1.1.2.3" xref="A1.Ex16.m1.1.1.2.3.cmml">forward</mi></munder><mrow id="A1.Ex16.m1.1.1.1.1" xref="A1.Ex16.m1.1.1.1.2.cmml"><mo lspace="0em" maxsize="90%" minsize="90%" id="A1.Ex16.m1.1.1.1.1.2" xref="A1.Ex16.m1.1.1.1.2.1.cmml">[</mo><mrow id="A1.Ex16.m1.1.1.1.1.1" xref="A1.Ex16.m1.1.1.1.1.1.cmml"><mrow id="A1.Ex16.m1.1.1.1.1.1.3" xref="A1.Ex16.m1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A1.Ex16.m1.1.1.1.1.1.3.1" xref="A1.Ex16.m1.1.1.1.1.1.3.1.cmml">ln</mi><mo lspace="0.167em" 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id="A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.1.3.2">𝑘</ci><ci id="A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply><apply id="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3"><and id="A1.Ex16.m1.1.1.1.1.1.1.1.1.3a.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3"></and><apply id="A1.Ex16.m1.1.1.1.1.1.1.1.1.3b.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3"><leq id="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.3"></leq><cn type="integer" id="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.2">1</cn><ci id="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.4.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.4">𝑡</ci></apply><apply id="A1.Ex16.m1.1.1.1.1.1.1.1.1.3c.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3"><leq id="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.5.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.5"></leq><share href="#A1.Ex16.m1.1.1.1.1.1.1.1.1.3.4.cmml" id="A1.Ex16.m1.1.1.1.1.1.1.1.1.3d.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3"></share><ci id="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.6.cmml" xref="A1.Ex16.m1.1.1.1.1.1.1.1.1.3.6">𝑇</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex16.m1.1c">\displaystyle\mathop{{\mathbb{E}}}_{\mathrm{forward}}[\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})]</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex16.m2.3" class="ltx_Math" alttext="\displaystyle\geq\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{t_{k}}\mid\mathbf{z}_{t-1})\right]" display="inline"><semantics id="A1.Ex16.m2.3a"><mrow id="A1.Ex16.m2.3.3" xref="A1.Ex16.m2.3.3.cmml"><mi id="A1.Ex16.m2.3.3.3" xref="A1.Ex16.m2.3.3.3.cmml"></mi><mo mathsize="90%" rspace="0.1389em" id="A1.Ex16.m2.3.3.2" xref="A1.Ex16.m2.3.3.2.cmml">≥</mo><mrow id="A1.Ex16.m2.3.3.1" xref="A1.Ex16.m2.3.3.1.cmml"><munder id="A1.Ex16.m2.3.3.1.2" xref="A1.Ex16.m2.3.3.1.2.cmml"><mo lspace="0.1389em" mathsize="108%" movablelimits="false" rspace="0.0835em" id="A1.Ex16.m2.3.3.1.2.2" xref="A1.Ex16.m2.3.3.1.2.2.cmml">𝔼</mo><mi mathsize="90%" id="A1.Ex16.m2.3.3.1.2.3" xref="A1.Ex16.m2.3.3.1.2.3.cmml">forward</mi></munder><mrow id="A1.Ex16.m2.3.3.1.1" xref="A1.Ex16.m2.3.3.1.1.cmml"><munder id="A1.Ex16.m2.3.3.1.1.2" 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id="A1.Ex16.m2.3c">\displaystyle\geq\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{t_{k}}\mid\mathbf{z}_{t-1})\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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xref="A1.Ex17.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex17.m1.6c">\displaystyle\quad=\sum_{t=1}^{T}\mathop{{\mathbb{E}}}_{&lt;t}\mathop{{\mathbb{E}}}_{k_{t}\overset{\mathrm{unif}}{\sim}\{0,1,\dots,K\}}\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{t_{k}}\mid\mathbf{z}_{t-1})\right].</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.1.p1.5" class="ltx_p"><span id="A1.SS2.1.p1.5.1" class="ltx_text" style="font-size:90%;">∎</span></p>
</div>
</div>
<div id="A1.SS2.p2" class="ltx_para">
<p id="A1.SS2.p2.1" class="ltx_p"><span id="A1.SS2.p2.1.1" class="ltx_text" style="font-size:90%;">We now unpack the terms obtained from the preceding claim.</span></p>
</div>
<div id="Thmclaim2" class="ltx_theorem ltx_theorem_claim">
<h6 class="ltx_title ltx_runin ltx_title_theorem" style="font-size:90%;">
<span class="ltx_tag ltx_tag_theorem"><span id="Thmclaim2.2.1.1" class="ltx_text ltx_font_bold">Claim 2</span></span><span id="Thmclaim2.3.2" class="ltx_text ltx_font_bold"> </span>(ELBO w.r.t. <math id="Thmclaim2.1.m1.1" class="ltx_Math" alttext="q" display="inline"><semantics id="Thmclaim2.1.m1.1b"><mi id="Thmclaim2.1.m1.1.1" xref="Thmclaim2.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="Thmclaim2.1.m1.1c"><ci id="Thmclaim2.1.m1.1.1.cmml" xref="Thmclaim2.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmclaim2.1.m1.1d">q</annotation></semantics></math>)<span id="Thmclaim2.4.3" class="ltx_text ltx_font_bold">.</span>
</h6>
<div id="Thmclaim2.p1" class="ltx_para">
<p id="Thmclaim2.p1.5" class="ltx_p"><span id="Thmclaim2.p1.5.1" class="ltx_text ltx_font_italic" style="font-size:90%;">It holds that</span></p>
<table id="A6.EGx19" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.Ex18.m1.5" class="ltx_Math" alttext="\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{z}_{t-1})\right]\geq C_{1}(\mathbf{x}_{0},k_{t})+\left[\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{0})}\right]." display="inline"><semantics id="A1.Ex18.m1.5a"><mrow id="A1.Ex18.m1.5.5.1" xref="A1.Ex18.m1.5.5.1.1.cmml"><mrow id="A1.Ex18.m1.5.5.1.1" xref="A1.Ex18.m1.5.5.1.1.cmml"><mrow id="A1.Ex18.m1.5.5.1.1.1" xref="A1.Ex18.m1.5.5.1.1.1.cmml"><munder id="A1.Ex18.m1.5.5.1.1.1.2" xref="A1.Ex18.m1.5.5.1.1.1.2.cmml"><mo mathsize="108%" movablelimits="false" id="A1.Ex18.m1.5.5.1.1.1.2.2" xref="A1.Ex18.m1.5.5.1.1.1.2.2.cmml">𝔼</mo><mrow 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id="A1.Ex18.m1.5.5.1.1.3.2.3" xref="A1.Ex18.m1.5.5.1.1.3.2.3.cmml">​</mo><mrow id="A1.Ex18.m1.5.5.1.1.3.2.2.2" xref="A1.Ex18.m1.5.5.1.1.3.2.2.3.cmml"><mo maxsize="90%" minsize="90%" id="A1.Ex18.m1.5.5.1.1.3.2.2.2.3" xref="A1.Ex18.m1.5.5.1.1.3.2.2.3.cmml">(</mo><msub id="A1.Ex18.m1.5.5.1.1.2.1.1.1.1" xref="A1.Ex18.m1.5.5.1.1.2.1.1.1.1.cmml"><mi mathsize="90%" id="A1.Ex18.m1.5.5.1.1.2.1.1.1.1.2" xref="A1.Ex18.m1.5.5.1.1.2.1.1.1.1.2.cmml">𝐱</mi><mn mathsize="90%" id="A1.Ex18.m1.5.5.1.1.2.1.1.1.1.3" xref="A1.Ex18.m1.5.5.1.1.2.1.1.1.1.3.cmml">0</mn></msub><mo mathsize="90%" id="A1.Ex18.m1.5.5.1.1.3.2.2.2.4" xref="A1.Ex18.m1.5.5.1.1.3.2.2.3.cmml">,</mo><msub id="A1.Ex18.m1.5.5.1.1.3.2.2.2.2" xref="A1.Ex18.m1.5.5.1.1.3.2.2.2.2.cmml"><mi mathsize="90%" id="A1.Ex18.m1.5.5.1.1.3.2.2.2.2.2" xref="A1.Ex18.m1.5.5.1.1.3.2.2.2.2.2.cmml">k</mi><mi mathsize="90%" id="A1.Ex18.m1.5.5.1.1.3.2.2.2.2.3" xref="A1.Ex18.m1.5.5.1.1.3.2.2.2.2.3.cmml">t</mi></msub><mo maxsize="90%" minsize="90%" 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id="A1.Ex18.m1.4.4.2.1.1.1.2.3.2.2.1.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.2.3.2.2">subscript</csymbol><ci id="A1.Ex18.m1.4.4.2.1.1.1.2.3.2.2.2.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.2.3.2.2.2">𝑘</ci><ci id="A1.Ex18.m1.4.4.2.1.1.1.2.3.2.2.3.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.2.3.2.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex18.m1.4.4.2.1.1.1.2.3.2.3.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.2.3.2.3">1</cn></apply><ci id="A1.Ex18.m1.4.4.2.1.1.1.2.3.3.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.2.3.3">𝐾</ci></apply></apply><apply id="A1.Ex18.m1.4.4.2.1.1.1.3.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex18.m1.4.4.2.1.1.1.3.1.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.3">superscript</csymbol><apply id="A1.Ex18.m1.4.4.2.1.1.1.3.2.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex18.m1.4.4.2.1.1.1.3.2.1.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.3">subscript</csymbol><ci id="A1.Ex18.m1.4.4.2.1.1.1.3.2.2.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.3.2.2">𝐱</ci><ci id="A1.Ex18.m1.4.4.2.1.1.1.3.2.3.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.3.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex18.m1.4.4.2.1.1.1.3.3.cmml" xref="A1.Ex18.m1.4.4.2.1.1.1.3.3">0</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex18.m1.5c">\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{z}_{t-1})\right]\geq C_{1}(\mathbf{x}_{0},k_{t})+\left[\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{0})}\right].</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
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xref="Thmclaim2.p1.1.1.m1.1.1.1.1.1.2.cmml">𝐱</mi><mn id="Thmclaim2.p1.1.1.m1.1.1.1.1.1.3" xref="Thmclaim2.p1.1.1.m1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="Thmclaim2.p1.1.1.m1.2.2.2.2.4" xref="Thmclaim2.p1.1.1.m1.2.2.2.3.cmml">,</mo><msub id="Thmclaim2.p1.1.1.m1.2.2.2.2.2" xref="Thmclaim2.p1.1.1.m1.2.2.2.2.2.cmml"><mi id="Thmclaim2.p1.1.1.m1.2.2.2.2.2.2" xref="Thmclaim2.p1.1.1.m1.2.2.2.2.2.2.cmml">k</mi><mi id="Thmclaim2.p1.1.1.m1.2.2.2.2.2.3" xref="Thmclaim2.p1.1.1.m1.2.2.2.2.2.3.cmml">t</mi></msub><mo stretchy="false" id="Thmclaim2.p1.1.1.m1.2.2.2.2.5" xref="Thmclaim2.p1.1.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmclaim2.p1.1.1.m1.2b"><apply id="Thmclaim2.p1.1.1.m1.2.2.cmml" xref="Thmclaim2.p1.1.1.m1.2.2"><times id="Thmclaim2.p1.1.1.m1.2.2.3.cmml" xref="Thmclaim2.p1.1.1.m1.2.2.3"></times><apply id="Thmclaim2.p1.1.1.m1.2.2.4.cmml" xref="Thmclaim2.p1.1.1.m1.2.2.4"><csymbol cd="ambiguous" id="Thmclaim2.p1.1.1.m1.2.2.4.1.cmml" xref="Thmclaim2.p1.1.1.m1.2.2.4">subscript</csymbol><ci id="Thmclaim2.p1.1.1.m1.2.2.4.2.cmml" xref="Thmclaim2.p1.1.1.m1.2.2.4.2">𝐶</ci><cn type="integer" id="Thmclaim2.p1.1.1.m1.2.2.4.3.cmml" xref="Thmclaim2.p1.1.1.m1.2.2.4.3">1</cn></apply><interval closure="open" id="Thmclaim2.p1.1.1.m1.2.2.2.3.cmml" xref="Thmclaim2.p1.1.1.m1.2.2.2.2"><apply id="Thmclaim2.p1.1.1.m1.1.1.1.1.1.cmml" xref="Thmclaim2.p1.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmclaim2.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmclaim2.p1.1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="Thmclaim2.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="Thmclaim2.p1.1.1.m1.1.1.1.1.1.2">𝐱</ci><cn type="integer" id="Thmclaim2.p1.1.1.m1.1.1.1.1.1.3.cmml" xref="Thmclaim2.p1.1.1.m1.1.1.1.1.1.3">0</cn></apply><apply id="Thmclaim2.p1.1.1.m1.2.2.2.2.2.cmml" xref="Thmclaim2.p1.1.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="Thmclaim2.p1.1.1.m1.2.2.2.2.2.1.cmml" xref="Thmclaim2.p1.1.1.m1.2.2.2.2.2">subscript</csymbol><ci id="Thmclaim2.p1.1.1.m1.2.2.2.2.2.2.cmml" xref="Thmclaim2.p1.1.1.m1.2.2.2.2.2.2">𝑘</ci><ci id="Thmclaim2.p1.1.1.m1.2.2.2.2.2.3.cmml" xref="Thmclaim2.p1.1.1.m1.2.2.2.2.2.3">𝑡</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmclaim2.p1.1.1.m1.2c">C_{1}(\mathbf{x}_{0},k_{t})</annotation></semantics></math> is a constant depending only on <math id="Thmclaim2.p1.2.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{0}" display="inline"><semantics id="Thmclaim2.p1.2.2.m2.1a"><msub id="Thmclaim2.p1.2.2.m2.1.1" xref="Thmclaim2.p1.2.2.m2.1.1.cmml"><mi id="Thmclaim2.p1.2.2.m2.1.1.2" xref="Thmclaim2.p1.2.2.m2.1.1.2.cmml">𝐱</mi><mn id="Thmclaim2.p1.2.2.m2.1.1.3" xref="Thmclaim2.p1.2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="Thmclaim2.p1.2.2.m2.1b"><apply id="Thmclaim2.p1.2.2.m2.1.1.cmml" xref="Thmclaim2.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmclaim2.p1.2.2.m2.1.1.1.cmml" xref="Thmclaim2.p1.2.2.m2.1.1">subscript</csymbol><ci id="Thmclaim2.p1.2.2.m2.1.1.2.cmml" xref="Thmclaim2.p1.2.2.m2.1.1.2">𝐱</ci><cn type="integer" id="Thmclaim2.p1.2.2.m2.1.1.3.cmml" xref="Thmclaim2.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmclaim2.p1.2.2.m2.1c">\mathbf{x}_{0}</annotation></semantics></math> and <math id="Thmclaim2.p1.3.3.m3.1" class="ltx_Math" alttext="k_{t}" display="inline"><semantics id="Thmclaim2.p1.3.3.m3.1a"><msub id="Thmclaim2.p1.3.3.m3.1.1" xref="Thmclaim2.p1.3.3.m3.1.1.cmml"><mi id="Thmclaim2.p1.3.3.m3.1.1.2" xref="Thmclaim2.p1.3.3.m3.1.1.2.cmml">k</mi><mi id="Thmclaim2.p1.3.3.m3.1.1.3" xref="Thmclaim2.p1.3.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="Thmclaim2.p1.3.3.m3.1b"><apply id="Thmclaim2.p1.3.3.m3.1.1.cmml" xref="Thmclaim2.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="Thmclaim2.p1.3.3.m3.1.1.1.cmml" xref="Thmclaim2.p1.3.3.m3.1.1">subscript</csymbol><ci id="Thmclaim2.p1.3.3.m3.1.1.2.cmml" xref="Thmclaim2.p1.3.3.m3.1.1.2">𝑘</ci><ci id="Thmclaim2.p1.3.3.m3.1.1.3.cmml" xref="Thmclaim2.p1.3.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmclaim2.p1.3.3.m3.1c">k_{t}</annotation></semantics></math>, and where the inequality holds with equality if and only if <math id="Thmclaim2.p1.4.4.m4.2" class="ltx_Math" alttext="q(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}})\equiv p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1})" display="inline"><semantics id="Thmclaim2.p1.4.4.m4.2a"><mrow id="Thmclaim2.p1.4.4.m4.2.2" xref="Thmclaim2.p1.4.4.m4.2.2.cmml"><mrow id="Thmclaim2.p1.4.4.m4.1.1.1" xref="Thmclaim2.p1.4.4.m4.1.1.1.cmml"><mi id="Thmclaim2.p1.4.4.m4.1.1.1.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.3.cmml">q</mi><mo lspace="0em" rspace="0em" id="Thmclaim2.p1.4.4.m4.1.1.1.2" xref="Thmclaim2.p1.4.4.m4.1.1.1.2.cmml">​</mo><mrow id="Thmclaim2.p1.4.4.m4.1.1.1.1.1" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.cmml"><mo stretchy="false" id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.2" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.cmml"><msubsup id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.cmml"><mi id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.2.2" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.2.2.cmml">𝐱</mi><mi id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.2.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.2.3.cmml">t</mi><mrow id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.cmml"><mrow id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.cmml"><msub id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.2" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.2.cmml"><mi id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.2.2" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.2.2.cmml">k</mi><mi id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.2.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.2.3.cmml">t</mi></msub><mo id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.1" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.1.cmml">+</mo><mn id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.2.3.cmml">1</mn></mrow><mo lspace="0.278em" rspace="0.278em" id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.1" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.1.cmml">:</mo><mi id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.2.3.3.cmml">T</mi></mrow></msubsup><mo id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.1" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.1.cmml">∣</mo><msubsup id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.cmml"><mi id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.2.2" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.2.2.cmml">𝐱</mi><mi id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.2.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.2.3.cmml">t</mi><msub id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.3.cmml"><mi id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.3.2" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.3.2.cmml">k</mi><mi id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.3.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.3.3.3.cmml">t</mi></msub></msubsup></mrow><mo stretchy="false" id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.3" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmclaim2.p1.4.4.m4.2.2.3" xref="Thmclaim2.p1.4.4.m4.2.2.3.cmml">≡</mo><mrow id="Thmclaim2.p1.4.4.m4.2.2.2" xref="Thmclaim2.p1.4.4.m4.2.2.2.cmml"><msub id="Thmclaim2.p1.4.4.m4.2.2.2.3" xref="Thmclaim2.p1.4.4.m4.2.2.2.3.cmml"><mi id="Thmclaim2.p1.4.4.m4.2.2.2.3.2" xref="Thmclaim2.p1.4.4.m4.2.2.2.3.2.cmml">p</mi><mi id="Thmclaim2.p1.4.4.m4.2.2.2.3.3" xref="Thmclaim2.p1.4.4.m4.2.2.2.3.3.cmml">𝛉</mi></msub><mo lspace="0em" rspace="0em" id="Thmclaim2.p1.4.4.m4.2.2.2.2" xref="Thmclaim2.p1.4.4.m4.2.2.2.2.cmml">​</mo><mrow id="Thmclaim2.p1.4.4.m4.2.2.2.1.1" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.cmml"><mo stretchy="false" id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.2" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.cmml">(</mo><mrow id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.cmml"><msubsup id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.cmml"><mi id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.2.2" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.2.2.cmml">𝐱</mi><mi id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.2.3" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.2.3.cmml">t</mi><mrow id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.cmml"><mrow id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.cmml"><msub id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.2" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.2.cmml"><mi id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.2.2" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.2.2.cmml">k</mi><mi id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.2.3" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.2.3.cmml">t</mi></msub><mo id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.1" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.1.cmml">+</mo><mn id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.3" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.2.3.cmml">1</mn></mrow><mo lspace="0.278em" rspace="0.278em" id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.1" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.1.cmml">:</mo><mi id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.3" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.4.3.3.cmml">T</mi></mrow></msubsup><mo id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.3" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.3.cmml">∣</mo><mrow id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.2.2" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.2.3.cmml"><msubsup id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.1.1.1" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.1.1.1.cmml"><mi id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.1.1.1.2.2" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.1.1.1.2.3" 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xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.2.2.2.3.1.cmml">−</mo><mn id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.2.2.2.3.3" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mo stretchy="false" id="Thmclaim2.p1.4.4.m4.2.2.2.1.1.3" xref="Thmclaim2.p1.4.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmclaim2.p1.4.4.m4.2b"><apply id="Thmclaim2.p1.4.4.m4.2.2.cmml" xref="Thmclaim2.p1.4.4.m4.2.2"><equivalent id="Thmclaim2.p1.4.4.m4.2.2.3.cmml" xref="Thmclaim2.p1.4.4.m4.2.2.3"></equivalent><apply id="Thmclaim2.p1.4.4.m4.1.1.1.cmml" xref="Thmclaim2.p1.4.4.m4.1.1.1"><times id="Thmclaim2.p1.4.4.m4.1.1.1.2.cmml" xref="Thmclaim2.p1.4.4.m4.1.1.1.2"></times><ci id="Thmclaim2.p1.4.4.m4.1.1.1.3.cmml" xref="Thmclaim2.p1.4.4.m4.1.1.1.3">𝑞</ci><apply id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="Thmclaim2.p1.4.4.m4.1.1.1.1.1"><csymbol cd="latexml" id="Thmclaim2.p1.4.4.m4.1.1.1.1.1.1.1.cmml" 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id="Thmclaim2.p1.4.4.m4.2c">q(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}})\equiv p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1})</annotation></semantics></math>.</span></p>
</div>
</div>
<div id="A1.SS2.2" class="ltx_proof">
<h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof" style="font-size:90%;">Proof.</h6>
<div id="A1.SS2.2.p1" class="ltx_para">
<p id="A1.SS2.2.p1.8" class="ltx_p"><span id="A1.SS2.2.p1.8.1" class="ltx_text" style="font-size:90%;">We have that</span></p>
<table id="A6.EGx20" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.Ex19.m1.2.2.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex19.m1.2c">\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{z}_{t-1})\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3"><ci id="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.1.cmml" xref="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.1">:</ci><apply id="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.cmml" xref="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2"><plus id="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.1.cmml" xref="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.1"></plus><apply id="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.2.cmml" xref="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.2"><csymbol cd="ambiguous" id="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.2.1.cmml" xref="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.2">subscript</csymbol><ci id="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.2.2.cmml" xref="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.2.2">𝑘</ci><ci id="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.2.3.cmml" xref="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.3.cmml" xref="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.2.3">1</cn></apply><ci id="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.3.cmml" xref="A1.Ex20.m1.2.2.1.1.1.1.1.1.4.2.3.3">𝐾</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex20.m1.2c">\displaystyle=\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\ln\int p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})\mathrm{d}\mathbf{x}_{t}^{k_{t}+1:K}\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.Ex21.m1.4.4.2.1.1.1.3.3.1.cmml" xref="A1.Ex21.m1.4.4.2.1.1.1.3.3">subscript</csymbol><ci id="A1.Ex21.m1.4.4.2.1.1.1.3.3.2.cmml" xref="A1.Ex21.m1.4.4.2.1.1.1.3.3.2">𝑘</ci><ci id="A1.Ex21.m1.4.4.2.1.1.1.3.3.3.cmml" xref="A1.Ex21.m1.4.4.2.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex21.m1.6c">\displaystyle=\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\ln\left(\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}+1:K}\sim q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\left[\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\right]\right)\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex22.m1.5" class="ltx_Math" alttext="\displaystyle\geq\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}+1:K}\sim q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\left[\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\right]\right]" display="inline"><semantics id="A1.Ex22.m1.5a"><mrow id="A1.Ex22.m1.5.5" xref="A1.Ex22.m1.5.5.cmml"><mi id="A1.Ex22.m1.5.5.3" xref="A1.Ex22.m1.5.5.3.cmml"></mi><mo mathsize="90%" rspace="0.1389em" id="A1.Ex22.m1.5.5.2" xref="A1.Ex22.m1.5.5.2.cmml">≥</mo><mrow id="A1.Ex22.m1.5.5.1" xref="A1.Ex22.m1.5.5.1.cmml"><munder id="A1.Ex22.m1.5.5.1.2" xref="A1.Ex22.m1.5.5.1.2.cmml"><mo lspace="0.1389em" mathsize="108%" movablelimits="false" rspace="0em" id="A1.Ex22.m1.5.5.1.2.2" xref="A1.Ex22.m1.5.5.1.2.2.cmml">𝔼</mo><mrow 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xref="A1.Ex22.m1.4.4.2.1.1.1.2.3.3">𝐾</ci></apply></apply><apply id="A1.Ex22.m1.4.4.2.1.1.1.3.cmml" xref="A1.Ex22.m1.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex22.m1.4.4.2.1.1.1.3.1.cmml" xref="A1.Ex22.m1.4.4.2.1.1.1.3">superscript</csymbol><apply id="A1.Ex22.m1.4.4.2.1.1.1.3.2.cmml" xref="A1.Ex22.m1.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex22.m1.4.4.2.1.1.1.3.2.1.cmml" xref="A1.Ex22.m1.4.4.2.1.1.1.3">subscript</csymbol><ci id="A1.Ex22.m1.4.4.2.1.1.1.3.2.2.cmml" xref="A1.Ex22.m1.4.4.2.1.1.1.3.2.2">𝐱</ci><ci id="A1.Ex22.m1.4.4.2.1.1.1.3.2.3.cmml" xref="A1.Ex22.m1.4.4.2.1.1.1.3.2.3">𝑡</ci></apply><apply id="A1.Ex22.m1.4.4.2.1.1.1.3.3.cmml" xref="A1.Ex22.m1.4.4.2.1.1.1.3.3"><csymbol cd="ambiguous" id="A1.Ex22.m1.4.4.2.1.1.1.3.3.1.cmml" xref="A1.Ex22.m1.4.4.2.1.1.1.3.3">subscript</csymbol><ci id="A1.Ex22.m1.4.4.2.1.1.1.3.3.2.cmml" xref="A1.Ex22.m1.4.4.2.1.1.1.3.3.2">𝑘</ci><ci id="A1.Ex22.m1.4.4.2.1.1.1.3.3.3.cmml" xref="A1.Ex22.m1.4.4.2.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex22.m1.5c">\displaystyle\geq\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}+1:K}\sim q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\left[\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\right]\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">((Jensen’s inequality))</span></td>
</tr></tbody>
<tbody id="A1.Ex23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.Ex23.m1.3.3.2.1.1.1.3.2.3.cmml" xref="A1.Ex23.m1.3.3.2.1.1.1.3.2.3">𝑡</ci></apply><apply id="A1.Ex23.m1.3.3.2.1.1.1.3.3.cmml" xref="A1.Ex23.m1.3.3.2.1.1.1.3.3"><csymbol cd="ambiguous" id="A1.Ex23.m1.3.3.2.1.1.1.3.3.1.cmml" xref="A1.Ex23.m1.3.3.2.1.1.1.3.3">subscript</csymbol><ci id="A1.Ex23.m1.3.3.2.1.1.1.3.3.2.cmml" xref="A1.Ex23.m1.3.3.2.1.1.1.3.3.2">𝑘</ci><ci id="A1.Ex23.m1.3.3.2.1.1.1.3.3.3.cmml" xref="A1.Ex23.m1.3.3.2.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex23.m1.4c">\displaystyle=\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\left[\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(Markov property of <math id="A1.Ex23.1.1.m1.1" class="ltx_Math" alttext="q(\cdot)" display="inline"><semantics id="A1.Ex23.1.1.m1.1b"><mrow id="A1.Ex23.1.1.m1.1.2" xref="A1.Ex23.1.1.m1.1.2.cmml"><mi id="A1.Ex23.1.1.m1.1.2.2" xref="A1.Ex23.1.1.m1.1.2.2.cmml">q</mi><mo lspace="0em" rspace="0em" id="A1.Ex23.1.1.m1.1.2.1" xref="A1.Ex23.1.1.m1.1.2.1.cmml">​</mo><mrow id="A1.Ex23.1.1.m1.1.2.3.2" xref="A1.Ex23.1.1.m1.1.2.cmml"><mo stretchy="false" id="A1.Ex23.1.1.m1.1.2.3.2.1" xref="A1.Ex23.1.1.m1.1.2.cmml">(</mo><mo lspace="0em" rspace="0em" id="A1.Ex23.1.1.m1.1.1" xref="A1.Ex23.1.1.m1.1.1.cmml">⋅</mo><mo stretchy="false" id="A1.Ex23.1.1.m1.1.2.3.2.2" xref="A1.Ex23.1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.Ex23.1.1.m1.1c"><apply id="A1.Ex23.1.1.m1.1.2.cmml" xref="A1.Ex23.1.1.m1.1.2"><times id="A1.Ex23.1.1.m1.1.2.1.cmml" xref="A1.Ex23.1.1.m1.1.2.1"></times><ci id="A1.Ex23.1.1.m1.1.2.2.cmml" xref="A1.Ex23.1.1.m1.1.2.2">𝑞</ci><ci id="A1.Ex23.1.1.m1.1.1.cmml" xref="A1.Ex23.1.1.m1.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex23.1.1.m1.1d">q(\cdot)</annotation></semantics></math>)</span></td>
</tr></tbody>
<tbody id="A1.Ex24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex24.m1.4" class="ltx_Math" alttext="\displaystyle=C_{1}(\mathbf{x}_{0},k_{t})+\left[\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{0})}\right]," display="inline"><semantics id="A1.Ex24.m1.4a"><mrow id="A1.Ex24.m1.4.4.1" xref="A1.Ex24.m1.4.4.1.1.cmml"><mrow id="A1.Ex24.m1.4.4.1.1" xref="A1.Ex24.m1.4.4.1.1.cmml"><mi id="A1.Ex24.m1.4.4.1.1.5" xref="A1.Ex24.m1.4.4.1.1.5.cmml"></mi><mo mathsize="90%" id="A1.Ex24.m1.4.4.1.1.4" xref="A1.Ex24.m1.4.4.1.1.4.cmml">=</mo><mrow id="A1.Ex24.m1.4.4.1.1.3" xref="A1.Ex24.m1.4.4.1.1.3.cmml"><mrow id="A1.Ex24.m1.4.4.1.1.2.2" xref="A1.Ex24.m1.4.4.1.1.2.2.cmml"><msub id="A1.Ex24.m1.4.4.1.1.2.2.4" xref="A1.Ex24.m1.4.4.1.1.2.2.4.cmml"><mi mathsize="90%" id="A1.Ex24.m1.4.4.1.1.2.2.4.2" 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xref="A1.Ex24.m1.3.3.2.1.1.1.2.3.3">𝐾</ci></apply></apply><apply id="A1.Ex24.m1.3.3.2.1.1.1.3.cmml" xref="A1.Ex24.m1.3.3.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex24.m1.3.3.2.1.1.1.3.1.cmml" xref="A1.Ex24.m1.3.3.2.1.1.1.3">superscript</csymbol><apply id="A1.Ex24.m1.3.3.2.1.1.1.3.2.cmml" xref="A1.Ex24.m1.3.3.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex24.m1.3.3.2.1.1.1.3.2.1.cmml" xref="A1.Ex24.m1.3.3.2.1.1.1.3">subscript</csymbol><ci id="A1.Ex24.m1.3.3.2.1.1.1.3.2.2.cmml" xref="A1.Ex24.m1.3.3.2.1.1.1.3.2.2">𝐱</ci><ci id="A1.Ex24.m1.3.3.2.1.1.1.3.2.3.cmml" xref="A1.Ex24.m1.3.3.2.1.1.1.3.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex24.m1.3.3.2.1.1.1.3.3.cmml" xref="A1.Ex24.m1.3.3.2.1.1.1.3.3">0</cn></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex24.m1.4c">\displaystyle=C_{1}(\mathbf{x}_{0},k_{t})+\left[\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{0})}\right],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
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encoding="application/x-tex" id="A1.SS2.2.p1.1.m1.5b">C_{1}(\mathbf{x}_{0},k_{t})=\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\left[\ln\frac{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{0}}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\right]</annotation></semantics></math><span id="A1.SS2.2.p1.4.2" class="ltx_text" style="font-size:90%;"> depends only on </span><math id="A1.SS2.2.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{0}" display="inline"><semantics id="A1.SS2.2.p1.2.m2.1a"><msub id="A1.SS2.2.p1.2.m2.1.1" xref="A1.SS2.2.p1.2.m2.1.1.cmml"><mi mathsize="90%" id="A1.SS2.2.p1.2.m2.1.1.2" xref="A1.SS2.2.p1.2.m2.1.1.2.cmml">𝐱</mi><mn mathsize="90%" id="A1.SS2.2.p1.2.m2.1.1.3" xref="A1.SS2.2.p1.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.2.m2.1b"><apply id="A1.SS2.2.p1.2.m2.1.1.cmml" xref="A1.SS2.2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p1.2.m2.1.1.1.cmml" xref="A1.SS2.2.p1.2.m2.1.1">subscript</csymbol><ci id="A1.SS2.2.p1.2.m2.1.1.2.cmml" xref="A1.SS2.2.p1.2.m2.1.1.2">𝐱</ci><cn type="integer" id="A1.SS2.2.p1.2.m2.1.1.3.cmml" xref="A1.SS2.2.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.2.m2.1c">\mathbf{x}_{0}</annotation></semantics></math><span id="A1.SS2.2.p1.4.3" class="ltx_text" style="font-size:90%;"> and </span><math id="A1.SS2.2.p1.3.m3.1" class="ltx_Math" alttext="k_{t}" display="inline"><semantics id="A1.SS2.2.p1.3.m3.1a"><msub id="A1.SS2.2.p1.3.m3.1.1" xref="A1.SS2.2.p1.3.m3.1.1.cmml"><mi mathsize="90%" id="A1.SS2.2.p1.3.m3.1.1.2" xref="A1.SS2.2.p1.3.m3.1.1.2.cmml">k</mi><mi mathsize="90%" id="A1.SS2.2.p1.3.m3.1.1.3" xref="A1.SS2.2.p1.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.3.m3.1b"><apply id="A1.SS2.2.p1.3.m3.1.1.cmml" xref="A1.SS2.2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A1.SS2.2.p1.3.m3.1.1.1.cmml" xref="A1.SS2.2.p1.3.m3.1.1">subscript</csymbol><ci id="A1.SS2.2.p1.3.m3.1.1.2.cmml" xref="A1.SS2.2.p1.3.m3.1.1.2">𝑘</ci><ci id="A1.SS2.2.p1.3.m3.1.1.3.cmml" xref="A1.SS2.2.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.3.m3.1c">k_{t}</annotation></semantics></math><span id="A1.SS2.2.p1.4.4" class="ltx_text" style="font-size:90%;">. 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id="A1.SS2.2.p1.4.m4.2.2.2.1.1.1.2.2.2.3.cmml" xref="A1.SS2.2.p1.4.m4.2.2.2.1.1.1.2.2.2.3"><minus id="A1.SS2.2.p1.4.m4.2.2.2.1.1.1.2.2.2.3.1.cmml" xref="A1.SS2.2.p1.4.m4.2.2.2.1.1.1.2.2.2.3.1"></minus><ci id="A1.SS2.2.p1.4.m4.2.2.2.1.1.1.2.2.2.3.2.cmml" xref="A1.SS2.2.p1.4.m4.2.2.2.1.1.1.2.2.2.3.2">𝑡</ci><cn type="integer" id="A1.SS2.2.p1.4.m4.2.2.2.1.1.1.2.2.2.3.3.cmml" xref="A1.SS2.2.p1.4.m4.2.2.2.1.1.1.2.2.2.3.3">1</cn></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.4.m4.2c">q(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}})\equiv p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1})</annotation></semantics></math><span id="A1.SS2.2.p1.4.5" class="ltx_text" style="font-size:90%;">, then</span></p>
<table id="A6.EGx21" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.Ex25.m1.3.3.2.1.1.1.3.2.2">𝐱</ci><ci id="A1.Ex25.m1.3.3.2.1.1.1.3.2.3.cmml" xref="A1.Ex25.m1.3.3.2.1.1.1.3.2.3">𝑡</ci></apply><apply id="A1.Ex25.m1.3.3.2.1.1.1.3.3.cmml" xref="A1.Ex25.m1.3.3.2.1.1.1.3.3"><csymbol cd="ambiguous" id="A1.Ex25.m1.3.3.2.1.1.1.3.3.1.cmml" xref="A1.Ex25.m1.3.3.2.1.1.1.3.3">subscript</csymbol><ci id="A1.Ex25.m1.3.3.2.1.1.1.3.3.2.cmml" xref="A1.Ex25.m1.3.3.2.1.1.1.3.3.2">𝑘</ci><ci id="A1.Ex25.m1.3.3.2.1.1.1.3.3.3.cmml" xref="A1.Ex25.m1.3.3.2.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex25.m1.4c">\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}+1:K}\sim q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\left[\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\right]</annotation></semantics></math></td>
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xref="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.2.2.cmml" xref="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.2.2">𝐱</ci><ci id="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.2.3.cmml" xref="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.2.3">𝑡</ci></apply><apply id="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.3.cmml" xref="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.3"><csymbol cd="ambiguous" id="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.3.1.cmml" xref="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.3">subscript</csymbol><ci id="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.3.2.cmml" xref="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.3.2">𝑘</ci><ci id="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.3.3.cmml" xref="A1.Ex25.m2.3.3.2.2.1.1.1.1.1.1.2.2.2.3.3">𝑡</ci></apply></apply></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex25.m2.3c">\displaystyle=\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{z}_{t-1})+\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}+1:K}\sim q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}})\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.2.p1.7" class="ltx_p"><span id="A1.SS2.2.p1.7.1" class="ltx_text" style="font-size:90%;">Since </span><math id="A1.SS2.2.p1.5.m1.2" class="ltx_Math" alttext="\ln(\cdot)" display="inline"><semantics id="A1.SS2.2.p1.5.m1.2a"><mrow id="A1.SS2.2.p1.5.m1.2.2.4" xref="A1.SS2.2.p1.5.m1.2.2.3.cmml"><mi mathsize="90%" id="A1.SS2.2.p1.5.m1.2.2.2.2" xref="A1.SS2.2.p1.5.m1.2.2.3.1.cmml">ln</mi><mo id="A1.SS2.2.p1.5.m1.2.2.4a" xref="A1.SS2.2.p1.5.m1.2.2.3.1.cmml">⁡</mo><mrow id="A1.SS2.2.p1.5.m1.2.2.4.1" xref="A1.SS2.2.p1.5.m1.2.2.3.cmml"><mo id="A1.SS2.2.p1.5.m1.2.2.4.1.1" xref="A1.SS2.2.p1.5.m1.2.2.3.1.cmml">(</mo><mo lspace="0em" mathsize="90%" rspace="0em" id="A1.SS2.2.p1.5.m1.1.1.1.1.1" xref="A1.SS2.2.p1.5.m1.1.1.1.1.1.cmml">⋅</mo><mo id="A1.SS2.2.p1.5.m1.2.2.4.1.2" xref="A1.SS2.2.p1.5.m1.2.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.2.p1.5.m1.2b"><apply id="A1.SS2.2.p1.5.m1.2.2.3.cmml" xref="A1.SS2.2.p1.5.m1.2.2.4"><ln id="A1.SS2.2.p1.5.m1.2.2.3.1.cmml" xref="A1.SS2.2.p1.5.m1.2.2.2.2"></ln><ci id="A1.SS2.2.p1.5.m1.1.1.1.1.1.cmml" xref="A1.SS2.2.p1.5.m1.1.1.1.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.5.m1.2c">\ln(\cdot)</annotation></semantics></math><span id="A1.SS2.2.p1.7.2" class="ltx_text" style="font-size:90%;"> is strictly concave, </span><math id="A1.SS2.2.p1.6.m2.2" class="ltx_Math" alttext="\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}+1:K}\sim q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{z}_{t-1})\right]=0" display="inline"><semantics id="A1.SS2.2.p1.6.m2.2a"><mrow id="A1.SS2.2.p1.6.m2.2.2" xref="A1.SS2.2.p1.6.m2.2.2.cmml"><mrow id="A1.SS2.2.p1.6.m2.2.2.1" xref="A1.SS2.2.p1.6.m2.2.2.1.cmml"><msub id="A1.SS2.2.p1.6.m2.2.2.1.2" xref="A1.SS2.2.p1.6.m2.2.2.1.2.cmml"><mo mathsize="108%" id="A1.SS2.2.p1.6.m2.2.2.1.2.2" xref="A1.SS2.2.p1.6.m2.2.2.1.2.2.cmml">𝔼</mo><mrow 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xref="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.2.2">𝐱</ci><ci id="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.2.3.cmml" xref="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.2.3">𝑡</ci></apply><apply id="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.3.cmml" xref="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.3"><csymbol cd="ambiguous" id="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.3.1.cmml" xref="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.3">subscript</csymbol><ci id="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.3.2.cmml" xref="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.3.2">𝑘</ci><ci id="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.3.3.cmml" xref="A1.SS2.2.p1.7.m3.2.2.2.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.2.p1.7.m3.2c">p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{z}_{t-1},\mathbf{x}_{t}^{k_{t}})=q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}_{t}^{k_{t}})</annotation></semantics></math><span id="A1.SS2.2.p1.7.4" class="ltx_text" style="font-size:90%;">.
∎</span></p>
</div>
</div>
<div id="Thmclaim3" class="ltx_theorem ltx_theorem_claim">
<h6 class="ltx_title ltx_runin ltx_title_theorem" style="font-size:90%;">
<span class="ltx_tag ltx_tag_theorem"><span id="Thmclaim3.1.1.1" class="ltx_text ltx_font_bold">Claim 3</span></span><span id="Thmclaim3.2.2" class="ltx_text ltx_font_bold"> </span>(Computing the expected ELBO)<span id="Thmclaim3.3.3" class="ltx_text ltx_font_bold">.</span>
</h6>
<div id="Thmclaim3.p1" class="ltx_para">
<table id="A6.EGx22" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.Ex26.m1.3.3.2.1.1.1.2"><csymbol cd="ambiguous" id="A1.Ex26.m1.3.3.2.1.1.1.2.2.1.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2">subscript</csymbol><ci id="A1.Ex26.m1.3.3.2.1.1.1.2.2.2.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.2.2">𝐱</ci><ci id="A1.Ex26.m1.3.3.2.1.1.1.2.2.3.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.2.3">𝑡</ci></apply><apply id="A1.Ex26.m1.3.3.2.1.1.1.2.3.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.3"><ci id="A1.Ex26.m1.3.3.2.1.1.1.2.3.1.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.3.1">:</ci><apply id="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.3.2"><plus id="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.1.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.1"></plus><apply id="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.2.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.2"><csymbol cd="ambiguous" id="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.2.1.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.2">subscript</csymbol><ci id="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.2.2.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.2.2">𝑘</ci><ci id="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.2.3.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.3.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.3.2.3">1</cn></apply><ci id="A1.Ex26.m1.3.3.2.1.1.1.2.3.3.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.2.3.3">𝐾</ci></apply></apply><apply id="A1.Ex26.m1.3.3.2.1.1.1.3.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex26.m1.3.3.2.1.1.1.3.1.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.3">subscript</csymbol><apply id="A1.Ex26.m1.3.3.2.1.1.1.3.2.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex26.m1.3.3.2.1.1.1.3.2.1.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.3">superscript</csymbol><ci id="A1.Ex26.m1.3.3.2.1.1.1.3.2.2.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.3.2.2">𝐱</ci><cn type="integer" id="A1.Ex26.m1.3.3.2.1.1.1.3.2.3.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.3.2.3">0</cn></apply><ci id="A1.Ex26.m1.3.3.2.1.1.1.3.3.cmml" xref="A1.Ex26.m1.3.3.2.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex26.m1.3c">\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}^{0}_{t})}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex27.m1.1" class="ltx_math_unparsed" alttext="\displaystyle=C_{3}(\mathbf{x}_{0},k_{t})+\mathbf{1}\{k_{t}=0\}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\mathbf{1}\{j\geq k_{t}\}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)," display="inline"><semantics id="A1.Ex27.m1.1a"><mrow id="A1.Ex27.m1.1b"><mo mathsize="90%" id="A1.Ex27.m1.1.1">=</mo><msub id="A1.Ex27.m1.1.2"><mi mathsize="90%" id="A1.Ex27.m1.1.2.2">C</mi><mn mathsize="90%" id="A1.Ex27.m1.1.2.3">3</mn></msub><mrow id="A1.Ex27.m1.1.3"><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.3.1">(</mo><msub id="A1.Ex27.m1.1.3.2"><mi mathsize="90%" id="A1.Ex27.m1.1.3.2.2">𝐱</mi><mn mathsize="90%" id="A1.Ex27.m1.1.3.2.3">0</mn></msub><mo mathsize="90%" id="A1.Ex27.m1.1.3.3">,</mo><msub id="A1.Ex27.m1.1.3.4"><mi mathsize="90%" id="A1.Ex27.m1.1.3.4.2">k</mi><mi mathsize="90%" id="A1.Ex27.m1.1.3.4.3">t</mi></msub><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.3.5">)</mo></mrow><mo mathsize="90%" id="A1.Ex27.m1.1.4">+</mo><mn mathsize="90%" id="A1.Ex27.m1.1.5">𝟏</mn><mrow id="A1.Ex27.m1.1.6"><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.6.1">{</mo><msub id="A1.Ex27.m1.1.6.2"><mi mathsize="90%" id="A1.Ex27.m1.1.6.2.2">k</mi><mi mathsize="90%" id="A1.Ex27.m1.1.6.2.3">t</mi></msub><mo mathsize="90%" id="A1.Ex27.m1.1.6.3">=</mo><mn mathsize="90%" id="A1.Ex27.m1.1.6.4">0</mn><mo maxsize="90%" minsize="90%" rspace="0.167em" id="A1.Ex27.m1.1.6.5">}</mo></mrow><mi mathsize="90%" id="A1.Ex27.m1.1.7">ln</mi><msub id="A1.Ex27.m1.1.8"><mi mathsize="90%" id="A1.Ex27.m1.1.8.2">p</mi><mi mathsize="90%" id="A1.Ex27.m1.1.8.3">𝜽</mi></msub><mrow id="A1.Ex27.m1.1.9"><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.9.1">(</mo><msubsup id="A1.Ex27.m1.1.9.2"><mi mathsize="90%" id="A1.Ex27.m1.1.9.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex27.m1.1.9.2.2.3">t</mi><mn mathsize="90%" id="A1.Ex27.m1.1.9.2.3">0</mn></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex27.m1.1.9.3">∣</mo><msubsup id="A1.Ex27.m1.1.9.4"><mi mathsize="90%" id="A1.Ex27.m1.1.9.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex27.m1.1.9.4.2.3">t</mi><mn mathsize="90%" id="A1.Ex27.m1.1.9.4.3">1</mn></msubsup><mo mathsize="90%" id="A1.Ex27.m1.1.9.5">,</mo><msub id="A1.Ex27.m1.1.9.6"><mi mathsize="90%" id="A1.Ex27.m1.1.9.6.2">𝐳</mi><mrow id="A1.Ex27.m1.1.9.6.3"><mi mathsize="90%" id="A1.Ex27.m1.1.9.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex27.m1.1.9.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex27.m1.1.9.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.9.7">)</mo></mrow><mo mathsize="90%" id="A1.Ex27.m1.1.10">+</mo><mstyle displaystyle="true" id="A1.Ex27.m1.1.11"><munderover id="A1.Ex27.m1.1.11a"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.Ex27.m1.1.11.2.2">∑</mo><mrow id="A1.Ex27.m1.1.11.2.3"><mi mathsize="90%" id="A1.Ex27.m1.1.11.2.3.2">j</mi><mo mathsize="90%" id="A1.Ex27.m1.1.11.2.3.1">=</mo><mn mathsize="90%" id="A1.Ex27.m1.1.11.2.3.3">1</mn></mrow><mrow id="A1.Ex27.m1.1.11.3"><mi mathsize="90%" id="A1.Ex27.m1.1.11.3.2">K</mi><mo mathsize="90%" id="A1.Ex27.m1.1.11.3.1">−</mo><mn mathsize="90%" id="A1.Ex27.m1.1.11.3.3">1</mn></mrow></munderover></mstyle><mn mathsize="90%" id="A1.Ex27.m1.1.12">𝟏</mn><mrow id="A1.Ex27.m1.1.13"><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.13.1">{</mo><mi mathsize="90%" id="A1.Ex27.m1.1.13.2">j</mi><mo mathsize="90%" id="A1.Ex27.m1.1.13.3">≥</mo><msub id="A1.Ex27.m1.1.13.4"><mi mathsize="90%" id="A1.Ex27.m1.1.13.4.2">k</mi><mi mathsize="90%" id="A1.Ex27.m1.1.13.4.3">t</mi></msub><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.13.5">}</mo></mrow><msub id="A1.Ex27.m1.1.14"><mi mathsize="90%" mathvariant="normal" id="A1.Ex27.m1.1.14.2">D</mi><mrow id="A1.Ex27.m1.1.14.3"><mi mathsize="90%" id="A1.Ex27.m1.1.14.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex27.m1.1.14.3.1">​</mo><mi mathsize="90%" id="A1.Ex27.m1.1.14.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex27.m1.1.15"><mo id="A1.Ex27.m1.1.15.1">(</mo><mi mathsize="90%" id="A1.Ex27.m1.1.15.2">q</mi><mrow id="A1.Ex27.m1.1.15.3"><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.15.3.1">(</mo><msubsup id="A1.Ex27.m1.1.15.3.2"><mi mathsize="90%" id="A1.Ex27.m1.1.15.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex27.m1.1.15.3.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex27.m1.1.15.3.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex27.m1.1.15.3.3">∣</mo><msubsup id="A1.Ex27.m1.1.15.3.4"><mi mathsize="90%" id="A1.Ex27.m1.1.15.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex27.m1.1.15.3.4.2.3">t</mi><mrow id="A1.Ex27.m1.1.15.3.4.3"><mi mathsize="90%" id="A1.Ex27.m1.1.15.3.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex27.m1.1.15.3.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex27.m1.1.15.3.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex27.m1.1.15.3.5">,</mo><msubsup id="A1.Ex27.m1.1.15.3.6"><mi mathsize="90%" id="A1.Ex27.m1.1.15.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex27.m1.1.15.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex27.m1.1.15.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.15.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex27.m1.1.15.4">∥</mo><msub id="A1.Ex27.m1.1.15.5"><mi mathsize="90%" id="A1.Ex27.m1.1.15.5.2">p</mi><mi mathsize="90%" id="A1.Ex27.m1.1.15.5.3">𝜽</mi></msub><mrow id="A1.Ex27.m1.1.15.6"><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.15.6.1">(</mo><msubsup id="A1.Ex27.m1.1.15.6.2"><mi mathsize="90%" id="A1.Ex27.m1.1.15.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex27.m1.1.15.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex27.m1.1.15.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex27.m1.1.15.6.3">∣</mo><msubsup id="A1.Ex27.m1.1.15.6.4"><mi mathsize="90%" id="A1.Ex27.m1.1.15.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex27.m1.1.15.6.4.2.3">t</mi><mrow id="A1.Ex27.m1.1.15.6.4.3"><mi mathsize="90%" id="A1.Ex27.m1.1.15.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex27.m1.1.15.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex27.m1.1.15.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex27.m1.1.15.6.5">,</mo><msub id="A1.Ex27.m1.1.15.6.6"><mi mathsize="90%" id="A1.Ex27.m1.1.15.6.6.2">𝐳</mi><mrow id="A1.Ex27.m1.1.15.6.6.3"><mi mathsize="90%" id="A1.Ex27.m1.1.15.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex27.m1.1.15.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex27.m1.1.15.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex27.m1.1.15.6.7">)</mo></mrow><mo id="A1.Ex27.m1.1.15.7">)</mo></mrow><mo mathsize="90%" id="A1.Ex27.m1.1.16">,</mo></mrow><annotation encoding="application/x-tex" id="A1.Ex27.m1.1c">\displaystyle=C_{3}(\mathbf{x}_{0},k_{t})+\mathbf{1}\{k_{t}=0\}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\mathbf{1}\{j\geq k_{t}\}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right),</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="Thmclaim3.p1.3" class="ltx_p"><span id="Thmclaim3.p1.3.3" class="ltx_text ltx_font_italic" style="font-size:90%;">where <math id="Thmclaim3.p1.1.1.m1.2" class="ltx_Math" alttext="C_{2}(\mathbf{x}_{0},k_{t})" display="inline"><semantics id="Thmclaim3.p1.1.1.m1.2a"><mrow id="Thmclaim3.p1.1.1.m1.2.2" xref="Thmclaim3.p1.1.1.m1.2.2.cmml"><msub id="Thmclaim3.p1.1.1.m1.2.2.4" xref="Thmclaim3.p1.1.1.m1.2.2.4.cmml"><mi id="Thmclaim3.p1.1.1.m1.2.2.4.2" xref="Thmclaim3.p1.1.1.m1.2.2.4.2.cmml">C</mi><mn id="Thmclaim3.p1.1.1.m1.2.2.4.3" xref="Thmclaim3.p1.1.1.m1.2.2.4.3.cmml">2</mn></msub><mo lspace="0em" rspace="0em" id="Thmclaim3.p1.1.1.m1.2.2.3" xref="Thmclaim3.p1.1.1.m1.2.2.3.cmml">​</mo><mrow id="Thmclaim3.p1.1.1.m1.2.2.2.2" xref="Thmclaim3.p1.1.1.m1.2.2.2.3.cmml"><mo stretchy="false" id="Thmclaim3.p1.1.1.m1.2.2.2.2.3" xref="Thmclaim3.p1.1.1.m1.2.2.2.3.cmml">(</mo><msub id="Thmclaim3.p1.1.1.m1.1.1.1.1.1" xref="Thmclaim3.p1.1.1.m1.1.1.1.1.1.cmml"><mi id="Thmclaim3.p1.1.1.m1.1.1.1.1.1.2" xref="Thmclaim3.p1.1.1.m1.1.1.1.1.1.2.cmml">𝐱</mi><mn id="Thmclaim3.p1.1.1.m1.1.1.1.1.1.3" xref="Thmclaim3.p1.1.1.m1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="Thmclaim3.p1.1.1.m1.2.2.2.2.4" xref="Thmclaim3.p1.1.1.m1.2.2.2.3.cmml">,</mo><msub id="Thmclaim3.p1.1.1.m1.2.2.2.2.2" xref="Thmclaim3.p1.1.1.m1.2.2.2.2.2.cmml"><mi id="Thmclaim3.p1.1.1.m1.2.2.2.2.2.2" xref="Thmclaim3.p1.1.1.m1.2.2.2.2.2.2.cmml">k</mi><mi id="Thmclaim3.p1.1.1.m1.2.2.2.2.2.3" xref="Thmclaim3.p1.1.1.m1.2.2.2.2.2.3.cmml">t</mi></msub><mo stretchy="false" id="Thmclaim3.p1.1.1.m1.2.2.2.2.5" xref="Thmclaim3.p1.1.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmclaim3.p1.1.1.m1.2b"><apply id="Thmclaim3.p1.1.1.m1.2.2.cmml" xref="Thmclaim3.p1.1.1.m1.2.2"><times id="Thmclaim3.p1.1.1.m1.2.2.3.cmml" xref="Thmclaim3.p1.1.1.m1.2.2.3"></times><apply id="Thmclaim3.p1.1.1.m1.2.2.4.cmml" xref="Thmclaim3.p1.1.1.m1.2.2.4"><csymbol cd="ambiguous" id="Thmclaim3.p1.1.1.m1.2.2.4.1.cmml" xref="Thmclaim3.p1.1.1.m1.2.2.4">subscript</csymbol><ci id="Thmclaim3.p1.1.1.m1.2.2.4.2.cmml" xref="Thmclaim3.p1.1.1.m1.2.2.4.2">𝐶</ci><cn type="integer" id="Thmclaim3.p1.1.1.m1.2.2.4.3.cmml" xref="Thmclaim3.p1.1.1.m1.2.2.4.3">2</cn></apply><interval closure="open" id="Thmclaim3.p1.1.1.m1.2.2.2.3.cmml" xref="Thmclaim3.p1.1.1.m1.2.2.2.2"><apply id="Thmclaim3.p1.1.1.m1.1.1.1.1.1.cmml" xref="Thmclaim3.p1.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmclaim3.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmclaim3.p1.1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="Thmclaim3.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="Thmclaim3.p1.1.1.m1.1.1.1.1.1.2">𝐱</ci><cn type="integer" id="Thmclaim3.p1.1.1.m1.1.1.1.1.1.3.cmml" xref="Thmclaim3.p1.1.1.m1.1.1.1.1.1.3">0</cn></apply><apply id="Thmclaim3.p1.1.1.m1.2.2.2.2.2.cmml" xref="Thmclaim3.p1.1.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="Thmclaim3.p1.1.1.m1.2.2.2.2.2.1.cmml" xref="Thmclaim3.p1.1.1.m1.2.2.2.2.2">subscript</csymbol><ci id="Thmclaim3.p1.1.1.m1.2.2.2.2.2.2.cmml" xref="Thmclaim3.p1.1.1.m1.2.2.2.2.2.2">𝑘</ci><ci id="Thmclaim3.p1.1.1.m1.2.2.2.2.2.3.cmml" xref="Thmclaim3.p1.1.1.m1.2.2.2.2.2.3">𝑡</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmclaim3.p1.1.1.m1.2c">C_{2}(\mathbf{x}_{0},k_{t})</annotation></semantics></math> is some other constant depending on <math id="Thmclaim3.p1.2.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{0}" display="inline"><semantics id="Thmclaim3.p1.2.2.m2.1a"><msub id="Thmclaim3.p1.2.2.m2.1.1" xref="Thmclaim3.p1.2.2.m2.1.1.cmml"><mi id="Thmclaim3.p1.2.2.m2.1.1.2" xref="Thmclaim3.p1.2.2.m2.1.1.2.cmml">𝐱</mi><mn id="Thmclaim3.p1.2.2.m2.1.1.3" xref="Thmclaim3.p1.2.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="Thmclaim3.p1.2.2.m2.1b"><apply id="Thmclaim3.p1.2.2.m2.1.1.cmml" xref="Thmclaim3.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmclaim3.p1.2.2.m2.1.1.1.cmml" xref="Thmclaim3.p1.2.2.m2.1.1">subscript</csymbol><ci id="Thmclaim3.p1.2.2.m2.1.1.2.cmml" xref="Thmclaim3.p1.2.2.m2.1.1.2">𝐱</ci><cn type="integer" id="Thmclaim3.p1.2.2.m2.1.1.3.cmml" xref="Thmclaim3.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmclaim3.p1.2.2.m2.1c">\mathbf{x}_{0}</annotation></semantics></math> and <math id="Thmclaim3.p1.3.3.m3.1" class="ltx_Math" alttext="k_{t}" display="inline"><semantics id="Thmclaim3.p1.3.3.m3.1a"><msub id="Thmclaim3.p1.3.3.m3.1.1" xref="Thmclaim3.p1.3.3.m3.1.1.cmml"><mi id="Thmclaim3.p1.3.3.m3.1.1.2" xref="Thmclaim3.p1.3.3.m3.1.1.2.cmml">k</mi><mi id="Thmclaim3.p1.3.3.m3.1.1.3" xref="Thmclaim3.p1.3.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="Thmclaim3.p1.3.3.m3.1b"><apply id="Thmclaim3.p1.3.3.m3.1.1.cmml" xref="Thmclaim3.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="Thmclaim3.p1.3.3.m3.1.1.1.cmml" xref="Thmclaim3.p1.3.3.m3.1.1">subscript</csymbol><ci id="Thmclaim3.p1.3.3.m3.1.1.2.cmml" xref="Thmclaim3.p1.3.3.m3.1.1.2">𝑘</ci><ci id="Thmclaim3.p1.3.3.m3.1.1.3.cmml" xref="Thmclaim3.p1.3.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmclaim3.p1.3.3.m3.1c">k_{t}</annotation></semantics></math>.</span></p>
</div>
</div>
<div id="A1.SS2.3" class="ltx_proof">
<h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof" style="font-size:90%;">Proof.</h6>
<div id="A1.SS2.3.p1" class="ltx_para">
<p id="A1.SS2.3.p1.1" class="ltx_p"><span id="A1.SS2.3.p1.1.1" class="ltx_text" style="font-size:90%;">The proof invokes similar manipulations to the standard ELBO derivation for diffusion, but with a few careful modifications to handle the fact that we only noise to level </span><math id="A1.SS2.3.p1.1.m1.1" class="ltx_Math" alttext="k_{t}" display="inline"><semantics id="A1.SS2.3.p1.1.m1.1a"><msub id="A1.SS2.3.p1.1.m1.1.1" xref="A1.SS2.3.p1.1.m1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.3.p1.1.m1.1.1.2" xref="A1.SS2.3.p1.1.m1.1.1.2.cmml">k</mi><mi mathsize="90%" id="A1.SS2.3.p1.1.m1.1.1.3" xref="A1.SS2.3.p1.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.3.p1.1.m1.1b"><apply id="A1.SS2.3.p1.1.m1.1.1.cmml" xref="A1.SS2.3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.SS2.3.p1.1.m1.1.1.1.cmml" xref="A1.SS2.3.p1.1.m1.1.1">subscript</csymbol><ci id="A1.SS2.3.p1.1.m1.1.1.2.cmml" xref="A1.SS2.3.p1.1.m1.1.1.2">𝑘</ci><ci id="A1.SS2.3.p1.1.m1.1.1.3.cmml" xref="A1.SS2.3.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.3.p1.1.m1.1c">k_{t}</annotation></semantics></math><span id="A1.SS2.3.p1.1.2" class="ltx_text" style="font-size:90%;">. As is standard, we require the identity</span></p>
<table id="A6.EGx23" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.E10.m1.3" class="ltx_Math" alttext="\displaystyle q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j-1},\mathbf{x}_{t}^{0})=q(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{j},\mathbf{x}_{t}^{0})\cdot\frac{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{0})}{q(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{0})}." display="inline"><semantics id="A1.E10.m1.3a"><mrow id="A1.E10.m1.3.3.1" xref="A1.E10.m1.3.3.1.1.cmml"><mrow id="A1.E10.m1.3.3.1.1" xref="A1.E10.m1.3.3.1.1.cmml"><mrow id="A1.E10.m1.3.3.1.1.1" xref="A1.E10.m1.3.3.1.1.1.cmml"><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.1.3" xref="A1.E10.m1.3.3.1.1.1.3.cmml">q</mi><mo lspace="0em" rspace="0em" id="A1.E10.m1.3.3.1.1.1.2" xref="A1.E10.m1.3.3.1.1.1.2.cmml">​</mo><mrow id="A1.E10.m1.3.3.1.1.1.1.1" xref="A1.E10.m1.3.3.1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.2" xref="A1.E10.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="A1.E10.m1.3.3.1.1.1.1.1.1" xref="A1.E10.m1.3.3.1.1.1.1.1.1.cmml"><msubsup id="A1.E10.m1.3.3.1.1.1.1.1.1.4" xref="A1.E10.m1.3.3.1.1.1.1.1.1.4.cmml"><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.4.2.2" xref="A1.E10.m1.3.3.1.1.1.1.1.1.4.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.4.2.3" xref="A1.E10.m1.3.3.1.1.1.1.1.1.4.2.3.cmml">t</mi><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.4.3" xref="A1.E10.m1.3.3.1.1.1.1.1.1.4.3.cmml">j</mi></msubsup><mo mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.3" xref="A1.E10.m1.3.3.1.1.1.1.1.1.3.cmml">∣</mo><mrow id="A1.E10.m1.3.3.1.1.1.1.1.1.2.2" xref="A1.E10.m1.3.3.1.1.1.1.1.1.2.3.cmml"><msubsup id="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1" xref="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.2.2" xref="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.2.3" xref="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.2.3.cmml">t</mi><mrow id="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.3" xref="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.3.2" xref="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.3.2.cmml">j</mi><mo mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.3.1" xref="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.3.3" xref="A1.E10.m1.3.3.1.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.2.2.3" xref="A1.E10.m1.3.3.1.1.1.1.1.1.2.3.cmml">,</mo><msubsup id="A1.E10.m1.3.3.1.1.1.1.1.1.2.2.2" xref="A1.E10.m1.3.3.1.1.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.2.2.2.2.2" xref="A1.E10.m1.3.3.1.1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.2.2.2.2.3" xref="A1.E10.m1.3.3.1.1.1.1.1.1.2.2.2.2.3.cmml">t</mi><mn mathsize="90%" id="A1.E10.m1.3.3.1.1.1.1.1.1.2.2.2.3" xref="A1.E10.m1.3.3.1.1.1.1.1.1.2.2.2.3.cmml">0</mn></msubsup></mrow></mrow><mo maxsize="90%" minsize="90%" 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id="A1.E10.m1.3.3.1.1.2.1.1.1.1.4.2.3" xref="A1.E10.m1.3.3.1.1.2.1.1.1.1.4.2.3.cmml">t</mi><mrow id="A1.E10.m1.3.3.1.1.2.1.1.1.1.4.3" xref="A1.E10.m1.3.3.1.1.2.1.1.1.1.4.3.cmml"><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.2.1.1.1.1.4.3.2" xref="A1.E10.m1.3.3.1.1.2.1.1.1.1.4.3.2.cmml">j</mi><mo mathsize="90%" id="A1.E10.m1.3.3.1.1.2.1.1.1.1.4.3.1" xref="A1.E10.m1.3.3.1.1.2.1.1.1.1.4.3.1.cmml">−</mo><mn mathsize="90%" id="A1.E10.m1.3.3.1.1.2.1.1.1.1.4.3.3" xref="A1.E10.m1.3.3.1.1.2.1.1.1.1.4.3.3.cmml">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.E10.m1.3.3.1.1.2.1.1.1.1.3" xref="A1.E10.m1.3.3.1.1.2.1.1.1.1.3.cmml">∣</mo><mrow id="A1.E10.m1.3.3.1.1.2.1.1.1.1.2.2" xref="A1.E10.m1.3.3.1.1.2.1.1.1.1.2.3.cmml"><msubsup id="A1.E10.m1.3.3.1.1.2.1.1.1.1.1.1.1" xref="A1.E10.m1.3.3.1.1.2.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.2.1.1.1.1.1.1.1.2.2" xref="A1.E10.m1.3.3.1.1.2.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E10.m1.3.3.1.1.2.1.1.1.1.1.1.1.2.3" 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q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j-1},\mathbf{x}_{t}^{0})=q(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{j},\mathbf{x}_{t}^{0})\cdot\frac{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{0})}{q(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{0})}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.10)</span></td>
</tr></tbody>
</table>
</div>
</div>
<section id="A1.SS2.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Part 1: Expanding the likelihood ratios</h5>

<div id="A1.SS2.SSS0.Px1.p1" class="ltx_para">
<p id="A1.SS2.SSS0.Px1.p1.6" class="ltx_p"><span id="A1.SS2.SSS0.Px1.p1.6.1" class="ltx_text" style="font-size:90%;">. Using the above identity, we obtain</span></p>
<table id="A6.EGx24" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex28"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.Ex28.m1.2c">\displaystyle\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}^{0}_{t})}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex29.m1.5" class="ltx_Math" alttext="\displaystyle=\ln p(\mathbf{x}_{t}^{K}\mid\mathbf{z}_{t-1})+\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1}\mid\mathbf{x}_{t}^{0})}+\sum_{j=k_{t}+2}^{K}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{j},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j-1},\mathbf{x}_{t}^{0})}" display="inline"><semantics id="A1.Ex29.m1.5a"><mrow id="A1.Ex29.m1.5.5" xref="A1.Ex29.m1.5.5.cmml"><mi id="A1.Ex29.m1.5.5.3" xref="A1.Ex29.m1.5.5.3.cmml"></mi><mo mathsize="90%" id="A1.Ex29.m1.5.5.2" xref="A1.Ex29.m1.5.5.2.cmml">=</mo><mrow id="A1.Ex29.m1.5.5.1" xref="A1.Ex29.m1.5.5.1.cmml"><mrow id="A1.Ex29.m1.5.5.1.1" xref="A1.Ex29.m1.5.5.1.1.cmml"><mrow id="A1.Ex29.m1.5.5.1.1.3" xref="A1.Ex29.m1.5.5.1.1.3.cmml"><mi mathsize="90%" id="A1.Ex29.m1.5.5.1.1.3.1" xref="A1.Ex29.m1.5.5.1.1.3.1.cmml">ln</mi><mo 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xref="A1.Ex29.m1.3.3.1.1.1.1.2.2.2.3.3">1</cn></apply></apply></list></apply></apply><apply id="A1.Ex29.m1.4.4.2.cmml" xref="A1.Ex29.m1.4.4.2"><times id="A1.Ex29.m1.4.4.2.2.cmml" xref="A1.Ex29.m1.4.4.2.2"></times><ci id="A1.Ex29.m1.4.4.2.3.cmml" xref="A1.Ex29.m1.4.4.2.3">𝑞</ci><apply id="A1.Ex29.m1.4.4.2.1.1.1.cmml" xref="A1.Ex29.m1.4.4.2.1.1"><csymbol cd="latexml" id="A1.Ex29.m1.4.4.2.1.1.1.3.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.3">conditional</csymbol><apply id="A1.Ex29.m1.4.4.2.1.1.1.4.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.4"><csymbol cd="ambiguous" id="A1.Ex29.m1.4.4.2.1.1.1.4.1.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.4">superscript</csymbol><apply id="A1.Ex29.m1.4.4.2.1.1.1.4.2.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.4"><csymbol cd="ambiguous" id="A1.Ex29.m1.4.4.2.1.1.1.4.2.1.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.4">subscript</csymbol><ci id="A1.Ex29.m1.4.4.2.1.1.1.4.2.2.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.4.2.2">𝐱</ci><ci id="A1.Ex29.m1.4.4.2.1.1.1.4.2.3.cmml" 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xref="A1.Ex29.m1.4.4.2.1.1.1.1.1.1.3.1"></minus><ci id="A1.Ex29.m1.4.4.2.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.1.1.1.3.2">𝑗</ci><cn type="integer" id="A1.Ex29.m1.4.4.2.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="A1.Ex29.m1.4.4.2.1.1.1.2.2.2.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex29.m1.4.4.2.1.1.1.2.2.2.1.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.2.2.2">superscript</csymbol><apply id="A1.Ex29.m1.4.4.2.1.1.1.2.2.2.2.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex29.m1.4.4.2.1.1.1.2.2.2.2.1.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.2.2.2">subscript</csymbol><ci id="A1.Ex29.m1.4.4.2.1.1.1.2.2.2.2.2.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.2.2.2.2.2">𝐱</ci><ci id="A1.Ex29.m1.4.4.2.1.1.1.2.2.2.2.3.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.2.2.2.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex29.m1.4.4.2.1.1.1.2.2.2.3.cmml" xref="A1.Ex29.m1.4.4.2.1.1.1.2.2.2.3">0</cn></apply></list></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex29.m1.5c">\displaystyle=\ln p(\mathbf{x}_{t}^{K}\mid\mathbf{z}_{t-1})+\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1}\mid\mathbf{x}_{t}^{0})}+\sum_{j=k_{t}+2}^{K}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{j},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j-1},\mathbf{x}_{t}^{0})}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.Ex30.m1.3.3.2.1.1.1.2.3.1.cmml">+</mo><mn mathsize="90%" id="A1.Ex30.m1.3.3.2.1.1.1.2.3.3" xref="A1.Ex30.m1.3.3.2.1.1.1.2.3.3.cmml">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex30.m1.3.3.2.1.1.1.1" xref="A1.Ex30.m1.3.3.2.1.1.1.1.cmml">∣</mo><msubsup id="A1.Ex30.m1.3.3.2.1.1.1.3" xref="A1.Ex30.m1.3.3.2.1.1.1.3.cmml"><mi mathsize="90%" id="A1.Ex30.m1.3.3.2.1.1.1.3.2.2" xref="A1.Ex30.m1.3.3.2.1.1.1.3.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.Ex30.m1.3.3.2.1.1.1.3.2.3" xref="A1.Ex30.m1.3.3.2.1.1.1.3.2.3.cmml">t</mi><mn mathsize="90%" id="A1.Ex30.m1.3.3.2.1.1.1.3.3" xref="A1.Ex30.m1.3.3.2.1.1.1.3.3.cmml">0</mn></msubsup></mrow><mo maxsize="90%" minsize="90%" id="A1.Ex30.m1.3.3.2.1.1.3" xref="A1.Ex30.m1.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo mathsize="90%" id="A1.Ex30.m1.9.9.3a" xref="A1.Ex30.m1.9.9.3.cmml">+</mo><mrow id="A1.Ex30.m1.9.9.2" xref="A1.Ex30.m1.9.9.2.cmml"><mstyle displaystyle="true" id="A1.Ex30.m1.9.9.2.2" xref="A1.Ex30.m1.9.9.2.2.cmml"><munderover id="A1.Ex30.m1.9.9.2.2a" xref="A1.Ex30.m1.9.9.2.2.cmml"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.Ex30.m1.9.9.2.2.2.2" xref="A1.Ex30.m1.9.9.2.2.2.2.cmml">∑</mo><mrow id="A1.Ex30.m1.9.9.2.2.2.3" xref="A1.Ex30.m1.9.9.2.2.2.3.cmml"><mi mathsize="90%" id="A1.Ex30.m1.9.9.2.2.2.3.2" xref="A1.Ex30.m1.9.9.2.2.2.3.2.cmml">j</mi><mo mathsize="90%" id="A1.Ex30.m1.9.9.2.2.2.3.1" xref="A1.Ex30.m1.9.9.2.2.2.3.1.cmml">=</mo><mrow id="A1.Ex30.m1.9.9.2.2.2.3.3" xref="A1.Ex30.m1.9.9.2.2.2.3.3.cmml"><msub id="A1.Ex30.m1.9.9.2.2.2.3.3.2" xref="A1.Ex30.m1.9.9.2.2.2.3.3.2.cmml"><mi mathsize="90%" id="A1.Ex30.m1.9.9.2.2.2.3.3.2.2" xref="A1.Ex30.m1.9.9.2.2.2.3.3.2.2.cmml">k</mi><mi mathsize="90%" id="A1.Ex30.m1.9.9.2.2.2.3.3.2.3" xref="A1.Ex30.m1.9.9.2.2.2.3.3.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A1.Ex30.m1.9.9.2.2.2.3.3.1" xref="A1.Ex30.m1.9.9.2.2.2.3.3.1.cmml">+</mo><mn mathsize="90%" id="A1.Ex30.m1.9.9.2.2.2.3.3.3" xref="A1.Ex30.m1.9.9.2.2.2.3.3.3.cmml">2</mn></mrow></mrow><mi mathsize="90%" id="A1.Ex30.m1.9.9.2.2.3" xref="A1.Ex30.m1.9.9.2.2.3.cmml">K</mi></munderover></mstyle><mrow id="A1.Ex30.m1.9.9.2.1.1" xref="A1.Ex30.m1.9.9.2.1.1.1.cmml"><mo id="A1.Ex30.m1.9.9.2.1.1.2" xref="A1.Ex30.m1.9.9.2.1.1.1.cmml">(</mo><mrow id="A1.Ex30.m1.9.9.2.1.1.1" xref="A1.Ex30.m1.9.9.2.1.1.1.cmml"><mrow id="A1.Ex30.m1.9.9.2.1.1.1.2" xref="A1.Ex30.m1.9.9.2.1.1.1.2.cmml"><mi mathsize="90%" id="A1.Ex30.m1.9.9.2.1.1.1.2.1" xref="A1.Ex30.m1.9.9.2.1.1.1.2.1.cmml">ln</mi><mo lspace="0.167em" id="A1.Ex30.m1.9.9.2.1.1.1.2a" xref="A1.Ex30.m1.9.9.2.1.1.1.2.cmml">⁡</mo><mstyle displaystyle="true" id="A1.Ex30.m1.5.5" xref="A1.Ex30.m1.5.5.cmml"><mfrac id="A1.Ex30.m1.5.5a" xref="A1.Ex30.m1.5.5.cmml"><mrow id="A1.Ex30.m1.4.4.1" xref="A1.Ex30.m1.4.4.1.cmml"><msub id="A1.Ex30.m1.4.4.1.3" xref="A1.Ex30.m1.4.4.1.3.cmml"><mi mathsize="90%" id="A1.Ex30.m1.4.4.1.3.2" xref="A1.Ex30.m1.4.4.1.3.2.cmml">p</mi><mi mathsize="90%" id="A1.Ex30.m1.4.4.1.3.3" xref="A1.Ex30.m1.4.4.1.3.3.cmml">𝜽</mi></msub><mo lspace="0em" rspace="0em" id="A1.Ex30.m1.4.4.1.2" xref="A1.Ex30.m1.4.4.1.2.cmml">​</mo><mrow id="A1.Ex30.m1.4.4.1.1.1" xref="A1.Ex30.m1.4.4.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.Ex30.m1.4.4.1.1.1.2" xref="A1.Ex30.m1.4.4.1.1.1.1.cmml">(</mo><mrow id="A1.Ex30.m1.4.4.1.1.1.1" xref="A1.Ex30.m1.4.4.1.1.1.1.cmml"><msubsup id="A1.Ex30.m1.4.4.1.1.1.1.4" xref="A1.Ex30.m1.4.4.1.1.1.1.4.cmml"><mi mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.4.2.2" xref="A1.Ex30.m1.4.4.1.1.1.1.4.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.4.2.3" xref="A1.Ex30.m1.4.4.1.1.1.1.4.2.3.cmml">t</mi><mrow id="A1.Ex30.m1.4.4.1.1.1.1.4.3" xref="A1.Ex30.m1.4.4.1.1.1.1.4.3.cmml"><mi mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.4.3.2" xref="A1.Ex30.m1.4.4.1.1.1.1.4.3.2.cmml">j</mi><mo mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.4.3.1" xref="A1.Ex30.m1.4.4.1.1.1.1.4.3.1.cmml">−</mo><mn mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.4.3.3" xref="A1.Ex30.m1.4.4.1.1.1.1.4.3.3.cmml">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.3" xref="A1.Ex30.m1.4.4.1.1.1.1.3.cmml">∣</mo><mrow id="A1.Ex30.m1.4.4.1.1.1.1.2.2" xref="A1.Ex30.m1.4.4.1.1.1.1.2.3.cmml"><msubsup id="A1.Ex30.m1.4.4.1.1.1.1.1.1.1" xref="A1.Ex30.m1.4.4.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.1.1.1.2.2" xref="A1.Ex30.m1.4.4.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.1.1.1.2.3" xref="A1.Ex30.m1.4.4.1.1.1.1.1.1.1.2.3.cmml">t</mi><mi mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.1.1.1.3" xref="A1.Ex30.m1.4.4.1.1.1.1.1.1.1.3.cmml">j</mi></msubsup><mo mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.2.2.3" xref="A1.Ex30.m1.4.4.1.1.1.1.2.3.cmml">,</mo><msub id="A1.Ex30.m1.4.4.1.1.1.1.2.2.2" xref="A1.Ex30.m1.4.4.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A1.Ex30.m1.4.4.1.1.1.1.2.2.2.2" xref="A1.Ex30.m1.4.4.1.1.1.1.2.2.2.2.cmml">𝐳</mi><mrow 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xref="A1.Ex30.m1.7.7.2.1.1.1.2.2.3">𝑡</ci></apply><ci id="A1.Ex30.m1.7.7.2.1.1.1.2.3.cmml" xref="A1.Ex30.m1.7.7.2.1.1.1.2.3">𝑗</ci></apply><apply id="A1.Ex30.m1.7.7.2.1.1.1.3.cmml" xref="A1.Ex30.m1.7.7.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex30.m1.7.7.2.1.1.1.3.1.cmml" xref="A1.Ex30.m1.7.7.2.1.1.1.3">superscript</csymbol><apply id="A1.Ex30.m1.7.7.2.1.1.1.3.2.cmml" xref="A1.Ex30.m1.7.7.2.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex30.m1.7.7.2.1.1.1.3.2.1.cmml" xref="A1.Ex30.m1.7.7.2.1.1.1.3">subscript</csymbol><ci id="A1.Ex30.m1.7.7.2.1.1.1.3.2.2.cmml" xref="A1.Ex30.m1.7.7.2.1.1.1.3.2.2">𝐱</ci><ci id="A1.Ex30.m1.7.7.2.1.1.1.3.2.3.cmml" xref="A1.Ex30.m1.7.7.2.1.1.1.3.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex30.m1.7.7.2.1.1.1.3.3.cmml" xref="A1.Ex30.m1.7.7.2.1.1.1.3.3">0</cn></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex30.m1.9c">\displaystyle\overset{(i)}{=}\ln p(\mathbf{x}_{t}^{K}\mid\mathbf{z}_{t-1})+\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1}\mid\mathbf{x}_{t}^{0})}+\sum_{j=k_{t}+2}^{K}\left(\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{j},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{j},\mathbf{x}_{t}^{k_{t}})}+\ln\frac{q(\mathbf{x}_{t}^{j-1}\mid\mathbf{x}_{t}^{0})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{0})}\right)</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex31.m1.8" class="ltx_Math" alttext="\displaystyle\overset{(ii)}{=}\ln p(\mathbf{x}_{t}^{K}\mid\mathbf{z}_{t-1})+\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1}\mid\mathbf{x}_{t}^{0})}+\ln\frac{q(\mathbf{x}_{t}^{k_{t}+1}\mid\mathbf{x}_{t}^{k_{t}})}{q(\mathbf{x}_{t}^{K}\mid\mathbf{x}_{t}^{k_{t}})}+\sum_{j=k_{t}+1}^{K-1}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})}" display="inline"><semantics id="A1.Ex31.m1.8a"><mrow id="A1.Ex31.m1.8.8" xref="A1.Ex31.m1.8.8.cmml"><mrow id="A1.Ex31.m1.8.8.1" xref="A1.Ex31.m1.8.8.1.cmml"><mover accent="true" id="A1.Ex31.m1.1.1" xref="A1.Ex31.m1.1.1.cmml"><mo mathsize="90%" id="A1.Ex31.m1.1.1.2" xref="A1.Ex31.m1.1.1.2.cmml">=</mo><mrow id="A1.Ex31.m1.1.1.1.1" xref="A1.Ex31.m1.1.1.1.1.1.cmml"><mo maxsize="90%" 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xref="A1.Ex31.m1.7.7.2.1.1.1.2.2.2.2.2">𝐱</ci><ci id="A1.Ex31.m1.7.7.2.1.1.1.2.2.2.2.3.cmml" xref="A1.Ex31.m1.7.7.2.1.1.1.2.2.2.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex31.m1.7.7.2.1.1.1.2.2.2.3.cmml" xref="A1.Ex31.m1.7.7.2.1.1.1.2.2.2.3">0</cn></apply></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex31.m1.8c">\displaystyle\overset{(ii)}{=}\ln p(\mathbf{x}_{t}^{K}\mid\mathbf{z}_{t-1})+\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1}\mid\mathbf{x}_{t}^{0})}+\ln\frac{q(\mathbf{x}_{t}^{k_{t}+1}\mid\mathbf{x}_{t}^{k_{t}})}{q(\mathbf{x}_{t}^{K}\mid\mathbf{x}_{t}^{k_{t}})}+\sum_{j=k_{t}+1}^{K-1}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.Ex32.m1.5.5.1.1.1.1.1.4.3.3" xref="A1.Ex32.m1.5.5.1.1.1.1.1.4.3.3.cmml">t</mi></msub></msubsup><mo mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.3" xref="A1.Ex32.m1.5.5.1.1.1.1.1.3.cmml">∣</mo><mrow id="A1.Ex32.m1.5.5.1.1.1.1.1.2.2" xref="A1.Ex32.m1.5.5.1.1.1.1.1.2.3.cmml"><msubsup id="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1" xref="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.2.2" xref="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.2.3" xref="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.2.3.cmml">t</mi><mrow id="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3" xref="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.cmml"><msub id="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.2" xref="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.2.cmml"><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.2.2" xref="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.2.2.cmml">k</mi><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.2.3" xref="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.1" xref="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.1.cmml">+</mo><mn mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.3" xref="A1.Ex32.m1.5.5.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.3" xref="A1.Ex32.m1.5.5.1.1.1.1.1.2.3.cmml">,</mo><msub id="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2" xref="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.2" xref="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.2.cmml">𝐳</mi><mrow id="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.3" xref="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.3.cmml"><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.3.2" xref="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.3.2.cmml">t</mi><mo mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.3.1" xref="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.3.1.cmml">−</mo><mn mathsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.3.3" xref="A1.Ex32.m1.5.5.1.1.1.1.1.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mo maxsize="90%" minsize="90%" id="A1.Ex32.m1.5.5.1.1.1.1.3" xref="A1.Ex32.m1.5.5.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo mathsize="90%" id="A1.Ex32.m1.5.5.1.2a" xref="A1.Ex32.m1.5.5.1.2.cmml">+</mo><mrow id="A1.Ex32.m1.5.5.1.3" xref="A1.Ex32.m1.5.5.1.3.cmml"><mstyle displaystyle="true" id="A1.Ex32.m1.5.5.1.3.1" xref="A1.Ex32.m1.5.5.1.3.1.cmml"><munderover id="A1.Ex32.m1.5.5.1.3.1a" xref="A1.Ex32.m1.5.5.1.3.1.cmml"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.Ex32.m1.5.5.1.3.1.2.2" xref="A1.Ex32.m1.5.5.1.3.1.2.2.cmml">∑</mo><mrow id="A1.Ex32.m1.5.5.1.3.1.2.3" xref="A1.Ex32.m1.5.5.1.3.1.2.3.cmml"><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.3.1.2.3.2" xref="A1.Ex32.m1.5.5.1.3.1.2.3.2.cmml">j</mi><mo mathsize="90%" id="A1.Ex32.m1.5.5.1.3.1.2.3.1" xref="A1.Ex32.m1.5.5.1.3.1.2.3.1.cmml">=</mo><mrow id="A1.Ex32.m1.5.5.1.3.1.2.3.3" xref="A1.Ex32.m1.5.5.1.3.1.2.3.3.cmml"><msub id="A1.Ex32.m1.5.5.1.3.1.2.3.3.2" xref="A1.Ex32.m1.5.5.1.3.1.2.3.3.2.cmml"><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.3.1.2.3.3.2.2" xref="A1.Ex32.m1.5.5.1.3.1.2.3.3.2.2.cmml">k</mi><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.3.1.2.3.3.2.3" xref="A1.Ex32.m1.5.5.1.3.1.2.3.3.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A1.Ex32.m1.5.5.1.3.1.2.3.3.1" xref="A1.Ex32.m1.5.5.1.3.1.2.3.3.1.cmml">+</mo><mn mathsize="90%" id="A1.Ex32.m1.5.5.1.3.1.2.3.3.3" xref="A1.Ex32.m1.5.5.1.3.1.2.3.3.3.cmml">1</mn></mrow></mrow><mrow id="A1.Ex32.m1.5.5.1.3.1.3" xref="A1.Ex32.m1.5.5.1.3.1.3.cmml"><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.3.1.3.2" xref="A1.Ex32.m1.5.5.1.3.1.3.2.cmml">K</mi><mo mathsize="90%" id="A1.Ex32.m1.5.5.1.3.1.3.1" xref="A1.Ex32.m1.5.5.1.3.1.3.1.cmml">−</mo><mn mathsize="90%" id="A1.Ex32.m1.5.5.1.3.1.3.3" xref="A1.Ex32.m1.5.5.1.3.1.3.3.cmml">1</mn></mrow></munderover></mstyle><mrow id="A1.Ex32.m1.5.5.1.3.2" xref="A1.Ex32.m1.5.5.1.3.2.cmml"><mi mathsize="90%" id="A1.Ex32.m1.5.5.1.3.2.1" xref="A1.Ex32.m1.5.5.1.3.2.1.cmml">ln</mi><mo lspace="0.167em" id="A1.Ex32.m1.5.5.1.3.2a" xref="A1.Ex32.m1.5.5.1.3.2.cmml">⁡</mo><mstyle displaystyle="true" id="A1.Ex32.m1.4.4" xref="A1.Ex32.m1.4.4.cmml"><mfrac id="A1.Ex32.m1.4.4a" xref="A1.Ex32.m1.4.4.cmml"><mrow id="A1.Ex32.m1.3.3.1" xref="A1.Ex32.m1.3.3.1.cmml"><msub id="A1.Ex32.m1.3.3.1.3" xref="A1.Ex32.m1.3.3.1.3.cmml"><mi mathsize="90%" id="A1.Ex32.m1.3.3.1.3.2" xref="A1.Ex32.m1.3.3.1.3.2.cmml">p</mi><mi mathsize="90%" id="A1.Ex32.m1.3.3.1.3.3" xref="A1.Ex32.m1.3.3.1.3.3.cmml">𝜽</mi></msub><mo lspace="0em" rspace="0em" id="A1.Ex32.m1.3.3.1.2" xref="A1.Ex32.m1.3.3.1.2.cmml">​</mo><mrow id="A1.Ex32.m1.3.3.1.1.1" xref="A1.Ex32.m1.3.3.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.Ex32.m1.3.3.1.1.1.2" xref="A1.Ex32.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="A1.Ex32.m1.3.3.1.1.1.1" xref="A1.Ex32.m1.3.3.1.1.1.1.cmml"><msubsup id="A1.Ex32.m1.3.3.1.1.1.1.4" xref="A1.Ex32.m1.3.3.1.1.1.1.4.cmml"><mi mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.4.2.2" xref="A1.Ex32.m1.3.3.1.1.1.1.4.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.4.2.3" xref="A1.Ex32.m1.3.3.1.1.1.1.4.2.3.cmml">t</mi><mi mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.4.3" xref="A1.Ex32.m1.3.3.1.1.1.1.4.3.cmml">j</mi></msubsup><mo mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.3" xref="A1.Ex32.m1.3.3.1.1.1.1.3.cmml">∣</mo><mrow id="A1.Ex32.m1.3.3.1.1.1.1.2.2" xref="A1.Ex32.m1.3.3.1.1.1.1.2.3.cmml"><msubsup id="A1.Ex32.m1.3.3.1.1.1.1.1.1.1" xref="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.2.2" xref="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.2.3" xref="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.2.3.cmml">t</mi><mrow id="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.3" xref="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.3.2" xref="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.3.2.cmml">j</mi><mo mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.3.1" xref="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.3.1.cmml">+</mo><mn mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.3.3" xref="A1.Ex32.m1.3.3.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.2.2.3" xref="A1.Ex32.m1.3.3.1.1.1.1.2.3.cmml">,</mo><msub id="A1.Ex32.m1.3.3.1.1.1.1.2.2.2" xref="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.2" xref="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.2.cmml">𝐳</mi><mrow id="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.3" xref="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.3.cmml"><mi mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.3.2" xref="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.3.2.cmml">t</mi><mo mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.3.1" xref="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.3.1.cmml">−</mo><mn mathsize="90%" id="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.3.3" xref="A1.Ex32.m1.3.3.1.1.1.1.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mo maxsize="90%" minsize="90%" id="A1.Ex32.m1.3.3.1.1.1.3" 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id="A1.Ex32.m1.4.4.2.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex32.m1.4.4.2.1.1.1.1.1.1.3.1"></plus><ci id="A1.Ex32.m1.4.4.2.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex32.m1.4.4.2.1.1.1.1.1.1.3.2">𝑗</ci><cn type="integer" id="A1.Ex32.m1.4.4.2.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex32.m1.4.4.2.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="A1.Ex32.m1.4.4.2.1.1.1.2.2.2.cmml" xref="A1.Ex32.m1.4.4.2.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex32.m1.4.4.2.1.1.1.2.2.2.1.cmml" xref="A1.Ex32.m1.4.4.2.1.1.1.2.2.2">superscript</csymbol><apply id="A1.Ex32.m1.4.4.2.1.1.1.2.2.2.2.cmml" xref="A1.Ex32.m1.4.4.2.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex32.m1.4.4.2.1.1.1.2.2.2.2.1.cmml" xref="A1.Ex32.m1.4.4.2.1.1.1.2.2.2">subscript</csymbol><ci id="A1.Ex32.m1.4.4.2.1.1.1.2.2.2.2.2.cmml" xref="A1.Ex32.m1.4.4.2.1.1.1.2.2.2.2.2">𝐱</ci><ci id="A1.Ex32.m1.4.4.2.1.1.1.2.2.2.2.3.cmml" xref="A1.Ex32.m1.4.4.2.1.1.1.2.2.2.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex32.m1.4.4.2.1.1.1.2.2.2.3.cmml" xref="A1.Ex32.m1.4.4.2.1.1.1.2.2.2.3">0</cn></apply></list></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex32.m1.5c">\displaystyle=\frac{\ln p(\mathbf{x}_{t}^{K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{K}\mid\mathbf{x}_{t}^{k_{t}})}+\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})+\sum_{j=k_{t}+1}^{K-1}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex33.m1.10" class="ltx_Math" alttext="\displaystyle=\ln(q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1})^{\mathbf{1}\{k_{t}\geq 1\}})+\frac{\ln p(\mathbf{x}_{t}^{K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{K}\mid\mathbf{x}_{t}^{k_{t}})}+\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1})^{\mathbf{1}\{k_{t}\geq 1\}}}+\sum_{j=k_{t}+1}^{K-1}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})}," display="inline"><semantics id="A1.Ex33.m1.10a"><mrow id="A1.Ex33.m1.10.10.1" xref="A1.Ex33.m1.10.10.1.1.cmml"><mrow id="A1.Ex33.m1.10.10.1.1" xref="A1.Ex33.m1.10.10.1.1.cmml"><mi id="A1.Ex33.m1.10.10.1.1.2" xref="A1.Ex33.m1.10.10.1.1.2.cmml"></mi><mo mathsize="90%" id="A1.Ex33.m1.10.10.1.1.1" xref="A1.Ex33.m1.10.10.1.1.1.cmml">=</mo><mrow 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xref="A1.Ex33.m1.6.6.2.1.1.3.cmml">𝟏</mn><mo lspace="0em" rspace="0em" id="A1.Ex33.m1.6.6.2.1.1.2" xref="A1.Ex33.m1.6.6.2.1.1.2.cmml">​</mo><mrow id="A1.Ex33.m1.6.6.2.1.1.1.1" xref="A1.Ex33.m1.6.6.2.1.1.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A1.Ex33.m1.6.6.2.1.1.1.1.2" xref="A1.Ex33.m1.6.6.2.1.1.1.2.cmml">{</mo><mrow id="A1.Ex33.m1.6.6.2.1.1.1.1.1" xref="A1.Ex33.m1.6.6.2.1.1.1.1.1.cmml"><msub id="A1.Ex33.m1.6.6.2.1.1.1.1.1.2" xref="A1.Ex33.m1.6.6.2.1.1.1.1.1.2.cmml"><mi mathsize="90%" id="A1.Ex33.m1.6.6.2.1.1.1.1.1.2.2" xref="A1.Ex33.m1.6.6.2.1.1.1.1.1.2.2.cmml">k</mi><mi mathsize="90%" id="A1.Ex33.m1.6.6.2.1.1.1.1.1.2.3" xref="A1.Ex33.m1.6.6.2.1.1.1.1.1.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A1.Ex33.m1.6.6.2.1.1.1.1.1.1" xref="A1.Ex33.m1.6.6.2.1.1.1.1.1.1.cmml">≥</mo><mn mathsize="90%" id="A1.Ex33.m1.6.6.2.1.1.1.1.1.3" xref="A1.Ex33.m1.6.6.2.1.1.1.1.1.3.cmml">1</mn></mrow><mo maxsize="90%" minsize="90%" id="A1.Ex33.m1.6.6.2.1.1.1.1.3" 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xref="A1.Ex33.m1.9.9.2.1.1.1.2.2"><apply id="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.1.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.1.1.1">superscript</csymbol><apply id="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.2.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.2.1.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.1.1.1">subscript</csymbol><ci id="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.2.2.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.2.2">𝐱</ci><ci id="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.2.3.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.3.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.3"><plus id="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.3.1"></plus><ci id="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.3.2">𝑗</ci><cn type="integer" id="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="A1.Ex33.m1.9.9.2.1.1.1.2.2.2.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex33.m1.9.9.2.1.1.1.2.2.2.1.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.2.2.2">superscript</csymbol><apply id="A1.Ex33.m1.9.9.2.1.1.1.2.2.2.2.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.Ex33.m1.9.9.2.1.1.1.2.2.2.2.1.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.2.2.2">subscript</csymbol><ci id="A1.Ex33.m1.9.9.2.1.1.1.2.2.2.2.2.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.2.2.2.2.2">𝐱</ci><ci id="A1.Ex33.m1.9.9.2.1.1.1.2.2.2.2.3.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.2.2.2.2.3">𝑡</ci></apply><cn type="integer" id="A1.Ex33.m1.9.9.2.1.1.1.2.2.2.3.cmml" xref="A1.Ex33.m1.9.9.2.1.1.1.2.2.2.3">0</cn></apply></list></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex33.m1.10c">\displaystyle=\ln(q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1})^{\mathbf{1}\{k_{t}\geq 1\}})+\frac{\ln p(\mathbf{x}_{t}^{K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{K}\mid\mathbf{x}_{t}^{k_{t}})}+\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1})^{\mathbf{1}\{k_{t}\geq 1\}}}+\sum_{j=k_{t}+1}^{K-1}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.SSS0.Px1.p1.2" class="ltx_p"><span id="A1.SS2.SSS0.Px1.p1.2.1" class="ltx_text" style="font-size:90%;">where </span><math id="A1.SS2.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="(i)" display="inline"><semantics id="A1.SS2.SSS0.Px1.p1.1.m1.1a"><mrow id="A1.SS2.SSS0.Px1.p1.1.m1.1.2.2"><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px1.p1.1.m1.1.2.2.1">(</mo><mi mathsize="90%" id="A1.SS2.SSS0.Px1.p1.1.m1.1.1" xref="A1.SS2.SSS0.Px1.p1.1.m1.1.1.cmml">i</mi><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px1.p1.1.m1.1.2.2.2">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.SSS0.Px1.p1.1.m1.1b"><ci id="A1.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" xref="A1.SS2.SSS0.Px1.p1.1.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px1.p1.1.m1.1c">(i)</annotation></semantics></math><span id="A1.SS2.SSS0.Px1.p1.2.2" class="ltx_text" style="font-size:90%;"> uses </span><a href="#A1.E10" title="Equation A.10 ‣ Proof. ‣ A.2 Proof of Theorem A.1 ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">A.10</span></a><span id="A1.SS2.SSS0.Px1.p1.2.3" class="ltx_text" style="font-size:90%;">, </span><math id="A1.SS2.SSS0.Px1.p1.2.m2.1" class="ltx_Math" alttext="(ii)" display="inline"><semantics id="A1.SS2.SSS0.Px1.p1.2.m2.1a"><mrow id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.2" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.cmml">(</mo><mrow id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.2" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.2.cmml">i</mi><mo lspace="0em" rspace="0em" id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.1" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.1.cmml">​</mo><mi mathsize="90%" id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.3" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.3.cmml">i</mi></mrow><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.3" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.SSS0.Px1.p1.2.m2.1b"><apply id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.cmml" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1"><times id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.1.cmml" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.1"></times><ci id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.2.cmml" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.2">𝑖</ci><ci id="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.3.cmml" xref="A1.SS2.SSS0.Px1.p1.2.m2.1.1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px1.p1.2.m2.1c">(ii)</annotation></semantics></math><span id="A1.SS2.SSS0.Px1.p1.2.4" class="ltx_text" style="font-size:90%;"> invokes a cancellation in the telescoping sum, and the final display follows from the computation</span></p>
<table id="A6.EGx25" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.E11.m1.2.2.2.2.2.1.2.2">𝑘</ci><ci id="A1.E11.m1.2.2.2.2.2.1.2.3.cmml" xref="A1.E11.m1.2.2.2.2.2.1.2.3">𝑡</ci></apply><cn type="integer" id="A1.E11.m1.2.2.2.2.2.1.3.cmml" xref="A1.E11.m1.2.2.2.2.2.1.3">0</cn></apply><apply id="A1.E11.m1.3.3.3.3.1.1.cmml" xref="A1.E11.m1.3.3.3.3.1.1"><times id="A1.E11.m1.3.3.3.3.1.1.2.cmml" xref="A1.E11.m1.3.3.3.3.1.1.2"></times><ci id="A1.E11.m1.3.3.3.3.1.1.3.cmml" xref="A1.E11.m1.3.3.3.3.1.1.3">𝑞</ci><apply id="A1.E11.m1.3.3.3.3.1.1.1.1.1.cmml" xref="A1.E11.m1.3.3.3.3.1.1.1.1"><csymbol cd="latexml" id="A1.E11.m1.3.3.3.3.1.1.1.1.1.1.cmml" xref="A1.E11.m1.3.3.3.3.1.1.1.1.1.1">conditional</csymbol><apply id="A1.E11.m1.3.3.3.3.1.1.1.1.1.2.cmml" xref="A1.E11.m1.3.3.3.3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A1.E11.m1.3.3.3.3.1.1.1.1.1.2.1.cmml" xref="A1.E11.m1.3.3.3.3.1.1.1.1.1.2">superscript</csymbol><apply id="A1.E11.m1.3.3.3.3.1.1.1.1.1.2.2.cmml" xref="A1.E11.m1.3.3.3.3.1.1.1.1.1.2"><csymbol cd="ambiguous" 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id="A1.E11.m1.3.3.3.3.1.1.1.1.1.3.3.3.cmml" xref="A1.E11.m1.3.3.3.3.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply><apply id="A1.E11.m1.4.4.4.4.2.1.cmml" xref="A1.E11.m1.4.4.4.4.2.1"><geq id="A1.E11.m1.4.4.4.4.2.1.1.cmml" xref="A1.E11.m1.4.4.4.4.2.1.1"></geq><apply id="A1.E11.m1.4.4.4.4.2.1.2.cmml" xref="A1.E11.m1.4.4.4.4.2.1.2"><csymbol cd="ambiguous" id="A1.E11.m1.4.4.4.4.2.1.2.1.cmml" xref="A1.E11.m1.4.4.4.4.2.1.2">subscript</csymbol><ci id="A1.E11.m1.4.4.4.4.2.1.2.2.cmml" xref="A1.E11.m1.4.4.4.4.2.1.2.2">𝑘</ci><ci id="A1.E11.m1.4.4.4.4.2.1.2.3.cmml" xref="A1.E11.m1.4.4.4.4.2.1.2.3">𝑡</ci></apply><cn type="integer" id="A1.E11.m1.4.4.4.4.2.1.3.cmml" xref="A1.E11.m1.4.4.4.4.2.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E11.m1.6c">\displaystyle q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1})^{\mathbf{1}\{k_{t}\geq 1\}}=\begin{cases}1&amp;k_{t}=0\\
q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1})&amp;k_{t}\geq 1\end{cases}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.11)</span></td>
</tr></tbody>
</table>
<p id="A1.SS2.SSS0.Px1.p1.5" class="ltx_p"><span id="A1.SS2.SSS0.Px1.p1.5.1" class="ltx_text" style="font-size:90%;">Observe that, because we don’t parameterize </span><math id="A1.SS2.SSS0.Px1.p1.3.m1.1" class="ltx_Math" alttext="p(\mathbf{x}_{t}^{K}\mid\mathbf{z}_{t-1})" display="inline"><semantics id="A1.SS2.SSS0.Px1.p1.3.m1.1a"><mrow id="A1.SS2.SSS0.Px1.p1.3.m1.1.1" xref="A1.SS2.SSS0.Px1.p1.3.m1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px1.p1.3.m1.1.1.3" xref="A1.SS2.SSS0.Px1.p1.3.m1.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="A1.SS2.SSS0.Px1.p1.3.m1.1.1.2" xref="A1.SS2.SSS0.Px1.p1.3.m1.1.1.2.cmml">​</mo><mrow id="A1.SS2.SSS0.Px1.p1.3.m1.1.1.1.1" xref="A1.SS2.SSS0.Px1.p1.3.m1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px1.p1.3.m1.1.1.1.1.2" xref="A1.SS2.SSS0.Px1.p1.3.m1.1.1.1.1.1.cmml">(</mo><mrow id="A1.SS2.SSS0.Px1.p1.3.m1.1.1.1.1.1" xref="A1.SS2.SSS0.Px1.p1.3.m1.1.1.1.1.1.cmml"><msubsup id="A1.SS2.SSS0.Px1.p1.3.m1.1.1.1.1.1.2" 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xref="A1.SS2.SSS0.Px1.p1.4.m2.4.4.2.1.1.1.3.3"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px1.p1.4.m2.4.4.2.1.1.1.3.3.1.cmml" xref="A1.SS2.SSS0.Px1.p1.4.m2.4.4.2.1.1.1.3.3">subscript</csymbol><ci id="A1.SS2.SSS0.Px1.p1.4.m2.4.4.2.1.1.1.3.3.2.cmml" xref="A1.SS2.SSS0.Px1.p1.4.m2.4.4.2.1.1.1.3.3.2">𝑘</ci><ci id="A1.SS2.SSS0.Px1.p1.4.m2.4.4.2.1.1.1.3.3.3.cmml" xref="A1.SS2.SSS0.Px1.p1.4.m2.4.4.2.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px1.p1.4.m2.4c">\ln(q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1})^{\mathbf{1}\{k_{t}\geq 1\}})+\frac{\ln p(\mathbf{x}_{t}^{K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{K}\mid\mathbf{x}_{t}^{k_{t}})}</annotation></semantics></math><span id="A1.SS2.SSS0.Px1.p1.5.3" class="ltx_text" style="font-size:90%;"> can be regarded as some constant </span><math id="A1.SS2.SSS0.Px1.p1.5.m3.3" class="ltx_Math" 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xref="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3.1.cmml" xref="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3">superscript</csymbol><apply id="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3.2.cmml" xref="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3.2.1.cmml" xref="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3">subscript</csymbol><ci id="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3.2.2.cmml" xref="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3.2.2">𝐱</ci><ci id="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3.2.3.cmml" xref="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3.2.3">𝑡</ci></apply><ci id="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3.3.cmml" xref="A1.SS2.SSS0.Px1.p1.5.m3.3.3.3.3.3.3">𝐾</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px1.p1.5.m3.3c">C^{\prime}(\mathbf{x}_{t}^{k_{t}},\mathbf{x}_{t}^{k_{t}+1},\mathbf{x}_{t}^{K})</annotation></semantics></math><span id="A1.SS2.SSS0.Px1.p1.5.4" class="ltx_text" style="font-size:90%;">. Thus,</span></p>
<table id="A6.EGx26" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.E12.m1.2.2.2.1.1.1.2.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.2.2.2.1.1.1.2.2.3" xref="A1.E12.m1.2.2.2.1.1.1.2.2.3.cmml">t</mi><mrow id="A1.E12.m1.2.2.2.1.1.1.2.3" xref="A1.E12.m1.2.2.2.1.1.1.2.3.cmml"><mrow id="A1.E12.m1.2.2.2.1.1.1.2.3.2" xref="A1.E12.m1.2.2.2.1.1.1.2.3.2.cmml"><msub id="A1.E12.m1.2.2.2.1.1.1.2.3.2.2" xref="A1.E12.m1.2.2.2.1.1.1.2.3.2.2.cmml"><mi mathsize="90%" id="A1.E12.m1.2.2.2.1.1.1.2.3.2.2.2" xref="A1.E12.m1.2.2.2.1.1.1.2.3.2.2.2.cmml">k</mi><mi mathsize="90%" id="A1.E12.m1.2.2.2.1.1.1.2.3.2.2.3" xref="A1.E12.m1.2.2.2.1.1.1.2.3.2.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A1.E12.m1.2.2.2.1.1.1.2.3.2.1" xref="A1.E12.m1.2.2.2.1.1.1.2.3.2.1.cmml">+</mo><mn mathsize="90%" id="A1.E12.m1.2.2.2.1.1.1.2.3.2.3" xref="A1.E12.m1.2.2.2.1.1.1.2.3.2.3.cmml">1</mn></mrow><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.E12.m1.2.2.2.1.1.1.2.3.1" xref="A1.E12.m1.2.2.2.1.1.1.2.3.1.cmml">:</mo><mi mathsize="90%" id="A1.E12.m1.2.2.2.1.1.1.2.3.3" xref="A1.E12.m1.2.2.2.1.1.1.2.3.3.cmml">K</mi></mrow></msubsup><mo mathsize="90%" id="A1.E12.m1.2.2.2.1.1.1.1" xref="A1.E12.m1.2.2.2.1.1.1.1.cmml">∣</mo><msubsup id="A1.E12.m1.2.2.2.1.1.1.3" xref="A1.E12.m1.2.2.2.1.1.1.3.cmml"><mi mathsize="90%" id="A1.E12.m1.2.2.2.1.1.1.3.2.2" xref="A1.E12.m1.2.2.2.1.1.1.3.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.2.2.2.1.1.1.3.3" xref="A1.E12.m1.2.2.2.1.1.1.3.3.cmml">t</mi><mn mathsize="90%" id="A1.E12.m1.2.2.2.1.1.1.3.2.3" xref="A1.E12.m1.2.2.2.1.1.1.3.2.3.cmml">0</mn></msubsup></mrow><mo maxsize="90%" minsize="90%" id="A1.E12.m1.2.2.2.1.1.3" xref="A1.E12.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mfrac></mstyle></mrow><mo mathsize="90%" id="A1.E12.m1.10.10.4" xref="A1.E12.m1.10.10.4.cmml">=</mo><mrow id="A1.E12.m1.10.10.3" xref="A1.E12.m1.10.10.3.cmml"><mrow id="A1.E12.m1.10.10.3.3" xref="A1.E12.m1.10.10.3.3.cmml"><msup id="A1.E12.m1.10.10.3.3.5" xref="A1.E12.m1.10.10.3.3.5.cmml"><mi mathsize="90%" id="A1.E12.m1.10.10.3.3.5.2" xref="A1.E12.m1.10.10.3.3.5.2.cmml">C</mi><mo mathsize="90%" id="A1.E12.m1.10.10.3.3.5.3" xref="A1.E12.m1.10.10.3.3.5.3.cmml">′</mo></msup><mo lspace="0em" rspace="0em" id="A1.E12.m1.10.10.3.3.4" xref="A1.E12.m1.10.10.3.3.4.cmml">​</mo><mrow id="A1.E12.m1.10.10.3.3.3.3" xref="A1.E12.m1.10.10.3.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A1.E12.m1.10.10.3.3.3.3.4" xref="A1.E12.m1.10.10.3.3.3.4.cmml">(</mo><msubsup id="A1.E12.m1.8.8.1.1.1.1.1" xref="A1.E12.m1.8.8.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.E12.m1.8.8.1.1.1.1.1.2.2" xref="A1.E12.m1.8.8.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.8.8.1.1.1.1.1.2.3" xref="A1.E12.m1.8.8.1.1.1.1.1.2.3.cmml">t</mi><msub id="A1.E12.m1.8.8.1.1.1.1.1.3" xref="A1.E12.m1.8.8.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A1.E12.m1.8.8.1.1.1.1.1.3.2" xref="A1.E12.m1.8.8.1.1.1.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A1.E12.m1.8.8.1.1.1.1.1.3.3" xref="A1.E12.m1.8.8.1.1.1.1.1.3.3.cmml">t</mi></msub></msubsup><mo mathsize="90%" id="A1.E12.m1.10.10.3.3.3.3.5" xref="A1.E12.m1.10.10.3.3.3.4.cmml">,</mo><msubsup id="A1.E12.m1.9.9.2.2.2.2.2" xref="A1.E12.m1.9.9.2.2.2.2.2.cmml"><mi mathsize="90%" id="A1.E12.m1.9.9.2.2.2.2.2.2.2" xref="A1.E12.m1.9.9.2.2.2.2.2.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.9.9.2.2.2.2.2.2.3" xref="A1.E12.m1.9.9.2.2.2.2.2.2.3.cmml">t</mi><mrow id="A1.E12.m1.9.9.2.2.2.2.2.3" xref="A1.E12.m1.9.9.2.2.2.2.2.3.cmml"><msub id="A1.E12.m1.9.9.2.2.2.2.2.3.2" xref="A1.E12.m1.9.9.2.2.2.2.2.3.2.cmml"><mi mathsize="90%" id="A1.E12.m1.9.9.2.2.2.2.2.3.2.2" xref="A1.E12.m1.9.9.2.2.2.2.2.3.2.2.cmml">k</mi><mi mathsize="90%" id="A1.E12.m1.9.9.2.2.2.2.2.3.2.3" xref="A1.E12.m1.9.9.2.2.2.2.2.3.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A1.E12.m1.9.9.2.2.2.2.2.3.1" xref="A1.E12.m1.9.9.2.2.2.2.2.3.1.cmml">+</mo><mn mathsize="90%" id="A1.E12.m1.9.9.2.2.2.2.2.3.3" xref="A1.E12.m1.9.9.2.2.2.2.2.3.3.cmml">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.E12.m1.10.10.3.3.3.3.6" xref="A1.E12.m1.10.10.3.3.3.4.cmml">,</mo><msubsup id="A1.E12.m1.10.10.3.3.3.3.3" xref="A1.E12.m1.10.10.3.3.3.3.3.cmml"><mi mathsize="90%" id="A1.E12.m1.10.10.3.3.3.3.3.2.2" xref="A1.E12.m1.10.10.3.3.3.3.3.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.10.10.3.3.3.3.3.2.3" xref="A1.E12.m1.10.10.3.3.3.3.3.2.3.cmml">t</mi><mi mathsize="90%" id="A1.E12.m1.10.10.3.3.3.3.3.3" xref="A1.E12.m1.10.10.3.3.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A1.E12.m1.10.10.3.3.3.3.7" xref="A1.E12.m1.10.10.3.3.3.4.cmml">)</mo></mrow></mrow><mo mathsize="90%" id="A1.E12.m1.10.10.3.4" xref="A1.E12.m1.10.10.3.4.cmml">+</mo><mrow id="A1.E12.m1.10.10.3.5" xref="A1.E12.m1.10.10.3.5.cmml"><mi mathsize="90%" id="A1.E12.m1.10.10.3.5.1" xref="A1.E12.m1.10.10.3.5.1.cmml">ln</mi><mo lspace="0.167em" id="A1.E12.m1.10.10.3.5a" xref="A1.E12.m1.10.10.3.5.cmml">⁡</mo><mstyle displaystyle="true" id="A1.E12.m1.5.5" xref="A1.E12.m1.5.5.cmml"><mfrac id="A1.E12.m1.5.5a" xref="A1.E12.m1.5.5.cmml"><mrow id="A1.E12.m1.3.3.1" xref="A1.E12.m1.3.3.1.cmml"><msub id="A1.E12.m1.3.3.1.3" xref="A1.E12.m1.3.3.1.3.cmml"><mi mathsize="90%" id="A1.E12.m1.3.3.1.3.2" xref="A1.E12.m1.3.3.1.3.2.cmml">p</mi><mi mathsize="90%" id="A1.E12.m1.3.3.1.3.3" xref="A1.E12.m1.3.3.1.3.3.cmml">𝜽</mi></msub><mo lspace="0em" rspace="0em" id="A1.E12.m1.3.3.1.2" xref="A1.E12.m1.3.3.1.2.cmml">​</mo><mrow id="A1.E12.m1.3.3.1.1.1" xref="A1.E12.m1.3.3.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.E12.m1.3.3.1.1.1.2" xref="A1.E12.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="A1.E12.m1.3.3.1.1.1.1" xref="A1.E12.m1.3.3.1.1.1.1.cmml"><msubsup id="A1.E12.m1.3.3.1.1.1.1.4" xref="A1.E12.m1.3.3.1.1.1.1.4.cmml"><mi mathsize="90%" id="A1.E12.m1.3.3.1.1.1.1.4.2.2" xref="A1.E12.m1.3.3.1.1.1.1.4.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.3.3.1.1.1.1.4.2.3" xref="A1.E12.m1.3.3.1.1.1.1.4.2.3.cmml">t</mi><msub id="A1.E12.m1.3.3.1.1.1.1.4.3" xref="A1.E12.m1.3.3.1.1.1.1.4.3.cmml"><mi mathsize="90%" id="A1.E12.m1.3.3.1.1.1.1.4.3.2" xref="A1.E12.m1.3.3.1.1.1.1.4.3.2.cmml">k</mi><mi mathsize="90%" id="A1.E12.m1.3.3.1.1.1.1.4.3.3" xref="A1.E12.m1.3.3.1.1.1.1.4.3.3.cmml">t</mi></msub></msubsup><mo mathsize="90%" id="A1.E12.m1.3.3.1.1.1.1.3" xref="A1.E12.m1.3.3.1.1.1.1.3.cmml">∣</mo><mrow id="A1.E12.m1.3.3.1.1.1.1.2.2" xref="A1.E12.m1.3.3.1.1.1.1.2.3.cmml"><msubsup id="A1.E12.m1.3.3.1.1.1.1.1.1.1" xref="A1.E12.m1.3.3.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.E12.m1.3.3.1.1.1.1.1.1.1.2.2" xref="A1.E12.m1.3.3.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.3.3.1.1.1.1.1.1.1.2.3" xref="A1.E12.m1.3.3.1.1.1.1.1.1.1.2.3.cmml">t</mi><mrow id="A1.E12.m1.3.3.1.1.1.1.1.1.1.3" xref="A1.E12.m1.3.3.1.1.1.1.1.1.1.3.cmml"><msub id="A1.E12.m1.3.3.1.1.1.1.1.1.1.3.2" xref="A1.E12.m1.3.3.1.1.1.1.1.1.1.3.2.cmml"><mi mathsize="90%" id="A1.E12.m1.3.3.1.1.1.1.1.1.1.3.2.2" xref="A1.E12.m1.3.3.1.1.1.1.1.1.1.3.2.2.cmml">k</mi><mi mathsize="90%" id="A1.E12.m1.3.3.1.1.1.1.1.1.1.3.2.3" 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maxsize="90%" minsize="90%" id="A1.E12.m1.3.3.1.1.1.3" xref="A1.E12.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="A1.E12.m1.5.5.3" xref="A1.E12.m1.5.5.3.cmml"><mi mathsize="90%" id="A1.E12.m1.5.5.3.4" xref="A1.E12.m1.5.5.3.4.cmml">q</mi><mo lspace="0em" rspace="0em" id="A1.E12.m1.5.5.3.3" xref="A1.E12.m1.5.5.3.3.cmml">​</mo><msup id="A1.E12.m1.5.5.3.2" xref="A1.E12.m1.5.5.3.2.cmml"><mrow id="A1.E12.m1.5.5.3.2.1.1" xref="A1.E12.m1.5.5.3.2.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.E12.m1.5.5.3.2.1.1.2" xref="A1.E12.m1.5.5.3.2.1.1.1.cmml">(</mo><mrow id="A1.E12.m1.5.5.3.2.1.1.1" xref="A1.E12.m1.5.5.3.2.1.1.1.cmml"><msubsup id="A1.E12.m1.5.5.3.2.1.1.1.2" xref="A1.E12.m1.5.5.3.2.1.1.1.2.cmml"><mi mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.2.2.2" xref="A1.E12.m1.5.5.3.2.1.1.1.2.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.2.2.3" xref="A1.E12.m1.5.5.3.2.1.1.1.2.2.3.cmml">t</mi><msub id="A1.E12.m1.5.5.3.2.1.1.1.2.3" xref="A1.E12.m1.5.5.3.2.1.1.1.2.3.cmml"><mi mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.2.3.2" xref="A1.E12.m1.5.5.3.2.1.1.1.2.3.2.cmml">k</mi><mi mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.2.3.3" xref="A1.E12.m1.5.5.3.2.1.1.1.2.3.3.cmml">t</mi></msub></msubsup><mo mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.1" xref="A1.E12.m1.5.5.3.2.1.1.1.1.cmml">∣</mo><msubsup id="A1.E12.m1.5.5.3.2.1.1.1.3" xref="A1.E12.m1.5.5.3.2.1.1.1.3.cmml"><mi mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.3.2.2" xref="A1.E12.m1.5.5.3.2.1.1.1.3.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.3.2.3" xref="A1.E12.m1.5.5.3.2.1.1.1.3.2.3.cmml">t</mi><mrow id="A1.E12.m1.5.5.3.2.1.1.1.3.3" xref="A1.E12.m1.5.5.3.2.1.1.1.3.3.cmml"><msub id="A1.E12.m1.5.5.3.2.1.1.1.3.3.2" xref="A1.E12.m1.5.5.3.2.1.1.1.3.3.2.cmml"><mi mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.3.3.2.2" xref="A1.E12.m1.5.5.3.2.1.1.1.3.3.2.2.cmml">k</mi><mi mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.3.3.2.3" xref="A1.E12.m1.5.5.3.2.1.1.1.3.3.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.3.3.1" xref="A1.E12.m1.5.5.3.2.1.1.1.3.3.1.cmml">+</mo><mn mathsize="90%" id="A1.E12.m1.5.5.3.2.1.1.1.3.3.3" xref="A1.E12.m1.5.5.3.2.1.1.1.3.3.3.cmml">1</mn></mrow></msubsup></mrow><mo maxsize="90%" minsize="90%" id="A1.E12.m1.5.5.3.2.1.1.3" xref="A1.E12.m1.5.5.3.2.1.1.1.cmml">)</mo></mrow><mrow id="A1.E12.m1.4.4.2.1.1" xref="A1.E12.m1.4.4.2.1.1.cmml"><mn mathsize="90%" id="A1.E12.m1.4.4.2.1.1.3" xref="A1.E12.m1.4.4.2.1.1.3.cmml">𝟏</mn><mo lspace="0em" rspace="0em" id="A1.E12.m1.4.4.2.1.1.2" xref="A1.E12.m1.4.4.2.1.1.2.cmml">​</mo><mrow id="A1.E12.m1.4.4.2.1.1.1.1" xref="A1.E12.m1.4.4.2.1.1.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A1.E12.m1.4.4.2.1.1.1.1.2" xref="A1.E12.m1.4.4.2.1.1.1.2.cmml">{</mo><mrow id="A1.E12.m1.4.4.2.1.1.1.1.1" xref="A1.E12.m1.4.4.2.1.1.1.1.1.cmml"><msub id="A1.E12.m1.4.4.2.1.1.1.1.1.2" xref="A1.E12.m1.4.4.2.1.1.1.1.1.2.cmml"><mi mathsize="90%" id="A1.E12.m1.4.4.2.1.1.1.1.1.2.2" xref="A1.E12.m1.4.4.2.1.1.1.1.1.2.2.cmml">k</mi><mi mathsize="90%" id="A1.E12.m1.4.4.2.1.1.1.1.1.2.3" xref="A1.E12.m1.4.4.2.1.1.1.1.1.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A1.E12.m1.4.4.2.1.1.1.1.1.1" xref="A1.E12.m1.4.4.2.1.1.1.1.1.1.cmml">≥</mo><mn mathsize="90%" id="A1.E12.m1.4.4.2.1.1.1.1.1.3" xref="A1.E12.m1.4.4.2.1.1.1.1.1.3.cmml">1</mn></mrow><mo maxsize="90%" minsize="90%" id="A1.E12.m1.4.4.2.1.1.1.1.3" xref="A1.E12.m1.4.4.2.1.1.1.2.cmml">}</mo></mrow></mrow></msup></mrow></mfrac></mstyle></mrow><mo mathsize="90%" id="A1.E12.m1.10.10.3.4a" xref="A1.E12.m1.10.10.3.4.cmml">+</mo><mrow id="A1.E12.m1.10.10.3.6" xref="A1.E12.m1.10.10.3.6.cmml"><mstyle displaystyle="true" id="A1.E12.m1.10.10.3.6.1" xref="A1.E12.m1.10.10.3.6.1.cmml"><munderover id="A1.E12.m1.10.10.3.6.1a" xref="A1.E12.m1.10.10.3.6.1.cmml"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.E12.m1.10.10.3.6.1.2.2" xref="A1.E12.m1.10.10.3.6.1.2.2.cmml">∑</mo><mrow id="A1.E12.m1.10.10.3.6.1.2.3" xref="A1.E12.m1.10.10.3.6.1.2.3.cmml"><mi mathsize="90%" id="A1.E12.m1.10.10.3.6.1.2.3.2" xref="A1.E12.m1.10.10.3.6.1.2.3.2.cmml">j</mi><mo mathsize="90%" id="A1.E12.m1.10.10.3.6.1.2.3.1" xref="A1.E12.m1.10.10.3.6.1.2.3.1.cmml">=</mo><mrow id="A1.E12.m1.10.10.3.6.1.2.3.3" xref="A1.E12.m1.10.10.3.6.1.2.3.3.cmml"><msub id="A1.E12.m1.10.10.3.6.1.2.3.3.2" xref="A1.E12.m1.10.10.3.6.1.2.3.3.2.cmml"><mi mathsize="90%" id="A1.E12.m1.10.10.3.6.1.2.3.3.2.2" xref="A1.E12.m1.10.10.3.6.1.2.3.3.2.2.cmml">k</mi><mi mathsize="90%" id="A1.E12.m1.10.10.3.6.1.2.3.3.2.3" xref="A1.E12.m1.10.10.3.6.1.2.3.3.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A1.E12.m1.10.10.3.6.1.2.3.3.1" xref="A1.E12.m1.10.10.3.6.1.2.3.3.1.cmml">+</mo><mn mathsize="90%" id="A1.E12.m1.10.10.3.6.1.2.3.3.3" xref="A1.E12.m1.10.10.3.6.1.2.3.3.3.cmml">1</mn></mrow></mrow><mrow id="A1.E12.m1.10.10.3.6.1.3" xref="A1.E12.m1.10.10.3.6.1.3.cmml"><mi mathsize="90%" id="A1.E12.m1.10.10.3.6.1.3.2" xref="A1.E12.m1.10.10.3.6.1.3.2.cmml">K</mi><mo mathsize="90%" id="A1.E12.m1.10.10.3.6.1.3.1" xref="A1.E12.m1.10.10.3.6.1.3.1.cmml">−</mo><mn mathsize="90%" id="A1.E12.m1.10.10.3.6.1.3.3" xref="A1.E12.m1.10.10.3.6.1.3.3.cmml">1</mn></mrow></munderover></mstyle><mrow id="A1.E12.m1.10.10.3.6.2" xref="A1.E12.m1.10.10.3.6.2.cmml"><mi mathsize="90%" id="A1.E12.m1.10.10.3.6.2.1" xref="A1.E12.m1.10.10.3.6.2.1.cmml">ln</mi><mo lspace="0.167em" id="A1.E12.m1.10.10.3.6.2a" xref="A1.E12.m1.10.10.3.6.2.cmml">⁡</mo><mstyle displaystyle="true" id="A1.E12.m1.7.7" xref="A1.E12.m1.7.7.cmml"><mfrac id="A1.E12.m1.7.7a" xref="A1.E12.m1.7.7.cmml"><mrow id="A1.E12.m1.6.6.1" xref="A1.E12.m1.6.6.1.cmml"><msub id="A1.E12.m1.6.6.1.3" xref="A1.E12.m1.6.6.1.3.cmml"><mi mathsize="90%" id="A1.E12.m1.6.6.1.3.2" xref="A1.E12.m1.6.6.1.3.2.cmml">p</mi><mi mathsize="90%" id="A1.E12.m1.6.6.1.3.3" xref="A1.E12.m1.6.6.1.3.3.cmml">𝜽</mi></msub><mo lspace="0em" rspace="0em" id="A1.E12.m1.6.6.1.2" xref="A1.E12.m1.6.6.1.2.cmml">​</mo><mrow id="A1.E12.m1.6.6.1.1.1" xref="A1.E12.m1.6.6.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.E12.m1.6.6.1.1.1.2" xref="A1.E12.m1.6.6.1.1.1.1.cmml">(</mo><mrow id="A1.E12.m1.6.6.1.1.1.1" xref="A1.E12.m1.6.6.1.1.1.1.cmml"><msubsup id="A1.E12.m1.6.6.1.1.1.1.4" xref="A1.E12.m1.6.6.1.1.1.1.4.cmml"><mi mathsize="90%" id="A1.E12.m1.6.6.1.1.1.1.4.2.2" xref="A1.E12.m1.6.6.1.1.1.1.4.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.6.6.1.1.1.1.4.2.3" xref="A1.E12.m1.6.6.1.1.1.1.4.2.3.cmml">t</mi><mi mathsize="90%" id="A1.E12.m1.6.6.1.1.1.1.4.3" xref="A1.E12.m1.6.6.1.1.1.1.4.3.cmml">j</mi></msubsup><mo mathsize="90%" id="A1.E12.m1.6.6.1.1.1.1.3" xref="A1.E12.m1.6.6.1.1.1.1.3.cmml">∣</mo><mrow id="A1.E12.m1.6.6.1.1.1.1.2.2" xref="A1.E12.m1.6.6.1.1.1.1.2.3.cmml"><msubsup id="A1.E12.m1.6.6.1.1.1.1.1.1.1" xref="A1.E12.m1.6.6.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.E12.m1.6.6.1.1.1.1.1.1.1.2.2" xref="A1.E12.m1.6.6.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.E12.m1.6.6.1.1.1.1.1.1.1.2.3" 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id="A1.E12.m1.10c">\displaystyle\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}^{0}_{t})}=C^{\prime}(\mathbf{x}_{t}^{k_{t}},\mathbf{x}_{t}^{k_{t}+1},\mathbf{x}_{t}^{K})+\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1})^{\mathbf{1}\{k_{t}\geq 1\}}}+\sum_{j=k_{t}+1}^{K-1}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.12)</span></td>
</tr></tbody>
</table>
</div>
</section>
<section id="A1.SS2.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Part 2: Taking expecations.</h5>

<div id="A1.SS2.SSS0.Px2.p1" class="ltx_para">
<p id="A1.SS2.SSS0.Px2.p1.4" class="ltx_p"><span id="A1.SS2.SSS0.Px2.p1.4.1" class="ltx_text" style="font-size:90%;">We can now simplify to taking expectations. Observe that</span></p>
<table id="A6.EGx27" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex34"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.Ex34.m1.3" class="ltx_math_unparsed" alttext="\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})}=\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)," display="inline"><semantics id="A1.Ex34.m1.3a"><mrow id="A1.Ex34.m1.3b"><munder id="A1.Ex34.m1.3.4"><mo mathsize="108%" movablelimits="false" id="A1.Ex34.m1.3.4.2">𝔼</mo><mrow id="A1.Ex34.m1.1.1.1"><msubsup id="A1.Ex34.m1.1.1.1.3"><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.3.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.3.2.3">t</mi><mrow id="A1.Ex34.m1.1.1.1.3.3"><msub id="A1.Ex34.m1.1.1.1.3.3.2"><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.3.3.2.2">k</mi><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.3.3.2.3">t</mi></msub><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.Ex34.m1.1.1.1.3.3.1">:</mo><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.3.3.3">K</mi></mrow></msubsup><mo mathsize="90%" id="A1.Ex34.m1.1.1.1.2">∼</mo><mrow id="A1.Ex34.m1.1.1.1.1"><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.1.3">q</mi><mo lspace="0em" rspace="0em" id="A1.Ex34.m1.1.1.1.1.2">​</mo><mrow id="A1.Ex34.m1.1.1.1.1.1.1"><mo maxsize="90%" minsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.2">(</mo><mrow id="A1.Ex34.m1.1.1.1.1.1.1.1"><msubsup id="A1.Ex34.m1.1.1.1.1.1.1.1.2"><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.1.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.1.2.2.3">t</mi><mrow id="A1.Ex34.m1.1.1.1.1.1.1.1.2.3"><msub id="A1.Ex34.m1.1.1.1.1.1.1.1.2.3.2"><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.1.2.3.2.2">k</mi><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.1.2.3.2.3">t</mi></msub><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.Ex34.m1.1.1.1.1.1.1.1.2.3.1">:</mo><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.1.2.3.3">K</mi></mrow></msubsup><mo mathsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.1.1">∣</mo><msubsup id="A1.Ex34.m1.1.1.1.1.1.1.1.3"><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.1.3.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.1.3.2.3">t</mi><mn mathsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.1.3.3">0</mn></msubsup></mrow><mo maxsize="90%" minsize="90%" id="A1.Ex34.m1.1.1.1.1.1.1.3">)</mo></mrow></mrow></mrow></munder><mi mathsize="90%" id="A1.Ex34.m1.3.5">ln</mi><mstyle displaystyle="true" id="A1.Ex34.m1.3.3"><mfrac id="A1.Ex34.m1.3.3a"><mrow id="A1.Ex34.m1.2.2.1"><msub id="A1.Ex34.m1.2.2.1.3"><mi mathsize="90%" id="A1.Ex34.m1.2.2.1.3.2">p</mi><mi mathsize="90%" id="A1.Ex34.m1.2.2.1.3.3">𝜽</mi></msub><mo lspace="0em" rspace="0em" id="A1.Ex34.m1.2.2.1.2">​</mo><mrow id="A1.Ex34.m1.2.2.1.1.1"><mo maxsize="90%" minsize="90%" id="A1.Ex34.m1.2.2.1.1.1.2">(</mo><mrow id="A1.Ex34.m1.2.2.1.1.1.1"><msubsup id="A1.Ex34.m1.2.2.1.1.1.1.4"><mi mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.4.2.3">t</mi><mi mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.4.3">j</mi></msubsup><mo mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.3">∣</mo><mrow id="A1.Ex34.m1.2.2.1.1.1.1.2.2"><msubsup id="A1.Ex34.m1.2.2.1.1.1.1.1.1.1"><mi mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.1.1.1.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.1.1.1.2.3">t</mi><mrow id="A1.Ex34.m1.2.2.1.1.1.1.1.1.1.3"><mi mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.1.1.1.3.2">j</mi><mo mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.1.1.1.3.1">+</mo><mn mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.1.1.1.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.2.2.3">,</mo><msub id="A1.Ex34.m1.2.2.1.1.1.1.2.2.2"><mi mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.2.2.2.2">𝐳</mi><mrow id="A1.Ex34.m1.2.2.1.1.1.1.2.2.2.3"><mi mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.2.2.2.3.2">t</mi><mo mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.2.2.2.3.1">−</mo><mn mathsize="90%" id="A1.Ex34.m1.2.2.1.1.1.1.2.2.2.3.3">1</mn></mrow></msub></mrow></mrow><mo maxsize="90%" minsize="90%" id="A1.Ex34.m1.2.2.1.1.1.3">)</mo></mrow></mrow><mrow id="A1.Ex34.m1.3.3.2"><mi mathsize="90%" id="A1.Ex34.m1.3.3.2.3">q</mi><mo lspace="0em" rspace="0em" id="A1.Ex34.m1.3.3.2.2">​</mo><mrow id="A1.Ex34.m1.3.3.2.1.1"><mo maxsize="90%" minsize="90%" id="A1.Ex34.m1.3.3.2.1.1.2">(</mo><mrow id="A1.Ex34.m1.3.3.2.1.1.1"><msubsup id="A1.Ex34.m1.3.3.2.1.1.1.4"><mi mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.4.2.3">t</mi><mi mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.4.3">j</mi></msubsup><mo mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.3">∣</mo><mrow id="A1.Ex34.m1.3.3.2.1.1.1.2.2"><msubsup id="A1.Ex34.m1.3.3.2.1.1.1.1.1.1"><mi mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.1.1.1.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.1.1.1.2.3">t</mi><mrow id="A1.Ex34.m1.3.3.2.1.1.1.1.1.1.3"><mi mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.1.1.1.3.2">j</mi><mo mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.1.1.1.3.1">+</mo><mn mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.1.1.1.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.2.2.3">,</mo><msubsup id="A1.Ex34.m1.3.3.2.1.1.1.2.2.2"><mi mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.2.2.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.2.2.2.2.3">t</mi><mn mathsize="90%" id="A1.Ex34.m1.3.3.2.1.1.1.2.2.2.3">0</mn></msubsup></mrow></mrow><mo maxsize="90%" minsize="90%" id="A1.Ex34.m1.3.3.2.1.1.3">)</mo></mrow></mrow></mfrac></mstyle><mo mathsize="90%" id="A1.Ex34.m1.3.6">=</mo><msub id="A1.Ex34.m1.3.7"><mi mathsize="90%" mathvariant="normal" id="A1.Ex34.m1.3.7.2">D</mi><mrow id="A1.Ex34.m1.3.7.3"><mi mathsize="90%" id="A1.Ex34.m1.3.7.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex34.m1.3.7.3.1">​</mo><mi mathsize="90%" id="A1.Ex34.m1.3.7.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex34.m1.3.8"><mo id="A1.Ex34.m1.3.8.1">(</mo><mi mathsize="90%" id="A1.Ex34.m1.3.8.2">q</mi><mrow id="A1.Ex34.m1.3.8.3"><mo maxsize="90%" minsize="90%" id="A1.Ex34.m1.3.8.3.1">(</mo><msubsup id="A1.Ex34.m1.3.8.3.2"><mi mathsize="90%" id="A1.Ex34.m1.3.8.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.3.8.3.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex34.m1.3.8.3.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex34.m1.3.8.3.3">∣</mo><msubsup id="A1.Ex34.m1.3.8.3.4"><mi mathsize="90%" id="A1.Ex34.m1.3.8.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.3.8.3.4.2.3">t</mi><mrow id="A1.Ex34.m1.3.8.3.4.3"><mi mathsize="90%" id="A1.Ex34.m1.3.8.3.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex34.m1.3.8.3.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex34.m1.3.8.3.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex34.m1.3.8.3.5">,</mo><msubsup id="A1.Ex34.m1.3.8.3.6"><mi mathsize="90%" id="A1.Ex34.m1.3.8.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.3.8.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex34.m1.3.8.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex34.m1.3.8.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex34.m1.3.8.4">∥</mo><msub id="A1.Ex34.m1.3.8.5"><mi mathsize="90%" id="A1.Ex34.m1.3.8.5.2">p</mi><mi mathsize="90%" id="A1.Ex34.m1.3.8.5.3">𝜽</mi></msub><mrow id="A1.Ex34.m1.3.8.6"><mo maxsize="90%" minsize="90%" id="A1.Ex34.m1.3.8.6.1">(</mo><msubsup id="A1.Ex34.m1.3.8.6.2"><mi mathsize="90%" id="A1.Ex34.m1.3.8.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.3.8.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex34.m1.3.8.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex34.m1.3.8.6.3">∣</mo><msubsup id="A1.Ex34.m1.3.8.6.4"><mi mathsize="90%" id="A1.Ex34.m1.3.8.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex34.m1.3.8.6.4.2.3">t</mi><mrow id="A1.Ex34.m1.3.8.6.4.3"><mi mathsize="90%" id="A1.Ex34.m1.3.8.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex34.m1.3.8.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex34.m1.3.8.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex34.m1.3.8.6.5">,</mo><msub id="A1.Ex34.m1.3.8.6.6"><mi mathsize="90%" id="A1.Ex34.m1.3.8.6.6.2">𝐳</mi><mrow id="A1.Ex34.m1.3.8.6.6.3"><mi mathsize="90%" id="A1.Ex34.m1.3.8.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex34.m1.3.8.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex34.m1.3.8.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex34.m1.3.8.6.7">)</mo></mrow><mo id="A1.Ex34.m1.3.8.7">)</mo></mrow><mo mathsize="90%" id="A1.Ex34.m1.3.9">,</mo></mrow><annotation encoding="application/x-tex" id="A1.Ex34.m1.3c">\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})}=\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right),</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.SSS0.Px2.p1.5" class="ltx_p"><span id="A1.SS2.SSS0.Px2.p1.5.1" class="ltx_text" style="font-size:90%;">and similarly,</span></p>
<table id="A6.EGx28" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A1.Ex35.m1.8" class="ltx_math_unparsed" alttext="\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1})^{\mathbf{1}\{k_{t}\geq 1\}}}=\begin{cases}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})&amp;k_{t}=0\\
\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)&amp;k_{t}\geq 1.\end{cases}" display="inline"><semantics id="A1.Ex35.m1.8a"><mrow id="A1.Ex35.m1.8.9"><mrow id="A1.Ex35.m1.8.9.2"><munder id="A1.Ex35.m1.8.9.2.1"><mo mathsize="108%" movablelimits="false" id="A1.Ex35.m1.8.9.2.1.2">𝔼</mo><mrow id="A1.Ex35.m1.5.5.1"><msubsup id="A1.Ex35.m1.5.5.1.3"><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.3.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.3.2.3">t</mi><mrow id="A1.Ex35.m1.5.5.1.3.3"><msub id="A1.Ex35.m1.5.5.1.3.3.2"><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.3.3.2.2">k</mi><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.3.3.2.3">t</mi></msub><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.Ex35.m1.5.5.1.3.3.1">:</mo><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.3.3.3">K</mi></mrow></msubsup><mo mathsize="90%" id="A1.Ex35.m1.5.5.1.2">∼</mo><mrow id="A1.Ex35.m1.5.5.1.1"><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.1.3">q</mi><mo lspace="0em" rspace="0em" id="A1.Ex35.m1.5.5.1.1.2">​</mo><mrow id="A1.Ex35.m1.5.5.1.1.1.1"><mo maxsize="90%" minsize="90%" id="A1.Ex35.m1.5.5.1.1.1.1.2">(</mo><mrow id="A1.Ex35.m1.5.5.1.1.1.1.1"><msubsup id="A1.Ex35.m1.5.5.1.1.1.1.1.2"><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.1.1.1.1.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.1.1.1.1.2.2.3">t</mi><mrow id="A1.Ex35.m1.5.5.1.1.1.1.1.2.3"><msub id="A1.Ex35.m1.5.5.1.1.1.1.1.2.3.2"><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.1.1.1.1.2.3.2.2">k</mi><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.1.1.1.1.2.3.2.3">t</mi></msub><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.Ex35.m1.5.5.1.1.1.1.1.2.3.1">:</mo><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.1.1.1.1.2.3.3">K</mi></mrow></msubsup><mo mathsize="90%" id="A1.Ex35.m1.5.5.1.1.1.1.1.1">∣</mo><msubsup id="A1.Ex35.m1.5.5.1.1.1.1.1.3"><mi mathsize="90%" id="A1.Ex35.m1.5.5.1.1.1.1.1.3.2.2">𝐱</mi><mi 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id="A1.Ex35.m1.3.3.3.3.1.1.2.5"><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.5.2">p</mi><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.5.3">𝜽</mi></msub><mrow id="A1.Ex35.m1.3.3.3.3.1.1.2.6"><mo maxsize="90%" minsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.1">(</mo><msubsup id="A1.Ex35.m1.3.3.3.3.1.1.2.6.2"><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.2.2.3">t</mi><msub id="A1.Ex35.m1.3.3.3.3.1.1.2.6.2.3"><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.2.3.2">k</mi><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.2.3.3">t</mi></msub></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.3">∣</mo><msubsup id="A1.Ex35.m1.3.3.3.3.1.1.2.6.4"><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.4.2.3">t</mi><mrow id="A1.Ex35.m1.3.3.3.3.1.1.2.6.4.3"><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.5">,</mo><msub id="A1.Ex35.m1.3.3.3.3.1.1.2.6.6"><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.6.2">𝐳</mi><mrow id="A1.Ex35.m1.3.3.3.3.1.1.2.6.6.3"><mi mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex35.m1.3.3.3.3.1.1.2.6.7">)</mo></mrow><mo id="A1.Ex35.m1.3.3.3.3.1.1.2.7">)</mo></mrow></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="A1.Ex35.m1.4.4.4af"><mrow id="A1.Ex35.m1.4.4.4.4.2.1.1"><mrow id="A1.Ex35.m1.4.4.4.4.2.1.1.1"><msub id="A1.Ex35.m1.4.4.4.4.2.1.1.1.2"><mi mathsize="90%" id="A1.Ex35.m1.4.4.4.4.2.1.1.1.2.2">k</mi><mi mathsize="90%" id="A1.Ex35.m1.4.4.4.4.2.1.1.1.2.3">t</mi></msub><mo mathsize="90%" id="A1.Ex35.m1.4.4.4.4.2.1.1.1.1">≥</mo><mn mathsize="90%" id="A1.Ex35.m1.4.4.4.4.2.1.1.1.3">1</mn></mrow><mo lspace="0em" mathsize="90%" id="A1.Ex35.m1.4.4.4.4.2.1.1.2">.</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex" id="A1.Ex35.m1.8b">\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1})^{\mathbf{1}\{k_{t}\geq 1\}}}=\begin{cases}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})&amp;k_{t}=0\\
\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{k_{t}+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)&amp;k_{t}\geq 1.\end{cases}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
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id="A1.SS2.SSS0.Px2.p1.2.m2.2" class="ltx_Math" alttext="C_{2}(k_{t},\mathbf{x}_{0})" display="inline"><semantics id="A1.SS2.SSS0.Px2.p1.2.m2.2a"><mrow id="A1.SS2.SSS0.Px2.p1.2.m2.2.2" xref="A1.SS2.SSS0.Px2.p1.2.m2.2.2.cmml"><msub id="A1.SS2.SSS0.Px2.p1.2.m2.2.2.4" xref="A1.SS2.SSS0.Px2.p1.2.m2.2.2.4.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px2.p1.2.m2.2.2.4.2" xref="A1.SS2.SSS0.Px2.p1.2.m2.2.2.4.2.cmml">C</mi><mn mathsize="90%" id="A1.SS2.SSS0.Px2.p1.2.m2.2.2.4.3" xref="A1.SS2.SSS0.Px2.p1.2.m2.2.2.4.3.cmml">2</mn></msub><mo lspace="0em" rspace="0em" id="A1.SS2.SSS0.Px2.p1.2.m2.2.2.3" xref="A1.SS2.SSS0.Px2.p1.2.m2.2.2.3.cmml">​</mo><mrow id="A1.SS2.SSS0.Px2.p1.2.m2.2.2.2.2" xref="A1.SS2.SSS0.Px2.p1.2.m2.2.2.2.3.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px2.p1.2.m2.2.2.2.2.3" xref="A1.SS2.SSS0.Px2.p1.2.m2.2.2.2.3.cmml">(</mo><msub id="A1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.1" xref="A1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px2.p1.2.m2.1.1.1.1.1.2" 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xref="A1.SS2.SSS0.Px2.p1.3.m3.1.1.1.1">subscript</csymbol><ci id="A1.SS2.SSS0.Px2.p1.3.m3.1.1.1.1.2.cmml" xref="A1.SS2.SSS0.Px2.p1.3.m3.1.1.1.1.2">𝑘</ci><ci id="A1.SS2.SSS0.Px2.p1.3.m3.1.1.1.1.3.cmml" xref="A1.SS2.SSS0.Px2.p1.3.m3.1.1.1.1.3">𝑡</ci></apply><apply id="A1.SS2.SSS0.Px2.p1.3.m3.2.2.2.2.cmml" xref="A1.SS2.SSS0.Px2.p1.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px2.p1.3.m3.2.2.2.2.1.cmml" xref="A1.SS2.SSS0.Px2.p1.3.m3.2.2.2.2">subscript</csymbol><ci id="A1.SS2.SSS0.Px2.p1.3.m3.2.2.2.2.2.cmml" xref="A1.SS2.SSS0.Px2.p1.3.m3.2.2.2.2.2">𝐱</ci><cn type="integer" id="A1.SS2.SSS0.Px2.p1.3.m3.2.2.2.2.3.cmml" xref="A1.SS2.SSS0.Px2.p1.3.m3.2.2.2.2.3">0</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px2.p1.3.m3.2c">k_{t},\mathbf{x}_{0}</annotation></semantics></math><span id="A1.SS2.SSS0.Px2.p1.3.4" class="ltx_text" style="font-size:90%;">.
Thus, from (</span><a href="#A1.E12" title="Equation A.12 ‣ Part 1: Expanding the likelihood ratios ‣ A.2 Proof of Theorem A.1 ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">A.12</span></a><span id="A1.SS2.SSS0.Px2.p1.3.5" class="ltx_text" style="font-size:90%;">)</span></p>
<table id="A6.EGx29" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.Ex36.m1.3.3.2.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex36.m1.3c">\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}:K}\sim q(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{x}_{t}^{0})}\ln\frac{p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}:K}\mid\mathbf{z}_{t-1})}{q(\mathbf{x}_{t}^{k_{t}+1:K}\mid\mathbf{x}^{0}_{t})}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex37"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex37.m1.3" class="ltx_math_unparsed" alttext="\displaystyle=C_{2}(k_{t},\mathbf{x}_{0})+\mathbf{1}\{k_{t}=0\}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=\max\{1,k_{t}\}}^{K-1}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)" display="inline"><semantics id="A1.Ex37.m1.3a"><mrow id="A1.Ex37.m1.3b"><mo mathsize="90%" id="A1.Ex37.m1.3.4">=</mo><msub id="A1.Ex37.m1.3.5"><mi mathsize="90%" id="A1.Ex37.m1.3.5.2">C</mi><mn mathsize="90%" id="A1.Ex37.m1.3.5.3">2</mn></msub><mrow id="A1.Ex37.m1.3.6"><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.6.1">(</mo><msub id="A1.Ex37.m1.3.6.2"><mi mathsize="90%" id="A1.Ex37.m1.3.6.2.2">k</mi><mi mathsize="90%" id="A1.Ex37.m1.3.6.2.3">t</mi></msub><mo mathsize="90%" id="A1.Ex37.m1.3.6.3">,</mo><msub id="A1.Ex37.m1.3.6.4"><mi mathsize="90%" id="A1.Ex37.m1.3.6.4.2">𝐱</mi><mn mathsize="90%" id="A1.Ex37.m1.3.6.4.3">0</mn></msub><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.6.5">)</mo></mrow><mo mathsize="90%" id="A1.Ex37.m1.3.7">+</mo><mn mathsize="90%" id="A1.Ex37.m1.3.8">𝟏</mn><mrow id="A1.Ex37.m1.3.9"><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.9.1">{</mo><msub id="A1.Ex37.m1.3.9.2"><mi mathsize="90%" id="A1.Ex37.m1.3.9.2.2">k</mi><mi mathsize="90%" id="A1.Ex37.m1.3.9.2.3">t</mi></msub><mo mathsize="90%" id="A1.Ex37.m1.3.9.3">=</mo><mn mathsize="90%" id="A1.Ex37.m1.3.9.4">0</mn><mo maxsize="90%" minsize="90%" rspace="0.167em" id="A1.Ex37.m1.3.9.5">}</mo></mrow><mi mathsize="90%" id="A1.Ex37.m1.3.10">ln</mi><msub id="A1.Ex37.m1.3.11"><mi mathsize="90%" id="A1.Ex37.m1.3.11.2">p</mi><mi mathsize="90%" id="A1.Ex37.m1.3.11.3">𝜽</mi></msub><mrow id="A1.Ex37.m1.3.12"><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.12.1">(</mo><msubsup id="A1.Ex37.m1.3.12.2"><mi mathsize="90%" id="A1.Ex37.m1.3.12.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex37.m1.3.12.2.2.3">t</mi><mn mathsize="90%" id="A1.Ex37.m1.3.12.2.3">0</mn></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex37.m1.3.12.3">∣</mo><msubsup id="A1.Ex37.m1.3.12.4"><mi mathsize="90%" id="A1.Ex37.m1.3.12.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex37.m1.3.12.4.2.3">t</mi><mn mathsize="90%" id="A1.Ex37.m1.3.12.4.3">1</mn></msubsup><mo mathsize="90%" id="A1.Ex37.m1.3.12.5">,</mo><msub id="A1.Ex37.m1.3.12.6"><mi mathsize="90%" id="A1.Ex37.m1.3.12.6.2">𝐳</mi><mrow id="A1.Ex37.m1.3.12.6.3"><mi mathsize="90%" id="A1.Ex37.m1.3.12.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex37.m1.3.12.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex37.m1.3.12.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.12.7">)</mo></mrow><mo mathsize="90%" id="A1.Ex37.m1.3.13">+</mo><mstyle displaystyle="true" id="A1.Ex37.m1.3.14"><munderover id="A1.Ex37.m1.3.14a"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.Ex37.m1.3.14.2.2">∑</mo><mrow id="A1.Ex37.m1.3.3.3"><mi mathsize="90%" id="A1.Ex37.m1.3.3.3.5">j</mi><mo mathsize="90%" id="A1.Ex37.m1.3.3.3.4">=</mo><mrow id="A1.Ex37.m1.3.3.3.3.1"><mi mathsize="90%" id="A1.Ex37.m1.1.1.1.1">max</mi><mo id="A1.Ex37.m1.3.3.3.3.1a">⁡</mo><mrow id="A1.Ex37.m1.3.3.3.3.1.1"><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.3.3.3.1.1.2">{</mo><mn mathsize="90%" id="A1.Ex37.m1.2.2.2.2">1</mn><mo mathsize="90%" id="A1.Ex37.m1.3.3.3.3.1.1.3">,</mo><msub id="A1.Ex37.m1.3.3.3.3.1.1.1"><mi mathsize="90%" id="A1.Ex37.m1.3.3.3.3.1.1.1.2">k</mi><mi mathsize="90%" id="A1.Ex37.m1.3.3.3.3.1.1.1.3">t</mi></msub><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.3.3.3.1.1.4">}</mo></mrow></mrow></mrow><mrow id="A1.Ex37.m1.3.14.3"><mi mathsize="90%" id="A1.Ex37.m1.3.14.3.2">K</mi><mo mathsize="90%" id="A1.Ex37.m1.3.14.3.1">−</mo><mn mathsize="90%" id="A1.Ex37.m1.3.14.3.3">1</mn></mrow></munderover></mstyle><msub id="A1.Ex37.m1.3.15"><mi mathsize="90%" mathvariant="normal" id="A1.Ex37.m1.3.15.2">D</mi><mrow id="A1.Ex37.m1.3.15.3"><mi mathsize="90%" id="A1.Ex37.m1.3.15.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex37.m1.3.15.3.1">​</mo><mi mathsize="90%" id="A1.Ex37.m1.3.15.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex37.m1.3.16"><mo id="A1.Ex37.m1.3.16.1">(</mo><mi mathsize="90%" id="A1.Ex37.m1.3.16.2">q</mi><mrow id="A1.Ex37.m1.3.16.3"><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.16.3.1">(</mo><msubsup id="A1.Ex37.m1.3.16.3.2"><mi mathsize="90%" id="A1.Ex37.m1.3.16.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex37.m1.3.16.3.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex37.m1.3.16.3.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex37.m1.3.16.3.3">∣</mo><msubsup id="A1.Ex37.m1.3.16.3.4"><mi mathsize="90%" id="A1.Ex37.m1.3.16.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex37.m1.3.16.3.4.2.3">t</mi><mrow id="A1.Ex37.m1.3.16.3.4.3"><mi mathsize="90%" id="A1.Ex37.m1.3.16.3.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex37.m1.3.16.3.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex37.m1.3.16.3.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex37.m1.3.16.3.5">,</mo><msubsup id="A1.Ex37.m1.3.16.3.6"><mi mathsize="90%" id="A1.Ex37.m1.3.16.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex37.m1.3.16.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex37.m1.3.16.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.16.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex37.m1.3.16.4">∥</mo><msub id="A1.Ex37.m1.3.16.5"><mi mathsize="90%" id="A1.Ex37.m1.3.16.5.2">p</mi><mi mathsize="90%" id="A1.Ex37.m1.3.16.5.3">𝜽</mi></msub><mrow id="A1.Ex37.m1.3.16.6"><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.16.6.1">(</mo><msubsup id="A1.Ex37.m1.3.16.6.2"><mi mathsize="90%" id="A1.Ex37.m1.3.16.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex37.m1.3.16.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex37.m1.3.16.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex37.m1.3.16.6.3">∣</mo><msubsup id="A1.Ex37.m1.3.16.6.4"><mi mathsize="90%" id="A1.Ex37.m1.3.16.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex37.m1.3.16.6.4.2.3">t</mi><mrow id="A1.Ex37.m1.3.16.6.4.3"><mi mathsize="90%" id="A1.Ex37.m1.3.16.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex37.m1.3.16.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex37.m1.3.16.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex37.m1.3.16.6.5">,</mo><msub id="A1.Ex37.m1.3.16.6.6"><mi mathsize="90%" id="A1.Ex37.m1.3.16.6.6.2">𝐳</mi><mrow id="A1.Ex37.m1.3.16.6.6.3"><mi mathsize="90%" id="A1.Ex37.m1.3.16.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex37.m1.3.16.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex37.m1.3.16.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex37.m1.3.16.6.7">)</mo></mrow><mo id="A1.Ex37.m1.3.16.7">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A1.Ex37.m1.3c">\displaystyle=C_{2}(k_{t},\mathbf{x}_{0})+\mathbf{1}\{k_{t}=0\}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=\max\{1,k_{t}\}}^{K-1}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex38"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex38.m1.1" class="ltx_math_unparsed" alttext="\displaystyle=C_{2}(k_{t},\mathbf{x}_{0})+\mathbf{1}\{k_{t}=0\}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\mathbf{1}\{j\geq k_{t}\}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)." display="inline"><semantics id="A1.Ex38.m1.1a"><mrow id="A1.Ex38.m1.1b"><mo mathsize="90%" id="A1.Ex38.m1.1.1">=</mo><msub id="A1.Ex38.m1.1.2"><mi mathsize="90%" id="A1.Ex38.m1.1.2.2">C</mi><mn mathsize="90%" id="A1.Ex38.m1.1.2.3">2</mn></msub><mrow id="A1.Ex38.m1.1.3"><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.3.1">(</mo><msub id="A1.Ex38.m1.1.3.2"><mi mathsize="90%" id="A1.Ex38.m1.1.3.2.2">k</mi><mi mathsize="90%" id="A1.Ex38.m1.1.3.2.3">t</mi></msub><mo mathsize="90%" id="A1.Ex38.m1.1.3.3">,</mo><msub id="A1.Ex38.m1.1.3.4"><mi mathsize="90%" id="A1.Ex38.m1.1.3.4.2">𝐱</mi><mn mathsize="90%" id="A1.Ex38.m1.1.3.4.3">0</mn></msub><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.3.5">)</mo></mrow><mo mathsize="90%" id="A1.Ex38.m1.1.4">+</mo><mn mathsize="90%" id="A1.Ex38.m1.1.5">𝟏</mn><mrow id="A1.Ex38.m1.1.6"><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.6.1">{</mo><msub id="A1.Ex38.m1.1.6.2"><mi mathsize="90%" id="A1.Ex38.m1.1.6.2.2">k</mi><mi mathsize="90%" id="A1.Ex38.m1.1.6.2.3">t</mi></msub><mo mathsize="90%" id="A1.Ex38.m1.1.6.3">=</mo><mn mathsize="90%" id="A1.Ex38.m1.1.6.4">0</mn><mo maxsize="90%" minsize="90%" rspace="0.167em" id="A1.Ex38.m1.1.6.5">}</mo></mrow><mi mathsize="90%" id="A1.Ex38.m1.1.7">ln</mi><msub id="A1.Ex38.m1.1.8"><mi mathsize="90%" id="A1.Ex38.m1.1.8.2">p</mi><mi mathsize="90%" id="A1.Ex38.m1.1.8.3">𝜽</mi></msub><mrow id="A1.Ex38.m1.1.9"><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.9.1">(</mo><msubsup id="A1.Ex38.m1.1.9.2"><mi mathsize="90%" id="A1.Ex38.m1.1.9.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex38.m1.1.9.2.2.3">t</mi><mn mathsize="90%" id="A1.Ex38.m1.1.9.2.3">0</mn></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex38.m1.1.9.3">∣</mo><msubsup id="A1.Ex38.m1.1.9.4"><mi mathsize="90%" id="A1.Ex38.m1.1.9.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex38.m1.1.9.4.2.3">t</mi><mn mathsize="90%" id="A1.Ex38.m1.1.9.4.3">1</mn></msubsup><mo mathsize="90%" id="A1.Ex38.m1.1.9.5">,</mo><msub id="A1.Ex38.m1.1.9.6"><mi mathsize="90%" id="A1.Ex38.m1.1.9.6.2">𝐳</mi><mrow id="A1.Ex38.m1.1.9.6.3"><mi mathsize="90%" id="A1.Ex38.m1.1.9.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex38.m1.1.9.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex38.m1.1.9.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.9.7">)</mo></mrow><mo mathsize="90%" id="A1.Ex38.m1.1.10">+</mo><mstyle displaystyle="true" id="A1.Ex38.m1.1.11"><munderover id="A1.Ex38.m1.1.11a"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.Ex38.m1.1.11.2.2">∑</mo><mrow id="A1.Ex38.m1.1.11.2.3"><mi mathsize="90%" id="A1.Ex38.m1.1.11.2.3.2">j</mi><mo mathsize="90%" id="A1.Ex38.m1.1.11.2.3.1">=</mo><mn mathsize="90%" id="A1.Ex38.m1.1.11.2.3.3">1</mn></mrow><mrow id="A1.Ex38.m1.1.11.3"><mi mathsize="90%" id="A1.Ex38.m1.1.11.3.2">K</mi><mo mathsize="90%" id="A1.Ex38.m1.1.11.3.1">−</mo><mn mathsize="90%" id="A1.Ex38.m1.1.11.3.3">1</mn></mrow></munderover></mstyle><mn mathsize="90%" id="A1.Ex38.m1.1.12">𝟏</mn><mrow id="A1.Ex38.m1.1.13"><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.13.1">{</mo><mi mathsize="90%" id="A1.Ex38.m1.1.13.2">j</mi><mo mathsize="90%" id="A1.Ex38.m1.1.13.3">≥</mo><msub id="A1.Ex38.m1.1.13.4"><mi mathsize="90%" id="A1.Ex38.m1.1.13.4.2">k</mi><mi mathsize="90%" id="A1.Ex38.m1.1.13.4.3">t</mi></msub><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.13.5">}</mo></mrow><msub id="A1.Ex38.m1.1.14"><mi mathsize="90%" mathvariant="normal" id="A1.Ex38.m1.1.14.2">D</mi><mrow id="A1.Ex38.m1.1.14.3"><mi mathsize="90%" id="A1.Ex38.m1.1.14.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex38.m1.1.14.3.1">​</mo><mi mathsize="90%" id="A1.Ex38.m1.1.14.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex38.m1.1.15"><mo id="A1.Ex38.m1.1.15.1">(</mo><mi mathsize="90%" id="A1.Ex38.m1.1.15.2">q</mi><mrow id="A1.Ex38.m1.1.15.3"><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.15.3.1">(</mo><msubsup id="A1.Ex38.m1.1.15.3.2"><mi mathsize="90%" id="A1.Ex38.m1.1.15.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex38.m1.1.15.3.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex38.m1.1.15.3.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex38.m1.1.15.3.3">∣</mo><msubsup id="A1.Ex38.m1.1.15.3.4"><mi mathsize="90%" id="A1.Ex38.m1.1.15.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex38.m1.1.15.3.4.2.3">t</mi><mrow id="A1.Ex38.m1.1.15.3.4.3"><mi mathsize="90%" id="A1.Ex38.m1.1.15.3.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex38.m1.1.15.3.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex38.m1.1.15.3.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex38.m1.1.15.3.5">,</mo><msubsup id="A1.Ex38.m1.1.15.3.6"><mi mathsize="90%" id="A1.Ex38.m1.1.15.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex38.m1.1.15.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex38.m1.1.15.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.15.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex38.m1.1.15.4">∥</mo><msub id="A1.Ex38.m1.1.15.5"><mi mathsize="90%" id="A1.Ex38.m1.1.15.5.2">p</mi><mi mathsize="90%" id="A1.Ex38.m1.1.15.5.3">𝜽</mi></msub><mrow id="A1.Ex38.m1.1.15.6"><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.15.6.1">(</mo><msubsup id="A1.Ex38.m1.1.15.6.2"><mi mathsize="90%" id="A1.Ex38.m1.1.15.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex38.m1.1.15.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex38.m1.1.15.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex38.m1.1.15.6.3">∣</mo><msubsup id="A1.Ex38.m1.1.15.6.4"><mi mathsize="90%" id="A1.Ex38.m1.1.15.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex38.m1.1.15.6.4.2.3">t</mi><mrow id="A1.Ex38.m1.1.15.6.4.3"><mi mathsize="90%" id="A1.Ex38.m1.1.15.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex38.m1.1.15.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex38.m1.1.15.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex38.m1.1.15.6.5">,</mo><msub id="A1.Ex38.m1.1.15.6.6"><mi mathsize="90%" id="A1.Ex38.m1.1.15.6.6.2">𝐳</mi><mrow id="A1.Ex38.m1.1.15.6.6.3"><mi mathsize="90%" id="A1.Ex38.m1.1.15.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex38.m1.1.15.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex38.m1.1.15.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex38.m1.1.15.6.7">)</mo></mrow><mo id="A1.Ex38.m1.1.15.7">)</mo></mrow><mo lspace="0em" mathsize="90%" id="A1.Ex38.m1.1.16">.</mo></mrow><annotation encoding="application/x-tex" id="A1.Ex38.m1.1c">\displaystyle=C_{2}(k_{t},\mathbf{x}_{0})+\mathbf{1}\{k_{t}=0\}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\mathbf{1}\{j\geq k_{t}\}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right).</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.SSS0.Px2.p1.6" class="ltx_p"><span id="A1.SS2.SSS0.Px2.p1.6.1" class="ltx_text" style="font-size:90%;">∎</span></p>
</div>
</section>
<section id="A1.SS2.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Completing the proof of the ELBO.</h5>

<div id="A1.SS2.SSS0.Px3.p1" class="ltx_para">
<p id="A1.SS2.SSS0.Px3.p1.11" class="ltx_p"><span id="A1.SS2.SSS0.Px3.p1.11.1" class="ltx_text" style="font-size:90%;">We are now ready to complete the proof. By combining the previous two claims, we have</span></p>
<table id="A6.EGx30" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex39"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.Ex39.m1.2.2.1.1.1.1.1.1.2.3.1.cmml" xref="A1.Ex39.m1.2.2.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="A1.Ex39.m1.2.2.1.1.1.1.1.1.2.3.2.cmml" xref="A1.Ex39.m1.2.2.1.1.1.1.1.1.2.3.2">𝑘</ci><ci id="A1.Ex39.m1.2.2.1.1.1.1.1.1.2.3.3.cmml" xref="A1.Ex39.m1.2.2.1.1.1.1.1.1.2.3.3">𝑡</ci></apply></apply><apply id="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.cmml" xref="A1.Ex39.m1.2.2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex39.m1.2.2.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.2">𝐳</ci><apply id="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.3"><minus id="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.3.1.cmml" xref="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.3.1"></minus><ci id="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.3.2.cmml" xref="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.3.2">𝑡</ci><cn type="integer" id="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.3.3.cmml" xref="A1.Ex39.m1.2.2.1.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex39.m1.2c">\displaystyle\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{z}_{t-1})\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex40"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex40.m1.1" class="ltx_math_unparsed" alttext="\displaystyle\quad\geq C_{3}(\mathbf{x}_{0},k_{t})+\mathbf{1}\{k_{t}=0\}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\mathbf{1}\{j\geq k_{t}\}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)," display="inline"><semantics id="A1.Ex40.m1.1a"><mrow id="A1.Ex40.m1.1b"><mo mathsize="90%" id="A1.Ex40.m1.1.1">≥</mo><msub id="A1.Ex40.m1.1.2"><mi mathsize="90%" id="A1.Ex40.m1.1.2.2">C</mi><mn mathsize="90%" id="A1.Ex40.m1.1.2.3">3</mn></msub><mrow id="A1.Ex40.m1.1.3"><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.3.1">(</mo><msub id="A1.Ex40.m1.1.3.2"><mi mathsize="90%" id="A1.Ex40.m1.1.3.2.2">𝐱</mi><mn mathsize="90%" id="A1.Ex40.m1.1.3.2.3">0</mn></msub><mo mathsize="90%" id="A1.Ex40.m1.1.3.3">,</mo><msub id="A1.Ex40.m1.1.3.4"><mi mathsize="90%" id="A1.Ex40.m1.1.3.4.2">k</mi><mi mathsize="90%" id="A1.Ex40.m1.1.3.4.3">t</mi></msub><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.3.5">)</mo></mrow><mo mathsize="90%" id="A1.Ex40.m1.1.4">+</mo><mn mathsize="90%" id="A1.Ex40.m1.1.5">𝟏</mn><mrow id="A1.Ex40.m1.1.6"><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.6.1">{</mo><msub id="A1.Ex40.m1.1.6.2"><mi mathsize="90%" id="A1.Ex40.m1.1.6.2.2">k</mi><mi mathsize="90%" id="A1.Ex40.m1.1.6.2.3">t</mi></msub><mo mathsize="90%" id="A1.Ex40.m1.1.6.3">=</mo><mn mathsize="90%" id="A1.Ex40.m1.1.6.4">0</mn><mo maxsize="90%" minsize="90%" rspace="0.167em" id="A1.Ex40.m1.1.6.5">}</mo></mrow><mi mathsize="90%" id="A1.Ex40.m1.1.7">ln</mi><msub id="A1.Ex40.m1.1.8"><mi mathsize="90%" id="A1.Ex40.m1.1.8.2">p</mi><mi mathsize="90%" id="A1.Ex40.m1.1.8.3">𝜽</mi></msub><mrow id="A1.Ex40.m1.1.9"><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.9.1">(</mo><msubsup id="A1.Ex40.m1.1.9.2"><mi mathsize="90%" id="A1.Ex40.m1.1.9.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex40.m1.1.9.2.2.3">t</mi><mn mathsize="90%" id="A1.Ex40.m1.1.9.2.3">0</mn></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex40.m1.1.9.3">∣</mo><msubsup id="A1.Ex40.m1.1.9.4"><mi mathsize="90%" id="A1.Ex40.m1.1.9.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex40.m1.1.9.4.2.3">t</mi><mn mathsize="90%" id="A1.Ex40.m1.1.9.4.3">1</mn></msubsup><mo mathsize="90%" id="A1.Ex40.m1.1.9.5">,</mo><msub id="A1.Ex40.m1.1.9.6"><mi mathsize="90%" id="A1.Ex40.m1.1.9.6.2">𝐳</mi><mrow id="A1.Ex40.m1.1.9.6.3"><mi mathsize="90%" id="A1.Ex40.m1.1.9.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex40.m1.1.9.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex40.m1.1.9.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.9.7">)</mo></mrow><mo mathsize="90%" id="A1.Ex40.m1.1.10">+</mo><mstyle displaystyle="true" id="A1.Ex40.m1.1.11"><munderover id="A1.Ex40.m1.1.11a"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.Ex40.m1.1.11.2.2">∑</mo><mrow id="A1.Ex40.m1.1.11.2.3"><mi mathsize="90%" id="A1.Ex40.m1.1.11.2.3.2">j</mi><mo mathsize="90%" id="A1.Ex40.m1.1.11.2.3.1">=</mo><mn mathsize="90%" id="A1.Ex40.m1.1.11.2.3.3">1</mn></mrow><mrow id="A1.Ex40.m1.1.11.3"><mi mathsize="90%" id="A1.Ex40.m1.1.11.3.2">K</mi><mo mathsize="90%" id="A1.Ex40.m1.1.11.3.1">−</mo><mn mathsize="90%" id="A1.Ex40.m1.1.11.3.3">1</mn></mrow></munderover></mstyle><mn mathsize="90%" id="A1.Ex40.m1.1.12">𝟏</mn><mrow id="A1.Ex40.m1.1.13"><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.13.1">{</mo><mi mathsize="90%" id="A1.Ex40.m1.1.13.2">j</mi><mo mathsize="90%" id="A1.Ex40.m1.1.13.3">≥</mo><msub id="A1.Ex40.m1.1.13.4"><mi mathsize="90%" id="A1.Ex40.m1.1.13.4.2">k</mi><mi mathsize="90%" id="A1.Ex40.m1.1.13.4.3">t</mi></msub><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.13.5">}</mo></mrow><msub id="A1.Ex40.m1.1.14"><mi mathsize="90%" mathvariant="normal" id="A1.Ex40.m1.1.14.2">D</mi><mrow id="A1.Ex40.m1.1.14.3"><mi mathsize="90%" id="A1.Ex40.m1.1.14.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex40.m1.1.14.3.1">​</mo><mi mathsize="90%" id="A1.Ex40.m1.1.14.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex40.m1.1.15"><mo id="A1.Ex40.m1.1.15.1">(</mo><mi mathsize="90%" id="A1.Ex40.m1.1.15.2">q</mi><mrow id="A1.Ex40.m1.1.15.3"><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.15.3.1">(</mo><msubsup id="A1.Ex40.m1.1.15.3.2"><mi mathsize="90%" id="A1.Ex40.m1.1.15.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex40.m1.1.15.3.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex40.m1.1.15.3.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex40.m1.1.15.3.3">∣</mo><msubsup id="A1.Ex40.m1.1.15.3.4"><mi mathsize="90%" id="A1.Ex40.m1.1.15.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex40.m1.1.15.3.4.2.3">t</mi><mrow id="A1.Ex40.m1.1.15.3.4.3"><mi mathsize="90%" id="A1.Ex40.m1.1.15.3.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex40.m1.1.15.3.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex40.m1.1.15.3.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex40.m1.1.15.3.5">,</mo><msubsup id="A1.Ex40.m1.1.15.3.6"><mi mathsize="90%" id="A1.Ex40.m1.1.15.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex40.m1.1.15.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex40.m1.1.15.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.15.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex40.m1.1.15.4">∥</mo><msub id="A1.Ex40.m1.1.15.5"><mi mathsize="90%" id="A1.Ex40.m1.1.15.5.2">p</mi><mi mathsize="90%" id="A1.Ex40.m1.1.15.5.3">𝜽</mi></msub><mrow id="A1.Ex40.m1.1.15.6"><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.15.6.1">(</mo><msubsup id="A1.Ex40.m1.1.15.6.2"><mi mathsize="90%" id="A1.Ex40.m1.1.15.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex40.m1.1.15.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex40.m1.1.15.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex40.m1.1.15.6.3">∣</mo><msubsup id="A1.Ex40.m1.1.15.6.4"><mi mathsize="90%" id="A1.Ex40.m1.1.15.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex40.m1.1.15.6.4.2.3">t</mi><mrow id="A1.Ex40.m1.1.15.6.4.3"><mi mathsize="90%" id="A1.Ex40.m1.1.15.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex40.m1.1.15.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex40.m1.1.15.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex40.m1.1.15.6.5">,</mo><msub id="A1.Ex40.m1.1.15.6.6"><mi mathsize="90%" id="A1.Ex40.m1.1.15.6.6.2">𝐳</mi><mrow id="A1.Ex40.m1.1.15.6.6.3"><mi mathsize="90%" id="A1.Ex40.m1.1.15.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex40.m1.1.15.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex40.m1.1.15.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex40.m1.1.15.6.7">)</mo></mrow><mo id="A1.Ex40.m1.1.15.7">)</mo></mrow><mo mathsize="90%" id="A1.Ex40.m1.1.16">,</mo></mrow><annotation encoding="application/x-tex" id="A1.Ex40.m1.1c">\displaystyle\quad\geq C_{3}(\mathbf{x}_{0},k_{t})+\mathbf{1}\{k_{t}=0\}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\mathbf{1}\{j\geq k_{t}\}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right),</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.SSS0.Px3.p1.3" class="ltx_p"><span id="A1.SS2.SSS0.Px3.p1.3.1" class="ltx_text" style="font-size:90%;">where </span><math id="A1.SS2.SSS0.Px3.p1.1.m1.6" class="ltx_Math" alttext="C_{3}(\mathbf{x}_{0},k_{t})=C_{1}(\mathbf{x}_{0},k_{t})+C_{2}(\mathbf{x}_{0},k_{t})" display="inline"><semantics id="A1.SS2.SSS0.Px3.p1.1.m1.6a"><mrow id="A1.SS2.SSS0.Px3.p1.1.m1.6.6" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.cmml"><mrow id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.cmml"><msub id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.2.cmml">C</mi><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.3.cmml">3</mn></msub><mo lspace="0em" rspace="0em" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.3.cmml">​</mo><mrow id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.3.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.3.cmml">(</mo><msub id="A1.SS2.SSS0.Px3.p1.1.m1.1.1.1.1.1.1" xref="A1.SS2.SSS0.Px3.p1.1.m1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.1.1.1.1.1.1.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.1.1.1.1.1.1.2.cmml">𝐱</mi><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.1.1.1.1.1.1.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2.4" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.3.cmml">,</mo><msub id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2.2.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2.2.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2.2.2.cmml">k</mi><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2.2.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2.2.3.cmml">t</mi></msub><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2.5" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.7" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.7.cmml">=</mo><mrow id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.cmml"><mrow id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.cmml"><msub id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.2.cmml">C</mi><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.3.cmml">1</mn></msub><mo lspace="0em" rspace="0em" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.3.cmml">​</mo><mrow id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.3.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.3.cmml">(</mo><msub id="A1.SS2.SSS0.Px3.p1.1.m1.3.3.3.1.1.1.1" xref="A1.SS2.SSS0.Px3.p1.1.m1.3.3.3.1.1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.3.3.3.1.1.1.1.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.3.3.3.1.1.1.1.2.cmml">𝐱</mi><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.3.3.3.1.1.1.1.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.3.3.3.1.1.1.1.3.cmml">0</mn></msub><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2.4" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.3.cmml">,</mo><msub id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2.2.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2.2.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2.2.2.cmml">k</mi><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2.2.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2.2.3.cmml">t</mi></msub><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2.5" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.5" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.5.cmml">+</mo><mrow id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.cmml"><msub id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.4" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.4.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.4.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.4.2.cmml">C</mi><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.4.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.4.3.cmml">2</mn></msub><mo lspace="0em" rspace="0em" id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.3.cmml">​</mo><mrow id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.3.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.2.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.3.cmml">(</mo><msub id="A1.SS2.SSS0.Px3.p1.1.m1.5.5.5.3.1.1.1" xref="A1.SS2.SSS0.Px3.p1.1.m1.5.5.5.3.1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.5.5.5.3.1.1.1.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.5.5.5.3.1.1.1.2.cmml">𝐱</mi><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.5.5.5.3.1.1.1.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.5.5.5.3.1.1.1.3.cmml">0</mn></msub><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.2.4" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.3.cmml">,</mo><msub id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.2.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.2.2.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.2.2.2" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.2.2.2.cmml">k</mi><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.2.2.3" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.2.2.3.cmml">t</mi></msub><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.2.5" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.4.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.SSS0.Px3.p1.1.m1.6b"><apply id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6"><eq id="A1.SS2.SSS0.Px3.p1.1.m1.6.6.7.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.7"></eq><apply id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2"><times id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.3.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.3"></times><apply id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.1.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4">subscript</csymbol><ci id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.2.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.2">𝐶</ci><cn type="integer" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.3.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.4.3">3</cn></apply><interval closure="open" id="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.3.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.2.2.2.2.2"><apply id="A1.SS2.SSS0.Px3.p1.1.m1.1.1.1.1.1.1.cmml" 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xref="A1.SS2.SSS0.Px3.p1.1.m1.6.6.6.5"></plus><apply id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2"><times id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.3.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.3"></times><apply id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.1.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4">subscript</csymbol><ci id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.2.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.2">𝐶</ci><cn type="integer" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.3.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.4.3">1</cn></apply><interval closure="open" id="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.3.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.4.4.4.2.2.2"><apply id="A1.SS2.SSS0.Px3.p1.1.m1.3.3.3.1.1.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.1.m1.3.3.3.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.1.m1.3.3.3.1.1.1.1.1.cmml" 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xref="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.1.1.1.3.2.cmml" xref="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.1.1.1.3.2">𝑘</ci><ci id="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.1.1.1.3.3.cmml" xref="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply><apply id="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.cmml" xref="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.1.cmml" xref="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2">subscript</csymbol><ci id="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.2.cmml" xref="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.2">𝐳</ci><apply id="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.3.cmml" xref="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.3"><minus id="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.3.1.cmml" xref="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.3.1"></minus><ci id="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.3.2.cmml" xref="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.3.2">𝑡</ci><cn type="integer" id="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.3.3.cmml" xref="A1.SS2.SSS0.Px3.p1.2.m2.2.2.2.1.1.1.2.2.2.3.3">1</cn></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px3.p1.2.m2.2c">q(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}})\equiv p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1})</annotation></semantics></math><span id="A1.SS2.SSS0.Px3.p1.3.3" class="ltx_text" style="font-size:90%;">. Taking an expectation over </span><math id="A1.SS2.SSS0.Px3.p1.3.m3.4" class="ltx_Math" alttext="k_{t}\overset{\mathrm{unif}}{\sim}\{0,1,\dots,K\}" display="inline"><semantics id="A1.SS2.SSS0.Px3.p1.3.m3.4a"><mrow id="A1.SS2.SSS0.Px3.p1.3.m3.4.5" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.cmml"><msub id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.2" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.2.cmml">k</mi><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.3" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.3.cmml">t</mi></msub><mo lspace="0em" rspace="0em" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.1" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.1.cmml">​</mo><mover accent="true" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3.cmml"><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3.2" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3.2.cmml">∼</mo><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3.1" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3.1.cmml">unif</mo></mover><mo lspace="0em" rspace="0em" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.1a" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.1.cmml">​</mo><mrow id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.2" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.2.1" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.1.cmml">{</mo><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.1.1" xref="A1.SS2.SSS0.Px3.p1.3.m3.1.1.cmml">0</mn><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.2.2" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.1.cmml">,</mo><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.2.2" xref="A1.SS2.SSS0.Px3.p1.3.m3.2.2.cmml">1</mn><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.2.3" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.1.cmml">,</mo><mi mathsize="90%" mathvariant="normal" id="A1.SS2.SSS0.Px3.p1.3.m3.3.3" xref="A1.SS2.SSS0.Px3.p1.3.m3.3.3.cmml">…</mi><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.2.4" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.1.cmml">,</mo><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.4.4" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.4.cmml">K</mi><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.2.5" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.SSS0.Px3.p1.3.m3.4b"><apply id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5"><times id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.1.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.1"></times><apply id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.1.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2">subscript</csymbol><ci id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.2.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.2">𝑘</ci><ci id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.3.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.2.3">𝑡</ci></apply><apply id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3"><ci id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3.1.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3.1">unif</ci><csymbol cd="latexml" id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3.2.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.3.2">similar-to</csymbol></apply><set id="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.1.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.5.4.2"><cn type="integer" id="A1.SS2.SSS0.Px3.p1.3.m3.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.1.1">0</cn><cn type="integer" id="A1.SS2.SSS0.Px3.p1.3.m3.2.2.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.2.2">1</cn><ci id="A1.SS2.SSS0.Px3.p1.3.m3.3.3.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.3.3">…</ci><ci id="A1.SS2.SSS0.Px3.p1.3.m3.4.4.cmml" xref="A1.SS2.SSS0.Px3.p1.3.m3.4.4">𝐾</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px3.p1.3.m3.4c">k_{t}\overset{\mathrm{unif}}{\sim}\{0,1,\dots,K\}</annotation></semantics></math><span id="A1.SS2.SSS0.Px3.p1.3.4" class="ltx_text" style="font-size:90%;">, we</span></p>
<table id="A6.EGx31" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.E13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.E13.m1.9.9.1.1.2.2.3.1.cmml" xref="A1.E13.m1.9.9.1.1.2.2.3"></divide><apply id="A1.E13.m1.9.9.1.1.2.2.3.2.cmml" xref="A1.E13.m1.9.9.1.1.2.2.3.2"><plus id="A1.E13.m1.9.9.1.1.2.2.3.2.1.cmml" xref="A1.E13.m1.9.9.1.1.2.2.3.2.1"></plus><ci id="A1.E13.m1.9.9.1.1.2.2.3.2.2.cmml" xref="A1.E13.m1.9.9.1.1.2.2.3.2.2">𝑗</ci><cn type="integer" id="A1.E13.m1.9.9.1.1.2.2.3.2.3.cmml" xref="A1.E13.m1.9.9.1.1.2.2.3.2.3">1</cn></apply><apply id="A1.E13.m1.9.9.1.1.2.2.3.3.cmml" xref="A1.E13.m1.9.9.1.1.2.2.3.3"><plus id="A1.E13.m1.9.9.1.1.2.2.3.3.1.cmml" xref="A1.E13.m1.9.9.1.1.2.2.3.3.1"></plus><ci id="A1.E13.m1.9.9.1.1.2.2.3.3.2.cmml" xref="A1.E13.m1.9.9.1.1.2.2.3.3.2">𝐾</ci><cn type="integer" id="A1.E13.m1.9.9.1.1.2.2.3.3.3.cmml" xref="A1.E13.m1.9.9.1.1.2.2.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E13.m1.9c">\displaystyle\mathop{{\mathbb{E}}}_{k_{t}\overset{\mathrm{unif}}{\sim}\{0,1,\dots,K\}}[\mathbf{1}\{k_{t}=0\}]=\frac{1}{K+1},\quad\mathop{{\mathbb{E}}}_{k_{t}\overset{\mathrm{unif}}{\sim}\{0,1,\dots,K\}}\mathbf{1}\{j\geq k_{t}\}=\frac{j+1}{K+1}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(A.13)</span></td>
</tr></tbody>
</table>
<p id="A1.SS2.SSS0.Px3.p1.12" class="ltx_p"><span id="A1.SS2.SSS0.Px3.p1.12.1" class="ltx_text" style="font-size:90%;">and consequently,</span></p>
<table id="A6.EGx32" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex41"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.Ex41.m1.6.6.2.2.2.3"></leq><cn type="integer" id="A1.Ex41.m1.6.6.2.2.2.2.cmml" xref="A1.Ex41.m1.6.6.2.2.2.2">1</cn><ci id="A1.Ex41.m1.6.6.2.2.2.4.cmml" xref="A1.Ex41.m1.6.6.2.2.2.4">𝑡</ci></apply><apply id="A1.Ex41.m1.6.6.2.2.2c.cmml" xref="A1.Ex41.m1.6.6.2.2.2"><leq id="A1.Ex41.m1.6.6.2.2.2.5.cmml" xref="A1.Ex41.m1.6.6.2.2.2.5"></leq><share href="#A1.Ex41.m1.6.6.2.2.2.4.cmml" id="A1.Ex41.m1.6.6.2.2.2d.cmml" xref="A1.Ex41.m1.6.6.2.2.2"></share><ci id="A1.Ex41.m1.6.6.2.2.2.6.cmml" xref="A1.Ex41.m1.6.6.2.2.2.6">𝑇</ci></apply></apply></apply></apply><apply id="A1.Ex41.m1.7.7.1.1.cmml" xref="A1.Ex41.m1.7.7.1.1"><times id="A1.Ex41.m1.7.7.1.1.2.cmml" xref="A1.Ex41.m1.7.7.1.1.2"></times><apply id="A1.Ex41.m1.7.7.1.1.3.cmml" xref="A1.Ex41.m1.7.7.1.1.3"><ln id="A1.Ex41.m1.7.7.1.1.3.1.cmml" xref="A1.Ex41.m1.7.7.1.1.3.1"></ln><apply id="A1.Ex41.m1.7.7.1.1.3.2.cmml" xref="A1.Ex41.m1.7.7.1.1.3.2"><csymbol cd="ambiguous" id="A1.Ex41.m1.7.7.1.1.3.2.1.cmml" xref="A1.Ex41.m1.7.7.1.1.3.2">subscript</csymbol><ci id="A1.Ex41.m1.7.7.1.1.3.2.2.cmml" xref="A1.Ex41.m1.7.7.1.1.3.2.2">𝑝</ci><ci id="A1.Ex41.m1.7.7.1.1.3.2.3.cmml" xref="A1.Ex41.m1.7.7.1.1.3.2.3">𝜽</ci></apply></apply><apply id="A1.Ex41.m1.7.7.1.1.1.1.1.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex41.m1.7.7.1.1.1.1.1.2.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1">subscript</csymbol><apply id="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.1.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.1.1">superscript</csymbol><apply id="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.2.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.2.1.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.1.1">subscript</csymbol><ci id="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.2.2.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.2.2">𝐱</ci><ci id="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.2.3.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.3.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.3.1.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.3.2.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.3.2">𝑘</ci><ci id="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.3.3.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply><apply id="A1.Ex41.m1.7.7.1.1.1.1.1.3.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.3"><and id="A1.Ex41.m1.7.7.1.1.1.1.1.3a.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.3"></and><apply id="A1.Ex41.m1.7.7.1.1.1.1.1.3b.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.3"><leq id="A1.Ex41.m1.7.7.1.1.1.1.1.3.3.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.3.3"></leq><cn type="integer" id="A1.Ex41.m1.7.7.1.1.1.1.1.3.2.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.3.2">1</cn><ci id="A1.Ex41.m1.7.7.1.1.1.1.1.3.4.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.3.4">𝑡</ci></apply><apply id="A1.Ex41.m1.7.7.1.1.1.1.1.3c.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.3"><leq id="A1.Ex41.m1.7.7.1.1.1.1.1.3.5.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.3.5"></leq><share href="#A1.Ex41.m1.7.7.1.1.1.1.1.3.4.cmml" id="A1.Ex41.m1.7.7.1.1.1.1.1.3d.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.3"></share><ci id="A1.Ex41.m1.7.7.1.1.1.1.1.3.6.cmml" xref="A1.Ex41.m1.7.7.1.1.1.1.1.3.6">𝑇</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex41.m1.7c">\displaystyle\mathop{{\mathbb{E}}}_{k_{t}\overset{\mathrm{unif}}{\sim}\{0,1,\dots,K\}}\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0}),1\leq t\leq T}\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex42"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex42.m1.1" class="ltx_math_unparsed" alttext="\displaystyle\quad\geq C_{4}(\mathbf{x}_{t}^{0})+\frac{1}{K+1}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\frac{j+1}{K+1}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)" display="inline"><semantics id="A1.Ex42.m1.1a"><mrow id="A1.Ex42.m1.1b"><mo mathsize="90%" id="A1.Ex42.m1.1.1">≥</mo><msub id="A1.Ex42.m1.1.2"><mi mathsize="90%" id="A1.Ex42.m1.1.2.2">C</mi><mn mathsize="90%" id="A1.Ex42.m1.1.2.3">4</mn></msub><mrow id="A1.Ex42.m1.1.3"><mo maxsize="90%" minsize="90%" id="A1.Ex42.m1.1.3.1">(</mo><msubsup id="A1.Ex42.m1.1.3.2"><mi mathsize="90%" id="A1.Ex42.m1.1.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex42.m1.1.3.2.2.3">t</mi><mn mathsize="90%" id="A1.Ex42.m1.1.3.2.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex42.m1.1.3.3">)</mo></mrow><mo mathsize="90%" id="A1.Ex42.m1.1.4">+</mo><mstyle displaystyle="true" id="A1.Ex42.m1.1.5"><mfrac id="A1.Ex42.m1.1.5a"><mn mathsize="90%" id="A1.Ex42.m1.1.5.2">1</mn><mrow id="A1.Ex42.m1.1.5.3"><mi mathsize="90%" id="A1.Ex42.m1.1.5.3.2">K</mi><mo mathsize="90%" id="A1.Ex42.m1.1.5.3.1">+</mo><mn mathsize="90%" id="A1.Ex42.m1.1.5.3.3">1</mn></mrow></mfrac></mstyle><mi mathsize="90%" id="A1.Ex42.m1.1.6">ln</mi><msub id="A1.Ex42.m1.1.7"><mi mathsize="90%" id="A1.Ex42.m1.1.7.2">p</mi><mi mathsize="90%" id="A1.Ex42.m1.1.7.3">𝜽</mi></msub><mrow id="A1.Ex42.m1.1.8"><mo maxsize="90%" minsize="90%" id="A1.Ex42.m1.1.8.1">(</mo><msubsup id="A1.Ex42.m1.1.8.2"><mi mathsize="90%" id="A1.Ex42.m1.1.8.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex42.m1.1.8.2.2.3">t</mi><mn mathsize="90%" id="A1.Ex42.m1.1.8.2.3">0</mn></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex42.m1.1.8.3">∣</mo><msubsup id="A1.Ex42.m1.1.8.4"><mi mathsize="90%" id="A1.Ex42.m1.1.8.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex42.m1.1.8.4.2.3">t</mi><mn mathsize="90%" id="A1.Ex42.m1.1.8.4.3">1</mn></msubsup><mo mathsize="90%" id="A1.Ex42.m1.1.8.5">,</mo><msub id="A1.Ex42.m1.1.8.6"><mi mathsize="90%" id="A1.Ex42.m1.1.8.6.2">𝐳</mi><mrow id="A1.Ex42.m1.1.8.6.3"><mi mathsize="90%" id="A1.Ex42.m1.1.8.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex42.m1.1.8.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex42.m1.1.8.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex42.m1.1.8.7">)</mo></mrow><mo mathsize="90%" id="A1.Ex42.m1.1.9">+</mo><mstyle displaystyle="true" id="A1.Ex42.m1.1.10"><munderover id="A1.Ex42.m1.1.10a"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.Ex42.m1.1.10.2.2">∑</mo><mrow id="A1.Ex42.m1.1.10.2.3"><mi mathsize="90%" id="A1.Ex42.m1.1.10.2.3.2">j</mi><mo mathsize="90%" id="A1.Ex42.m1.1.10.2.3.1">=</mo><mn mathsize="90%" id="A1.Ex42.m1.1.10.2.3.3">1</mn></mrow><mrow id="A1.Ex42.m1.1.10.3"><mi mathsize="90%" id="A1.Ex42.m1.1.10.3.2">K</mi><mo mathsize="90%" id="A1.Ex42.m1.1.10.3.1">−</mo><mn mathsize="90%" id="A1.Ex42.m1.1.10.3.3">1</mn></mrow></munderover></mstyle><mstyle displaystyle="true" id="A1.Ex42.m1.1.11"><mfrac id="A1.Ex42.m1.1.11a"><mrow id="A1.Ex42.m1.1.11.2"><mi mathsize="90%" id="A1.Ex42.m1.1.11.2.2">j</mi><mo mathsize="90%" id="A1.Ex42.m1.1.11.2.1">+</mo><mn mathsize="90%" id="A1.Ex42.m1.1.11.2.3">1</mn></mrow><mrow id="A1.Ex42.m1.1.11.3"><mi mathsize="90%" id="A1.Ex42.m1.1.11.3.2">K</mi><mo mathsize="90%" id="A1.Ex42.m1.1.11.3.1">+</mo><mn mathsize="90%" id="A1.Ex42.m1.1.11.3.3">1</mn></mrow></mfrac></mstyle><msub id="A1.Ex42.m1.1.12"><mi mathsize="90%" mathvariant="normal" id="A1.Ex42.m1.1.12.2">D</mi><mrow id="A1.Ex42.m1.1.12.3"><mi mathsize="90%" id="A1.Ex42.m1.1.12.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex42.m1.1.12.3.1">​</mo><mi mathsize="90%" id="A1.Ex42.m1.1.12.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex42.m1.1.13"><mo id="A1.Ex42.m1.1.13.1">(</mo><mi mathsize="90%" id="A1.Ex42.m1.1.13.2">q</mi><mrow id="A1.Ex42.m1.1.13.3"><mo maxsize="90%" minsize="90%" id="A1.Ex42.m1.1.13.3.1">(</mo><msubsup id="A1.Ex42.m1.1.13.3.2"><mi mathsize="90%" id="A1.Ex42.m1.1.13.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex42.m1.1.13.3.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex42.m1.1.13.3.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex42.m1.1.13.3.3">∣</mo><msubsup id="A1.Ex42.m1.1.13.3.4"><mi mathsize="90%" id="A1.Ex42.m1.1.13.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex42.m1.1.13.3.4.2.3">t</mi><mrow id="A1.Ex42.m1.1.13.3.4.3"><mi mathsize="90%" id="A1.Ex42.m1.1.13.3.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex42.m1.1.13.3.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex42.m1.1.13.3.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex42.m1.1.13.3.5">,</mo><msubsup id="A1.Ex42.m1.1.13.3.6"><mi mathsize="90%" id="A1.Ex42.m1.1.13.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex42.m1.1.13.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex42.m1.1.13.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex42.m1.1.13.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex42.m1.1.13.4">∥</mo><msub id="A1.Ex42.m1.1.13.5"><mi mathsize="90%" id="A1.Ex42.m1.1.13.5.2">p</mi><mi mathsize="90%" id="A1.Ex42.m1.1.13.5.3">𝜽</mi></msub><mrow id="A1.Ex42.m1.1.13.6"><mo maxsize="90%" minsize="90%" id="A1.Ex42.m1.1.13.6.1">(</mo><msubsup id="A1.Ex42.m1.1.13.6.2"><mi mathsize="90%" id="A1.Ex42.m1.1.13.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex42.m1.1.13.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex42.m1.1.13.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex42.m1.1.13.6.3">∣</mo><msubsup id="A1.Ex42.m1.1.13.6.4"><mi mathsize="90%" id="A1.Ex42.m1.1.13.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex42.m1.1.13.6.4.2.3">t</mi><mrow id="A1.Ex42.m1.1.13.6.4.3"><mi mathsize="90%" id="A1.Ex42.m1.1.13.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex42.m1.1.13.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex42.m1.1.13.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex42.m1.1.13.6.5">,</mo><msub id="A1.Ex42.m1.1.13.6.6"><mi mathsize="90%" id="A1.Ex42.m1.1.13.6.6.2">𝐳</mi><mrow id="A1.Ex42.m1.1.13.6.6.3"><mi mathsize="90%" id="A1.Ex42.m1.1.13.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex42.m1.1.13.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex42.m1.1.13.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex42.m1.1.13.6.7">)</mo></mrow><mo id="A1.Ex42.m1.1.13.7">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A1.Ex42.m1.1c">\displaystyle\quad\geq C_{4}(\mathbf{x}_{t}^{0})+\frac{1}{K+1}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\frac{j+1}{K+1}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.SSS0.Px3.p1.13" class="ltx_p"><span id="A1.SS2.SSS0.Px3.p1.13.1" class="ltx_text" style="font-size:90%;">Invoking </span><a href="#Thmclaim1" title="Claim 1 (Expanding the latents). ‣ A.2 Proof of Theorem A.1 ‣ Appendix A Theoretical Justification ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">1</span></a><span id="A1.SS2.SSS0.Px3.p1.13.2" class="ltx_text" style="font-size:90%;">,</span></p>
<table id="A6.EGx33" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex43"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex44"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.Ex44.m1.6c">\displaystyle\geq\sum_{t=1}^{T}\mathop{{\mathbb{E}}}_{&lt;t}\mathop{{\mathbb{E}}}_{k_{t}\overset{\mathrm{unif}}{\sim}\{0,1,\dots,K\}}\mathop{{\mathbb{E}}}_{\mathbf{x}_{t}^{k_{t}}\sim q(\mathbf{x}_{t}^{k_{t}}\mid\mathbf{x}_{t}^{0})}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{t_{k}}\mid\mathbf{z}_{t-1})\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex45"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A1.Ex45.m1.1.2.2.3.3">1</mn></mrow><mi mathsize="90%" id="A1.Ex45.m1.1.2.3">T</mi></munderover></mstyle><munder id="A1.Ex45.m1.1.3"><mo lspace="0.167em" mathsize="108%" movablelimits="false" rspace="0em" id="A1.Ex45.m1.1.3.2">𝔼</mo><mrow id="A1.Ex45.m1.1.3.3"><mi id="A1.Ex45.m1.1.3.3.2"></mi><mo mathsize="90%" id="A1.Ex45.m1.1.3.3.1">&lt;</mo><mi mathsize="90%" id="A1.Ex45.m1.1.3.3.3">t</mi></mrow></munder><mrow id="A1.Ex45.m1.1.4"><mo id="A1.Ex45.m1.1.4.1">[</mo><msub id="A1.Ex45.m1.1.4.2"><mi mathsize="90%" id="A1.Ex45.m1.1.4.2.2">C</mi><mn mathsize="90%" id="A1.Ex45.m1.1.4.2.3">4</mn></msub><mrow id="A1.Ex45.m1.1.4.3"><mo maxsize="90%" minsize="90%" id="A1.Ex45.m1.1.4.3.1">(</mo><msubsup id="A1.Ex45.m1.1.4.3.2"><mi mathsize="90%" id="A1.Ex45.m1.1.4.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.3.2.2.3">t</mi><mn mathsize="90%" id="A1.Ex45.m1.1.4.3.2.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex45.m1.1.4.3.3">)</mo></mrow><mo mathsize="90%" id="A1.Ex45.m1.1.4.4">+</mo><mstyle displaystyle="true" id="A1.Ex45.m1.1.4.5"><mfrac id="A1.Ex45.m1.1.4.5a"><mn mathsize="90%" id="A1.Ex45.m1.1.4.5.2">1</mn><mrow id="A1.Ex45.m1.1.4.5.3"><mi mathsize="90%" id="A1.Ex45.m1.1.4.5.3.2">K</mi><mo mathsize="90%" id="A1.Ex45.m1.1.4.5.3.1">+</mo><mn mathsize="90%" id="A1.Ex45.m1.1.4.5.3.3">1</mn></mrow></mfrac></mstyle><mi mathsize="90%" id="A1.Ex45.m1.1.4.6">ln</mi><msub id="A1.Ex45.m1.1.4.7"><mi mathsize="90%" id="A1.Ex45.m1.1.4.7.2">p</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.7.3">𝜽</mi></msub><mrow id="A1.Ex45.m1.1.4.8"><mo maxsize="90%" minsize="90%" id="A1.Ex45.m1.1.4.8.1">(</mo><msubsup id="A1.Ex45.m1.1.4.8.2"><mi mathsize="90%" id="A1.Ex45.m1.1.4.8.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.8.2.2.3">t</mi><mn mathsize="90%" id="A1.Ex45.m1.1.4.8.2.3">0</mn></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex45.m1.1.4.8.3">∣</mo><msubsup id="A1.Ex45.m1.1.4.8.4"><mi mathsize="90%" id="A1.Ex45.m1.1.4.8.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.8.4.2.3">t</mi><mn mathsize="90%" id="A1.Ex45.m1.1.4.8.4.3">1</mn></msubsup><mo mathsize="90%" id="A1.Ex45.m1.1.4.8.5">,</mo><msub id="A1.Ex45.m1.1.4.8.6"><mi mathsize="90%" id="A1.Ex45.m1.1.4.8.6.2">𝐳</mi><mrow id="A1.Ex45.m1.1.4.8.6.3"><mi mathsize="90%" id="A1.Ex45.m1.1.4.8.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex45.m1.1.4.8.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex45.m1.1.4.8.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex45.m1.1.4.8.7">)</mo></mrow><mo mathsize="90%" id="A1.Ex45.m1.1.4.9">+</mo><mstyle displaystyle="true" id="A1.Ex45.m1.1.4.10"><munderover id="A1.Ex45.m1.1.4.10a"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.Ex45.m1.1.4.10.2.2">∑</mo><mrow id="A1.Ex45.m1.1.4.10.2.3"><mi mathsize="90%" id="A1.Ex45.m1.1.4.10.2.3.2">j</mi><mo mathsize="90%" id="A1.Ex45.m1.1.4.10.2.3.1">=</mo><mn mathsize="90%" id="A1.Ex45.m1.1.4.10.2.3.3">1</mn></mrow><mrow id="A1.Ex45.m1.1.4.10.3"><mi mathsize="90%" id="A1.Ex45.m1.1.4.10.3.2">K</mi><mo mathsize="90%" id="A1.Ex45.m1.1.4.10.3.1">−</mo><mn mathsize="90%" id="A1.Ex45.m1.1.4.10.3.3">1</mn></mrow></munderover></mstyle><mstyle displaystyle="true" id="A1.Ex45.m1.1.4.11"><mfrac id="A1.Ex45.m1.1.4.11a"><mrow id="A1.Ex45.m1.1.4.11.2"><mi mathsize="90%" id="A1.Ex45.m1.1.4.11.2.2">j</mi><mo mathsize="90%" id="A1.Ex45.m1.1.4.11.2.1">+</mo><mn mathsize="90%" id="A1.Ex45.m1.1.4.11.2.3">1</mn></mrow><mrow id="A1.Ex45.m1.1.4.11.3"><mi mathsize="90%" id="A1.Ex45.m1.1.4.11.3.2">K</mi><mo mathsize="90%" id="A1.Ex45.m1.1.4.11.3.1">+</mo><mn mathsize="90%" id="A1.Ex45.m1.1.4.11.3.3">1</mn></mrow></mfrac></mstyle><msub id="A1.Ex45.m1.1.4.12"><mi mathsize="90%" mathvariant="normal" id="A1.Ex45.m1.1.4.12.2">D</mi><mrow id="A1.Ex45.m1.1.4.12.3"><mi mathsize="90%" id="A1.Ex45.m1.1.4.12.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex45.m1.1.4.12.3.1">​</mo><mi mathsize="90%" id="A1.Ex45.m1.1.4.12.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex45.m1.1.4.13"><mo id="A1.Ex45.m1.1.4.13.1">(</mo><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.2">q</mi><mrow id="A1.Ex45.m1.1.4.13.3"><mo maxsize="90%" minsize="90%" id="A1.Ex45.m1.1.4.13.3.1">(</mo><msubsup id="A1.Ex45.m1.1.4.13.3.2"><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.3.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.3.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex45.m1.1.4.13.3.3">∣</mo><msubsup id="A1.Ex45.m1.1.4.13.3.4"><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.3.4.2.3">t</mi><mrow id="A1.Ex45.m1.1.4.13.3.4.3"><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.3.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex45.m1.1.4.13.3.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex45.m1.1.4.13.3.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex45.m1.1.4.13.3.5">,</mo><msubsup id="A1.Ex45.m1.1.4.13.3.6"><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex45.m1.1.4.13.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex45.m1.1.4.13.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex45.m1.1.4.13.4">∥</mo><msub id="A1.Ex45.m1.1.4.13.5"><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.5.2">p</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.5.3">𝜽</mi></msub><mrow id="A1.Ex45.m1.1.4.13.6"><mo maxsize="90%" minsize="90%" id="A1.Ex45.m1.1.4.13.6.1">(</mo><msubsup id="A1.Ex45.m1.1.4.13.6.2"><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex45.m1.1.4.13.6.3">∣</mo><msubsup id="A1.Ex45.m1.1.4.13.6.4"><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.6.4.2.3">t</mi><mrow id="A1.Ex45.m1.1.4.13.6.4.3"><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex45.m1.1.4.13.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex45.m1.1.4.13.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex45.m1.1.4.13.6.5">,</mo><msub id="A1.Ex45.m1.1.4.13.6.6"><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.6.6.2">𝐳</mi><mrow id="A1.Ex45.m1.1.4.13.6.6.3"><mi mathsize="90%" id="A1.Ex45.m1.1.4.13.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex45.m1.1.4.13.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex45.m1.1.4.13.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex45.m1.1.4.13.6.7">)</mo></mrow><mo id="A1.Ex45.m1.1.4.13.7">)</mo></mrow><mo id="A1.Ex45.m1.1.4.14">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="A1.Ex45.m1.1c">\displaystyle=\sum_{t=1}^{T}\mathop{{\mathbb{E}}}_{&lt;t}\left[C_{4}(\mathbf{x}_{t}^{0})+\frac{1}{K+1}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\frac{j+1}{K+1}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.SSS0.Px3.p1.5" class="ltx_p"><span id="A1.SS2.SSS0.Px3.p1.5.1" class="ltx_text" style="font-size:90%;">We conclude by observing that </span><math id="A1.SS2.SSS0.Px3.p1.4.m1.1" class="ltx_Math" alttext="\sum_{t=1}^{T}\mathop{{\mathbb{E}}}_{&lt;t}\left[C_{4}(\mathbf{x}_{t}^{0})\right]" display="inline"><semantics id="A1.SS2.SSS0.Px3.p1.4.m1.1a"><mrow id="A1.SS2.SSS0.Px3.p1.4.m1.1.1" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.cmml"><msubsup id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.cmml"><mo maxsize="90%" minsize="90%" stretchy="true" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.2.2" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.2.2.cmml">∑</mo><mrow id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.2.3" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.2.3.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.2.3.2" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.2.3.2.cmml">t</mi><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.2.3.1" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.2.3.1.cmml">=</mo><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.2.3.3" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.2.3.3.cmml">1</mn></mrow><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.3" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.2.3.cmml">T</mi></msubsup><mrow id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.cmml"><msub id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.cmml"><mo lspace="0.167em" mathsize="108%" rspace="0em" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.2" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.2.cmml">𝔼</mo><mrow id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.3" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.3.cmml"><mi id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.3.2" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.3.2.cmml"></mi><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.3.1" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.3.1.cmml">&lt;</mo><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.3.3" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.2.3.3.cmml">t</mi></mrow></msub><mrow 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xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1"><csymbol cd="latexml" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.2.1.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1"><times id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.2.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.2"></times><apply id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.3.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.3.1.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.3.2.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.3.2">𝐶</ci><cn type="integer" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.3.3.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.3.3">4</cn></apply><apply id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1">superscript</csymbol><apply id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1.1.2.2">𝐱</ci><ci id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1.1.2.3">𝑡</ci></apply><cn type="integer" id="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A1.SS2.SSS0.Px3.p1.4.m1.1.1.1.1.1.1.1.1.1.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px3.p1.4.m1.1c">\sum_{t=1}^{T}\mathop{{\mathbb{E}}}_{&lt;t}\left[C_{4}(\mathbf{x}_{t}^{0})\right]</annotation></semantics></math><span id="A1.SS2.SSS0.Px3.p1.5.2" class="ltx_text" style="font-size:90%;"> is a constant </span><math id="A1.SS2.SSS0.Px3.p1.5.m2.1" class="ltx_Math" alttext="C(\mathbf{x}_{1:T}^{0})" display="inline"><semantics id="A1.SS2.SSS0.Px3.p1.5.m2.1a"><mrow id="A1.SS2.SSS0.Px3.p1.5.m2.1.1" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.3" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.3.cmml">C</mi><mo lspace="0em" rspace="0em" id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.2" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.2.cmml">​</mo><mrow id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.2" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.cmml">(</mo><msubsup id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.2" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.2.cmml">𝐱</mi><mrow 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xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.2"></times><ci id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.3.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.3">𝐶</ci><apply id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1">superscript</csymbol><apply id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.1.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1">subscript</csymbol><ci id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.2.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.2">𝐱</ci><apply id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.3.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.3"><ci id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.3.1.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.3.1">:</ci><cn type="integer" id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.3.2.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.3.2">1</cn><ci id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.3.3.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.2.3.3">𝑇</ci></apply></apply><cn type="integer" id="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.3.cmml" xref="A1.SS2.SSS0.Px3.p1.5.m2.1.1.1.1.1.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px3.p1.5.m2.1c">C(\mathbf{x}_{1:T}^{0})</annotation></semantics></math><span id="A1.SS2.SSS0.Px3.p1.5.3" class="ltx_text" style="font-size:90%;">, and that</span></p>
<table id="A6.EGx34" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex46"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex46.m1.4" class="ltx_Math" alttext="\displaystyle\mathop{{\mathbb{E}}}_{&lt;t}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})\right]=\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})\right]" display="inline"><semantics id="A1.Ex46.m1.4a"><mrow id="A1.Ex46.m1.4.4" xref="A1.Ex46.m1.4.4.cmml"><mrow id="A1.Ex46.m1.3.3.1" xref="A1.Ex46.m1.3.3.1.cmml"><munder id="A1.Ex46.m1.3.3.1.2" xref="A1.Ex46.m1.3.3.1.2.cmml"><mo mathsize="108%" movablelimits="false" id="A1.Ex46.m1.3.3.1.2.2" xref="A1.Ex46.m1.3.3.1.2.2.cmml">𝔼</mo><mrow id="A1.Ex46.m1.3.3.1.2.3" xref="A1.Ex46.m1.3.3.1.2.3.cmml"><mi id="A1.Ex46.m1.3.3.1.2.3.2" xref="A1.Ex46.m1.3.3.1.2.3.2.cmml"></mi><mo mathsize="90%" id="A1.Ex46.m1.3.3.1.2.3.1" xref="A1.Ex46.m1.3.3.1.2.3.1.cmml">&lt;</mo><mi mathsize="90%" 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id="A1.Ex46.m1.4c">\displaystyle\mathop{{\mathbb{E}}}_{&lt;t}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})\right]=\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex47"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex47.m1.1" class="ltx_math_unparsed" alttext="\displaystyle\mathop{{\mathbb{E}}}_{&lt;t}\left[\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)\right]" display="inline"><semantics id="A1.Ex47.m1.1a"><mrow id="A1.Ex47.m1.1b"><munder id="A1.Ex47.m1.1.1"><mo mathsize="108%" movablelimits="false" id="A1.Ex47.m1.1.1.2">𝔼</mo><mrow id="A1.Ex47.m1.1.1.3"><mi id="A1.Ex47.m1.1.1.3.2"></mi><mo mathsize="90%" id="A1.Ex47.m1.1.1.3.1">&lt;</mo><mi mathsize="90%" id="A1.Ex47.m1.1.1.3.3">t</mi></mrow></munder><mrow id="A1.Ex47.m1.1.2"><mo lspace="0em" id="A1.Ex47.m1.1.2.1">[</mo><msub id="A1.Ex47.m1.1.2.2"><mi mathsize="90%" mathvariant="normal" id="A1.Ex47.m1.1.2.2.2">D</mi><mrow id="A1.Ex47.m1.1.2.2.3"><mi mathsize="90%" id="A1.Ex47.m1.1.2.2.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex47.m1.1.2.2.3.1">​</mo><mi mathsize="90%" id="A1.Ex47.m1.1.2.2.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex47.m1.1.2.3"><mo id="A1.Ex47.m1.1.2.3.1">(</mo><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.2">q</mi><mrow id="A1.Ex47.m1.1.2.3.3"><mo maxsize="90%" minsize="90%" id="A1.Ex47.m1.1.2.3.3.1">(</mo><msubsup id="A1.Ex47.m1.1.2.3.3.2"><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.3.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.3.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex47.m1.1.2.3.3.3">∣</mo><msubsup id="A1.Ex47.m1.1.2.3.3.4"><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.3.4.2.3">t</mi><mrow id="A1.Ex47.m1.1.2.3.3.4.3"><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.3.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex47.m1.1.2.3.3.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex47.m1.1.2.3.3.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex47.m1.1.2.3.3.5">,</mo><msubsup id="A1.Ex47.m1.1.2.3.3.6"><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex47.m1.1.2.3.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex47.m1.1.2.3.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex47.m1.1.2.3.4">∥</mo><msub id="A1.Ex47.m1.1.2.3.5"><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.5.2">p</mi><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.5.3">𝜽</mi></msub><mrow id="A1.Ex47.m1.1.2.3.6"><mo maxsize="90%" minsize="90%" id="A1.Ex47.m1.1.2.3.6.1">(</mo><msubsup id="A1.Ex47.m1.1.2.3.6.2"><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex47.m1.1.2.3.6.3">∣</mo><msubsup id="A1.Ex47.m1.1.2.3.6.4"><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.6.4.2.3">t</mi><mrow id="A1.Ex47.m1.1.2.3.6.4.3"><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex47.m1.1.2.3.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex47.m1.1.2.3.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex47.m1.1.2.3.6.5">,</mo><msub id="A1.Ex47.m1.1.2.3.6.6"><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.6.6.2">𝐳</mi><mrow id="A1.Ex47.m1.1.2.3.6.6.3"><mi mathsize="90%" id="A1.Ex47.m1.1.2.3.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex47.m1.1.2.3.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex47.m1.1.2.3.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex47.m1.1.2.3.6.7">)</mo></mrow><mo id="A1.Ex47.m1.1.2.3.7">)</mo></mrow><mo id="A1.Ex47.m1.1.2.4">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="A1.Ex47.m1.1c">\displaystyle\mathop{{\mathbb{E}}}_{&lt;t}\left[\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex48"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex48.m1.2" class="ltx_math_unparsed" alttext="\displaystyle=\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)\right]," display="inline"><semantics id="A1.Ex48.m1.2a"><mrow id="A1.Ex48.m1.2b"><mo mathsize="90%" rspace="0.1389em" id="A1.Ex48.m1.2.3">=</mo><munder id="A1.Ex48.m1.2.4"><mo lspace="0.1389em" mathsize="108%" movablelimits="false" rspace="0.0835em" id="A1.Ex48.m1.2.4.2">𝔼</mo><mi mathsize="90%" id="A1.Ex48.m1.2.4.3">forward</mi></munder><munder id="A1.Ex48.m1.2.5"><mo lspace="0.0835em" mathsize="108%" movablelimits="false" rspace="0em" id="A1.Ex48.m1.2.5.2">𝔼</mo><mrow id="A1.Ex48.m1.2.2.2.2"><mi mathsize="90%" id="A1.Ex48.m1.1.1.1.1">p</mi><mo mathsize="90%" id="A1.Ex48.m1.2.2.2.2.2">,</mo><msub id="A1.Ex48.m1.2.2.2.2.1"><mi mathsize="90%" id="A1.Ex48.m1.2.2.2.2.1.2">𝐳</mi><mrow id="A1.Ex48.m1.2.2.2.2.1.3"><mn mathsize="90%" id="A1.Ex48.m1.2.2.2.2.1.3.2">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.Ex48.m1.2.2.2.2.1.3.1">:</mo><mi mathsize="90%" id="A1.Ex48.m1.2.2.2.2.1.3.3">T</mi></mrow></msub></mrow></munder><mrow id="A1.Ex48.m1.2.6"><mo id="A1.Ex48.m1.2.6.1">[</mo><msub id="A1.Ex48.m1.2.6.2"><mi mathsize="90%" mathvariant="normal" id="A1.Ex48.m1.2.6.2.2">D</mi><mrow id="A1.Ex48.m1.2.6.2.3"><mi mathsize="90%" id="A1.Ex48.m1.2.6.2.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex48.m1.2.6.2.3.1">​</mo><mi mathsize="90%" id="A1.Ex48.m1.2.6.2.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex48.m1.2.6.3"><mo id="A1.Ex48.m1.2.6.3.1">(</mo><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.2">q</mi><mrow id="A1.Ex48.m1.2.6.3.3"><mo maxsize="90%" minsize="90%" id="A1.Ex48.m1.2.6.3.3.1">(</mo><msubsup id="A1.Ex48.m1.2.6.3.3.2"><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.3.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.3.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex48.m1.2.6.3.3.3">∣</mo><msubsup id="A1.Ex48.m1.2.6.3.3.4"><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.3.4.2.3">t</mi><mrow id="A1.Ex48.m1.2.6.3.3.4.3"><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.3.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex48.m1.2.6.3.3.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex48.m1.2.6.3.3.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex48.m1.2.6.3.3.5">,</mo><msubsup id="A1.Ex48.m1.2.6.3.3.6"><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.3.6.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.3.6.2.3">t</mi><mn mathsize="90%" id="A1.Ex48.m1.2.6.3.3.6.3">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A1.Ex48.m1.2.6.3.3.7">)</mo></mrow><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex48.m1.2.6.3.4">∥</mo><msub id="A1.Ex48.m1.2.6.3.5"><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.5.2">p</mi><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.5.3">𝜽</mi></msub><mrow id="A1.Ex48.m1.2.6.3.6"><mo maxsize="90%" minsize="90%" id="A1.Ex48.m1.2.6.3.6.1">(</mo><msubsup id="A1.Ex48.m1.2.6.3.6.2"><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.6.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.6.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.6.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex48.m1.2.6.3.6.3">∣</mo><msubsup id="A1.Ex48.m1.2.6.3.6.4"><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.6.4.2.3">t</mi><mrow id="A1.Ex48.m1.2.6.3.6.4.3"><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex48.m1.2.6.3.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex48.m1.2.6.3.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex48.m1.2.6.3.6.5">,</mo><msub id="A1.Ex48.m1.2.6.3.6.6"><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.6.6.2">𝐳</mi><mrow id="A1.Ex48.m1.2.6.3.6.6.3"><mi mathsize="90%" id="A1.Ex48.m1.2.6.3.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex48.m1.2.6.3.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex48.m1.2.6.3.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex48.m1.2.6.3.6.7">)</mo></mrow><mo id="A1.Ex48.m1.2.6.3.7">)</mo></mrow><mo id="A1.Ex48.m1.2.6.4">]</mo></mrow><mo mathsize="90%" id="A1.Ex48.m1.2.7">,</mo></mrow><annotation encoding="application/x-tex" id="A1.Ex48.m1.2c">\displaystyle=\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)\right],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.SSS0.Px3.p1.7" class="ltx_p"><span id="A1.SS2.SSS0.Px3.p1.7.1" class="ltx_text" style="font-size:90%;">since both terms only depend on </span><math id="A1.SS2.SSS0.Px3.p1.6.m1.2" class="ltx_Math" alttext="k_{1:t-1},(\mathbf{x}_{s}^{k_{s}})_{1\leq s\leq t-1}" display="inline"><semantics id="A1.SS2.SSS0.Px3.p1.6.m1.2a"><mrow id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.3.cmml"><msub id="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1" xref="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.2" xref="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.2.cmml">k</mi><mrow id="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3" xref="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.cmml"><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.2" xref="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.1" xref="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.1.cmml">:</mo><mrow id="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.3" xref="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.3.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.3.2" xref="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.3.2.cmml">t</mi><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.3.1" xref="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.3.1.cmml">−</mo><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.3.3" xref="A1.SS2.SSS0.Px3.p1.6.m1.1.1.1.1.3.3.3.cmml">1</mn></mrow></mrow></msub><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.3" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.3.cmml">,</mo><msub id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.cmml"><mrow id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.2" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.cmml">(</mo><msubsup id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.2.2" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.2.3" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.2.3.cmml">s</mi><msub id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.3" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.3.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.3.2" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.3.3" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.3.3.cmml">s</mi></msub></msubsup><mo maxsize="90%" minsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.3" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.3" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.3.cmml"><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.3.2" xref="A1.SS2.SSS0.Px3.p1.6.m1.2.2.2.2.3.2.cmml">1</mn><mo mathsize="90%" 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id="A1.SS2.SSS0.Px3.p1.6.m1.2c">k_{1:t-1},(\mathbf{x}_{s}^{k_{s}})_{1\leq s\leq t-1}</annotation></semantics></math><span id="A1.SS2.SSS0.Px3.p1.7.2" class="ltx_text" style="font-size:90%;"> and </span><math id="A1.SS2.SSS0.Px3.p1.7.m2.1" class="ltx_Math" alttext="\mathbf{z}_{1:t-1}" display="inline"><semantics id="A1.SS2.SSS0.Px3.p1.7.m2.1a"><msub id="A1.SS2.SSS0.Px3.p1.7.m2.1.1" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.2" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.2.cmml">𝐳</mi><mrow id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.cmml"><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.2" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.1" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.1.cmml">:</mo><mrow id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.2" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.2.cmml">t</mi><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.1" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.1.cmml">−</mo><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.3" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="A1.SS2.SSS0.Px3.p1.7.m2.1b"><apply id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1">subscript</csymbol><ci id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.2.cmml" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.2">𝐳</ci><apply id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.cmml" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3"><ci id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.1.cmml" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.1">:</ci><cn type="integer" id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.2.cmml" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.2">1</cn><apply id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.cmml" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3"><minus id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.1.cmml" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.1"></minus><ci id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.2.cmml" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.2">𝑡</ci><cn type="integer" id="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.3.cmml" xref="A1.SS2.SSS0.Px3.p1.7.m2.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px3.p1.7.m2.1c">\mathbf{z}_{1:t-1}</annotation></semantics></math><span id="A1.SS2.SSS0.Px3.p1.7.3" class="ltx_text" style="font-size:90%;">. We conclude then that</span></p>
<table id="A6.EGx35" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A1.Ex49"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A1.Ex49.m1.1.1.1.1.1.1.1.1.1.3.6">𝑇</ci></apply></apply></apply></apply></apply></apply><apply id="A1.Ex49.m1.2.2.2.cmml" xref="A1.Ex49.m1.2.2.2"><times id="A1.Ex49.m1.2.2.2.2.cmml" xref="A1.Ex49.m1.2.2.2.2"></times><ci id="A1.Ex49.m1.2.2.2.3.cmml" xref="A1.Ex49.m1.2.2.2.3">𝐶</ci><apply id="A1.Ex49.m1.2.2.2.1.1.1.cmml" xref="A1.Ex49.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="A1.Ex49.m1.2.2.2.1.1.1.1.cmml" xref="A1.Ex49.m1.2.2.2.1.1">superscript</csymbol><apply id="A1.Ex49.m1.2.2.2.1.1.1.2.cmml" xref="A1.Ex49.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="A1.Ex49.m1.2.2.2.1.1.1.2.1.cmml" xref="A1.Ex49.m1.2.2.2.1.1">subscript</csymbol><ci id="A1.Ex49.m1.2.2.2.1.1.1.2.2.cmml" xref="A1.Ex49.m1.2.2.2.1.1.1.2.2">𝐱</ci><apply id="A1.Ex49.m1.2.2.2.1.1.1.2.3.cmml" xref="A1.Ex49.m1.2.2.2.1.1.1.2.3"><ci id="A1.Ex49.m1.2.2.2.1.1.1.2.3.1.cmml" xref="A1.Ex49.m1.2.2.2.1.1.1.2.3.1">:</ci><cn type="integer" id="A1.Ex49.m1.2.2.2.1.1.1.2.3.2.cmml" xref="A1.Ex49.m1.2.2.2.1.1.1.2.3.2">1</cn><ci id="A1.Ex49.m1.2.2.2.1.1.1.2.3.3.cmml" xref="A1.Ex49.m1.2.2.2.1.1.1.2.3.3">𝑇</ci></apply></apply><cn type="integer" id="A1.Ex49.m1.2.2.2.1.1.1.3.cmml" xref="A1.Ex49.m1.2.2.2.1.1.1.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.Ex49.m1.2c">\displaystyle\mathop{{\mathbb{E}}}_{\mathrm{forward}}[\ln p_{\bm{\theta}}((\mathbf{x}_{t}^{k_{t}})_{1\leq t\leq T})]\geq C(\mathbf{x}_{1:T}^{0})</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A1.Ex50"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A1.Ex50.m1.2" class="ltx_math_unparsed" alttext="\displaystyle+\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\left(\frac{1}{K+1}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\frac{j+1}{K+1}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)\right)\right]," display="inline"><semantics id="A1.Ex50.m1.2a"><mrow id="A1.Ex50.m1.2b"><mo mathsize="90%" id="A1.Ex50.m1.2.3">+</mo><munder id="A1.Ex50.m1.2.4"><mo lspace="0em" mathsize="108%" movablelimits="false" rspace="0.0835em" id="A1.Ex50.m1.2.4.2">𝔼</mo><mi mathsize="90%" id="A1.Ex50.m1.2.4.3">forward</mi></munder><munder id="A1.Ex50.m1.2.5"><mo lspace="0.0835em" mathsize="108%" movablelimits="false" rspace="0em" id="A1.Ex50.m1.2.5.2">𝔼</mo><mrow 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id="A1.Ex50.m1.2.6.3"><mo id="A1.Ex50.m1.2.6.3.1">(</mo><mstyle displaystyle="true" id="A1.Ex50.m1.2.6.3.2"><mfrac id="A1.Ex50.m1.2.6.3.2a"><mn mathsize="90%" id="A1.Ex50.m1.2.6.3.2.2">1</mn><mrow id="A1.Ex50.m1.2.6.3.2.3"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.2.3.2">K</mi><mo mathsize="90%" id="A1.Ex50.m1.2.6.3.2.3.1">+</mo><mn mathsize="90%" id="A1.Ex50.m1.2.6.3.2.3.3">1</mn></mrow></mfrac></mstyle><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.3">ln</mi><msub id="A1.Ex50.m1.2.6.3.4"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.4.2">p</mi><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.4.3">𝜽</mi></msub><mrow id="A1.Ex50.m1.2.6.3.5"><mo maxsize="90%" minsize="90%" id="A1.Ex50.m1.2.6.3.5.1">(</mo><msubsup id="A1.Ex50.m1.2.6.3.5.2"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.5.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.5.2.2.3">t</mi><mn mathsize="90%" id="A1.Ex50.m1.2.6.3.5.2.3">0</mn></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex50.m1.2.6.3.5.3">∣</mo><msubsup id="A1.Ex50.m1.2.6.3.5.4"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.5.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.5.4.2.3">t</mi><mn mathsize="90%" id="A1.Ex50.m1.2.6.3.5.4.3">1</mn></msubsup><mo mathsize="90%" id="A1.Ex50.m1.2.6.3.5.5">,</mo><msub id="A1.Ex50.m1.2.6.3.5.6"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.5.6.2">𝐳</mi><mrow id="A1.Ex50.m1.2.6.3.5.6.3"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.5.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex50.m1.2.6.3.5.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex50.m1.2.6.3.5.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex50.m1.2.6.3.5.7">)</mo></mrow><mo mathsize="90%" id="A1.Ex50.m1.2.6.3.6">+</mo><mstyle displaystyle="true" id="A1.Ex50.m1.2.6.3.7"><munderover id="A1.Ex50.m1.2.6.3.7a"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A1.Ex50.m1.2.6.3.7.2.2">∑</mo><mrow id="A1.Ex50.m1.2.6.3.7.2.3"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.7.2.3.2">j</mi><mo mathsize="90%" 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id="A1.Ex50.m1.2.6.3.9.3.2">𝕂</mi><mo lspace="0em" rspace="0em" id="A1.Ex50.m1.2.6.3.9.3.1">​</mo><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.9.3.3">𝕃</mi></mrow></msub><mrow id="A1.Ex50.m1.2.6.3.10"><mo id="A1.Ex50.m1.2.6.3.10.1">(</mo><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.2">q</mi><mrow id="A1.Ex50.m1.2.6.3.10.3"><mo maxsize="90%" minsize="90%" id="A1.Ex50.m1.2.6.3.10.3.1">(</mo><msubsup id="A1.Ex50.m1.2.6.3.10.3.2"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.3.2.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.3.2.2.3">t</mi><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.3.2.3">j</mi></msubsup><mo lspace="0em" mathsize="90%" rspace="0.167em" id="A1.Ex50.m1.2.6.3.10.3.3">∣</mo><msubsup id="A1.Ex50.m1.2.6.3.10.3.4"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.3.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.3.4.2.3">t</mi><mrow id="A1.Ex50.m1.2.6.3.10.3.4.3"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.3.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex50.m1.2.6.3.10.3.4.3.1">+</mo><mn 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rspace="0.167em" id="A1.Ex50.m1.2.6.3.10.6.3">∣</mo><msubsup id="A1.Ex50.m1.2.6.3.10.6.4"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.6.4.2.2">𝐱</mi><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.6.4.2.3">t</mi><mrow id="A1.Ex50.m1.2.6.3.10.6.4.3"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.6.4.3.2">j</mi><mo mathsize="90%" id="A1.Ex50.m1.2.6.3.10.6.4.3.1">+</mo><mn mathsize="90%" id="A1.Ex50.m1.2.6.3.10.6.4.3.3">1</mn></mrow></msubsup><mo mathsize="90%" id="A1.Ex50.m1.2.6.3.10.6.5">,</mo><msub id="A1.Ex50.m1.2.6.3.10.6.6"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.6.6.2">𝐳</mi><mrow id="A1.Ex50.m1.2.6.3.10.6.6.3"><mi mathsize="90%" id="A1.Ex50.m1.2.6.3.10.6.6.3.2">t</mi><mo mathsize="90%" id="A1.Ex50.m1.2.6.3.10.6.6.3.1">−</mo><mn mathsize="90%" id="A1.Ex50.m1.2.6.3.10.6.6.3.3">1</mn></mrow></msub><mo maxsize="90%" minsize="90%" id="A1.Ex50.m1.2.6.3.10.6.7">)</mo></mrow><mo id="A1.Ex50.m1.2.6.3.10.7">)</mo></mrow><mo id="A1.Ex50.m1.2.6.3.11">)</mo></mrow><mo id="A1.Ex50.m1.2.6.4">]</mo></mrow><mo mathsize="90%" id="A1.Ex50.m1.2.7">,</mo></mrow><annotation encoding="application/x-tex" id="A1.Ex50.m1.2c">\displaystyle+\mathop{{\mathbb{E}}}_{\mathrm{forward}}\mathop{{\mathbb{E}}}_{p,\mathbf{z}_{1:T}}\left[\sum_{t=1}^{T}\left(\frac{1}{K+1}\ln p_{\bm{\theta}}(\mathbf{x}_{t}^{0}\mid\mathbf{x}_{t}^{1},\mathbf{z}_{t-1})+\sum_{j=1}^{K-1}\frac{j+1}{K+1}\mathrm{D}_{\mathbb{KL}}\left(q(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{x}_{t}^{0})\parallel p_{\bm{\theta}}(\mathbf{x}_{t}^{j}\mid\mathbf{x}_{t}^{j+1},\mathbf{z}_{t-1})\right)\right)\right],</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A1.SS2.SSS0.Px3.p1.10" class="ltx_p"><span id="A1.SS2.SSS0.Px3.p1.10.1" class="ltx_text" style="font-size:90%;">as needed. 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id="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.cmml" xref="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.1.cmml" xref="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2">subscript</csymbol><ci id="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.2.cmml" xref="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.2">𝐳</ci><apply id="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.3.cmml" xref="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.3"><minus id="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.3.1.cmml" xref="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.3.1"></minus><ci id="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.3.2.cmml" xref="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.3.2">𝑡</ci><cn type="integer" id="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.3.3.cmml" xref="A1.SS2.SSS0.Px3.p1.9.m2.2.2.2.1.1.1.2.2.2.3.3">1</cn></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px3.p1.9.m2.2c">q(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}})\equiv p_{\bm{\theta}}(\mathbf{x}_{t}^{k_{t}+1:T}\mid\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1})</annotation></semantics></math><span id="A1.SS2.SSS0.Px3.p1.10.5" class="ltx_text" style="font-size:90%;">, and we reindex </span><math id="A1.SS2.SSS0.Px3.p1.10.m3.1" class="ltx_Math" alttext="j\leftarrow j+1" display="inline"><semantics id="A1.SS2.SSS0.Px3.p1.10.m3.1a"><mrow id="A1.SS2.SSS0.Px3.p1.10.m3.1.1" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.2" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.2.cmml">j</mi><mo mathsize="90%" stretchy="false" id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.1" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.1.cmml">←</mo><mrow id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.cmml"><mi mathsize="90%" id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.2" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.2.cmml">j</mi><mo mathsize="90%" id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.1" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.1.cmml">+</mo><mn mathsize="90%" id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.3" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.SSS0.Px3.p1.10.m3.1b"><apply id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1"><ci id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.1.cmml" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.1">←</ci><ci id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.2.cmml" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.2">𝑗</ci><apply id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.cmml" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3"><plus id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.1.cmml" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.1"></plus><ci id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.2.cmml" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.2">𝑗</ci><cn type="integer" id="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.3.cmml" xref="A1.SS2.SSS0.Px3.p1.10.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px3.p1.10.m3.1c">j\leftarrow j+1</annotation></semantics></math><span id="A1.SS2.SSS0.Px3.p1.10.6" class="ltx_text" style="font-size:90%;"> to ensure consistency with indexing in standard expositions of the diffusion ELBO.</span></p>
</div>
<div id="A1.SS2.SSS0.Px3.p2" class="ltx_para">
<p id="A1.SS2.SSS0.Px3.p2.1" class="ltx_p"><span id="A1.SS2.SSS0.Px3.p2.1.1" class="ltx_text" style="font-size:90%;">∎</span></p>
</div>
</section>
</section>
</section>
<section id="A2" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix" style="font-size:90%;">
<span class="ltx_tag ltx_tag_appendix">Appendix B </span>Additional Intuitions and Explainations</h2>

<section id="A2.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">B.1 </span>Guidance as planning</h3>

<div id="A2.SS1.p1" class="ltx_para">
<p id="A2.SS1.p1.6" class="ltx_p"><span id="A2.SS1.p1.6.1" class="ltx_text" style="font-size:90%;">As stated in Section </span><a href="#S2" title="2 Related Work and Preliminaries ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">2</span></a><span id="A2.SS1.p1.6.2" class="ltx_text" style="font-size:90%;">, one can use the gradient of the logarithmic of a classifer </span><math id="A2.SS1.p1.1.m1.1" class="ltx_Math" alttext="\log c(y|\mathbf{x}_{t}^{k})" display="inline"><semantics id="A2.SS1.p1.1.m1.1a"><mrow id="A2.SS1.p1.1.m1.1.1" xref="A2.SS1.p1.1.m1.1.1.cmml"><mrow id="A2.SS1.p1.1.m1.1.1.3" xref="A2.SS1.p1.1.m1.1.1.3.cmml"><mi mathsize="90%" id="A2.SS1.p1.1.m1.1.1.3.1" xref="A2.SS1.p1.1.m1.1.1.3.1.cmml">log</mi><mo lspace="0.167em" id="A2.SS1.p1.1.m1.1.1.3a" xref="A2.SS1.p1.1.m1.1.1.3.cmml">⁡</mo><mi mathsize="90%" id="A2.SS1.p1.1.m1.1.1.3.2" xref="A2.SS1.p1.1.m1.1.1.3.2.cmml">c</mi></mrow><mo lspace="0em" rspace="0em" id="A2.SS1.p1.1.m1.1.1.2" xref="A2.SS1.p1.1.m1.1.1.2.cmml">​</mo><mrow id="A2.SS1.p1.1.m1.1.1.1.1" xref="A2.SS1.p1.1.m1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS1.p1.1.m1.1.1.1.1.2" xref="A2.SS1.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="A2.SS1.p1.1.m1.1.1.1.1.1" xref="A2.SS1.p1.1.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.SS1.p1.1.m1.1.1.1.1.1.2" xref="A2.SS1.p1.1.m1.1.1.1.1.1.2.cmml">y</mi><mo fence="false" mathsize="90%" id="A2.SS1.p1.1.m1.1.1.1.1.1.1" xref="A2.SS1.p1.1.m1.1.1.1.1.1.1.cmml">|</mo><msubsup id="A2.SS1.p1.1.m1.1.1.1.1.1.3" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A2.SS1.p1.1.m1.1.1.1.1.1.3.2.2" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A2.SS1.p1.1.m1.1.1.1.1.1.3.2.3" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3.2.3.cmml">t</mi><mi mathsize="90%" id="A2.SS1.p1.1.m1.1.1.1.1.1.3.3" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3.3.cmml">k</mi></msubsup></mrow><mo maxsize="90%" minsize="90%" id="A2.SS1.p1.1.m1.1.1.1.1.3" xref="A2.SS1.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.1.m1.1b"><apply id="A2.SS1.p1.1.m1.1.1.cmml" xref="A2.SS1.p1.1.m1.1.1"><times id="A2.SS1.p1.1.m1.1.1.2.cmml" xref="A2.SS1.p1.1.m1.1.1.2"></times><apply id="A2.SS1.p1.1.m1.1.1.3.cmml" xref="A2.SS1.p1.1.m1.1.1.3"><log id="A2.SS1.p1.1.m1.1.1.3.1.cmml" xref="A2.SS1.p1.1.m1.1.1.3.1"></log><ci id="A2.SS1.p1.1.m1.1.1.3.2.cmml" xref="A2.SS1.p1.1.m1.1.1.3.2">𝑐</ci></apply><apply id="A2.SS1.p1.1.m1.1.1.1.1.1.cmml" xref="A2.SS1.p1.1.m1.1.1.1.1"><csymbol cd="latexml" id="A2.SS1.p1.1.m1.1.1.1.1.1.1.cmml" xref="A2.SS1.p1.1.m1.1.1.1.1.1.1">conditional</csymbol><ci id="A2.SS1.p1.1.m1.1.1.1.1.1.2.cmml" xref="A2.SS1.p1.1.m1.1.1.1.1.1.2">𝑦</ci><apply id="A2.SS1.p1.1.m1.1.1.1.1.1.3.cmml" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.SS1.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3">superscript</csymbol><apply id="A2.SS1.p1.1.m1.1.1.1.1.1.3.2.cmml" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.SS1.p1.1.m1.1.1.1.1.1.3.2.1.cmml" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3">subscript</csymbol><ci id="A2.SS1.p1.1.m1.1.1.1.1.1.3.2.2.cmml" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3.2.2">𝐱</ci><ci id="A2.SS1.p1.1.m1.1.1.1.1.1.3.2.3.cmml" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3.2.3">𝑡</ci></apply><ci id="A2.SS1.p1.1.m1.1.1.1.1.1.3.3.cmml" xref="A2.SS1.p1.1.m1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.1.m1.1c">\log c(y|\mathbf{x}_{t}^{k})</annotation></semantics></math><span id="A2.SS1.p1.6.3" class="ltx_text" style="font-size:90%;"> to guide the sampling process of diffusion model towards samples with a desired attribute </span><math id="A2.SS1.p1.2.m2.1" class="ltx_Math" alttext="y" display="inline"><semantics id="A2.SS1.p1.2.m2.1a"><mi mathsize="90%" id="A2.SS1.p1.2.m2.1.1" xref="A2.SS1.p1.2.m2.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.2.m2.1b"><ci id="A2.SS1.p1.2.m2.1.1.cmml" xref="A2.SS1.p1.2.m2.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.2.m2.1c">y</annotation></semantics></math><span id="A2.SS1.p1.6.4" class="ltx_text" style="font-size:90%;">. For example, </span><math id="A2.SS1.p1.3.m3.1" class="ltx_Math" alttext="y" display="inline"><semantics id="A2.SS1.p1.3.m3.1a"><mi mathsize="90%" id="A2.SS1.p1.3.m3.1.1" xref="A2.SS1.p1.3.m3.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.3.m3.1b"><ci id="A2.SS1.p1.3.m3.1.1.cmml" xref="A2.SS1.p1.3.m3.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.3.m3.1c">y</annotation></semantics></math><span id="A2.SS1.p1.6.5" class="ltx_text" style="font-size:90%;"> can refer to the indicator of a success event. However, we can consider the logarithmic of a more general </span><em id="A2.SS1.p1.6.6" class="ltx_emph ltx_font_italic" style="font-size:90%;">energy function</em><span id="A2.SS1.p1.6.7" class="ltx_text" style="font-size:90%;"> </span><math id="A2.SS1.p1.4.m4.1" class="ltx_Math" alttext="c(\mathbf{x}_{t}^{k})" display="inline"><semantics id="A2.SS1.p1.4.m4.1a"><mrow id="A2.SS1.p1.4.m4.1.1" xref="A2.SS1.p1.4.m4.1.1.cmml"><mi mathsize="90%" id="A2.SS1.p1.4.m4.1.1.3" xref="A2.SS1.p1.4.m4.1.1.3.cmml">c</mi><mo lspace="0em" rspace="0em" id="A2.SS1.p1.4.m4.1.1.2" xref="A2.SS1.p1.4.m4.1.1.2.cmml">​</mo><mrow id="A2.SS1.p1.4.m4.1.1.1.1" xref="A2.SS1.p1.4.m4.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS1.p1.4.m4.1.1.1.1.2" xref="A2.SS1.p1.4.m4.1.1.1.1.1.cmml">(</mo><msubsup id="A2.SS1.p1.4.m4.1.1.1.1.1" xref="A2.SS1.p1.4.m4.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.SS1.p1.4.m4.1.1.1.1.1.2.2" xref="A2.SS1.p1.4.m4.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A2.SS1.p1.4.m4.1.1.1.1.1.2.3" xref="A2.SS1.p1.4.m4.1.1.1.1.1.2.3.cmml">t</mi><mi mathsize="90%" id="A2.SS1.p1.4.m4.1.1.1.1.1.3" xref="A2.SS1.p1.4.m4.1.1.1.1.1.3.cmml">k</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.SS1.p1.4.m4.1.1.1.1.3" xref="A2.SS1.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.4.m4.1b"><apply id="A2.SS1.p1.4.m4.1.1.cmml" xref="A2.SS1.p1.4.m4.1.1"><times id="A2.SS1.p1.4.m4.1.1.2.cmml" xref="A2.SS1.p1.4.m4.1.1.2"></times><ci id="A2.SS1.p1.4.m4.1.1.3.cmml" xref="A2.SS1.p1.4.m4.1.1.3">𝑐</ci><apply id="A2.SS1.p1.4.m4.1.1.1.1.1.cmml" xref="A2.SS1.p1.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p1.4.m4.1.1.1.1.1.1.cmml" xref="A2.SS1.p1.4.m4.1.1.1.1">superscript</csymbol><apply id="A2.SS1.p1.4.m4.1.1.1.1.1.2.cmml" xref="A2.SS1.p1.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p1.4.m4.1.1.1.1.1.2.1.cmml" xref="A2.SS1.p1.4.m4.1.1.1.1">subscript</csymbol><ci id="A2.SS1.p1.4.m4.1.1.1.1.1.2.2.cmml" xref="A2.SS1.p1.4.m4.1.1.1.1.1.2.2">𝐱</ci><ci id="A2.SS1.p1.4.m4.1.1.1.1.1.2.3.cmml" xref="A2.SS1.p1.4.m4.1.1.1.1.1.2.3">𝑡</ci></apply><ci id="A2.SS1.p1.4.m4.1.1.1.1.1.3.cmml" xref="A2.SS1.p1.4.m4.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.4.m4.1c">c(\mathbf{x}_{t}^{k})</annotation></semantics></math><span id="A2.SS1.p1.6.8" class="ltx_text" style="font-size:90%;">. This has the interpretation as </span><math id="A2.SS1.p1.5.m5.2" class="ltx_Math" alttext="\Pr(y|\mathbf{x}_{t}^{k})" display="inline"><semantics id="A2.SS1.p1.5.m5.2a"><mrow id="A2.SS1.p1.5.m5.2.2.4" xref="A2.SS1.p1.5.m5.2.2.3.cmml"><mi mathsize="90%" id="A2.SS1.p1.5.m5.2.2.2.2" xref="A2.SS1.p1.5.m5.2.2.3.1.cmml">Pr</mi><mo id="A2.SS1.p1.5.m5.2.2.4a" xref="A2.SS1.p1.5.m5.2.2.3.1.cmml">⁡</mo><mrow id="A2.SS1.p1.5.m5.2.2.4.1" xref="A2.SS1.p1.5.m5.2.2.3.cmml"><mo id="A2.SS1.p1.5.m5.2.2.4.1.1" xref="A2.SS1.p1.5.m5.2.2.3.1.cmml">(</mo><mrow id="A2.SS1.p1.5.m5.1.1.1.1.1" xref="A2.SS1.p1.5.m5.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.SS1.p1.5.m5.1.1.1.1.1.2" xref="A2.SS1.p1.5.m5.1.1.1.1.1.2.cmml">y</mi><mo fence="false" mathsize="90%" id="A2.SS1.p1.5.m5.1.1.1.1.1.1" xref="A2.SS1.p1.5.m5.1.1.1.1.1.1.cmml">|</mo><msubsup id="A2.SS1.p1.5.m5.1.1.1.1.1.3" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A2.SS1.p1.5.m5.1.1.1.1.1.3.2.2" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A2.SS1.p1.5.m5.1.1.1.1.1.3.2.3" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3.2.3.cmml">t</mi><mi mathsize="90%" id="A2.SS1.p1.5.m5.1.1.1.1.1.3.3" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3.3.cmml">k</mi></msubsup></mrow><mo id="A2.SS1.p1.5.m5.2.2.4.1.2" xref="A2.SS1.p1.5.m5.2.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.5.m5.2b"><apply id="A2.SS1.p1.5.m5.2.2.3.cmml" xref="A2.SS1.p1.5.m5.2.2.4"><csymbol cd="latexml" id="A2.SS1.p1.5.m5.2.2.3.1.cmml" xref="A2.SS1.p1.5.m5.2.2.2.2">probability</csymbol><apply id="A2.SS1.p1.5.m5.1.1.1.1.1.cmml" xref="A2.SS1.p1.5.m5.1.1.1.1.1"><csymbol cd="latexml" id="A2.SS1.p1.5.m5.1.1.1.1.1.1.cmml" xref="A2.SS1.p1.5.m5.1.1.1.1.1.1">conditional</csymbol><ci id="A2.SS1.p1.5.m5.1.1.1.1.1.2.cmml" xref="A2.SS1.p1.5.m5.1.1.1.1.1.2">𝑦</ci><apply id="A2.SS1.p1.5.m5.1.1.1.1.1.3.cmml" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.SS1.p1.5.m5.1.1.1.1.1.3.1.cmml" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3">superscript</csymbol><apply id="A2.SS1.p1.5.m5.1.1.1.1.1.3.2.cmml" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.SS1.p1.5.m5.1.1.1.1.1.3.2.1.cmml" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3">subscript</csymbol><ci id="A2.SS1.p1.5.m5.1.1.1.1.1.3.2.2.cmml" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3.2.2">𝐱</ci><ci id="A2.SS1.p1.5.m5.1.1.1.1.1.3.2.3.cmml" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3.2.3">𝑡</ci></apply><ci id="A2.SS1.p1.5.m5.1.1.1.1.1.3.3.cmml" xref="A2.SS1.p1.5.m5.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.5.m5.2c">\Pr(y|\mathbf{x}_{t}^{k})</annotation></semantics></math><span id="A2.SS1.p1.6.9" class="ltx_text" style="font-size:90%;">, where </span><math id="A2.SS1.p1.6.m6.3" class="ltx_Math" alttext="\Pr[y=1\mid\mathbf{x}_{t}^{k}]=e^{c(\mathbf{x}_{t}^{k})}" display="inline"><semantics id="A2.SS1.p1.6.m6.3a"><mrow id="A2.SS1.p1.6.m6.3.3" 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id="A2.SS1.p1.6.m6.3.3.2.cmml" xref="A2.SS1.p1.6.m6.3.3.2"></eq><apply id="A2.SS1.p1.6.m6.3.3.1.2.cmml" xref="A2.SS1.p1.6.m6.3.3.1.1"><csymbol cd="latexml" id="A2.SS1.p1.6.m6.2.2.cmml" xref="A2.SS1.p1.6.m6.2.2">probability</csymbol><apply id="A2.SS1.p1.6.m6.3.3.1.1.1.1.cmml" xref="A2.SS1.p1.6.m6.3.3.1.1.1.1"><eq id="A2.SS1.p1.6.m6.3.3.1.1.1.1.1.cmml" xref="A2.SS1.p1.6.m6.3.3.1.1.1.1.1"></eq><ci id="A2.SS1.p1.6.m6.3.3.1.1.1.1.2.cmml" xref="A2.SS1.p1.6.m6.3.3.1.1.1.1.2">𝑦</ci><apply id="A2.SS1.p1.6.m6.3.3.1.1.1.1.3.cmml" xref="A2.SS1.p1.6.m6.3.3.1.1.1.1.3"><csymbol cd="latexml" id="A2.SS1.p1.6.m6.3.3.1.1.1.1.3.1.cmml" xref="A2.SS1.p1.6.m6.3.3.1.1.1.1.3.1">conditional</csymbol><cn type="integer" id="A2.SS1.p1.6.m6.3.3.1.1.1.1.3.2.cmml" xref="A2.SS1.p1.6.m6.3.3.1.1.1.1.3.2">1</cn><apply id="A2.SS1.p1.6.m6.3.3.1.1.1.1.3.3.cmml" xref="A2.SS1.p1.6.m6.3.3.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A2.SS1.p1.6.m6.3.3.1.1.1.1.3.3.1.cmml" 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id="A2.SS1.p1.6.m6.1.1.1.3.cmml" xref="A2.SS1.p1.6.m6.1.1.1.3">𝑐</ci><apply id="A2.SS1.p1.6.m6.1.1.1.1.1.1.cmml" xref="A2.SS1.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="A2.SS1.p1.6.m6.1.1.1.1.1">superscript</csymbol><apply id="A2.SS1.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="A2.SS1.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p1.6.m6.1.1.1.1.1.1.2.1.cmml" xref="A2.SS1.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="A2.SS1.p1.6.m6.1.1.1.1.1.1.2.2.cmml" xref="A2.SS1.p1.6.m6.1.1.1.1.1.1.2.2">𝐱</ci><ci id="A2.SS1.p1.6.m6.1.1.1.1.1.1.2.3.cmml" xref="A2.SS1.p1.6.m6.1.1.1.1.1.1.2.3">𝑡</ci></apply><ci id="A2.SS1.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="A2.SS1.p1.6.m6.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.6.m6.3c">\Pr[y=1\mid\mathbf{x}_{t}^{k}]=e^{c(\mathbf{x}_{t}^{k})}</annotation></semantics></math><span id="A2.SS1.p1.6.10" class="ltx_text" style="font-size:90%;">. Some popular cadidate energies include</span></p>
<table id="A6.EGx36" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A2.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A2.E1.m1.1" class="ltx_Math" alttext="\displaystyle c(\mathbf{x}_{t}^{k})=\mathop{{\mathbb{E}}}[\sum_{t^{\prime}&gt;t}\mathbf{r}_{{}^{\prime}}(\mathbf{x}_{t^{\prime}}^{k_{t^{\prime}}})\mid\mathbf{x}_{t}^{k}]," display="inline"><semantics id="A2.E1.m1.1a"><mrow id="A2.E1.m1.1.1.1" xref="A2.E1.m1.1.1.1.1.cmml"><mrow id="A2.E1.m1.1.1.1.1" xref="A2.E1.m1.1.1.1.1.cmml"><mrow id="A2.E1.m1.1.1.1.1.1" xref="A2.E1.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.E1.m1.1.1.1.1.1.3" xref="A2.E1.m1.1.1.1.1.1.3.cmml">c</mi><mo lspace="0em" rspace="0em" id="A2.E1.m1.1.1.1.1.1.2" xref="A2.E1.m1.1.1.1.1.1.2.cmml">​</mo><mrow id="A2.E1.m1.1.1.1.1.1.1.1" xref="A2.E1.m1.1.1.1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.E1.m1.1.1.1.1.1.1.1.2" xref="A2.E1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="A2.E1.m1.1.1.1.1.1.1.1.1" xref="A2.E1.m1.1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.E1.m1.1.1.1.1.1.1.1.1.2.2" xref="A2.E1.m1.1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A2.E1.m1.1.1.1.1.1.1.1.1.2.3" xref="A2.E1.m1.1.1.1.1.1.1.1.1.2.3.cmml">t</mi><mi mathsize="90%" id="A2.E1.m1.1.1.1.1.1.1.1.1.3" xref="A2.E1.m1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.E1.m1.1.1.1.1.1.1.1.3" xref="A2.E1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo mathsize="90%" rspace="0.1389em" id="A2.E1.m1.1.1.1.1.3" xref="A2.E1.m1.1.1.1.1.3.cmml">=</mo><mrow id="A2.E1.m1.1.1.1.1.2" xref="A2.E1.m1.1.1.1.1.2.cmml"><mo lspace="0.1389em" mathsize="108%" movablelimits="false" rspace="0em" id="A2.E1.m1.1.1.1.1.2.2" xref="A2.E1.m1.1.1.1.1.2.2.cmml">𝔼</mo><mrow id="A2.E1.m1.1.1.1.1.2.1.1" xref="A2.E1.m1.1.1.1.1.2.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A2.E1.m1.1.1.1.1.2.1.1.2" xref="A2.E1.m1.1.1.1.1.2.1.2.1.cmml">[</mo><mrow id="A2.E1.m1.1.1.1.1.2.1.1.1" xref="A2.E1.m1.1.1.1.1.2.1.1.1.cmml"><mrow id="A2.E1.m1.1.1.1.1.2.1.1.1.1" xref="A2.E1.m1.1.1.1.1.2.1.1.1.1.cmml"><mstyle displaystyle="true" id="A2.E1.m1.1.1.1.1.2.1.1.1.1.2" xref="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.cmml"><munder id="A2.E1.m1.1.1.1.1.2.1.1.1.1.2a" xref="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.cmml"><mo maxsize="90%" minsize="90%" movablelimits="false" stretchy="true" id="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.2" xref="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.2.cmml">∑</mo><mrow id="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3" xref="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.cmml"><msup id="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.2" xref="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.2.cmml"><mi mathsize="90%" id="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.2.2" xref="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.2.2.cmml">t</mi><mo mathsize="90%" id="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.2.3" xref="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.2.3.cmml">′</mo></msup><mo mathsize="90%" id="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.1" xref="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.1.cmml">&gt;</mo><mi mathsize="90%" id="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.3" xref="A2.E1.m1.1.1.1.1.2.1.1.1.1.2.3.3.cmml">t</mi></mrow></munder></mstyle><mrow 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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(B.1)</span></td>
</tr></tbody>
</table>
<p id="A2.SS1.p1.8" class="ltx_p"><span id="A2.SS1.p1.8.1" class="ltx_text" style="font-size:90%;">corresponding to a cost-to-go; we can obtain unbiased estiames of this gradient by using cumulative reward </span><math id="A2.SS1.p1.7.m1.2" class="ltx_Math" alttext="\tilde{c}(\mathbf{x}_{t}^{k})=\sum_{t^{\prime}&gt;t}\mathbf{r}_{{}^{\prime}}(\mathbf{x}_{t^{\prime}}^{k_{t^{\prime}}})" display="inline"><semantics id="A2.SS1.p1.7.m1.2a"><mrow id="A2.SS1.p1.7.m1.2.2" xref="A2.SS1.p1.7.m1.2.2.cmml"><mrow id="A2.SS1.p1.7.m1.1.1.1" xref="A2.SS1.p1.7.m1.1.1.1.cmml"><mover accent="true" id="A2.SS1.p1.7.m1.1.1.1.3" xref="A2.SS1.p1.7.m1.1.1.1.3.cmml"><mi mathsize="90%" id="A2.SS1.p1.7.m1.1.1.1.3.2" xref="A2.SS1.p1.7.m1.1.1.1.3.2.cmml">c</mi><mo mathsize="90%" id="A2.SS1.p1.7.m1.1.1.1.3.1" xref="A2.SS1.p1.7.m1.1.1.1.3.1.cmml">~</mo></mover><mo lspace="0em" rspace="0em" id="A2.SS1.p1.7.m1.1.1.1.2" xref="A2.SS1.p1.7.m1.1.1.1.2.cmml">​</mo><mrow id="A2.SS1.p1.7.m1.1.1.1.1.1" 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We can also use goal distance </span><math id="A2.SS1.p1.8.m2.1" class="ltx_Math" alttext="c=-\|\mathbf{x}_{T}^{k_{T}}-\mathbf{g}\|^{2}" display="inline"><semantics id="A2.SS1.p1.8.m2.1a"><mrow id="A2.SS1.p1.8.m2.1.1" xref="A2.SS1.p1.8.m2.1.1.cmml"><mi mathsize="90%" id="A2.SS1.p1.8.m2.1.1.3" xref="A2.SS1.p1.8.m2.1.1.3.cmml">c</mi><mo mathsize="90%" id="A2.SS1.p1.8.m2.1.1.2" xref="A2.SS1.p1.8.m2.1.1.2.cmml">=</mo><mrow id="A2.SS1.p1.8.m2.1.1.1" xref="A2.SS1.p1.8.m2.1.1.1.cmml"><mo mathsize="90%" id="A2.SS1.p1.8.m2.1.1.1a" xref="A2.SS1.p1.8.m2.1.1.1.cmml">−</mo><msup id="A2.SS1.p1.8.m2.1.1.1.1" xref="A2.SS1.p1.8.m2.1.1.1.1.cmml"><mrow id="A2.SS1.p1.8.m2.1.1.1.1.1.1" xref="A2.SS1.p1.8.m2.1.1.1.1.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.2" xref="A2.SS1.p1.8.m2.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.cmml"><msubsup id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.cmml"><mi mathsize="90%" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.2.2" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.2.3" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.2.3.cmml">T</mi><msub id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.cmml"><mi mathsize="90%" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.2" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.2.cmml">k</mi><mi mathsize="90%" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.3" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.3.cmml">T</mi></msub></msubsup><mo mathsize="90%" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.1" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.1.cmml">−</mo><mi mathsize="90%" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.3" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.3.cmml">𝐠</mi></mrow><mo maxsize="90%" minsize="90%" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.3" xref="A2.SS1.p1.8.m2.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn mathsize="90%" id="A2.SS1.p1.8.m2.1.1.1.1.3" xref="A2.SS1.p1.8.m2.1.1.1.1.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS1.p1.8.m2.1b"><apply id="A2.SS1.p1.8.m2.1.1.cmml" xref="A2.SS1.p1.8.m2.1.1"><eq id="A2.SS1.p1.8.m2.1.1.2.cmml" xref="A2.SS1.p1.8.m2.1.1.2"></eq><ci id="A2.SS1.p1.8.m2.1.1.3.cmml" xref="A2.SS1.p1.8.m2.1.1.3">𝑐</ci><apply id="A2.SS1.p1.8.m2.1.1.1.cmml" xref="A2.SS1.p1.8.m2.1.1.1"><minus id="A2.SS1.p1.8.m2.1.1.1.2.cmml" xref="A2.SS1.p1.8.m2.1.1.1"></minus><apply id="A2.SS1.p1.8.m2.1.1.1.1.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS1.p1.8.m2.1.1.1.1.2.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1">superscript</csymbol><apply id="A2.SS1.p1.8.m2.1.1.1.1.1.2.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1"><csymbol cd="latexml" id="A2.SS1.p1.8.m2.1.1.1.1.1.2.1.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.2">norm</csymbol><apply id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1"><minus id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.1.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.1"></minus><apply id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.1.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2">superscript</csymbol><apply id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.2.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.2.1.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.2.2.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.2.2">𝐱</ci><ci id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.2.3.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.2.3">𝑇</ci></apply><apply id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.1.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.2.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.2">𝑘</ci><ci id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.3.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.2.3.3">𝑇</ci></apply></apply><ci id="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.3.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.1.1.1.3">𝐠</ci></apply></apply><cn type="integer" id="A2.SS1.p1.8.m2.1.1.1.1.3.cmml" xref="A2.SS1.p1.8.m2.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS1.p1.8.m2.1c">c=-\|\mathbf{x}_{T}^{k_{T}}-\mathbf{g}\|^{2}</annotation></semantics></math><span id="A2.SS1.p1.8.3" class="ltx_text" style="font-size:90%;"> as a terminal reward. We provide details about the guidance function deployed in the maze2d planning experiment in Appendix </span><a href="#A4.SS5.SSS0.Px2" title="Guidance for maze planning ‣ D.5 Implementation Details of Sampling with Guidance ‣ Appendix D Additional Method Details ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">D.5</span></a><span id="A2.SS1.p1.8.4" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
</section>
<section id="A2.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">B.2 </span>Noising and stabilizing long-horizon generations</h3>

<div id="A2.SS2.p1" class="ltx_para">
<p id="A2.SS2.p1.11" class="ltx_p"><span id="A2.SS2.p1.11.1" class="ltx_text" style="font-size:90%;">Here explain in detail how we use noising to stabilize long horizon generation. At each time </span><math id="A2.SS2.p1.1.m1.1" class="ltx_Math" alttext="t" display="inline"><semantics id="A2.SS2.p1.1.m1.1a"><mi mathsize="90%" id="A2.SS2.p1.1.m1.1.1" xref="A2.SS2.p1.1.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.1.m1.1b"><ci id="A2.SS2.p1.1.m1.1.1.cmml" xref="A2.SS2.p1.1.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.1.m1.1c">t</annotation></semantics></math><span id="A2.SS2.p1.11.2" class="ltx_text" style="font-size:90%;">, during the denosing, we maintain a latent </span><math id="A2.SS2.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}^{k_{\mathrm{small}}}" display="inline"><semantics id="A2.SS2.p1.2.m2.1a"><msubsup id="A2.SS2.p1.2.m2.1.1" xref="A2.SS2.p1.2.m2.1.1.cmml"><mi mathsize="90%" id="A2.SS2.p1.2.m2.1.1.2.2" xref="A2.SS2.p1.2.m2.1.1.2.2.cmml">𝐳</mi><mrow id="A2.SS2.p1.2.m2.1.1.2.3" xref="A2.SS2.p1.2.m2.1.1.2.3.cmml"><mi mathsize="90%" id="A2.SS2.p1.2.m2.1.1.2.3.2" xref="A2.SS2.p1.2.m2.1.1.2.3.2.cmml">t</mi><mo mathsize="90%" id="A2.SS2.p1.2.m2.1.1.2.3.1" xref="A2.SS2.p1.2.m2.1.1.2.3.1.cmml">−</mo><mn mathsize="90%" id="A2.SS2.p1.2.m2.1.1.2.3.3" xref="A2.SS2.p1.2.m2.1.1.2.3.3.cmml">1</mn></mrow><msub id="A2.SS2.p1.2.m2.1.1.3" xref="A2.SS2.p1.2.m2.1.1.3.cmml"><mi mathsize="90%" id="A2.SS2.p1.2.m2.1.1.3.2" xref="A2.SS2.p1.2.m2.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A2.SS2.p1.2.m2.1.1.3.3" xref="A2.SS2.p1.2.m2.1.1.3.3.cmml">small</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.2.m2.1b"><apply id="A2.SS2.p1.2.m2.1.1.cmml" xref="A2.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.2.m2.1.1.1.cmml" xref="A2.SS2.p1.2.m2.1.1">superscript</csymbol><apply id="A2.SS2.p1.2.m2.1.1.2.cmml" xref="A2.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.2.m2.1.1.2.1.cmml" xref="A2.SS2.p1.2.m2.1.1">subscript</csymbol><ci id="A2.SS2.p1.2.m2.1.1.2.2.cmml" xref="A2.SS2.p1.2.m2.1.1.2.2">𝐳</ci><apply id="A2.SS2.p1.2.m2.1.1.2.3.cmml" xref="A2.SS2.p1.2.m2.1.1.2.3"><minus id="A2.SS2.p1.2.m2.1.1.2.3.1.cmml" xref="A2.SS2.p1.2.m2.1.1.2.3.1"></minus><ci id="A2.SS2.p1.2.m2.1.1.2.3.2.cmml" xref="A2.SS2.p1.2.m2.1.1.2.3.2">𝑡</ci><cn type="integer" id="A2.SS2.p1.2.m2.1.1.2.3.3.cmml" xref="A2.SS2.p1.2.m2.1.1.2.3.3">1</cn></apply></apply><apply id="A2.SS2.p1.2.m2.1.1.3.cmml" xref="A2.SS2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A2.SS2.p1.2.m2.1.1.3.1.cmml" xref="A2.SS2.p1.2.m2.1.1.3">subscript</csymbol><ci id="A2.SS2.p1.2.m2.1.1.3.2.cmml" xref="A2.SS2.p1.2.m2.1.1.3.2">𝑘</ci><ci id="A2.SS2.p1.2.m2.1.1.3.3.cmml" xref="A2.SS2.p1.2.m2.1.1.3.3">small</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.2.m2.1c">\mathbf{z}_{t-1}^{k_{\mathrm{small}}}</annotation></semantics></math><span id="A2.SS2.p1.11.3" class="ltx_text" style="font-size:90%;"> from the previous time step, with </span><math id="A2.SS2.p1.3.m3.1" class="ltx_Math" alttext="0&lt;k_{\mathrm{small}}\ll K" display="inline"><semantics id="A2.SS2.p1.3.m3.1a"><mrow id="A2.SS2.p1.3.m3.1.1" xref="A2.SS2.p1.3.m3.1.1.cmml"><mn mathsize="90%" id="A2.SS2.p1.3.m3.1.1.2" xref="A2.SS2.p1.3.m3.1.1.2.cmml">0</mn><mo mathsize="90%" id="A2.SS2.p1.3.m3.1.1.3" xref="A2.SS2.p1.3.m3.1.1.3.cmml">&lt;</mo><msub id="A2.SS2.p1.3.m3.1.1.4" xref="A2.SS2.p1.3.m3.1.1.4.cmml"><mi mathsize="90%" id="A2.SS2.p1.3.m3.1.1.4.2" xref="A2.SS2.p1.3.m3.1.1.4.2.cmml">k</mi><mi mathsize="90%" id="A2.SS2.p1.3.m3.1.1.4.3" xref="A2.SS2.p1.3.m3.1.1.4.3.cmml">small</mi></msub><mo mathsize="90%" id="A2.SS2.p1.3.m3.1.1.5" xref="A2.SS2.p1.3.m3.1.1.5.cmml">≪</mo><mi mathsize="90%" id="A2.SS2.p1.3.m3.1.1.6" xref="A2.SS2.p1.3.m3.1.1.6.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.3.m3.1b"><apply id="A2.SS2.p1.3.m3.1.1.cmml" xref="A2.SS2.p1.3.m3.1.1"><and id="A2.SS2.p1.3.m3.1.1a.cmml" xref="A2.SS2.p1.3.m3.1.1"></and><apply id="A2.SS2.p1.3.m3.1.1b.cmml" xref="A2.SS2.p1.3.m3.1.1"><lt id="A2.SS2.p1.3.m3.1.1.3.cmml" xref="A2.SS2.p1.3.m3.1.1.3"></lt><cn type="integer" id="A2.SS2.p1.3.m3.1.1.2.cmml" xref="A2.SS2.p1.3.m3.1.1.2">0</cn><apply id="A2.SS2.p1.3.m3.1.1.4.cmml" xref="A2.SS2.p1.3.m3.1.1.4"><csymbol cd="ambiguous" id="A2.SS2.p1.3.m3.1.1.4.1.cmml" xref="A2.SS2.p1.3.m3.1.1.4">subscript</csymbol><ci id="A2.SS2.p1.3.m3.1.1.4.2.cmml" xref="A2.SS2.p1.3.m3.1.1.4.2">𝑘</ci><ci id="A2.SS2.p1.3.m3.1.1.4.3.cmml" xref="A2.SS2.p1.3.m3.1.1.4.3">small</ci></apply></apply><apply id="A2.SS2.p1.3.m3.1.1c.cmml" xref="A2.SS2.p1.3.m3.1.1"><csymbol cd="latexml" id="A2.SS2.p1.3.m3.1.1.5.cmml" xref="A2.SS2.p1.3.m3.1.1.5">much-less-than</csymbol><share href="#A2.SS2.p1.3.m3.1.1.4.cmml" id="A2.SS2.p1.3.m3.1.1d.cmml" xref="A2.SS2.p1.3.m3.1.1"></share><ci id="A2.SS2.p1.3.m3.1.1.6.cmml" xref="A2.SS2.p1.3.m3.1.1.6">𝐾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.3.m3.1c">0&lt;k_{\mathrm{small}}\ll K</annotation></semantics></math><span id="A2.SS2.p1.11.4" class="ltx_text" style="font-size:90%;"> corresponding to some small amount of noise. We then do </span><em id="A2.SS2.p1.11.5" class="ltx_emph ltx_font_italic" style="font-size:90%;">next token</em><span id="A2.SS2.p1.11.6" class="ltx_text" style="font-size:90%;"> diffusion to diffuse the token </span><math id="A2.SS2.p1.4.m4.1" class="ltx_Math" alttext="\mathbf{x}_{t}" display="inline"><semantics id="A2.SS2.p1.4.m4.1a"><msub id="A2.SS2.p1.4.m4.1.1" xref="A2.SS2.p1.4.m4.1.1.cmml"><mi mathsize="90%" id="A2.SS2.p1.4.m4.1.1.2" xref="A2.SS2.p1.4.m4.1.1.2.cmml">𝐱</mi><mi mathsize="90%" id="A2.SS2.p1.4.m4.1.1.3" xref="A2.SS2.p1.4.m4.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.4.m4.1b"><apply id="A2.SS2.p1.4.m4.1.1.cmml" xref="A2.SS2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.4.m4.1.1.1.cmml" xref="A2.SS2.p1.4.m4.1.1">subscript</csymbol><ci id="A2.SS2.p1.4.m4.1.1.2.cmml" xref="A2.SS2.p1.4.m4.1.1.2">𝐱</ci><ci id="A2.SS2.p1.4.m4.1.1.3.cmml" xref="A2.SS2.p1.4.m4.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.4.m4.1c">\mathbf{x}_{t}</annotation></semantics></math><span id="A2.SS2.p1.11.7" class="ltx_text" style="font-size:90%;"> across noise levels </span><math id="A2.SS2.p1.5.m5.4" class="ltx_Math" alttext="\mathbf{x}_{t}^{K},\mathbf{x}_{t}^{K-1},\dots,\mathbf{x}_{t}^{0}" display="inline"><semantics id="A2.SS2.p1.5.m5.4a"><mrow id="A2.SS2.p1.5.m5.4.4.3" xref="A2.SS2.p1.5.m5.4.4.4.cmml"><msubsup id="A2.SS2.p1.5.m5.2.2.1.1" xref="A2.SS2.p1.5.m5.2.2.1.1.cmml"><mi mathsize="90%" id="A2.SS2.p1.5.m5.2.2.1.1.2.2" xref="A2.SS2.p1.5.m5.2.2.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A2.SS2.p1.5.m5.2.2.1.1.2.3" xref="A2.SS2.p1.5.m5.2.2.1.1.2.3.cmml">t</mi><mi mathsize="90%" id="A2.SS2.p1.5.m5.2.2.1.1.3" xref="A2.SS2.p1.5.m5.2.2.1.1.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.SS2.p1.5.m5.4.4.3.4" xref="A2.SS2.p1.5.m5.4.4.4.cmml">,</mo><msubsup id="A2.SS2.p1.5.m5.3.3.2.2" xref="A2.SS2.p1.5.m5.3.3.2.2.cmml"><mi mathsize="90%" 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cd="ambiguous" id="A2.SS2.p1.5.m5.4.4.3.3.1.cmml" xref="A2.SS2.p1.5.m5.4.4.3.3">superscript</csymbol><apply id="A2.SS2.p1.5.m5.4.4.3.3.2.cmml" xref="A2.SS2.p1.5.m5.4.4.3.3"><csymbol cd="ambiguous" id="A2.SS2.p1.5.m5.4.4.3.3.2.1.cmml" xref="A2.SS2.p1.5.m5.4.4.3.3">subscript</csymbol><ci id="A2.SS2.p1.5.m5.4.4.3.3.2.2.cmml" xref="A2.SS2.p1.5.m5.4.4.3.3.2.2">𝐱</ci><ci id="A2.SS2.p1.5.m5.4.4.3.3.2.3.cmml" xref="A2.SS2.p1.5.m5.4.4.3.3.2.3">𝑡</ci></apply><cn type="integer" id="A2.SS2.p1.5.m5.4.4.3.3.3.cmml" xref="A2.SS2.p1.5.m5.4.4.3.3.3">0</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.5.m5.4c">\mathbf{x}_{t}^{K},\mathbf{x}_{t}^{K-1},\dots,\mathbf{x}_{t}^{0}</annotation></semantics></math><span id="A2.SS2.p1.11.8" class="ltx_text" style="font-size:90%;"> (corresponding to </span><a href="#alg2" title="In Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a><span id="A2.SS2.p1.11.9" class="ltx_text" style="font-size:90%;"> with horizon </span><math id="A2.SS2.p1.6.m6.1" class="ltx_Math" alttext="T=1" display="inline"><semantics id="A2.SS2.p1.6.m6.1a"><mrow id="A2.SS2.p1.6.m6.1.1" xref="A2.SS2.p1.6.m6.1.1.cmml"><mi mathsize="90%" id="A2.SS2.p1.6.m6.1.1.2" xref="A2.SS2.p1.6.m6.1.1.2.cmml">T</mi><mo mathsize="90%" id="A2.SS2.p1.6.m6.1.1.1" xref="A2.SS2.p1.6.m6.1.1.1.cmml">=</mo><mn mathsize="90%" id="A2.SS2.p1.6.m6.1.1.3" xref="A2.SS2.p1.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.6.m6.1b"><apply id="A2.SS2.p1.6.m6.1.1.cmml" xref="A2.SS2.p1.6.m6.1.1"><eq id="A2.SS2.p1.6.m6.1.1.1.cmml" xref="A2.SS2.p1.6.m6.1.1.1"></eq><ci id="A2.SS2.p1.6.m6.1.1.2.cmml" xref="A2.SS2.p1.6.m6.1.1.2">𝑇</ci><cn type="integer" id="A2.SS2.p1.6.m6.1.1.3.cmml" xref="A2.SS2.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.6.m6.1c">T=1</annotation></semantics></math><span id="A2.SS2.p1.11.10" class="ltx_text" style="font-size:90%;">, initial latent </span><math id="A2.SS2.p1.7.m7.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}^{k}" display="inline"><semantics id="A2.SS2.p1.7.m7.1a"><msubsup id="A2.SS2.p1.7.m7.1.1" xref="A2.SS2.p1.7.m7.1.1.cmml"><mi mathsize="90%" id="A2.SS2.p1.7.m7.1.1.2.2" xref="A2.SS2.p1.7.m7.1.1.2.2.cmml">𝐳</mi><mrow id="A2.SS2.p1.7.m7.1.1.2.3" xref="A2.SS2.p1.7.m7.1.1.2.3.cmml"><mi mathsize="90%" id="A2.SS2.p1.7.m7.1.1.2.3.2" xref="A2.SS2.p1.7.m7.1.1.2.3.2.cmml">t</mi><mo mathsize="90%" id="A2.SS2.p1.7.m7.1.1.2.3.1" xref="A2.SS2.p1.7.m7.1.1.2.3.1.cmml">−</mo><mn mathsize="90%" id="A2.SS2.p1.7.m7.1.1.2.3.3" xref="A2.SS2.p1.7.m7.1.1.2.3.3.cmml">1</mn></mrow><mi mathsize="90%" id="A2.SS2.p1.7.m7.1.1.3" xref="A2.SS2.p1.7.m7.1.1.3.cmml">k</mi></msubsup><annotation-xml 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id="A2.SS2.p1.7.m7.1c">\mathbf{z}_{t-1}^{k}</annotation></semantics></math><span id="A2.SS2.p1.11.11" class="ltx_text" style="font-size:90%;">, and noise schedule </span><math id="A2.SS2.p1.8.m8.2" class="ltx_Math" alttext="\mathcal{K}_{m,1}=m" display="inline"><semantics id="A2.SS2.p1.8.m8.2a"><mrow id="A2.SS2.p1.8.m8.2.3" xref="A2.SS2.p1.8.m8.2.3.cmml"><msub id="A2.SS2.p1.8.m8.2.3.2" xref="A2.SS2.p1.8.m8.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A2.SS2.p1.8.m8.2.3.2.2" xref="A2.SS2.p1.8.m8.2.3.2.2.cmml">𝒦</mi><mrow id="A2.SS2.p1.8.m8.2.2.2.4" xref="A2.SS2.p1.8.m8.2.2.2.3.cmml"><mi mathsize="90%" id="A2.SS2.p1.8.m8.1.1.1.1" xref="A2.SS2.p1.8.m8.1.1.1.1.cmml">m</mi><mo mathsize="90%" id="A2.SS2.p1.8.m8.2.2.2.4.1" xref="A2.SS2.p1.8.m8.2.2.2.3.cmml">,</mo><mn mathsize="90%" id="A2.SS2.p1.8.m8.2.2.2.2" xref="A2.SS2.p1.8.m8.2.2.2.2.cmml">1</mn></mrow></msub><mo mathsize="90%" id="A2.SS2.p1.8.m8.2.3.1" xref="A2.SS2.p1.8.m8.2.3.1.cmml">=</mo><mi mathsize="90%" id="A2.SS2.p1.8.m8.2.3.3" xref="A2.SS2.p1.8.m8.2.3.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.8.m8.2b"><apply id="A2.SS2.p1.8.m8.2.3.cmml" xref="A2.SS2.p1.8.m8.2.3"><eq id="A2.SS2.p1.8.m8.2.3.1.cmml" xref="A2.SS2.p1.8.m8.2.3.1"></eq><apply id="A2.SS2.p1.8.m8.2.3.2.cmml" xref="A2.SS2.p1.8.m8.2.3.2"><csymbol cd="ambiguous" id="A2.SS2.p1.8.m8.2.3.2.1.cmml" xref="A2.SS2.p1.8.m8.2.3.2">subscript</csymbol><ci id="A2.SS2.p1.8.m8.2.3.2.2.cmml" xref="A2.SS2.p1.8.m8.2.3.2.2">𝒦</ci><list id="A2.SS2.p1.8.m8.2.2.2.3.cmml" xref="A2.SS2.p1.8.m8.2.2.2.4"><ci id="A2.SS2.p1.8.m8.1.1.1.1.cmml" xref="A2.SS2.p1.8.m8.1.1.1.1">𝑚</ci><cn type="integer" id="A2.SS2.p1.8.m8.2.2.2.2.cmml" xref="A2.SS2.p1.8.m8.2.2.2.2">1</cn></list></apply><ci id="A2.SS2.p1.8.m8.2.3.3.cmml" xref="A2.SS2.p1.8.m8.2.3.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.8.m8.2c">\mathcal{K}_{m,1}=m</annotation></semantics></math><span id="A2.SS2.p1.11.12" class="ltx_text" style="font-size:90%;">); this process also produces latents </span><math id="A2.SS2.p1.9.m9.4" class="ltx_Math" alttext="\mathbf{z}_{t}^{K},\mathbf{z}_{t}^{K-1},\dots,\mathbf{z}_{t}^{0}" display="inline"><semantics id="A2.SS2.p1.9.m9.4a"><mrow id="A2.SS2.p1.9.m9.4.4.3" xref="A2.SS2.p1.9.m9.4.4.4.cmml"><msubsup id="A2.SS2.p1.9.m9.2.2.1.1" xref="A2.SS2.p1.9.m9.2.2.1.1.cmml"><mi mathsize="90%" id="A2.SS2.p1.9.m9.2.2.1.1.2.2" xref="A2.SS2.p1.9.m9.2.2.1.1.2.2.cmml">𝐳</mi><mi mathsize="90%" id="A2.SS2.p1.9.m9.2.2.1.1.2.3" xref="A2.SS2.p1.9.m9.2.2.1.1.2.3.cmml">t</mi><mi mathsize="90%" id="A2.SS2.p1.9.m9.2.2.1.1.3" xref="A2.SS2.p1.9.m9.2.2.1.1.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.SS2.p1.9.m9.4.4.3.4" xref="A2.SS2.p1.9.m9.4.4.4.cmml">,</mo><msubsup id="A2.SS2.p1.9.m9.3.3.2.2" xref="A2.SS2.p1.9.m9.3.3.2.2.cmml"><mi mathsize="90%" id="A2.SS2.p1.9.m9.3.3.2.2.2.2" xref="A2.SS2.p1.9.m9.3.3.2.2.2.2.cmml">𝐳</mi><mi mathsize="90%" id="A2.SS2.p1.9.m9.3.3.2.2.2.3" xref="A2.SS2.p1.9.m9.3.3.2.2.2.3.cmml">t</mi><mrow id="A2.SS2.p1.9.m9.3.3.2.2.3" xref="A2.SS2.p1.9.m9.3.3.2.2.3.cmml"><mi mathsize="90%" id="A2.SS2.p1.9.m9.3.3.2.2.3.2" xref="A2.SS2.p1.9.m9.3.3.2.2.3.2.cmml">K</mi><mo mathsize="90%" id="A2.SS2.p1.9.m9.3.3.2.2.3.1" xref="A2.SS2.p1.9.m9.3.3.2.2.3.1.cmml">−</mo><mn mathsize="90%" id="A2.SS2.p1.9.m9.3.3.2.2.3.3" xref="A2.SS2.p1.9.m9.3.3.2.2.3.3.cmml">1</mn></mrow></msubsup><mo mathsize="90%" id="A2.SS2.p1.9.m9.4.4.3.5" xref="A2.SS2.p1.9.m9.4.4.4.cmml">,</mo><mi mathsize="90%" mathvariant="normal" id="A2.SS2.p1.9.m9.1.1" xref="A2.SS2.p1.9.m9.1.1.cmml">…</mi><mo mathsize="90%" id="A2.SS2.p1.9.m9.4.4.3.6" xref="A2.SS2.p1.9.m9.4.4.4.cmml">,</mo><msubsup id="A2.SS2.p1.9.m9.4.4.3.3" xref="A2.SS2.p1.9.m9.4.4.3.3.cmml"><mi mathsize="90%" id="A2.SS2.p1.9.m9.4.4.3.3.2.2" xref="A2.SS2.p1.9.m9.4.4.3.3.2.2.cmml">𝐳</mi><mi mathsize="90%" id="A2.SS2.p1.9.m9.4.4.3.3.2.3" xref="A2.SS2.p1.9.m9.4.4.3.3.2.3.cmml">t</mi><mn mathsize="90%" id="A2.SS2.p1.9.m9.4.4.3.3.3" xref="A2.SS2.p1.9.m9.4.4.3.3.3.cmml">0</mn></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.9.m9.4b"><list id="A2.SS2.p1.9.m9.4.4.4.cmml" xref="A2.SS2.p1.9.m9.4.4.3"><apply id="A2.SS2.p1.9.m9.2.2.1.1.cmml" xref="A2.SS2.p1.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.9.m9.2.2.1.1.1.cmml" xref="A2.SS2.p1.9.m9.2.2.1.1">superscript</csymbol><apply id="A2.SS2.p1.9.m9.2.2.1.1.2.cmml" xref="A2.SS2.p1.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.9.m9.2.2.1.1.2.1.cmml" xref="A2.SS2.p1.9.m9.2.2.1.1">subscript</csymbol><ci id="A2.SS2.p1.9.m9.2.2.1.1.2.2.cmml" xref="A2.SS2.p1.9.m9.2.2.1.1.2.2">𝐳</ci><ci id="A2.SS2.p1.9.m9.2.2.1.1.2.3.cmml" xref="A2.SS2.p1.9.m9.2.2.1.1.2.3">𝑡</ci></apply><ci id="A2.SS2.p1.9.m9.2.2.1.1.3.cmml" xref="A2.SS2.p1.9.m9.2.2.1.1.3">𝐾</ci></apply><apply id="A2.SS2.p1.9.m9.3.3.2.2.cmml" xref="A2.SS2.p1.9.m9.3.3.2.2"><csymbol cd="ambiguous" id="A2.SS2.p1.9.m9.3.3.2.2.1.cmml" xref="A2.SS2.p1.9.m9.3.3.2.2">superscript</csymbol><apply id="A2.SS2.p1.9.m9.3.3.2.2.2.cmml" xref="A2.SS2.p1.9.m9.3.3.2.2"><csymbol cd="ambiguous" id="A2.SS2.p1.9.m9.3.3.2.2.2.1.cmml" xref="A2.SS2.p1.9.m9.3.3.2.2">subscript</csymbol><ci id="A2.SS2.p1.9.m9.3.3.2.2.2.2.cmml" xref="A2.SS2.p1.9.m9.3.3.2.2.2.2">𝐳</ci><ci id="A2.SS2.p1.9.m9.3.3.2.2.2.3.cmml" xref="A2.SS2.p1.9.m9.3.3.2.2.2.3">𝑡</ci></apply><apply id="A2.SS2.p1.9.m9.3.3.2.2.3.cmml" xref="A2.SS2.p1.9.m9.3.3.2.2.3"><minus id="A2.SS2.p1.9.m9.3.3.2.2.3.1.cmml" xref="A2.SS2.p1.9.m9.3.3.2.2.3.1"></minus><ci id="A2.SS2.p1.9.m9.3.3.2.2.3.2.cmml" xref="A2.SS2.p1.9.m9.3.3.2.2.3.2">𝐾</ci><cn type="integer" id="A2.SS2.p1.9.m9.3.3.2.2.3.3.cmml" xref="A2.SS2.p1.9.m9.3.3.2.2.3.3">1</cn></apply></apply><ci id="A2.SS2.p1.9.m9.1.1.cmml" xref="A2.SS2.p1.9.m9.1.1">…</ci><apply id="A2.SS2.p1.9.m9.4.4.3.3.cmml" xref="A2.SS2.p1.9.m9.4.4.3.3"><csymbol cd="ambiguous" id="A2.SS2.p1.9.m9.4.4.3.3.1.cmml" xref="A2.SS2.p1.9.m9.4.4.3.3">superscript</csymbol><apply id="A2.SS2.p1.9.m9.4.4.3.3.2.cmml" xref="A2.SS2.p1.9.m9.4.4.3.3"><csymbol cd="ambiguous" id="A2.SS2.p1.9.m9.4.4.3.3.2.1.cmml" xref="A2.SS2.p1.9.m9.4.4.3.3">subscript</csymbol><ci id="A2.SS2.p1.9.m9.4.4.3.3.2.2.cmml" xref="A2.SS2.p1.9.m9.4.4.3.3.2.2">𝐳</ci><ci id="A2.SS2.p1.9.m9.4.4.3.3.2.3.cmml" xref="A2.SS2.p1.9.m9.4.4.3.3.2.3">𝑡</ci></apply><cn type="integer" id="A2.SS2.p1.9.m9.4.4.3.3.3.cmml" xref="A2.SS2.p1.9.m9.4.4.3.3.3">0</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.9.m9.4c">\mathbf{z}_{t}^{K},\mathbf{z}_{t}^{K-1},\dots,\mathbf{z}_{t}^{0}</annotation></semantics></math><span id="A2.SS2.p1.11.13" class="ltx_text" style="font-size:90%;"> associated with each noise level. From these, we use the latent </span><math id="A2.SS2.p1.10.m10.1" class="ltx_Math" alttext="\mathbf{z}_{t}^{k_{\mathrm{small}}}" display="inline"><semantics id="A2.SS2.p1.10.m10.1a"><msubsup id="A2.SS2.p1.10.m10.1.1" xref="A2.SS2.p1.10.m10.1.1.cmml"><mi mathsize="90%" id="A2.SS2.p1.10.m10.1.1.2.2" xref="A2.SS2.p1.10.m10.1.1.2.2.cmml">𝐳</mi><mi mathsize="90%" id="A2.SS2.p1.10.m10.1.1.2.3" xref="A2.SS2.p1.10.m10.1.1.2.3.cmml">t</mi><msub id="A2.SS2.p1.10.m10.1.1.3" xref="A2.SS2.p1.10.m10.1.1.3.cmml"><mi mathsize="90%" id="A2.SS2.p1.10.m10.1.1.3.2" xref="A2.SS2.p1.10.m10.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A2.SS2.p1.10.m10.1.1.3.3" xref="A2.SS2.p1.10.m10.1.1.3.3.cmml">small</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.10.m10.1b"><apply id="A2.SS2.p1.10.m10.1.1.cmml" xref="A2.SS2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.10.m10.1.1.1.cmml" xref="A2.SS2.p1.10.m10.1.1">superscript</csymbol><apply id="A2.SS2.p1.10.m10.1.1.2.cmml" xref="A2.SS2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.10.m10.1.1.2.1.cmml" xref="A2.SS2.p1.10.m10.1.1">subscript</csymbol><ci id="A2.SS2.p1.10.m10.1.1.2.2.cmml" xref="A2.SS2.p1.10.m10.1.1.2.2">𝐳</ci><ci id="A2.SS2.p1.10.m10.1.1.2.3.cmml" xref="A2.SS2.p1.10.m10.1.1.2.3">𝑡</ci></apply><apply id="A2.SS2.p1.10.m10.1.1.3.cmml" xref="A2.SS2.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="A2.SS2.p1.10.m10.1.1.3.1.cmml" xref="A2.SS2.p1.10.m10.1.1.3">subscript</csymbol><ci id="A2.SS2.p1.10.m10.1.1.3.2.cmml" xref="A2.SS2.p1.10.m10.1.1.3.2">𝑘</ci><ci id="A2.SS2.p1.10.m10.1.1.3.3.cmml" xref="A2.SS2.p1.10.m10.1.1.3.3">small</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.10.m10.1c">\mathbf{z}_{t}^{k_{\mathrm{small}}}</annotation></semantics></math><span id="A2.SS2.p1.11.14" class="ltx_text" style="font-size:90%;"> to repreat the process. This noised latent can be interpreted as an implementation of conditioning on </span><math id="A2.SS2.p1.11.m11.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{\mathrm{small}}}" display="inline"><semantics id="A2.SS2.p1.11.m11.1a"><msubsup id="A2.SS2.p1.11.m11.1.1" xref="A2.SS2.p1.11.m11.1.1.cmml"><mi mathsize="90%" id="A2.SS2.p1.11.m11.1.1.2.2" xref="A2.SS2.p1.11.m11.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A2.SS2.p1.11.m11.1.1.2.3" xref="A2.SS2.p1.11.m11.1.1.2.3.cmml">t</mi><msub id="A2.SS2.p1.11.m11.1.1.3" xref="A2.SS2.p1.11.m11.1.1.3.cmml"><mi mathsize="90%" id="A2.SS2.p1.11.m11.1.1.3.2" xref="A2.SS2.p1.11.m11.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A2.SS2.p1.11.m11.1.1.3.3" xref="A2.SS2.p1.11.m11.1.1.3.3.cmml">small</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A2.SS2.p1.11.m11.1b"><apply id="A2.SS2.p1.11.m11.1.1.cmml" xref="A2.SS2.p1.11.m11.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.11.m11.1.1.1.cmml" xref="A2.SS2.p1.11.m11.1.1">superscript</csymbol><apply id="A2.SS2.p1.11.m11.1.1.2.cmml" xref="A2.SS2.p1.11.m11.1.1"><csymbol cd="ambiguous" id="A2.SS2.p1.11.m11.1.1.2.1.cmml" xref="A2.SS2.p1.11.m11.1.1">subscript</csymbol><ci id="A2.SS2.p1.11.m11.1.1.2.2.cmml" xref="A2.SS2.p1.11.m11.1.1.2.2">𝐱</ci><ci id="A2.SS2.p1.11.m11.1.1.2.3.cmml" xref="A2.SS2.p1.11.m11.1.1.2.3">𝑡</ci></apply><apply id="A2.SS2.p1.11.m11.1.1.3.cmml" xref="A2.SS2.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="A2.SS2.p1.11.m11.1.1.3.1.cmml" xref="A2.SS2.p1.11.m11.1.1.3">subscript</csymbol><ci id="A2.SS2.p1.11.m11.1.1.3.2.cmml" xref="A2.SS2.p1.11.m11.1.1.3.2">𝑘</ci><ci id="A2.SS2.p1.11.m11.1.1.3.3.cmml" xref="A2.SS2.p1.11.m11.1.1.3.3">small</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p1.11.m11.1c">\mathbf{x}_{t}^{k_{\mathrm{small}}}</annotation></semantics></math><span id="A2.SS2.p1.11.15" class="ltx_text" style="font-size:90%;"> in an autoregressive process.</span></p>
</div>
<div id="A2.SS2.p2" class="ltx_para">
<p id="A2.SS2.p2.1" class="ltx_p"><span id="A2.SS2.p2.1.1" class="ltx_text" style="font-size:90%;">It is widely appreciated that adding noise to data ameliorates long term compounding error in behavior cloning applications </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A2.SS2.p2.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib38" title="" class="ltx_ref">38</a>, <a href="#bib.bib41" title="" class="ltx_ref">41</a><span id="A2.SS2.p2.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A2.SS2.p2.1.4" class="ltx_text" style="font-size:90%;">, and even induces robustness to non-sequential adversarial attacks </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A2.SS2.p2.1.5.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib13" title="" class="ltx_ref">13</a><span id="A2.SS2.p2.1.6.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A2.SS2.p2.1.7" class="ltx_text" style="font-size:90%;">. In autoregressive video generation, the noised </span><math id="A2.SS2.p2.1.m1.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{\mathrm{small}}}" display="inline"><semantics id="A2.SS2.p2.1.m1.1a"><msubsup id="A2.SS2.p2.1.m1.1.1" xref="A2.SS2.p2.1.m1.1.1.cmml"><mi mathsize="90%" id="A2.SS2.p2.1.m1.1.1.2.2" xref="A2.SS2.p2.1.m1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A2.SS2.p2.1.m1.1.1.2.3" xref="A2.SS2.p2.1.m1.1.1.2.3.cmml">t</mi><msub id="A2.SS2.p2.1.m1.1.1.3" xref="A2.SS2.p2.1.m1.1.1.3.cmml"><mi mathsize="90%" id="A2.SS2.p2.1.m1.1.1.3.2" xref="A2.SS2.p2.1.m1.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A2.SS2.p2.1.m1.1.1.3.3" xref="A2.SS2.p2.1.m1.1.1.3.3.cmml">small</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A2.SS2.p2.1.m1.1b"><apply id="A2.SS2.p2.1.m1.1.1.cmml" xref="A2.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A2.SS2.p2.1.m1.1.1.1.cmml" xref="A2.SS2.p2.1.m1.1.1">superscript</csymbol><apply id="A2.SS2.p2.1.m1.1.1.2.cmml" xref="A2.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A2.SS2.p2.1.m1.1.1.2.1.cmml" xref="A2.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="A2.SS2.p2.1.m1.1.1.2.2.cmml" xref="A2.SS2.p2.1.m1.1.1.2.2">𝐱</ci><ci id="A2.SS2.p2.1.m1.1.1.2.3.cmml" xref="A2.SS2.p2.1.m1.1.1.2.3">𝑡</ci></apply><apply id="A2.SS2.p2.1.m1.1.1.3.cmml" xref="A2.SS2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="A2.SS2.p2.1.m1.1.1.3.1.cmml" xref="A2.SS2.p2.1.m1.1.1.3">subscript</csymbol><ci id="A2.SS2.p2.1.m1.1.1.3.2.cmml" xref="A2.SS2.p2.1.m1.1.1.3.2">𝑘</ci><ci id="A2.SS2.p2.1.m1.1.1.3.3.cmml" xref="A2.SS2.p2.1.m1.1.1.3.3">small</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS2.p2.1.m1.1c">\mathbf{x}_{t}^{k_{\mathrm{small}}}</annotation></semantics></math><span id="A2.SS2.p2.1.8" class="ltx_text" style="font-size:90%;"> is in-distribution for training, because Diffusion Forcing trains from noisy past observation in its training objective.
Hence, this method can be interpreted as a special case of the DART algorithm for behavior cloning </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A2.SS2.p2.1.9.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib41" title="" class="ltx_ref">41</a><span id="A2.SS2.p2.1.10.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A2.SS2.p2.1.11" class="ltx_text" style="font-size:90%;">, where the imitiator (in our case, video generator) is given actions (in our case, next video frames) from noisy observations (in our case, noised previous frames). Somewhat more precisely, because we are both tokens at training time to train Diffusion Forcing, and using slighlty noised tokens for autoregression at test time, our approach inherits the theoretical guarantess of the HINT algorithm </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A2.SS2.p2.1.12.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib5" title="" class="ltx_ref">5</a><span id="A2.SS2.p2.1.13.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A2.SS2.p2.1.14" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
</section>
<section id="A2.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">B.3 </span>Why Monte Carlo Tree Guidance relies on Diffusion Forcing</h3>

<div id="A2.SS3.p1" class="ltx_para">
<p id="A2.SS3.p1.1" class="ltx_p"><span id="A2.SS3.p1.1.1" class="ltx_text" style="font-size:90%;">Monte Carlo Tree Guidance provides substantial variance reduction in our estimate of a cost-to-go guidance (</span><a href="#A2.E1" title="Equation B.1 ‣ B.1 Guidance as planning ‣ Appendix B Additional Intuitions and Explainations ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">B.1</span></a><span id="A2.SS3.p1.1.2" class="ltx_text" style="font-size:90%;">). This technique crucially relies on the ability to rollout future tokens from current ones to use these sample rollouts to get Monte Carlo estimates for gradients. This is not feasible with full-sequence diffusion, because this requires denoising all tokens in tandem; thus, for a given fixed noise level, there is no obvious source of randomness to use for the Monte Carlo estimate. It may be possible to achieve variable horizon via the trick proposed in the following subsection to simulate future rollouts, but to our knowledge, this approach is nonstandard.</span></p>
</div>
</section>
<section id="A2.SS4" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">B.4 </span>Does the replacement technique lead to flexible horizons in full-sequence diffusion?</h3>

<div id="A2.SS4.p1" class="ltx_para">
<p id="A2.SS4.p1.8" class="ltx_p"><span id="A2.SS4.p1.8.1" class="ltx_text" style="font-size:90%;">A naive way to obtain flexible horizon generation in full-sequence diffusion is via the “replacement trick”: consider a full sequence model trained to diffuse </span><math id="A2.SS4.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}_{1:T}" display="inline"><semantics id="A2.SS4.p1.1.m1.1a"><msub id="A2.SS4.p1.1.m1.1.1" xref="A2.SS4.p1.1.m1.1.1.cmml"><mi mathsize="90%" id="A2.SS4.p1.1.m1.1.1.2" xref="A2.SS4.p1.1.m1.1.1.2.cmml">𝐱</mi><mrow id="A2.SS4.p1.1.m1.1.1.3" xref="A2.SS4.p1.1.m1.1.1.3.cmml"><mn mathsize="90%" id="A2.SS4.p1.1.m1.1.1.3.2" xref="A2.SS4.p1.1.m1.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A2.SS4.p1.1.m1.1.1.3.1" xref="A2.SS4.p1.1.m1.1.1.3.1.cmml">:</mo><mi mathsize="90%" id="A2.SS4.p1.1.m1.1.1.3.3" xref="A2.SS4.p1.1.m1.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A2.SS4.p1.1.m1.1b"><apply id="A2.SS4.p1.1.m1.1.1.cmml" xref="A2.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A2.SS4.p1.1.m1.1.1.1.cmml" xref="A2.SS4.p1.1.m1.1.1">subscript</csymbol><ci id="A2.SS4.p1.1.m1.1.1.2.cmml" xref="A2.SS4.p1.1.m1.1.1.2">𝐱</ci><apply id="A2.SS4.p1.1.m1.1.1.3.cmml" xref="A2.SS4.p1.1.m1.1.1.3"><ci id="A2.SS4.p1.1.m1.1.1.3.1.cmml" xref="A2.SS4.p1.1.m1.1.1.3.1">:</ci><cn type="integer" id="A2.SS4.p1.1.m1.1.1.3.2.cmml" xref="A2.SS4.p1.1.m1.1.1.3.2">1</cn><ci id="A2.SS4.p1.1.m1.1.1.3.3.cmml" xref="A2.SS4.p1.1.m1.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS4.p1.1.m1.1c">\mathbf{x}_{1:T}</annotation></semantics></math><span id="A2.SS4.p1.8.2" class="ltx_text" style="font-size:90%;">, which we partition into </span><math id="A2.SS4.p1.2.m2.1" class="ltx_math_unparsed" alttext="\mathbf{x}_{1:t-1},\mathbf{x}_{t:T}]" display="inline"><semantics id="A2.SS4.p1.2.m2.1a"><mrow id="A2.SS4.p1.2.m2.1b"><msub id="A2.SS4.p1.2.m2.1.1"><mi mathsize="90%" id="A2.SS4.p1.2.m2.1.1.2">𝐱</mi><mrow id="A2.SS4.p1.2.m2.1.1.3"><mn mathsize="90%" id="A2.SS4.p1.2.m2.1.1.3.2">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A2.SS4.p1.2.m2.1.1.3.1">:</mo><mrow id="A2.SS4.p1.2.m2.1.1.3.3"><mi mathsize="90%" id="A2.SS4.p1.2.m2.1.1.3.3.2">t</mi><mo mathsize="90%" id="A2.SS4.p1.2.m2.1.1.3.3.1">−</mo><mn mathsize="90%" id="A2.SS4.p1.2.m2.1.1.3.3.3">1</mn></mrow></mrow></msub><mo mathsize="90%" id="A2.SS4.p1.2.m2.1.2">,</mo><msub id="A2.SS4.p1.2.m2.1.3"><mi mathsize="90%" id="A2.SS4.p1.2.m2.1.3.2">𝐱</mi><mrow id="A2.SS4.p1.2.m2.1.3.3"><mi mathsize="90%" id="A2.SS4.p1.2.m2.1.3.3.2">t</mi><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A2.SS4.p1.2.m2.1.3.3.1">:</mo><mi mathsize="90%" id="A2.SS4.p1.2.m2.1.3.3.3">T</mi></mrow></msub><mo maxsize="90%" minsize="90%" id="A2.SS4.p1.2.m2.1.4">]</mo></mrow><annotation encoding="application/x-tex" id="A2.SS4.p1.2.m2.1c">\mathbf{x}_{1:t-1},\mathbf{x}_{t:T}]</annotation></semantics></math><span id="A2.SS4.p1.8.3" class="ltx_text" style="font-size:90%;">. Having diffused tokens </span><math id="A2.SS4.p1.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{1:t-1}" display="inline"><semantics id="A2.SS4.p1.3.m3.1a"><msub id="A2.SS4.p1.3.m3.1.1" xref="A2.SS4.p1.3.m3.1.1.cmml"><mi mathsize="90%" id="A2.SS4.p1.3.m3.1.1.2" xref="A2.SS4.p1.3.m3.1.1.2.cmml">𝐱</mi><mrow id="A2.SS4.p1.3.m3.1.1.3" xref="A2.SS4.p1.3.m3.1.1.3.cmml"><mn mathsize="90%" id="A2.SS4.p1.3.m3.1.1.3.2" xref="A2.SS4.p1.3.m3.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A2.SS4.p1.3.m3.1.1.3.1" xref="A2.SS4.p1.3.m3.1.1.3.1.cmml">:</mo><mrow id="A2.SS4.p1.3.m3.1.1.3.3" xref="A2.SS4.p1.3.m3.1.1.3.3.cmml"><mi mathsize="90%" id="A2.SS4.p1.3.m3.1.1.3.3.2" xref="A2.SS4.p1.3.m3.1.1.3.3.2.cmml">t</mi><mo mathsize="90%" id="A2.SS4.p1.3.m3.1.1.3.3.1" xref="A2.SS4.p1.3.m3.1.1.3.3.1.cmml">−</mo><mn mathsize="90%" id="A2.SS4.p1.3.m3.1.1.3.3.3" xref="A2.SS4.p1.3.m3.1.1.3.3.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" 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xref="A2.SS4.p1.4.m4.1.1.1.1.3">𝑘</ci></apply><apply id="A2.SS4.p1.4.m4.2.2.2.2.cmml" xref="A2.SS4.p1.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="A2.SS4.p1.4.m4.2.2.2.2.1.cmml" xref="A2.SS4.p1.4.m4.2.2.2.2">superscript</csymbol><apply id="A2.SS4.p1.4.m4.2.2.2.2.2.cmml" xref="A2.SS4.p1.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="A2.SS4.p1.4.m4.2.2.2.2.2.1.cmml" xref="A2.SS4.p1.4.m4.2.2.2.2">subscript</csymbol><ci id="A2.SS4.p1.4.m4.2.2.2.2.2.2.cmml" xref="A2.SS4.p1.4.m4.2.2.2.2.2.2">𝐱</ci><apply id="A2.SS4.p1.4.m4.2.2.2.2.2.3.cmml" xref="A2.SS4.p1.4.m4.2.2.2.2.2.3"><ci id="A2.SS4.p1.4.m4.2.2.2.2.2.3.1.cmml" xref="A2.SS4.p1.4.m4.2.2.2.2.2.3.1">:</ci><ci id="A2.SS4.p1.4.m4.2.2.2.2.2.3.2.cmml" xref="A2.SS4.p1.4.m4.2.2.2.2.2.3.2">𝑡</ci><ci id="A2.SS4.p1.4.m4.2.2.2.2.2.3.3.cmml" xref="A2.SS4.p1.4.m4.2.2.2.2.2.3.3">𝑇</ci></apply></apply><ci id="A2.SS4.p1.4.m4.2.2.2.2.3.cmml" xref="A2.SS4.p1.4.m4.2.2.2.2.3">𝑘</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="A2.SS4.p1.4.m4.2c">[\tilde{\mathbf{x}}_{1:t-1}^{k},\mathbf{x}_{t:T}^{k}]</annotation></semantics></math><span id="A2.SS4.p1.8.5" class="ltx_text" style="font-size:90%;">, where we </span><em id="A2.SS4.p1.8.6" class="ltx_emph ltx_font_italic" style="font-size:90%;">fix</em><span id="A2.SS4.p1.8.7" class="ltx_text" style="font-size:90%;"> </span><math id="A2.SS4.p1.5.m5.1" class="ltx_Math" alttext="\tilde{\mathbf{x}}_{1:t-1}^{k}=\mathbf{x}_{1:t-1}" display="inline"><semantics id="A2.SS4.p1.5.m5.1a"><mrow id="A2.SS4.p1.5.m5.1.1" xref="A2.SS4.p1.5.m5.1.1.cmml"><msubsup id="A2.SS4.p1.5.m5.1.1.2" xref="A2.SS4.p1.5.m5.1.1.2.cmml"><mover accent="true" id="A2.SS4.p1.5.m5.1.1.2.2.2" xref="A2.SS4.p1.5.m5.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.SS4.p1.5.m5.1.1.2.2.2.2" xref="A2.SS4.p1.5.m5.1.1.2.2.2.2.cmml">𝐱</mi><mo mathsize="90%" id="A2.SS4.p1.5.m5.1.1.2.2.2.1" xref="A2.SS4.p1.5.m5.1.1.2.2.2.1.cmml">~</mo></mover><mrow id="A2.SS4.p1.5.m5.1.1.2.2.3" xref="A2.SS4.p1.5.m5.1.1.2.2.3.cmml"><mn mathsize="90%" id="A2.SS4.p1.5.m5.1.1.2.2.3.2" xref="A2.SS4.p1.5.m5.1.1.2.2.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A2.SS4.p1.5.m5.1.1.2.2.3.1" xref="A2.SS4.p1.5.m5.1.1.2.2.3.1.cmml">:</mo><mrow id="A2.SS4.p1.5.m5.1.1.2.2.3.3" xref="A2.SS4.p1.5.m5.1.1.2.2.3.3.cmml"><mi mathsize="90%" id="A2.SS4.p1.5.m5.1.1.2.2.3.3.2" xref="A2.SS4.p1.5.m5.1.1.2.2.3.3.2.cmml">t</mi><mo mathsize="90%" id="A2.SS4.p1.5.m5.1.1.2.2.3.3.1" xref="A2.SS4.p1.5.m5.1.1.2.2.3.3.1.cmml">−</mo><mn mathsize="90%" id="A2.SS4.p1.5.m5.1.1.2.2.3.3.3" xref="A2.SS4.p1.5.m5.1.1.2.2.3.3.3.cmml">1</mn></mrow></mrow><mi mathsize="90%" id="A2.SS4.p1.5.m5.1.1.2.3" xref="A2.SS4.p1.5.m5.1.1.2.3.cmml">k</mi></msubsup><mo mathsize="90%" id="A2.SS4.p1.5.m5.1.1.1" xref="A2.SS4.p1.5.m5.1.1.1.cmml">=</mo><msub id="A2.SS4.p1.5.m5.1.1.3" xref="A2.SS4.p1.5.m5.1.1.3.cmml"><mi mathsize="90%" id="A2.SS4.p1.5.m5.1.1.3.2" xref="A2.SS4.p1.5.m5.1.1.3.2.cmml">𝐱</mi><mrow id="A2.SS4.p1.5.m5.1.1.3.3" xref="A2.SS4.p1.5.m5.1.1.3.3.cmml"><mn mathsize="90%" id="A2.SS4.p1.5.m5.1.1.3.3.2" xref="A2.SS4.p1.5.m5.1.1.3.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A2.SS4.p1.5.m5.1.1.3.3.1" xref="A2.SS4.p1.5.m5.1.1.3.3.1.cmml">:</mo><mrow id="A2.SS4.p1.5.m5.1.1.3.3.3" xref="A2.SS4.p1.5.m5.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.SS4.p1.5.m5.1.1.3.3.3.2" xref="A2.SS4.p1.5.m5.1.1.3.3.3.2.cmml">t</mi><mo mathsize="90%" id="A2.SS4.p1.5.m5.1.1.3.3.3.1" xref="A2.SS4.p1.5.m5.1.1.3.3.3.1.cmml">−</mo><mn mathsize="90%" id="A2.SS4.p1.5.m5.1.1.3.3.3.3" xref="A2.SS4.p1.5.m5.1.1.3.3.3.3.cmml">1</mn></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.SS4.p1.5.m5.1b"><apply id="A2.SS4.p1.5.m5.1.1.cmml" xref="A2.SS4.p1.5.m5.1.1"><eq id="A2.SS4.p1.5.m5.1.1.1.cmml" xref="A2.SS4.p1.5.m5.1.1.1"></eq><apply id="A2.SS4.p1.5.m5.1.1.2.cmml" xref="A2.SS4.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="A2.SS4.p1.5.m5.1.1.2.1.cmml" xref="A2.SS4.p1.5.m5.1.1.2">superscript</csymbol><apply id="A2.SS4.p1.5.m5.1.1.2.2.cmml" xref="A2.SS4.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="A2.SS4.p1.5.m5.1.1.2.2.1.cmml" xref="A2.SS4.p1.5.m5.1.1.2">subscript</csymbol><apply id="A2.SS4.p1.5.m5.1.1.2.2.2.cmml" xref="A2.SS4.p1.5.m5.1.1.2.2.2"><ci id="A2.SS4.p1.5.m5.1.1.2.2.2.1.cmml" xref="A2.SS4.p1.5.m5.1.1.2.2.2.1">~</ci><ci id="A2.SS4.p1.5.m5.1.1.2.2.2.2.cmml" xref="A2.SS4.p1.5.m5.1.1.2.2.2.2">𝐱</ci></apply><apply id="A2.SS4.p1.5.m5.1.1.2.2.3.cmml" xref="A2.SS4.p1.5.m5.1.1.2.2.3"><ci id="A2.SS4.p1.5.m5.1.1.2.2.3.1.cmml" xref="A2.SS4.p1.5.m5.1.1.2.2.3.1">:</ci><cn type="integer" id="A2.SS4.p1.5.m5.1.1.2.2.3.2.cmml" xref="A2.SS4.p1.5.m5.1.1.2.2.3.2">1</cn><apply id="A2.SS4.p1.5.m5.1.1.2.2.3.3.cmml" xref="A2.SS4.p1.5.m5.1.1.2.2.3.3"><minus id="A2.SS4.p1.5.m5.1.1.2.2.3.3.1.cmml" xref="A2.SS4.p1.5.m5.1.1.2.2.3.3.1"></minus><ci id="A2.SS4.p1.5.m5.1.1.2.2.3.3.2.cmml" xref="A2.SS4.p1.5.m5.1.1.2.2.3.3.2">𝑡</ci><cn type="integer" id="A2.SS4.p1.5.m5.1.1.2.2.3.3.3.cmml" xref="A2.SS4.p1.5.m5.1.1.2.2.3.3.3">1</cn></apply></apply></apply><ci id="A2.SS4.p1.5.m5.1.1.2.3.cmml" xref="A2.SS4.p1.5.m5.1.1.2.3">𝑘</ci></apply><apply id="A2.SS4.p1.5.m5.1.1.3.cmml" xref="A2.SS4.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="A2.SS4.p1.5.m5.1.1.3.1.cmml" xref="A2.SS4.p1.5.m5.1.1.3">subscript</csymbol><ci id="A2.SS4.p1.5.m5.1.1.3.2.cmml" xref="A2.SS4.p1.5.m5.1.1.3.2">𝐱</ci><apply id="A2.SS4.p1.5.m5.1.1.3.3.cmml" xref="A2.SS4.p1.5.m5.1.1.3.3"><ci id="A2.SS4.p1.5.m5.1.1.3.3.1.cmml" xref="A2.SS4.p1.5.m5.1.1.3.3.1">:</ci><cn type="integer" id="A2.SS4.p1.5.m5.1.1.3.3.2.cmml" xref="A2.SS4.p1.5.m5.1.1.3.3.2">1</cn><apply id="A2.SS4.p1.5.m5.1.1.3.3.3.cmml" xref="A2.SS4.p1.5.m5.1.1.3.3.3"><minus id="A2.SS4.p1.5.m5.1.1.3.3.3.1.cmml" xref="A2.SS4.p1.5.m5.1.1.3.3.3.1"></minus><ci id="A2.SS4.p1.5.m5.1.1.3.3.3.2.cmml" xref="A2.SS4.p1.5.m5.1.1.3.3.3.2">𝑡</ci><cn type="integer" id="A2.SS4.p1.5.m5.1.1.3.3.3.3.cmml" xref="A2.SS4.p1.5.m5.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS4.p1.5.m5.1c">\tilde{\mathbf{x}}_{1:t-1}^{k}=\mathbf{x}_{1:t-1}</annotation></semantics></math><span id="A2.SS4.p1.8.8" class="ltx_text" style="font-size:90%;"> to be the previously generated token, and only have score gradients update the remaining </span><math id="A2.SS4.p1.6.m6.1" class="ltx_Math" alttext="\mathbf{x}_{t:T}^{k}" display="inline"><semantics id="A2.SS4.p1.6.m6.1a"><msubsup id="A2.SS4.p1.6.m6.1.1" xref="A2.SS4.p1.6.m6.1.1.cmml"><mi mathsize="90%" id="A2.SS4.p1.6.m6.1.1.2.2" xref="A2.SS4.p1.6.m6.1.1.2.2.cmml">𝐱</mi><mrow id="A2.SS4.p1.6.m6.1.1.2.3" xref="A2.SS4.p1.6.m6.1.1.2.3.cmml"><mi mathsize="90%" id="A2.SS4.p1.6.m6.1.1.2.3.2" xref="A2.SS4.p1.6.m6.1.1.2.3.2.cmml">t</mi><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A2.SS4.p1.6.m6.1.1.2.3.1" xref="A2.SS4.p1.6.m6.1.1.2.3.1.cmml">:</mo><mi mathsize="90%" id="A2.SS4.p1.6.m6.1.1.2.3.3" xref="A2.SS4.p1.6.m6.1.1.2.3.3.cmml">T</mi></mrow><mi mathsize="90%" id="A2.SS4.p1.6.m6.1.1.3" xref="A2.SS4.p1.6.m6.1.1.3.cmml">k</mi></msubsup><annotation-xml encoding="MathML-Content" id="A2.SS4.p1.6.m6.1b"><apply id="A2.SS4.p1.6.m6.1.1.cmml" xref="A2.SS4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A2.SS4.p1.6.m6.1.1.1.cmml" xref="A2.SS4.p1.6.m6.1.1">superscript</csymbol><apply id="A2.SS4.p1.6.m6.1.1.2.cmml" xref="A2.SS4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A2.SS4.p1.6.m6.1.1.2.1.cmml" xref="A2.SS4.p1.6.m6.1.1">subscript</csymbol><ci id="A2.SS4.p1.6.m6.1.1.2.2.cmml" xref="A2.SS4.p1.6.m6.1.1.2.2">𝐱</ci><apply id="A2.SS4.p1.6.m6.1.1.2.3.cmml" xref="A2.SS4.p1.6.m6.1.1.2.3"><ci id="A2.SS4.p1.6.m6.1.1.2.3.1.cmml" xref="A2.SS4.p1.6.m6.1.1.2.3.1">:</ci><ci id="A2.SS4.p1.6.m6.1.1.2.3.2.cmml" xref="A2.SS4.p1.6.m6.1.1.2.3.2">𝑡</ci><ci id="A2.SS4.p1.6.m6.1.1.2.3.3.cmml" xref="A2.SS4.p1.6.m6.1.1.2.3.3">𝑇</ci></apply></apply><ci id="A2.SS4.p1.6.m6.1.1.3.cmml" xref="A2.SS4.p1.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS4.p1.6.m6.1c">\mathbf{x}_{t:T}^{k}</annotation></semantics></math><span id="A2.SS4.p1.8.9" class="ltx_text" style="font-size:90%;">. One clear disadvantage of this method is inefficiency - one still needs to diffuse the whole sequence even when there is one step left at </span><math id="A2.SS4.p1.7.m7.1" class="ltx_Math" alttext="t=T-1" display="inline"><semantics id="A2.SS4.p1.7.m7.1a"><mrow id="A2.SS4.p1.7.m7.1.1" xref="A2.SS4.p1.7.m7.1.1.cmml"><mi mathsize="90%" id="A2.SS4.p1.7.m7.1.1.2" xref="A2.SS4.p1.7.m7.1.1.2.cmml">t</mi><mo mathsize="90%" id="A2.SS4.p1.7.m7.1.1.1" xref="A2.SS4.p1.7.m7.1.1.1.cmml">=</mo><mrow id="A2.SS4.p1.7.m7.1.1.3" xref="A2.SS4.p1.7.m7.1.1.3.cmml"><mi mathsize="90%" id="A2.SS4.p1.7.m7.1.1.3.2" xref="A2.SS4.p1.7.m7.1.1.3.2.cmml">T</mi><mo mathsize="90%" id="A2.SS4.p1.7.m7.1.1.3.1" xref="A2.SS4.p1.7.m7.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A2.SS4.p1.7.m7.1.1.3.3" xref="A2.SS4.p1.7.m7.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS4.p1.7.m7.1b"><apply id="A2.SS4.p1.7.m7.1.1.cmml" xref="A2.SS4.p1.7.m7.1.1"><eq id="A2.SS4.p1.7.m7.1.1.1.cmml" xref="A2.SS4.p1.7.m7.1.1.1"></eq><ci id="A2.SS4.p1.7.m7.1.1.2.cmml" xref="A2.SS4.p1.7.m7.1.1.2">𝑡</ci><apply id="A2.SS4.p1.7.m7.1.1.3.cmml" xref="A2.SS4.p1.7.m7.1.1.3"><minus id="A2.SS4.p1.7.m7.1.1.3.1.cmml" xref="A2.SS4.p1.7.m7.1.1.3.1"></minus><ci id="A2.SS4.p1.7.m7.1.1.3.2.cmml" xref="A2.SS4.p1.7.m7.1.1.3.2">𝑇</ci><cn type="integer" id="A2.SS4.p1.7.m7.1.1.3.3.cmml" xref="A2.SS4.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS4.p1.7.m7.1c">t=T-1</annotation></semantics></math><span id="A2.SS4.p1.8.10" class="ltx_text" style="font-size:90%;">. What’s more, </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A2.SS4.p1.8.11.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib31" title="" class="ltx_ref">31</a><span id="A2.SS4.p1.8.12.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A2.SS4.p1.8.13" class="ltx_text" style="font-size:90%;"> points out that this approach of conditioning, named “conditioning by replacement”, is both mathematically unprincipled and can lead to inconsistency in the generated sequence. The best fix proposed by </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A2.SS4.p1.8.14.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib31" title="" class="ltx_ref">31</a><span id="A2.SS4.p1.8.15.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A2.SS4.p1.8.16" class="ltx_text" style="font-size:90%;"> incorporates an additional gradient term with respect to </span><math id="A2.SS4.p1.8.m8.1" class="ltx_Math" alttext="\mathbf{x}_{t:T}" display="inline"><semantics id="A2.SS4.p1.8.m8.1a"><msub id="A2.SS4.p1.8.m8.1.1" xref="A2.SS4.p1.8.m8.1.1.cmml"><mi mathsize="90%" id="A2.SS4.p1.8.m8.1.1.2" xref="A2.SS4.p1.8.m8.1.1.2.cmml">𝐱</mi><mrow id="A2.SS4.p1.8.m8.1.1.3" xref="A2.SS4.p1.8.m8.1.1.3.cmml"><mi mathsize="90%" id="A2.SS4.p1.8.m8.1.1.3.2" xref="A2.SS4.p1.8.m8.1.1.3.2.cmml">t</mi><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A2.SS4.p1.8.m8.1.1.3.1" xref="A2.SS4.p1.8.m8.1.1.3.1.cmml">:</mo><mi mathsize="90%" id="A2.SS4.p1.8.m8.1.1.3.3" xref="A2.SS4.p1.8.m8.1.1.3.3.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A2.SS4.p1.8.m8.1b"><apply id="A2.SS4.p1.8.m8.1.1.cmml" xref="A2.SS4.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A2.SS4.p1.8.m8.1.1.1.cmml" xref="A2.SS4.p1.8.m8.1.1">subscript</csymbol><ci id="A2.SS4.p1.8.m8.1.1.2.cmml" xref="A2.SS4.p1.8.m8.1.1.2">𝐱</ci><apply id="A2.SS4.p1.8.m8.1.1.3.cmml" xref="A2.SS4.p1.8.m8.1.1.3"><ci id="A2.SS4.p1.8.m8.1.1.3.1.cmml" xref="A2.SS4.p1.8.m8.1.1.3.1">:</ci><ci id="A2.SS4.p1.8.m8.1.1.3.2.cmml" xref="A2.SS4.p1.8.m8.1.1.3.2">𝑡</ci><ci id="A2.SS4.p1.8.m8.1.1.3.3.cmml" xref="A2.SS4.p1.8.m8.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS4.p1.8.m8.1c">\mathbf{x}_{t:T}</annotation></semantics></math><span id="A2.SS4.p1.8.17" class="ltx_text" style="font-size:90%;"> at every diffusion step; this is still an incomplete fix and suffers the computation cost of an extra backward propagation for every sampling step.</span></p>
</div>
</section>
<section id="A2.SS5" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">B.5 </span>Further connection to Bayesian filtering</h3>

<div id="A2.SS5.p1" class="ltx_para">
<p id="A2.SS5.p1.12" class="ltx_p"><span id="A2.SS5.p1.12.1" class="ltx_text" style="font-size:90%;">The core idea of Diffusion Forcing can be interpreted as using diffusion to construct an interpolation between prior distribution and posterior distribution of a Bayes filter. Consider the hybrid distribution </span><math id="A2.SS5.p1.1.m1.1" class="ltx_Math" alttext="p(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k})" display="inline"><semantics id="A2.SS5.p1.1.m1.1a"><mrow id="A2.SS5.p1.1.m1.1.1" xref="A2.SS5.p1.1.m1.1.1.cmml"><mi mathsize="90%" id="A2.SS5.p1.1.m1.1.1.3" xref="A2.SS5.p1.1.m1.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="A2.SS5.p1.1.m1.1.1.2" xref="A2.SS5.p1.1.m1.1.1.2.cmml">​</mo><mrow id="A2.SS5.p1.1.m1.1.1.1.1" xref="A2.SS5.p1.1.m1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS5.p1.1.m1.1.1.1.1.2" xref="A2.SS5.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="A2.SS5.p1.1.m1.1.1.1.1.1" xref="A2.SS5.p1.1.m1.1.1.1.1.1.cmml"><msub id="A2.SS5.p1.1.m1.1.1.1.1.1.4" xref="A2.SS5.p1.1.m1.1.1.1.1.1.4.cmml"><mi mathsize="90%" id="A2.SS5.p1.1.m1.1.1.1.1.1.4.2" xref="A2.SS5.p1.1.m1.1.1.1.1.1.4.2.cmml">𝐳</mi><mi mathsize="90%" id="A2.SS5.p1.1.m1.1.1.1.1.1.4.3" xref="A2.SS5.p1.1.m1.1.1.1.1.1.4.3.cmml">t</mi></msub><mo fence="false" 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id="A2.SS5.p1.1.m1.1c">p(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t}^{k})</annotation></semantics></math><span id="A2.SS5.p1.12.2" class="ltx_text" style="font-size:90%;">. When </span><math id="A2.SS5.p1.2.m2.1" class="ltx_Math" alttext="k=0" display="inline"><semantics id="A2.SS5.p1.2.m2.1a"><mrow id="A2.SS5.p1.2.m2.1.1" xref="A2.SS5.p1.2.m2.1.1.cmml"><mi mathsize="90%" id="A2.SS5.p1.2.m2.1.1.2" xref="A2.SS5.p1.2.m2.1.1.2.cmml">k</mi><mo mathsize="90%" id="A2.SS5.p1.2.m2.1.1.1" xref="A2.SS5.p1.2.m2.1.1.1.cmml">=</mo><mn mathsize="90%" id="A2.SS5.p1.2.m2.1.1.3" xref="A2.SS5.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.2.m2.1b"><apply id="A2.SS5.p1.2.m2.1.1.cmml" xref="A2.SS5.p1.2.m2.1.1"><eq id="A2.SS5.p1.2.m2.1.1.1.cmml" xref="A2.SS5.p1.2.m2.1.1.1"></eq><ci id="A2.SS5.p1.2.m2.1.1.2.cmml" xref="A2.SS5.p1.2.m2.1.1.2">𝑘</ci><cn type="integer" id="A2.SS5.p1.2.m2.1.1.3.cmml" xref="A2.SS5.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS5.p1.2.m2.1c">k=0</annotation></semantics></math><span id="A2.SS5.p1.12.3" class="ltx_text" style="font-size:90%;">, this hybrid distribution becomes the posterior </span><math id="A2.SS5.p1.3.m3.1" class="ltx_Math" alttext="p(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t})" display="inline"><semantics id="A2.SS5.p1.3.m3.1a"><mrow id="A2.SS5.p1.3.m3.1.1" xref="A2.SS5.p1.3.m3.1.1.cmml"><mi mathsize="90%" id="A2.SS5.p1.3.m3.1.1.3" xref="A2.SS5.p1.3.m3.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="A2.SS5.p1.3.m3.1.1.2" xref="A2.SS5.p1.3.m3.1.1.2.cmml">​</mo><mrow id="A2.SS5.p1.3.m3.1.1.1.1" xref="A2.SS5.p1.3.m3.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.2" xref="A2.SS5.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="A2.SS5.p1.3.m3.1.1.1.1.1" xref="A2.SS5.p1.3.m3.1.1.1.1.1.cmml"><msub id="A2.SS5.p1.3.m3.1.1.1.1.1.4" xref="A2.SS5.p1.3.m3.1.1.1.1.1.4.cmml"><mi mathsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.1.4.2" xref="A2.SS5.p1.3.m3.1.1.1.1.1.4.2.cmml">𝐳</mi><mi mathsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.1.4.3" xref="A2.SS5.p1.3.m3.1.1.1.1.1.4.3.cmml">t</mi></msub><mo fence="false" mathsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.1.3" xref="A2.SS5.p1.3.m3.1.1.1.1.1.3.cmml">|</mo><mrow id="A2.SS5.p1.3.m3.1.1.1.1.1.2.2" xref="A2.SS5.p1.3.m3.1.1.1.1.1.2.3.cmml"><msub id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.2" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.2" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.1" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.3" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo mathsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.3" xref="A2.SS5.p1.3.m3.1.1.1.1.1.2.3.cmml">,</mo><msub id="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2" xref="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.2" xref="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.3" xref="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.3.cmml">t</mi></msub></mrow></mrow><mo maxsize="90%" minsize="90%" id="A2.SS5.p1.3.m3.1.1.1.1.3" xref="A2.SS5.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.3.m3.1b"><apply id="A2.SS5.p1.3.m3.1.1.cmml" xref="A2.SS5.p1.3.m3.1.1"><times id="A2.SS5.p1.3.m3.1.1.2.cmml" xref="A2.SS5.p1.3.m3.1.1.2"></times><ci id="A2.SS5.p1.3.m3.1.1.3.cmml" xref="A2.SS5.p1.3.m3.1.1.3">𝑝</ci><apply id="A2.SS5.p1.3.m3.1.1.1.1.1.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1"><csymbol cd="latexml" id="A2.SS5.p1.3.m3.1.1.1.1.1.3.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.3">conditional</csymbol><apply id="A2.SS5.p1.3.m3.1.1.1.1.1.4.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.4"><csymbol cd="ambiguous" id="A2.SS5.p1.3.m3.1.1.1.1.1.4.1.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.4">subscript</csymbol><ci id="A2.SS5.p1.3.m3.1.1.1.1.1.4.2.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.4.2">𝐳</ci><ci id="A2.SS5.p1.3.m3.1.1.1.1.1.4.3.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.4.3">𝑡</ci></apply><list id="A2.SS5.p1.3.m3.1.1.1.1.1.2.3.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.2.2"><apply id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.1.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.2.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.2">𝐳</ci><apply id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3"><minus id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.1.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.1"></minus><ci id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.2.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.2">𝑡</ci><cn type="integer" id="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.3.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.1.1.1.3.3">1</cn></apply></apply><apply id="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.1.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.2.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.2">𝐱</ci><ci id="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.3.cmml" xref="A2.SS5.p1.3.m3.1.1.1.1.1.2.2.2.3">𝑡</ci></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS5.p1.3.m3.1c">p(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}_{t})</annotation></semantics></math><span id="A2.SS5.p1.12.4" class="ltx_text" style="font-size:90%;">. On the other hand, when </span><math id="A2.SS5.p1.4.m4.1" class="ltx_Math" alttext="k=K" display="inline"><semantics id="A2.SS5.p1.4.m4.1a"><mrow id="A2.SS5.p1.4.m4.1.1" xref="A2.SS5.p1.4.m4.1.1.cmml"><mi mathsize="90%" id="A2.SS5.p1.4.m4.1.1.2" xref="A2.SS5.p1.4.m4.1.1.2.cmml">k</mi><mo mathsize="90%" id="A2.SS5.p1.4.m4.1.1.1" xref="A2.SS5.p1.4.m4.1.1.1.cmml">=</mo><mi mathsize="90%" id="A2.SS5.p1.4.m4.1.1.3" xref="A2.SS5.p1.4.m4.1.1.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.4.m4.1b"><apply id="A2.SS5.p1.4.m4.1.1.cmml" xref="A2.SS5.p1.4.m4.1.1"><eq id="A2.SS5.p1.4.m4.1.1.1.cmml" xref="A2.SS5.p1.4.m4.1.1.1"></eq><ci id="A2.SS5.p1.4.m4.1.1.2.cmml" xref="A2.SS5.p1.4.m4.1.1.2">𝑘</ci><ci id="A2.SS5.p1.4.m4.1.1.3.cmml" xref="A2.SS5.p1.4.m4.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS5.p1.4.m4.1c">k=K</annotation></semantics></math><span id="A2.SS5.p1.12.5" class="ltx_text" style="font-size:90%;">, the hybrid distribution becomes </span><math id="A2.SS5.p1.5.m5.2" class="ltx_Math" alttext="p(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{n})" display="inline"><semantics id="A2.SS5.p1.5.m5.2a"><mrow id="A2.SS5.p1.5.m5.2.2" xref="A2.SS5.p1.5.m5.2.2.cmml"><mi mathsize="90%" id="A2.SS5.p1.5.m5.2.2.3" xref="A2.SS5.p1.5.m5.2.2.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="A2.SS5.p1.5.m5.2.2.2" xref="A2.SS5.p1.5.m5.2.2.2.cmml">​</mo><mrow id="A2.SS5.p1.5.m5.2.2.1.1" xref="A2.SS5.p1.5.m5.2.2.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS5.p1.5.m5.2.2.1.1.2" xref="A2.SS5.p1.5.m5.2.2.1.1.1.cmml">(</mo><mrow id="A2.SS5.p1.5.m5.2.2.1.1.1" xref="A2.SS5.p1.5.m5.2.2.1.1.1.cmml"><msub id="A2.SS5.p1.5.m5.2.2.1.1.1.3" xref="A2.SS5.p1.5.m5.2.2.1.1.1.3.cmml"><mi mathsize="90%" id="A2.SS5.p1.5.m5.2.2.1.1.1.3.2" xref="A2.SS5.p1.5.m5.2.2.1.1.1.3.2.cmml">𝐳</mi><mi mathsize="90%" id="A2.SS5.p1.5.m5.2.2.1.1.1.3.3" xref="A2.SS5.p1.5.m5.2.2.1.1.1.3.3.cmml">t</mi></msub><mo fence="false" mathsize="90%" id="A2.SS5.p1.5.m5.2.2.1.1.1.2" xref="A2.SS5.p1.5.m5.2.2.1.1.1.2.cmml">|</mo><mrow id="A2.SS5.p1.5.m5.2.2.1.1.1.1.1" xref="A2.SS5.p1.5.m5.2.2.1.1.1.1.2.cmml"><msub id="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1" xref="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.2" xref="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.3" xref="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.3.2" xref="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.3.1" xref="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.3.3" xref="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo mathsize="90%" id="A2.SS5.p1.5.m5.2.2.1.1.1.1.1.2" xref="A2.SS5.p1.5.m5.2.2.1.1.1.1.2.cmml">,</mo><mi mathsize="90%" id="A2.SS5.p1.5.m5.1.1" xref="A2.SS5.p1.5.m5.1.1.cmml">𝐧</mi></mrow></mrow><mo maxsize="90%" minsize="90%" id="A2.SS5.p1.5.m5.2.2.1.1.3" xref="A2.SS5.p1.5.m5.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.5.m5.2b"><apply id="A2.SS5.p1.5.m5.2.2.cmml" xref="A2.SS5.p1.5.m5.2.2"><times id="A2.SS5.p1.5.m5.2.2.2.cmml" xref="A2.SS5.p1.5.m5.2.2.2"></times><ci id="A2.SS5.p1.5.m5.2.2.3.cmml" xref="A2.SS5.p1.5.m5.2.2.3">𝑝</ci><apply id="A2.SS5.p1.5.m5.2.2.1.1.1.cmml" xref="A2.SS5.p1.5.m5.2.2.1.1"><csymbol cd="latexml" id="A2.SS5.p1.5.m5.2.2.1.1.1.2.cmml" xref="A2.SS5.p1.5.m5.2.2.1.1.1.2">conditional</csymbol><apply id="A2.SS5.p1.5.m5.2.2.1.1.1.3.cmml" xref="A2.SS5.p1.5.m5.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A2.SS5.p1.5.m5.2.2.1.1.1.3.1.cmml" xref="A2.SS5.p1.5.m5.2.2.1.1.1.3">subscript</csymbol><ci id="A2.SS5.p1.5.m5.2.2.1.1.1.3.2.cmml" xref="A2.SS5.p1.5.m5.2.2.1.1.1.3.2">𝐳</ci><ci id="A2.SS5.p1.5.m5.2.2.1.1.1.3.3.cmml" xref="A2.SS5.p1.5.m5.2.2.1.1.1.3.3">𝑡</ci></apply><list id="A2.SS5.p1.5.m5.2.2.1.1.1.1.2.cmml" 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id="A2.SS5.p1.5.m5.2c">p(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{n})</annotation></semantics></math><span id="A2.SS5.p1.12.6" class="ltx_text" style="font-size:90%;"> for </span><math id="A2.SS5.p1.6.m6.2" class="ltx_Math" alttext="\mathbf{n}\sim\mathcal{N}(0,\mathbf{I})" display="inline"><semantics id="A2.SS5.p1.6.m6.2a"><mrow id="A2.SS5.p1.6.m6.2.3" xref="A2.SS5.p1.6.m6.2.3.cmml"><mi mathsize="90%" id="A2.SS5.p1.6.m6.2.3.2" xref="A2.SS5.p1.6.m6.2.3.2.cmml">𝐧</mi><mo mathsize="90%" id="A2.SS5.p1.6.m6.2.3.1" xref="A2.SS5.p1.6.m6.2.3.1.cmml">∼</mo><mrow id="A2.SS5.p1.6.m6.2.3.3" xref="A2.SS5.p1.6.m6.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A2.SS5.p1.6.m6.2.3.3.2" xref="A2.SS5.p1.6.m6.2.3.3.2.cmml">𝒩</mi><mo lspace="0em" rspace="0em" id="A2.SS5.p1.6.m6.2.3.3.1" xref="A2.SS5.p1.6.m6.2.3.3.1.cmml">​</mo><mrow id="A2.SS5.p1.6.m6.2.3.3.3.2" xref="A2.SS5.p1.6.m6.2.3.3.3.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS5.p1.6.m6.2.3.3.3.2.1" xref="A2.SS5.p1.6.m6.2.3.3.3.1.cmml">(</mo><mn mathsize="90%" id="A2.SS5.p1.6.m6.1.1" xref="A2.SS5.p1.6.m6.1.1.cmml">0</mn><mo mathsize="90%" id="A2.SS5.p1.6.m6.2.3.3.3.2.2" xref="A2.SS5.p1.6.m6.2.3.3.3.1.cmml">,</mo><mi mathsize="90%" id="A2.SS5.p1.6.m6.2.2" xref="A2.SS5.p1.6.m6.2.2.cmml">𝐈</mi><mo maxsize="90%" minsize="90%" id="A2.SS5.p1.6.m6.2.3.3.3.2.3" xref="A2.SS5.p1.6.m6.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.6.m6.2b"><apply id="A2.SS5.p1.6.m6.2.3.cmml" xref="A2.SS5.p1.6.m6.2.3"><csymbol cd="latexml" id="A2.SS5.p1.6.m6.2.3.1.cmml" xref="A2.SS5.p1.6.m6.2.3.1">similar-to</csymbol><ci id="A2.SS5.p1.6.m6.2.3.2.cmml" xref="A2.SS5.p1.6.m6.2.3.2">𝐧</ci><apply id="A2.SS5.p1.6.m6.2.3.3.cmml" xref="A2.SS5.p1.6.m6.2.3.3"><times id="A2.SS5.p1.6.m6.2.3.3.1.cmml" xref="A2.SS5.p1.6.m6.2.3.3.1"></times><ci id="A2.SS5.p1.6.m6.2.3.3.2.cmml" xref="A2.SS5.p1.6.m6.2.3.3.2">𝒩</ci><interval closure="open" id="A2.SS5.p1.6.m6.2.3.3.3.1.cmml" xref="A2.SS5.p1.6.m6.2.3.3.3.2"><cn type="integer" id="A2.SS5.p1.6.m6.1.1.cmml" xref="A2.SS5.p1.6.m6.1.1">0</cn><ci id="A2.SS5.p1.6.m6.2.2.cmml" xref="A2.SS5.p1.6.m6.2.2">𝐈</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS5.p1.6.m6.2c">\mathbf{n}\sim\mathcal{N}(0,\mathbf{I})</annotation></semantics></math><span id="A2.SS5.p1.12.7" class="ltx_text" style="font-size:90%;">. Since the independent Gaussian noise term </span><math id="A2.SS5.p1.7.m7.1" class="ltx_Math" alttext="\mathbf{n}" display="inline"><semantics id="A2.SS5.p1.7.m7.1a"><mi mathsize="90%" id="A2.SS5.p1.7.m7.1.1" xref="A2.SS5.p1.7.m7.1.1.cmml">𝐧</mi><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.7.m7.1b"><ci id="A2.SS5.p1.7.m7.1.1.cmml" xref="A2.SS5.p1.7.m7.1.1">𝐧</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS5.p1.7.m7.1c">\mathbf{n}</annotation></semantics></math><span id="A2.SS5.p1.12.8" class="ltx_text" style="font-size:90%;"> contains no information about </span><math id="A2.SS5.p1.8.m8.1" class="ltx_Math" alttext="\mathbf{z}" display="inline"><semantics id="A2.SS5.p1.8.m8.1a"><mi mathsize="90%" id="A2.SS5.p1.8.m8.1.1" xref="A2.SS5.p1.8.m8.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.8.m8.1b"><ci id="A2.SS5.p1.8.m8.1.1.cmml" xref="A2.SS5.p1.8.m8.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS5.p1.8.m8.1c">\mathbf{z}</annotation></semantics></math><span id="A2.SS5.p1.12.9" class="ltx_text" style="font-size:90%;">, this is exactly the prior distribution </span><math id="A2.SS5.p1.9.m9.1" class="ltx_Math" alttext="p(\mathbf{z}_{t}|\mathbf{z}_{t-1})" display="inline"><semantics id="A2.SS5.p1.9.m9.1a"><mrow id="A2.SS5.p1.9.m9.1.1" xref="A2.SS5.p1.9.m9.1.1.cmml"><mi mathsize="90%" id="A2.SS5.p1.9.m9.1.1.3" xref="A2.SS5.p1.9.m9.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="A2.SS5.p1.9.m9.1.1.2" xref="A2.SS5.p1.9.m9.1.1.2.cmml">​</mo><mrow id="A2.SS5.p1.9.m9.1.1.1.1" xref="A2.SS5.p1.9.m9.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS5.p1.9.m9.1.1.1.1.2" xref="A2.SS5.p1.9.m9.1.1.1.1.1.cmml">(</mo><mrow id="A2.SS5.p1.9.m9.1.1.1.1.1" xref="A2.SS5.p1.9.m9.1.1.1.1.1.cmml"><msub id="A2.SS5.p1.9.m9.1.1.1.1.1.2" xref="A2.SS5.p1.9.m9.1.1.1.1.1.2.cmml"><mi mathsize="90%" id="A2.SS5.p1.9.m9.1.1.1.1.1.2.2" xref="A2.SS5.p1.9.m9.1.1.1.1.1.2.2.cmml">𝐳</mi><mi mathsize="90%" id="A2.SS5.p1.9.m9.1.1.1.1.1.2.3" xref="A2.SS5.p1.9.m9.1.1.1.1.1.2.3.cmml">t</mi></msub><mo fence="false" mathsize="90%" id="A2.SS5.p1.9.m9.1.1.1.1.1.1" xref="A2.SS5.p1.9.m9.1.1.1.1.1.1.cmml">|</mo><msub id="A2.SS5.p1.9.m9.1.1.1.1.1.3" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A2.SS5.p1.9.m9.1.1.1.1.1.3.2" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.2.cmml">𝐳</mi><mrow id="A2.SS5.p1.9.m9.1.1.1.1.1.3.3" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.cmml"><mi mathsize="90%" id="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.2" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.2.cmml">t</mi><mo mathsize="90%" id="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.1" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.1.cmml">−</mo><mn mathsize="90%" id="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.3" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><mo maxsize="90%" minsize="90%" id="A2.SS5.p1.9.m9.1.1.1.1.3" xref="A2.SS5.p1.9.m9.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.9.m9.1b"><apply id="A2.SS5.p1.9.m9.1.1.cmml" xref="A2.SS5.p1.9.m9.1.1"><times id="A2.SS5.p1.9.m9.1.1.2.cmml" xref="A2.SS5.p1.9.m9.1.1.2"></times><ci id="A2.SS5.p1.9.m9.1.1.3.cmml" xref="A2.SS5.p1.9.m9.1.1.3">𝑝</ci><apply id="A2.SS5.p1.9.m9.1.1.1.1.1.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1"><csymbol cd="latexml" id="A2.SS5.p1.9.m9.1.1.1.1.1.1.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.1">conditional</csymbol><apply id="A2.SS5.p1.9.m9.1.1.1.1.1.2.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.SS5.p1.9.m9.1.1.1.1.1.2.1.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.2">subscript</csymbol><ci id="A2.SS5.p1.9.m9.1.1.1.1.1.2.2.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.2.2">𝐳</ci><ci id="A2.SS5.p1.9.m9.1.1.1.1.1.2.3.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="A2.SS5.p1.9.m9.1.1.1.1.1.3.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.SS5.p1.9.m9.1.1.1.1.1.3.1.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3">subscript</csymbol><ci id="A2.SS5.p1.9.m9.1.1.1.1.1.3.2.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.2">𝐳</ci><apply id="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.3"><minus id="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.1.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.1"></minus><ci id="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.2.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.2">𝑡</ci><cn type="integer" id="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.3.cmml" xref="A2.SS5.p1.9.m9.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS5.p1.9.m9.1c">p(\mathbf{z}_{t}|\mathbf{z}_{t-1})</annotation></semantics></math><span id="A2.SS5.p1.12.10" class="ltx_text" style="font-size:90%;">. By varying </span><math id="A2.SS5.p1.10.m10.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.SS5.p1.10.m10.1a"><mi mathsize="90%" id="A2.SS5.p1.10.m10.1.1" xref="A2.SS5.p1.10.m10.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.10.m10.1b"><ci id="A2.SS5.p1.10.m10.1.1.cmml" xref="A2.SS5.p1.10.m10.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS5.p1.10.m10.1c">k</annotation></semantics></math><span id="A2.SS5.p1.12.11" class="ltx_text" style="font-size:90%;"> between </span><math id="A2.SS5.p1.11.m11.1" class="ltx_Math" alttext="K" display="inline"><semantics id="A2.SS5.p1.11.m11.1a"><mi mathsize="90%" id="A2.SS5.p1.11.m11.1.1" xref="A2.SS5.p1.11.m11.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.11.m11.1b"><ci id="A2.SS5.p1.11.m11.1.1.cmml" xref="A2.SS5.p1.11.m11.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS5.p1.11.m11.1c">K</annotation></semantics></math><span id="A2.SS5.p1.12.12" class="ltx_text" style="font-size:90%;"> and </span><math id="A2.SS5.p1.12.m12.1" class="ltx_Math" alttext="0" display="inline"><semantics id="A2.SS5.p1.12.m12.1a"><mn mathsize="90%" id="A2.SS5.p1.12.m12.1.1" xref="A2.SS5.p1.12.m12.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A2.SS5.p1.12.m12.1b"><cn type="integer" id="A2.SS5.p1.12.m12.1.1.cmml" xref="A2.SS5.p1.12.m12.1.1">0</cn></annotation-xml></semantics></math><span id="A2.SS5.p1.12.13" class="ltx_text" style="font-size:90%;">, the same neural network can parameterize everything between prior and posterior.</span></p>
</div>
<figure id="A2.F5" class="ltx_figure"><img src="/html/2407.01392/assets/x7.png" id="A2.F5.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="368" height="230" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:90%;"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Diffusion Forcing is trained on independent level of noises at different timesteps. As a result, we can control the noise level <math id="A2.F5.2.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.F5.2.m1.1b"><mi id="A2.F5.2.m1.1.1" xref="A2.F5.2.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.F5.2.m1.1c"><ci id="A2.F5.2.m1.1.1.cmml" xref="A2.F5.2.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.F5.2.m1.1d">k</annotation></semantics></math> to achieve different effects on conditioning and prediction.</figcaption>
</figure>
</section>
<section id="A2.SS6" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">B.6 </span>Connection to other sequence training schemes</h3>

<div id="A2.SS6.p1" class="ltx_para">
<p id="A2.SS6.p1.4" class="ltx_p"><span id="A2.SS6.p1.4.1" class="ltx_text" style="font-size:90%;">Noise as masking provides a unified view of different sequence training schemes. The following exposition uses a length </span><math id="A2.SS6.p1.1.m1.1" class="ltx_Math" alttext="3" display="inline"><semantics id="A2.SS6.p1.1.m1.1a"><mn mathsize="90%" id="A2.SS6.p1.1.m1.1.1" xref="A2.SS6.p1.1.m1.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="A2.SS6.p1.1.m1.1b"><cn type="integer" id="A2.SS6.p1.1.m1.1.1.cmml" xref="A2.SS6.p1.1.m1.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="A2.SS6.p1.1.m1.1c">3</annotation></semantics></math><span id="A2.SS6.p1.4.2" class="ltx_text" style="font-size:90%;"> sequence as an example: We always start with fully masked sequence </span><math id="A2.SS6.p1.2.m2.3" class="ltx_Math" alttext="[\mathbf{x}_{1}^{K},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]" display="inline"><semantics id="A2.SS6.p1.2.m2.3a"><mrow id="A2.SS6.p1.2.m2.3.3.3" xref="A2.SS6.p1.2.m2.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS6.p1.2.m2.3.3.3.4" xref="A2.SS6.p1.2.m2.3.3.4.cmml">[</mo><msubsup id="A2.SS6.p1.2.m2.1.1.1.1" 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Assume all diffusions are sampled with </span><math id="A2.SS6.p1.4.m4.1" class="ltx_Math" alttext="3" display="inline"><semantics id="A2.SS6.p1.4.m4.1a"><mn mathsize="90%" id="A2.SS6.p1.4.m4.1.1" xref="A2.SS6.p1.4.m4.1.1.cmml">3</mn><annotation-xml encoding="MathML-Content" id="A2.SS6.p1.4.m4.1b"><cn type="integer" id="A2.SS6.p1.4.m4.1.1.cmml" xref="A2.SS6.p1.4.m4.1.1">3</cn></annotation-xml><annotation encoding="application/x-tex" id="A2.SS6.p1.4.m4.1c">3</annotation></semantics></math><span id="A2.SS6.p1.4.5" class="ltx_text" style="font-size:90%;">-step DDIM.</span></p>
</div>
<section id="A2.SS6.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Autoregressive.</h5>

<div id="A2.SS6.SSS0.Px1.p1" class="ltx_para">
<p id="A2.SS6.SSS0.Px1.p1.2" class="ltx_p"><span id="A2.SS6.SSS0.Px1.p1.2.1" class="ltx_text" style="font-size:90%;">In teacher forcing, one trains a model to predict the next token conditioned on prior observations. One can train next-token diffusion models with teacher forcing such as  </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A2.SS6.SSS0.Px1.p1.2.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib48" title="" class="ltx_ref">48</a><span id="A2.SS6.SSS0.Px1.p1.2.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A2.SS6.SSS0.Px1.p1.2.4" class="ltx_text" style="font-size:90%;">: feed neural network with past observations as well as a current observation and ask it to predict clean current observation. A typical training pair can have the input of </span><math id="A2.SS6.SSS0.Px1.p1.1.m1.3" class="ltx_Math" alttext="[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{K}]^{\top}" display="inline"><semantics id="A2.SS6.SSS0.Px1.p1.1.m1.3a"><msup id="A2.SS6.SSS0.Px1.p1.1.m1.3.3" xref="A2.SS6.SSS0.Px1.p1.1.m1.3.3.cmml"><mrow id="A2.SS6.SSS0.Px1.p1.1.m1.3.3.3.3" xref="A2.SS6.SSS0.Px1.p1.1.m1.3.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS6.SSS0.Px1.p1.1.m1.3.3.3.3.4" xref="A2.SS6.SSS0.Px1.p1.1.m1.3.3.3.4.cmml">[</mo><msubsup id="A2.SS6.SSS0.Px1.p1.1.m1.1.1.1.1.1" xref="A2.SS6.SSS0.Px1.p1.1.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.SS6.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.2" xref="A2.SS6.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.SS6.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3" xref="A2.SS6.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.SS6.SSS0.Px1.p1.1.m1.1.1.1.1.1.3" xref="A2.SS6.SSS0.Px1.p1.1.m1.1.1.1.1.1.3.cmml">0</mn></msubsup><mo 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xref="A2.SS6.SSS0.Px1.p1.1.m1.3.3.3.3.3.2.3">3</cn></apply><ci id="A2.SS6.SSS0.Px1.p1.1.m1.3.3.3.3.3.3.cmml" xref="A2.SS6.SSS0.Px1.p1.1.m1.3.3.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.SS6.SSS0.Px1.p1.1.m1.3.3.5.cmml" xref="A2.SS6.SSS0.Px1.p1.1.m1.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS6.SSS0.Px1.p1.1.m1.3c">[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{K}]^{\top}</annotation></semantics></math><span id="A2.SS6.SSS0.Px1.p1.2.5" class="ltx_text" style="font-size:90%;"> and target of </span><math id="A2.SS6.SSS0.Px1.p1.2.m2.3" class="ltx_Math" alttext="[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{0}]^{\top}" display="inline"><semantics id="A2.SS6.SSS0.Px1.p1.2.m2.3a"><msup id="A2.SS6.SSS0.Px1.p1.2.m2.3.3" xref="A2.SS6.SSS0.Px1.p1.2.m2.3.3.cmml"><mrow id="A2.SS6.SSS0.Px1.p1.2.m2.3.3.3.3" xref="A2.SS6.SSS0.Px1.p1.2.m2.3.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.SS6.SSS0.Px1.p1.2.m2.3.3.3.3.4" xref="A2.SS6.SSS0.Px1.p1.2.m2.3.3.3.4.cmml">[</mo><msubsup id="A2.SS6.SSS0.Px1.p1.2.m2.1.1.1.1.1" xref="A2.SS6.SSS0.Px1.p1.2.m2.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.SS6.SSS0.Px1.p1.2.m2.1.1.1.1.1.2.2" xref="A2.SS6.SSS0.Px1.p1.2.m2.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.SS6.SSS0.Px1.p1.2.m2.1.1.1.1.1.2.3" xref="A2.SS6.SSS0.Px1.p1.2.m2.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.SS6.SSS0.Px1.p1.2.m2.1.1.1.1.1.3" xref="A2.SS6.SSS0.Px1.p1.2.m2.1.1.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.SS6.SSS0.Px1.p1.2.m2.3.3.3.3.5" xref="A2.SS6.SSS0.Px1.p1.2.m2.3.3.3.4.cmml">,</mo><msubsup id="A2.SS6.SSS0.Px1.p1.2.m2.2.2.2.2.2" xref="A2.SS6.SSS0.Px1.p1.2.m2.2.2.2.2.2.cmml"><mi mathsize="90%" id="A2.SS6.SSS0.Px1.p1.2.m2.2.2.2.2.2.2.2" xref="A2.SS6.SSS0.Px1.p1.2.m2.2.2.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.SS6.SSS0.Px1.p1.2.m2.2.2.2.2.2.2.3" xref="A2.SS6.SSS0.Px1.p1.2.m2.2.2.2.2.2.2.3.cmml">2</mn><mn mathsize="90%" 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id="A2.SS6.SSS0.Px1.p1.2.m2.3c">[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{0}]^{\top}</annotation></semantics></math><span id="A2.SS6.SSS0.Px1.p1.2.6" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
<div id="A2.SS6.SSS0.Px1.p2" class="ltx_para">
<p id="A2.SS6.SSS0.Px1.p2.1" class="ltx_p"><span id="A2.SS6.SSS0.Px1.p2.1.1" class="ltx_text" style="font-size:90%;">At sampling time, one fully diffuses the next token before adding the diffused observation to history to perform an autoregressive rollout. The diffusion process would thus look like</span></p>
<table id="A6.EGx37" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A2.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex1.m1.3" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{K},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}" display="inline"><semantics id="A2.Ex1.m1.3a"><msup id="A2.Ex1.m1.3.3" xref="A2.Ex1.m1.3.3.cmml"><mrow id="A2.Ex1.m1.3.3.3.3" xref="A2.Ex1.m1.3.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex1.m1.3.3.3.3.4" xref="A2.Ex1.m1.3.3.3.4.cmml">[</mo><msubsup id="A2.Ex1.m1.1.1.1.1.1" xref="A2.Ex1.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex1.m1.1.1.1.1.1.2.2" xref="A2.Ex1.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex1.m1.1.1.1.1.1.2.3" xref="A2.Ex1.m1.1.1.1.1.1.2.3.cmml">1</mn><mi mathsize="90%" id="A2.Ex1.m1.1.1.1.1.1.3" xref="A2.Ex1.m1.1.1.1.1.1.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex1.m1.3.3.3.3.5" xref="A2.Ex1.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex1.m1.2.2.2.2.2" xref="A2.Ex1.m1.2.2.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex1.m1.2.2.2.2.2.2.2" xref="A2.Ex1.m1.2.2.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex1.m1.2.2.2.2.2.2.3" xref="A2.Ex1.m1.2.2.2.2.2.2.3.cmml">2</mn><mi mathsize="90%" id="A2.Ex1.m1.2.2.2.2.2.3" xref="A2.Ex1.m1.2.2.2.2.2.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex1.m1.3.3.3.3.6" xref="A2.Ex1.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex1.m1.3.3.3.3.3" xref="A2.Ex1.m1.3.3.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex1.m1.3.3.3.3.3.2.2" xref="A2.Ex1.m1.3.3.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex1.m1.3.3.3.3.3.2.3" xref="A2.Ex1.m1.3.3.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex1.m1.3.3.3.3.3.3" xref="A2.Ex1.m1.3.3.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex1.m1.3.3.3.3.7" xref="A2.Ex1.m1.3.3.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex1.m1.3.3.5" xref="A2.Ex1.m1.3.3.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="A2.Ex1.m1.3b"><apply id="A2.Ex1.m1.3.3.cmml" xref="A2.Ex1.m1.3.3"><csymbol cd="ambiguous" id="A2.Ex1.m1.3.3.4.cmml" xref="A2.Ex1.m1.3.3">superscript</csymbol><list id="A2.Ex1.m1.3.3.3.4.cmml" xref="A2.Ex1.m1.3.3.3.3"><apply id="A2.Ex1.m1.1.1.1.1.1.cmml" xref="A2.Ex1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex1.m1.1.1.1.1.1.1.cmml" xref="A2.Ex1.m1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex1.m1.1.1.1.1.1.2.cmml" xref="A2.Ex1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex1.m1.1.1.1.1.1.2.1.cmml" xref="A2.Ex1.m1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex1.m1.1.1.1.1.1.2.2.cmml" xref="A2.Ex1.m1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex1.m1.1.1.1.1.1.2.3.cmml" xref="A2.Ex1.m1.1.1.1.1.1.2.3">1</cn></apply><ci id="A2.Ex1.m1.1.1.1.1.1.3.cmml" xref="A2.Ex1.m1.1.1.1.1.1.3">𝐾</ci></apply><apply id="A2.Ex1.m1.2.2.2.2.2.cmml" xref="A2.Ex1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex1.m1.2.2.2.2.2.1.cmml" xref="A2.Ex1.m1.2.2.2.2.2">superscript</csymbol><apply id="A2.Ex1.m1.2.2.2.2.2.2.cmml" xref="A2.Ex1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex1.m1.2.2.2.2.2.2.1.cmml" xref="A2.Ex1.m1.2.2.2.2.2">subscript</csymbol><ci id="A2.Ex1.m1.2.2.2.2.2.2.2.cmml" xref="A2.Ex1.m1.2.2.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex1.m1.2.2.2.2.2.2.3.cmml" xref="A2.Ex1.m1.2.2.2.2.2.2.3">2</cn></apply><ci id="A2.Ex1.m1.2.2.2.2.2.3.cmml" xref="A2.Ex1.m1.2.2.2.2.2.3">𝐾</ci></apply><apply id="A2.Ex1.m1.3.3.3.3.3.cmml" xref="A2.Ex1.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex1.m1.3.3.3.3.3.1.cmml" xref="A2.Ex1.m1.3.3.3.3.3">superscript</csymbol><apply id="A2.Ex1.m1.3.3.3.3.3.2.cmml" xref="A2.Ex1.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex1.m1.3.3.3.3.3.2.1.cmml" xref="A2.Ex1.m1.3.3.3.3.3">subscript</csymbol><ci id="A2.Ex1.m1.3.3.3.3.3.2.2.cmml" xref="A2.Ex1.m1.3.3.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex1.m1.3.3.3.3.3.2.3.cmml" xref="A2.Ex1.m1.3.3.3.3.3.2.3">3</cn></apply><ci id="A2.Ex1.m1.3.3.3.3.3.3.cmml" xref="A2.Ex1.m1.3.3.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex1.m1.3.3.5.cmml" xref="A2.Ex1.m1.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex1.m1.3c">\displaystyle[\mathbf{x}_{1}^{K},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex2.m1.4" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{K//2},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}," display="inline"><semantics id="A2.Ex2.m1.4a"><mrow id="A2.Ex2.m1.4.4.1" xref="A2.Ex2.m1.4.4.1.1.cmml"><msup id="A2.Ex2.m1.4.4.1.1" xref="A2.Ex2.m1.4.4.1.1.cmml"><mrow id="A2.Ex2.m1.4.4.1.1.3.3" xref="A2.Ex2.m1.4.4.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex2.m1.4.4.1.1.3.3.4" xref="A2.Ex2.m1.4.4.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex2.m1.4.4.1.1.1.1.1" xref="A2.Ex2.m1.4.4.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex2.m1.4.4.1.1.1.1.1.2.2" xref="A2.Ex2.m1.4.4.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex2.m1.4.4.1.1.1.1.1.2.3" xref="A2.Ex2.m1.4.4.1.1.1.1.1.2.3.cmml">1</mn><mrow id="A2.Ex2.m1.3.3.3.3" xref="A2.Ex2.m1.3.3.3.4.cmml"><mi mathsize="90%" id="A2.Ex2.m1.1.1.1.1" xref="A2.Ex2.m1.1.1.1.1.cmml">K</mi><mo id="A2.Ex2.m1.3.3.3.3.2" xref="A2.Ex2.m1.3.3.3.4.cmml">⁣</mo><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex2.m1.2.2.2.2" xref="A2.Ex2.m1.2.2.2.2.cmml">/</mo><mo id="A2.Ex2.m1.3.3.3.3.3" xref="A2.Ex2.m1.3.3.3.4.cmml">⁣</mo><mrow id="A2.Ex2.m1.3.3.3.3.1" xref="A2.Ex2.m1.3.3.3.3.1.cmml"><mi id="A2.Ex2.m1.3.3.3.3.1.2" xref="A2.Ex2.m1.3.3.3.3.1.2.cmml"></mi><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex2.m1.3.3.3.3.1.1" xref="A2.Ex2.m1.3.3.3.3.1.1.cmml">/</mo><mn mathsize="90%" id="A2.Ex2.m1.3.3.3.3.1.3" xref="A2.Ex2.m1.3.3.3.3.1.3.cmml">2</mn></mrow></mrow></msubsup><mo mathsize="90%" id="A2.Ex2.m1.4.4.1.1.3.3.5" xref="A2.Ex2.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex2.m1.4.4.1.1.2.2.2" xref="A2.Ex2.m1.4.4.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex2.m1.4.4.1.1.2.2.2.2.2" xref="A2.Ex2.m1.4.4.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex2.m1.4.4.1.1.2.2.2.2.3" xref="A2.Ex2.m1.4.4.1.1.2.2.2.2.3.cmml">2</mn><mi mathsize="90%" id="A2.Ex2.m1.4.4.1.1.2.2.2.3" xref="A2.Ex2.m1.4.4.1.1.2.2.2.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex2.m1.4.4.1.1.3.3.6" xref="A2.Ex2.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex2.m1.4.4.1.1.3.3.3" xref="A2.Ex2.m1.4.4.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex2.m1.4.4.1.1.3.3.3.2.2" xref="A2.Ex2.m1.4.4.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex2.m1.4.4.1.1.3.3.3.2.3" xref="A2.Ex2.m1.4.4.1.1.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex2.m1.4.4.1.1.3.3.3.3" xref="A2.Ex2.m1.4.4.1.1.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex2.m1.4.4.1.1.3.3.7" xref="A2.Ex2.m1.4.4.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex2.m1.4.4.1.1.5" xref="A2.Ex2.m1.4.4.1.1.5.cmml">⊤</mo></msup><mo mathsize="90%" id="A2.Ex2.m1.4.4.1.2" xref="A2.Ex2.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex2.m1.4b"><apply id="A2.Ex2.m1.4.4.1.1.cmml" xref="A2.Ex2.m1.4.4.1"><csymbol cd="ambiguous" id="A2.Ex2.m1.4.4.1.1.4.cmml" xref="A2.Ex2.m1.4.4.1">superscript</csymbol><list id="A2.Ex2.m1.4.4.1.1.3.4.cmml" xref="A2.Ex2.m1.4.4.1.1.3.3"><apply id="A2.Ex2.m1.4.4.1.1.1.1.1.cmml" xref="A2.Ex2.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex2.m1.4.4.1.1.1.1.1.1.cmml" xref="A2.Ex2.m1.4.4.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex2.m1.4.4.1.1.1.1.1.2.cmml" xref="A2.Ex2.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex2.m1.4.4.1.1.1.1.1.2.1.cmml" xref="A2.Ex2.m1.4.4.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex2.m1.4.4.1.1.1.1.1.2.2.cmml" xref="A2.Ex2.m1.4.4.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex2.m1.4.4.1.1.1.1.1.2.3.cmml" xref="A2.Ex2.m1.4.4.1.1.1.1.1.2.3">1</cn></apply><list id="A2.Ex2.m1.3.3.3.4.cmml" xref="A2.Ex2.m1.3.3.3.3"><ci id="A2.Ex2.m1.1.1.1.1.cmml" xref="A2.Ex2.m1.1.1.1.1">𝐾</ci><divide id="A2.Ex2.m1.2.2.2.2.cmml" xref="A2.Ex2.m1.2.2.2.2"></divide><apply id="A2.Ex2.m1.3.3.3.3.1.cmml" xref="A2.Ex2.m1.3.3.3.3.1"><divide id="A2.Ex2.m1.3.3.3.3.1.1.cmml" xref="A2.Ex2.m1.3.3.3.3.1.1"></divide><csymbol cd="latexml" id="A2.Ex2.m1.3.3.3.3.1.2.cmml" xref="A2.Ex2.m1.3.3.3.3.1.2">absent</csymbol><cn type="integer" id="A2.Ex2.m1.3.3.3.3.1.3.cmml" xref="A2.Ex2.m1.3.3.3.3.1.3">2</cn></apply></list></apply><apply id="A2.Ex2.m1.4.4.1.1.2.2.2.cmml" xref="A2.Ex2.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex2.m1.4.4.1.1.2.2.2.1.cmml" xref="A2.Ex2.m1.4.4.1.1.2.2.2">superscript</csymbol><apply id="A2.Ex2.m1.4.4.1.1.2.2.2.2.cmml" xref="A2.Ex2.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex2.m1.4.4.1.1.2.2.2.2.1.cmml" xref="A2.Ex2.m1.4.4.1.1.2.2.2">subscript</csymbol><ci id="A2.Ex2.m1.4.4.1.1.2.2.2.2.2.cmml" xref="A2.Ex2.m1.4.4.1.1.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex2.m1.4.4.1.1.2.2.2.2.3.cmml" xref="A2.Ex2.m1.4.4.1.1.2.2.2.2.3">2</cn></apply><ci id="A2.Ex2.m1.4.4.1.1.2.2.2.3.cmml" xref="A2.Ex2.m1.4.4.1.1.2.2.2.3">𝐾</ci></apply><apply id="A2.Ex2.m1.4.4.1.1.3.3.3.cmml" xref="A2.Ex2.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex2.m1.4.4.1.1.3.3.3.1.cmml" xref="A2.Ex2.m1.4.4.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex2.m1.4.4.1.1.3.3.3.2.cmml" xref="A2.Ex2.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex2.m1.4.4.1.1.3.3.3.2.1.cmml" xref="A2.Ex2.m1.4.4.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex2.m1.4.4.1.1.3.3.3.2.2.cmml" xref="A2.Ex2.m1.4.4.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex2.m1.4.4.1.1.3.3.3.2.3.cmml" xref="A2.Ex2.m1.4.4.1.1.3.3.3.2.3">3</cn></apply><ci id="A2.Ex2.m1.4.4.1.1.3.3.3.3.cmml" xref="A2.Ex2.m1.4.4.1.1.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex2.m1.4.4.1.1.5.cmml" xref="A2.Ex2.m1.4.4.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex2.m1.4c">\displaystyle[\mathbf{x}_{1}^{K//2},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex3.m1.1" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}," display="inline"><semantics id="A2.Ex3.m1.1a"><mrow id="A2.Ex3.m1.1.1.1" xref="A2.Ex3.m1.1.1.1.1.cmml"><msup id="A2.Ex3.m1.1.1.1.1" xref="A2.Ex3.m1.1.1.1.1.cmml"><mrow id="A2.Ex3.m1.1.1.1.1.3.3" xref="A2.Ex3.m1.1.1.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex3.m1.1.1.1.1.3.3.4" xref="A2.Ex3.m1.1.1.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex3.m1.1.1.1.1.1.1.1" xref="A2.Ex3.m1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex3.m1.1.1.1.1.1.1.1.2.2" xref="A2.Ex3.m1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex3.m1.1.1.1.1.1.1.1.2.3" xref="A2.Ex3.m1.1.1.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex3.m1.1.1.1.1.1.1.1.3" xref="A2.Ex3.m1.1.1.1.1.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex3.m1.1.1.1.1.3.3.5" xref="A2.Ex3.m1.1.1.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex3.m1.1.1.1.1.2.2.2" xref="A2.Ex3.m1.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex3.m1.1.1.1.1.2.2.2.2.2" xref="A2.Ex3.m1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex3.m1.1.1.1.1.2.2.2.2.3" xref="A2.Ex3.m1.1.1.1.1.2.2.2.2.3.cmml">2</mn><mi mathsize="90%" id="A2.Ex3.m1.1.1.1.1.2.2.2.3" xref="A2.Ex3.m1.1.1.1.1.2.2.2.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex3.m1.1.1.1.1.3.3.6" xref="A2.Ex3.m1.1.1.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex3.m1.1.1.1.1.3.3.3" xref="A2.Ex3.m1.1.1.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex3.m1.1.1.1.1.3.3.3.2.2" xref="A2.Ex3.m1.1.1.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex3.m1.1.1.1.1.3.3.3.2.3" xref="A2.Ex3.m1.1.1.1.1.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex3.m1.1.1.1.1.3.3.3.3" xref="A2.Ex3.m1.1.1.1.1.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex3.m1.1.1.1.1.3.3.7" xref="A2.Ex3.m1.1.1.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex3.m1.1.1.1.1.5" xref="A2.Ex3.m1.1.1.1.1.5.cmml">⊤</mo></msup><mo mathsize="90%" id="A2.Ex3.m1.1.1.1.2" xref="A2.Ex3.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex3.m1.1b"><apply id="A2.Ex3.m1.1.1.1.1.cmml" xref="A2.Ex3.m1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex3.m1.1.1.1.1.4.cmml" xref="A2.Ex3.m1.1.1.1">superscript</csymbol><list id="A2.Ex3.m1.1.1.1.1.3.4.cmml" xref="A2.Ex3.m1.1.1.1.1.3.3"><apply id="A2.Ex3.m1.1.1.1.1.1.1.1.cmml" xref="A2.Ex3.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex3.m1.1.1.1.1.1.1.1.1.cmml" xref="A2.Ex3.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex3.m1.1.1.1.1.1.1.1.2.cmml" xref="A2.Ex3.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex3.m1.1.1.1.1.1.1.1.2.1.cmml" xref="A2.Ex3.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex3.m1.1.1.1.1.1.1.1.2.2.cmml" xref="A2.Ex3.m1.1.1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex3.m1.1.1.1.1.1.1.1.2.3.cmml" xref="A2.Ex3.m1.1.1.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex3.m1.1.1.1.1.1.1.1.3.cmml" xref="A2.Ex3.m1.1.1.1.1.1.1.1.3">0</cn></apply><apply id="A2.Ex3.m1.1.1.1.1.2.2.2.cmml" xref="A2.Ex3.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex3.m1.1.1.1.1.2.2.2.1.cmml" xref="A2.Ex3.m1.1.1.1.1.2.2.2">superscript</csymbol><apply id="A2.Ex3.m1.1.1.1.1.2.2.2.2.cmml" xref="A2.Ex3.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex3.m1.1.1.1.1.2.2.2.2.1.cmml" xref="A2.Ex3.m1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A2.Ex3.m1.1.1.1.1.2.2.2.2.2.cmml" xref="A2.Ex3.m1.1.1.1.1.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex3.m1.1.1.1.1.2.2.2.2.3.cmml" xref="A2.Ex3.m1.1.1.1.1.2.2.2.2.3">2</cn></apply><ci id="A2.Ex3.m1.1.1.1.1.2.2.2.3.cmml" xref="A2.Ex3.m1.1.1.1.1.2.2.2.3">𝐾</ci></apply><apply id="A2.Ex3.m1.1.1.1.1.3.3.3.cmml" xref="A2.Ex3.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex3.m1.1.1.1.1.3.3.3.1.cmml" xref="A2.Ex3.m1.1.1.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex3.m1.1.1.1.1.3.3.3.2.cmml" xref="A2.Ex3.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex3.m1.1.1.1.1.3.3.3.2.1.cmml" xref="A2.Ex3.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex3.m1.1.1.1.1.3.3.3.2.2.cmml" xref="A2.Ex3.m1.1.1.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex3.m1.1.1.1.1.3.3.3.2.3.cmml" xref="A2.Ex3.m1.1.1.1.1.3.3.3.2.3">3</cn></apply><ci id="A2.Ex3.m1.1.1.1.1.3.3.3.3.cmml" xref="A2.Ex3.m1.1.1.1.1.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex3.m1.1.1.1.1.5.cmml" xref="A2.Ex3.m1.1.1.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex3.m1.1c">\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex4.m1.6" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{K//2},\mathbf{x}_{3}^{K}]^{\top}" display="inline"><semantics id="A2.Ex4.m1.6a"><msup id="A2.Ex4.m1.6.6" xref="A2.Ex4.m1.6.6.cmml"><mrow id="A2.Ex4.m1.6.6.3.3" xref="A2.Ex4.m1.6.6.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex4.m1.6.6.3.3.4" xref="A2.Ex4.m1.6.6.3.4.cmml">[</mo><msubsup id="A2.Ex4.m1.4.4.1.1.1" xref="A2.Ex4.m1.4.4.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex4.m1.4.4.1.1.1.2.2" xref="A2.Ex4.m1.4.4.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex4.m1.4.4.1.1.1.2.3" xref="A2.Ex4.m1.4.4.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex4.m1.4.4.1.1.1.3" xref="A2.Ex4.m1.4.4.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex4.m1.6.6.3.3.5" xref="A2.Ex4.m1.6.6.3.4.cmml">,</mo><msubsup id="A2.Ex4.m1.5.5.2.2.2" xref="A2.Ex4.m1.5.5.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex4.m1.5.5.2.2.2.2.2" xref="A2.Ex4.m1.5.5.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex4.m1.5.5.2.2.2.2.3" xref="A2.Ex4.m1.5.5.2.2.2.2.3.cmml">2</mn><mrow id="A2.Ex4.m1.3.3.3.3" xref="A2.Ex4.m1.3.3.3.4.cmml"><mi mathsize="90%" id="A2.Ex4.m1.1.1.1.1" xref="A2.Ex4.m1.1.1.1.1.cmml">K</mi><mo id="A2.Ex4.m1.3.3.3.3.2" xref="A2.Ex4.m1.3.3.3.4.cmml">⁣</mo><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex4.m1.2.2.2.2" xref="A2.Ex4.m1.2.2.2.2.cmml">/</mo><mo id="A2.Ex4.m1.3.3.3.3.3" xref="A2.Ex4.m1.3.3.3.4.cmml">⁣</mo><mrow id="A2.Ex4.m1.3.3.3.3.1" xref="A2.Ex4.m1.3.3.3.3.1.cmml"><mi id="A2.Ex4.m1.3.3.3.3.1.2" xref="A2.Ex4.m1.3.3.3.3.1.2.cmml"></mi><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex4.m1.3.3.3.3.1.1" xref="A2.Ex4.m1.3.3.3.3.1.1.cmml">/</mo><mn mathsize="90%" id="A2.Ex4.m1.3.3.3.3.1.3" xref="A2.Ex4.m1.3.3.3.3.1.3.cmml">2</mn></mrow></mrow></msubsup><mo mathsize="90%" id="A2.Ex4.m1.6.6.3.3.6" xref="A2.Ex4.m1.6.6.3.4.cmml">,</mo><msubsup id="A2.Ex4.m1.6.6.3.3.3" xref="A2.Ex4.m1.6.6.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex4.m1.6.6.3.3.3.2.2" xref="A2.Ex4.m1.6.6.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex4.m1.6.6.3.3.3.2.3" xref="A2.Ex4.m1.6.6.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex4.m1.6.6.3.3.3.3" xref="A2.Ex4.m1.6.6.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex4.m1.6.6.3.3.7" xref="A2.Ex4.m1.6.6.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex4.m1.6.6.5" xref="A2.Ex4.m1.6.6.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="A2.Ex4.m1.6b"><apply id="A2.Ex4.m1.6.6.cmml" xref="A2.Ex4.m1.6.6"><csymbol cd="ambiguous" id="A2.Ex4.m1.6.6.4.cmml" xref="A2.Ex4.m1.6.6">superscript</csymbol><list id="A2.Ex4.m1.6.6.3.4.cmml" xref="A2.Ex4.m1.6.6.3.3"><apply id="A2.Ex4.m1.4.4.1.1.1.cmml" xref="A2.Ex4.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="A2.Ex4.m1.4.4.1.1.1.1.cmml" xref="A2.Ex4.m1.4.4.1.1.1">superscript</csymbol><apply id="A2.Ex4.m1.4.4.1.1.1.2.cmml" xref="A2.Ex4.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="A2.Ex4.m1.4.4.1.1.1.2.1.cmml" xref="A2.Ex4.m1.4.4.1.1.1">subscript</csymbol><ci id="A2.Ex4.m1.4.4.1.1.1.2.2.cmml" xref="A2.Ex4.m1.4.4.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex4.m1.4.4.1.1.1.2.3.cmml" xref="A2.Ex4.m1.4.4.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex4.m1.4.4.1.1.1.3.cmml" xref="A2.Ex4.m1.4.4.1.1.1.3">0</cn></apply><apply id="A2.Ex4.m1.5.5.2.2.2.cmml" xref="A2.Ex4.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="A2.Ex4.m1.5.5.2.2.2.1.cmml" xref="A2.Ex4.m1.5.5.2.2.2">superscript</csymbol><apply id="A2.Ex4.m1.5.5.2.2.2.2.cmml" xref="A2.Ex4.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="A2.Ex4.m1.5.5.2.2.2.2.1.cmml" xref="A2.Ex4.m1.5.5.2.2.2">subscript</csymbol><ci id="A2.Ex4.m1.5.5.2.2.2.2.2.cmml" xref="A2.Ex4.m1.5.5.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex4.m1.5.5.2.2.2.2.3.cmml" xref="A2.Ex4.m1.5.5.2.2.2.2.3">2</cn></apply><list id="A2.Ex4.m1.3.3.3.4.cmml" xref="A2.Ex4.m1.3.3.3.3"><ci id="A2.Ex4.m1.1.1.1.1.cmml" xref="A2.Ex4.m1.1.1.1.1">𝐾</ci><divide id="A2.Ex4.m1.2.2.2.2.cmml" xref="A2.Ex4.m1.2.2.2.2"></divide><apply id="A2.Ex4.m1.3.3.3.3.1.cmml" xref="A2.Ex4.m1.3.3.3.3.1"><divide id="A2.Ex4.m1.3.3.3.3.1.1.cmml" xref="A2.Ex4.m1.3.3.3.3.1.1"></divide><csymbol cd="latexml" id="A2.Ex4.m1.3.3.3.3.1.2.cmml" xref="A2.Ex4.m1.3.3.3.3.1.2">absent</csymbol><cn type="integer" id="A2.Ex4.m1.3.3.3.3.1.3.cmml" xref="A2.Ex4.m1.3.3.3.3.1.3">2</cn></apply></list></apply><apply id="A2.Ex4.m1.6.6.3.3.3.cmml" xref="A2.Ex4.m1.6.6.3.3.3"><csymbol cd="ambiguous" id="A2.Ex4.m1.6.6.3.3.3.1.cmml" xref="A2.Ex4.m1.6.6.3.3.3">superscript</csymbol><apply id="A2.Ex4.m1.6.6.3.3.3.2.cmml" xref="A2.Ex4.m1.6.6.3.3.3"><csymbol cd="ambiguous" id="A2.Ex4.m1.6.6.3.3.3.2.1.cmml" xref="A2.Ex4.m1.6.6.3.3.3">subscript</csymbol><ci id="A2.Ex4.m1.6.6.3.3.3.2.2.cmml" xref="A2.Ex4.m1.6.6.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex4.m1.6.6.3.3.3.2.3.cmml" xref="A2.Ex4.m1.6.6.3.3.3.2.3">3</cn></apply><ci id="A2.Ex4.m1.6.6.3.3.3.3.cmml" xref="A2.Ex4.m1.6.6.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex4.m1.6.6.5.cmml" xref="A2.Ex4.m1.6.6.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex4.m1.6c">\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{K//2},\mathbf{x}_{3}^{K}]^{\top}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex5.m1.1" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{K}]^{\top}," display="inline"><semantics id="A2.Ex5.m1.1a"><mrow id="A2.Ex5.m1.1.1.1" xref="A2.Ex5.m1.1.1.1.1.cmml"><msup id="A2.Ex5.m1.1.1.1.1" xref="A2.Ex5.m1.1.1.1.1.cmml"><mrow id="A2.Ex5.m1.1.1.1.1.3.3" xref="A2.Ex5.m1.1.1.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex5.m1.1.1.1.1.3.3.4" xref="A2.Ex5.m1.1.1.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex5.m1.1.1.1.1.1.1.1" xref="A2.Ex5.m1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex5.m1.1.1.1.1.1.1.1.2.2" xref="A2.Ex5.m1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex5.m1.1.1.1.1.1.1.1.2.3" xref="A2.Ex5.m1.1.1.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex5.m1.1.1.1.1.1.1.1.3" xref="A2.Ex5.m1.1.1.1.1.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex5.m1.1.1.1.1.3.3.5" xref="A2.Ex5.m1.1.1.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex5.m1.1.1.1.1.2.2.2" xref="A2.Ex5.m1.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex5.m1.1.1.1.1.2.2.2.2.2" xref="A2.Ex5.m1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex5.m1.1.1.1.1.2.2.2.2.3" xref="A2.Ex5.m1.1.1.1.1.2.2.2.2.3.cmml">2</mn><mn mathsize="90%" id="A2.Ex5.m1.1.1.1.1.2.2.2.3" xref="A2.Ex5.m1.1.1.1.1.2.2.2.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex5.m1.1.1.1.1.3.3.6" xref="A2.Ex5.m1.1.1.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex5.m1.1.1.1.1.3.3.3" xref="A2.Ex5.m1.1.1.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex5.m1.1.1.1.1.3.3.3.2.2" xref="A2.Ex5.m1.1.1.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex5.m1.1.1.1.1.3.3.3.2.3" xref="A2.Ex5.m1.1.1.1.1.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex5.m1.1.1.1.1.3.3.3.3" xref="A2.Ex5.m1.1.1.1.1.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex5.m1.1.1.1.1.3.3.7" xref="A2.Ex5.m1.1.1.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex5.m1.1.1.1.1.5" xref="A2.Ex5.m1.1.1.1.1.5.cmml">⊤</mo></msup><mo mathsize="90%" id="A2.Ex5.m1.1.1.1.2" xref="A2.Ex5.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex5.m1.1b"><apply id="A2.Ex5.m1.1.1.1.1.cmml" xref="A2.Ex5.m1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex5.m1.1.1.1.1.4.cmml" xref="A2.Ex5.m1.1.1.1">superscript</csymbol><list id="A2.Ex5.m1.1.1.1.1.3.4.cmml" xref="A2.Ex5.m1.1.1.1.1.3.3"><apply id="A2.Ex5.m1.1.1.1.1.1.1.1.cmml" xref="A2.Ex5.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex5.m1.1.1.1.1.1.1.1.1.cmml" xref="A2.Ex5.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex5.m1.1.1.1.1.1.1.1.2.cmml" xref="A2.Ex5.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex5.m1.1.1.1.1.1.1.1.2.1.cmml" xref="A2.Ex5.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex5.m1.1.1.1.1.1.1.1.2.2.cmml" xref="A2.Ex5.m1.1.1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex5.m1.1.1.1.1.1.1.1.2.3.cmml" xref="A2.Ex5.m1.1.1.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex5.m1.1.1.1.1.1.1.1.3.cmml" xref="A2.Ex5.m1.1.1.1.1.1.1.1.3">0</cn></apply><apply id="A2.Ex5.m1.1.1.1.1.2.2.2.cmml" xref="A2.Ex5.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex5.m1.1.1.1.1.2.2.2.1.cmml" xref="A2.Ex5.m1.1.1.1.1.2.2.2">superscript</csymbol><apply id="A2.Ex5.m1.1.1.1.1.2.2.2.2.cmml" xref="A2.Ex5.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex5.m1.1.1.1.1.2.2.2.2.1.cmml" xref="A2.Ex5.m1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A2.Ex5.m1.1.1.1.1.2.2.2.2.2.cmml" xref="A2.Ex5.m1.1.1.1.1.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex5.m1.1.1.1.1.2.2.2.2.3.cmml" xref="A2.Ex5.m1.1.1.1.1.2.2.2.2.3">2</cn></apply><cn type="integer" id="A2.Ex5.m1.1.1.1.1.2.2.2.3.cmml" xref="A2.Ex5.m1.1.1.1.1.2.2.2.3">0</cn></apply><apply id="A2.Ex5.m1.1.1.1.1.3.3.3.cmml" xref="A2.Ex5.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex5.m1.1.1.1.1.3.3.3.1.cmml" xref="A2.Ex5.m1.1.1.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex5.m1.1.1.1.1.3.3.3.2.cmml" xref="A2.Ex5.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex5.m1.1.1.1.1.3.3.3.2.1.cmml" xref="A2.Ex5.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex5.m1.1.1.1.1.3.3.3.2.2.cmml" xref="A2.Ex5.m1.1.1.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex5.m1.1.1.1.1.3.3.3.2.3.cmml" xref="A2.Ex5.m1.1.1.1.1.3.3.3.2.3">3</cn></apply><ci id="A2.Ex5.m1.1.1.1.1.3.3.3.3.cmml" xref="A2.Ex5.m1.1.1.1.1.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex5.m1.1.1.1.1.5.cmml" xref="A2.Ex5.m1.1.1.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex5.m1.1c">\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{K}]^{\top},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex6.m1.4" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{K//2}]^{\top}," display="inline"><semantics id="A2.Ex6.m1.4a"><mrow id="A2.Ex6.m1.4.4.1" xref="A2.Ex6.m1.4.4.1.1.cmml"><msup id="A2.Ex6.m1.4.4.1.1" xref="A2.Ex6.m1.4.4.1.1.cmml"><mrow id="A2.Ex6.m1.4.4.1.1.3.3" xref="A2.Ex6.m1.4.4.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex6.m1.4.4.1.1.3.3.4" xref="A2.Ex6.m1.4.4.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex6.m1.4.4.1.1.1.1.1" xref="A2.Ex6.m1.4.4.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex6.m1.4.4.1.1.1.1.1.2.2" xref="A2.Ex6.m1.4.4.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex6.m1.4.4.1.1.1.1.1.2.3" xref="A2.Ex6.m1.4.4.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex6.m1.4.4.1.1.1.1.1.3" xref="A2.Ex6.m1.4.4.1.1.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex6.m1.4.4.1.1.3.3.5" xref="A2.Ex6.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex6.m1.4.4.1.1.2.2.2" xref="A2.Ex6.m1.4.4.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex6.m1.4.4.1.1.2.2.2.2.2" xref="A2.Ex6.m1.4.4.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex6.m1.4.4.1.1.2.2.2.2.3" xref="A2.Ex6.m1.4.4.1.1.2.2.2.2.3.cmml">2</mn><mn mathsize="90%" id="A2.Ex6.m1.4.4.1.1.2.2.2.3" xref="A2.Ex6.m1.4.4.1.1.2.2.2.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex6.m1.4.4.1.1.3.3.6" xref="A2.Ex6.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex6.m1.4.4.1.1.3.3.3" xref="A2.Ex6.m1.4.4.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex6.m1.4.4.1.1.3.3.3.2.2" xref="A2.Ex6.m1.4.4.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex6.m1.4.4.1.1.3.3.3.2.3" xref="A2.Ex6.m1.4.4.1.1.3.3.3.2.3.cmml">3</mn><mrow id="A2.Ex6.m1.3.3.3.3" xref="A2.Ex6.m1.3.3.3.4.cmml"><mi mathsize="90%" id="A2.Ex6.m1.1.1.1.1" xref="A2.Ex6.m1.1.1.1.1.cmml">K</mi><mo id="A2.Ex6.m1.3.3.3.3.2" xref="A2.Ex6.m1.3.3.3.4.cmml">⁣</mo><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex6.m1.2.2.2.2" xref="A2.Ex6.m1.2.2.2.2.cmml">/</mo><mo id="A2.Ex6.m1.3.3.3.3.3" xref="A2.Ex6.m1.3.3.3.4.cmml">⁣</mo><mrow id="A2.Ex6.m1.3.3.3.3.1" xref="A2.Ex6.m1.3.3.3.3.1.cmml"><mi id="A2.Ex6.m1.3.3.3.3.1.2" xref="A2.Ex6.m1.3.3.3.3.1.2.cmml"></mi><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex6.m1.3.3.3.3.1.1" xref="A2.Ex6.m1.3.3.3.3.1.1.cmml">/</mo><mn mathsize="90%" id="A2.Ex6.m1.3.3.3.3.1.3" xref="A2.Ex6.m1.3.3.3.3.1.3.cmml">2</mn></mrow></mrow></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex6.m1.4.4.1.1.3.3.7" xref="A2.Ex6.m1.4.4.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex6.m1.4.4.1.1.5" xref="A2.Ex6.m1.4.4.1.1.5.cmml">⊤</mo></msup><mo mathsize="90%" id="A2.Ex6.m1.4.4.1.2" xref="A2.Ex6.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex6.m1.4b"><apply id="A2.Ex6.m1.4.4.1.1.cmml" xref="A2.Ex6.m1.4.4.1"><csymbol cd="ambiguous" id="A2.Ex6.m1.4.4.1.1.4.cmml" xref="A2.Ex6.m1.4.4.1">superscript</csymbol><list id="A2.Ex6.m1.4.4.1.1.3.4.cmml" xref="A2.Ex6.m1.4.4.1.1.3.3"><apply id="A2.Ex6.m1.4.4.1.1.1.1.1.cmml" xref="A2.Ex6.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex6.m1.4.4.1.1.1.1.1.1.cmml" xref="A2.Ex6.m1.4.4.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex6.m1.4.4.1.1.1.1.1.2.cmml" xref="A2.Ex6.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex6.m1.4.4.1.1.1.1.1.2.1.cmml" xref="A2.Ex6.m1.4.4.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex6.m1.4.4.1.1.1.1.1.2.2.cmml" xref="A2.Ex6.m1.4.4.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex6.m1.4.4.1.1.1.1.1.2.3.cmml" xref="A2.Ex6.m1.4.4.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex6.m1.4.4.1.1.1.1.1.3.cmml" xref="A2.Ex6.m1.4.4.1.1.1.1.1.3">0</cn></apply><apply id="A2.Ex6.m1.4.4.1.1.2.2.2.cmml" xref="A2.Ex6.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex6.m1.4.4.1.1.2.2.2.1.cmml" xref="A2.Ex6.m1.4.4.1.1.2.2.2">superscript</csymbol><apply id="A2.Ex6.m1.4.4.1.1.2.2.2.2.cmml" xref="A2.Ex6.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex6.m1.4.4.1.1.2.2.2.2.1.cmml" xref="A2.Ex6.m1.4.4.1.1.2.2.2">subscript</csymbol><ci id="A2.Ex6.m1.4.4.1.1.2.2.2.2.2.cmml" xref="A2.Ex6.m1.4.4.1.1.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex6.m1.4.4.1.1.2.2.2.2.3.cmml" xref="A2.Ex6.m1.4.4.1.1.2.2.2.2.3">2</cn></apply><cn type="integer" id="A2.Ex6.m1.4.4.1.1.2.2.2.3.cmml" xref="A2.Ex6.m1.4.4.1.1.2.2.2.3">0</cn></apply><apply id="A2.Ex6.m1.4.4.1.1.3.3.3.cmml" xref="A2.Ex6.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex6.m1.4.4.1.1.3.3.3.1.cmml" xref="A2.Ex6.m1.4.4.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex6.m1.4.4.1.1.3.3.3.2.cmml" xref="A2.Ex6.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex6.m1.4.4.1.1.3.3.3.2.1.cmml" xref="A2.Ex6.m1.4.4.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex6.m1.4.4.1.1.3.3.3.2.2.cmml" xref="A2.Ex6.m1.4.4.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex6.m1.4.4.1.1.3.3.3.2.3.cmml" xref="A2.Ex6.m1.4.4.1.1.3.3.3.2.3">3</cn></apply><list id="A2.Ex6.m1.3.3.3.4.cmml" xref="A2.Ex6.m1.3.3.3.3"><ci id="A2.Ex6.m1.1.1.1.1.cmml" xref="A2.Ex6.m1.1.1.1.1">𝐾</ci><divide id="A2.Ex6.m1.2.2.2.2.cmml" xref="A2.Ex6.m1.2.2.2.2"></divide><apply id="A2.Ex6.m1.3.3.3.3.1.cmml" xref="A2.Ex6.m1.3.3.3.3.1"><divide id="A2.Ex6.m1.3.3.3.3.1.1.cmml" xref="A2.Ex6.m1.3.3.3.3.1.1"></divide><csymbol cd="latexml" id="A2.Ex6.m1.3.3.3.3.1.2.cmml" xref="A2.Ex6.m1.3.3.3.3.1.2">absent</csymbol><cn type="integer" id="A2.Ex6.m1.3.3.3.3.1.3.cmml" xref="A2.Ex6.m1.3.3.3.3.1.3">2</cn></apply></list></apply></list><csymbol cd="latexml" id="A2.Ex6.m1.4.4.1.1.5.cmml" xref="A2.Ex6.m1.4.4.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex6.m1.4c">\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{K//2}]^{\top},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex7.m1.1" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{0}]^{\top}." display="inline"><semantics id="A2.Ex7.m1.1a"><mrow id="A2.Ex7.m1.1.1.1" xref="A2.Ex7.m1.1.1.1.1.cmml"><msup id="A2.Ex7.m1.1.1.1.1" xref="A2.Ex7.m1.1.1.1.1.cmml"><mrow id="A2.Ex7.m1.1.1.1.1.3.3" xref="A2.Ex7.m1.1.1.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex7.m1.1.1.1.1.3.3.4" xref="A2.Ex7.m1.1.1.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex7.m1.1.1.1.1.1.1.1" xref="A2.Ex7.m1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex7.m1.1.1.1.1.1.1.1.2.2" xref="A2.Ex7.m1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex7.m1.1.1.1.1.1.1.1.2.3" xref="A2.Ex7.m1.1.1.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex7.m1.1.1.1.1.1.1.1.3" xref="A2.Ex7.m1.1.1.1.1.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex7.m1.1.1.1.1.3.3.5" xref="A2.Ex7.m1.1.1.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex7.m1.1.1.1.1.2.2.2" xref="A2.Ex7.m1.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex7.m1.1.1.1.1.2.2.2.2.2" xref="A2.Ex7.m1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex7.m1.1.1.1.1.2.2.2.2.3" xref="A2.Ex7.m1.1.1.1.1.2.2.2.2.3.cmml">2</mn><mn mathsize="90%" id="A2.Ex7.m1.1.1.1.1.2.2.2.3" xref="A2.Ex7.m1.1.1.1.1.2.2.2.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex7.m1.1.1.1.1.3.3.6" xref="A2.Ex7.m1.1.1.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex7.m1.1.1.1.1.3.3.3" xref="A2.Ex7.m1.1.1.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex7.m1.1.1.1.1.3.3.3.2.2" xref="A2.Ex7.m1.1.1.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex7.m1.1.1.1.1.3.3.3.2.3" xref="A2.Ex7.m1.1.1.1.1.3.3.3.2.3.cmml">3</mn><mn mathsize="90%" id="A2.Ex7.m1.1.1.1.1.3.3.3.3" xref="A2.Ex7.m1.1.1.1.1.3.3.3.3.cmml">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex7.m1.1.1.1.1.3.3.7" xref="A2.Ex7.m1.1.1.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex7.m1.1.1.1.1.5" xref="A2.Ex7.m1.1.1.1.1.5.cmml">⊤</mo></msup><mo lspace="0em" mathsize="90%" id="A2.Ex7.m1.1.1.1.2" xref="A2.Ex7.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex7.m1.1b"><apply id="A2.Ex7.m1.1.1.1.1.cmml" xref="A2.Ex7.m1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex7.m1.1.1.1.1.4.cmml" xref="A2.Ex7.m1.1.1.1">superscript</csymbol><list id="A2.Ex7.m1.1.1.1.1.3.4.cmml" xref="A2.Ex7.m1.1.1.1.1.3.3"><apply id="A2.Ex7.m1.1.1.1.1.1.1.1.cmml" xref="A2.Ex7.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex7.m1.1.1.1.1.1.1.1.1.cmml" xref="A2.Ex7.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex7.m1.1.1.1.1.1.1.1.2.cmml" xref="A2.Ex7.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex7.m1.1.1.1.1.1.1.1.2.1.cmml" xref="A2.Ex7.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex7.m1.1.1.1.1.1.1.1.2.2.cmml" xref="A2.Ex7.m1.1.1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex7.m1.1.1.1.1.1.1.1.2.3.cmml" xref="A2.Ex7.m1.1.1.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex7.m1.1.1.1.1.1.1.1.3.cmml" xref="A2.Ex7.m1.1.1.1.1.1.1.1.3">0</cn></apply><apply id="A2.Ex7.m1.1.1.1.1.2.2.2.cmml" xref="A2.Ex7.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex7.m1.1.1.1.1.2.2.2.1.cmml" xref="A2.Ex7.m1.1.1.1.1.2.2.2">superscript</csymbol><apply id="A2.Ex7.m1.1.1.1.1.2.2.2.2.cmml" xref="A2.Ex7.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex7.m1.1.1.1.1.2.2.2.2.1.cmml" xref="A2.Ex7.m1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A2.Ex7.m1.1.1.1.1.2.2.2.2.2.cmml" xref="A2.Ex7.m1.1.1.1.1.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex7.m1.1.1.1.1.2.2.2.2.3.cmml" xref="A2.Ex7.m1.1.1.1.1.2.2.2.2.3">2</cn></apply><cn type="integer" id="A2.Ex7.m1.1.1.1.1.2.2.2.3.cmml" xref="A2.Ex7.m1.1.1.1.1.2.2.2.3">0</cn></apply><apply id="A2.Ex7.m1.1.1.1.1.3.3.3.cmml" xref="A2.Ex7.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex7.m1.1.1.1.1.3.3.3.1.cmml" xref="A2.Ex7.m1.1.1.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex7.m1.1.1.1.1.3.3.3.2.cmml" xref="A2.Ex7.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex7.m1.1.1.1.1.3.3.3.2.1.cmml" xref="A2.Ex7.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex7.m1.1.1.1.1.3.3.3.2.2.cmml" xref="A2.Ex7.m1.1.1.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex7.m1.1.1.1.1.3.3.3.2.3.cmml" xref="A2.Ex7.m1.1.1.1.1.3.3.3.2.3">3</cn></apply><cn type="integer" id="A2.Ex7.m1.1.1.1.1.3.3.3.3.cmml" xref="A2.Ex7.m1.1.1.1.1.3.3.3.3">0</cn></apply></list><csymbol cd="latexml" id="A2.Ex7.m1.1.1.1.1.5.cmml" xref="A2.Ex7.m1.1.1.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex7.m1.1c">\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{0}]^{\top}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A2.SS6.SSS0.Px1.p2.2" class="ltx_p"><span id="A2.SS6.SSS0.Px1.p2.2.1" class="ltx_text" style="font-size:90%;">Notably, Diffusion Forcing can also perform this sampling scheme at sampling time for applications like imitation learning, when one wants to diffuse the next action as fast as possible.</span></p>
</div>
</section>
<section id="A2.SS6.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Full Sequence Diffusion.</h5>

<div id="A2.SS6.SSS0.Px2.p1" class="ltx_para">
<p id="A2.SS6.SSS0.Px2.p1.1" class="ltx_p"><span id="A2.SS6.SSS0.Px2.p1.1.1" class="ltx_text" style="font-size:90%;">Full sequence diffusion models accepts a noisy sequence and denoises level-by-level</span></p>
<table id="A6.EGx38" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A2.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex8.m1.3" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{K},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}" display="inline"><semantics id="A2.Ex8.m1.3a"><msup id="A2.Ex8.m1.3.3" xref="A2.Ex8.m1.3.3.cmml"><mrow id="A2.Ex8.m1.3.3.3.3" xref="A2.Ex8.m1.3.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex8.m1.3.3.3.3.4" xref="A2.Ex8.m1.3.3.3.4.cmml">[</mo><msubsup id="A2.Ex8.m1.1.1.1.1.1" xref="A2.Ex8.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex8.m1.1.1.1.1.1.2.2" xref="A2.Ex8.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex8.m1.1.1.1.1.1.2.3" xref="A2.Ex8.m1.1.1.1.1.1.2.3.cmml">1</mn><mi mathsize="90%" id="A2.Ex8.m1.1.1.1.1.1.3" xref="A2.Ex8.m1.1.1.1.1.1.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex8.m1.3.3.3.3.5" xref="A2.Ex8.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex8.m1.2.2.2.2.2" xref="A2.Ex8.m1.2.2.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex8.m1.2.2.2.2.2.2.2" xref="A2.Ex8.m1.2.2.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex8.m1.2.2.2.2.2.2.3" xref="A2.Ex8.m1.2.2.2.2.2.2.3.cmml">2</mn><mi mathsize="90%" id="A2.Ex8.m1.2.2.2.2.2.3" xref="A2.Ex8.m1.2.2.2.2.2.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex8.m1.3.3.3.3.6" xref="A2.Ex8.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex8.m1.3.3.3.3.3" xref="A2.Ex8.m1.3.3.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex8.m1.3.3.3.3.3.2.2" xref="A2.Ex8.m1.3.3.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex8.m1.3.3.3.3.3.2.3" xref="A2.Ex8.m1.3.3.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex8.m1.3.3.3.3.3.3" xref="A2.Ex8.m1.3.3.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex8.m1.3.3.3.3.7" xref="A2.Ex8.m1.3.3.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex8.m1.3.3.5" xref="A2.Ex8.m1.3.3.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="A2.Ex8.m1.3b"><apply id="A2.Ex8.m1.3.3.cmml" xref="A2.Ex8.m1.3.3"><csymbol cd="ambiguous" id="A2.Ex8.m1.3.3.4.cmml" xref="A2.Ex8.m1.3.3">superscript</csymbol><list id="A2.Ex8.m1.3.3.3.4.cmml" xref="A2.Ex8.m1.3.3.3.3"><apply id="A2.Ex8.m1.1.1.1.1.1.cmml" xref="A2.Ex8.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex8.m1.1.1.1.1.1.1.cmml" xref="A2.Ex8.m1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex8.m1.1.1.1.1.1.2.cmml" xref="A2.Ex8.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex8.m1.1.1.1.1.1.2.1.cmml" xref="A2.Ex8.m1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex8.m1.1.1.1.1.1.2.2.cmml" xref="A2.Ex8.m1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex8.m1.1.1.1.1.1.2.3.cmml" xref="A2.Ex8.m1.1.1.1.1.1.2.3">1</cn></apply><ci id="A2.Ex8.m1.1.1.1.1.1.3.cmml" xref="A2.Ex8.m1.1.1.1.1.1.3">𝐾</ci></apply><apply id="A2.Ex8.m1.2.2.2.2.2.cmml" xref="A2.Ex8.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex8.m1.2.2.2.2.2.1.cmml" xref="A2.Ex8.m1.2.2.2.2.2">superscript</csymbol><apply id="A2.Ex8.m1.2.2.2.2.2.2.cmml" xref="A2.Ex8.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex8.m1.2.2.2.2.2.2.1.cmml" xref="A2.Ex8.m1.2.2.2.2.2">subscript</csymbol><ci id="A2.Ex8.m1.2.2.2.2.2.2.2.cmml" xref="A2.Ex8.m1.2.2.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex8.m1.2.2.2.2.2.2.3.cmml" xref="A2.Ex8.m1.2.2.2.2.2.2.3">2</cn></apply><ci id="A2.Ex8.m1.2.2.2.2.2.3.cmml" xref="A2.Ex8.m1.2.2.2.2.2.3">𝐾</ci></apply><apply id="A2.Ex8.m1.3.3.3.3.3.cmml" xref="A2.Ex8.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex8.m1.3.3.3.3.3.1.cmml" xref="A2.Ex8.m1.3.3.3.3.3">superscript</csymbol><apply id="A2.Ex8.m1.3.3.3.3.3.2.cmml" xref="A2.Ex8.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex8.m1.3.3.3.3.3.2.1.cmml" xref="A2.Ex8.m1.3.3.3.3.3">subscript</csymbol><ci id="A2.Ex8.m1.3.3.3.3.3.2.2.cmml" xref="A2.Ex8.m1.3.3.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex8.m1.3.3.3.3.3.2.3.cmml" xref="A2.Ex8.m1.3.3.3.3.3.2.3">3</cn></apply><ci id="A2.Ex8.m1.3.3.3.3.3.3.cmml" xref="A2.Ex8.m1.3.3.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex8.m1.3.3.5.cmml" xref="A2.Ex8.m1.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex8.m1.3c">\displaystyle[\mathbf{x}_{1}^{K},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex9.m1.10" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{K//2},\mathbf{x}_{2}^{K//2},\mathbf{x}_{3}^{K//2}]^{\top}," display="inline"><semantics id="A2.Ex9.m1.10a"><mrow id="A2.Ex9.m1.10.10.1" xref="A2.Ex9.m1.10.10.1.1.cmml"><msup id="A2.Ex9.m1.10.10.1.1" xref="A2.Ex9.m1.10.10.1.1.cmml"><mrow id="A2.Ex9.m1.10.10.1.1.3.3" xref="A2.Ex9.m1.10.10.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex9.m1.10.10.1.1.3.3.4" xref="A2.Ex9.m1.10.10.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex9.m1.10.10.1.1.1.1.1" xref="A2.Ex9.m1.10.10.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex9.m1.10.10.1.1.1.1.1.2.2" xref="A2.Ex9.m1.10.10.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex9.m1.10.10.1.1.1.1.1.2.3" xref="A2.Ex9.m1.10.10.1.1.1.1.1.2.3.cmml">1</mn><mrow id="A2.Ex9.m1.3.3.3.3" xref="A2.Ex9.m1.3.3.3.4.cmml"><mi mathsize="90%" id="A2.Ex9.m1.1.1.1.1" xref="A2.Ex9.m1.1.1.1.1.cmml">K</mi><mo id="A2.Ex9.m1.3.3.3.3.2" 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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex10.m1.1" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{0}]^{\top}." display="inline"><semantics id="A2.Ex10.m1.1a"><mrow id="A2.Ex10.m1.1.1.1" xref="A2.Ex10.m1.1.1.1.1.cmml"><msup id="A2.Ex10.m1.1.1.1.1" xref="A2.Ex10.m1.1.1.1.1.cmml"><mrow id="A2.Ex10.m1.1.1.1.1.3.3" xref="A2.Ex10.m1.1.1.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex10.m1.1.1.1.1.3.3.4" xref="A2.Ex10.m1.1.1.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex10.m1.1.1.1.1.1.1.1" xref="A2.Ex10.m1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex10.m1.1.1.1.1.1.1.1.2.2" xref="A2.Ex10.m1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex10.m1.1.1.1.1.1.1.1.2.3" xref="A2.Ex10.m1.1.1.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex10.m1.1.1.1.1.1.1.1.3" xref="A2.Ex10.m1.1.1.1.1.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex10.m1.1.1.1.1.3.3.5" xref="A2.Ex10.m1.1.1.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex10.m1.1.1.1.1.2.2.2" xref="A2.Ex10.m1.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex10.m1.1.1.1.1.2.2.2.2.2" xref="A2.Ex10.m1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex10.m1.1.1.1.1.2.2.2.2.3" xref="A2.Ex10.m1.1.1.1.1.2.2.2.2.3.cmml">2</mn><mn mathsize="90%" id="A2.Ex10.m1.1.1.1.1.2.2.2.3" xref="A2.Ex10.m1.1.1.1.1.2.2.2.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex10.m1.1.1.1.1.3.3.6" xref="A2.Ex10.m1.1.1.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex10.m1.1.1.1.1.3.3.3" xref="A2.Ex10.m1.1.1.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex10.m1.1.1.1.1.3.3.3.2.2" xref="A2.Ex10.m1.1.1.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex10.m1.1.1.1.1.3.3.3.2.3" xref="A2.Ex10.m1.1.1.1.1.3.3.3.2.3.cmml">3</mn><mn mathsize="90%" id="A2.Ex10.m1.1.1.1.1.3.3.3.3" xref="A2.Ex10.m1.1.1.1.1.3.3.3.3.cmml">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex10.m1.1.1.1.1.3.3.7" xref="A2.Ex10.m1.1.1.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex10.m1.1.1.1.1.5" xref="A2.Ex10.m1.1.1.1.1.5.cmml">⊤</mo></msup><mo lspace="0em" mathsize="90%" id="A2.Ex10.m1.1.1.1.2" xref="A2.Ex10.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex10.m1.1b"><apply id="A2.Ex10.m1.1.1.1.1.cmml" xref="A2.Ex10.m1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex10.m1.1.1.1.1.4.cmml" xref="A2.Ex10.m1.1.1.1">superscript</csymbol><list id="A2.Ex10.m1.1.1.1.1.3.4.cmml" xref="A2.Ex10.m1.1.1.1.1.3.3"><apply id="A2.Ex10.m1.1.1.1.1.1.1.1.cmml" xref="A2.Ex10.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex10.m1.1.1.1.1.1.1.1.1.cmml" xref="A2.Ex10.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex10.m1.1.1.1.1.1.1.1.2.cmml" xref="A2.Ex10.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex10.m1.1.1.1.1.1.1.1.2.1.cmml" xref="A2.Ex10.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex10.m1.1.1.1.1.1.1.1.2.2.cmml" xref="A2.Ex10.m1.1.1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex10.m1.1.1.1.1.1.1.1.2.3.cmml" xref="A2.Ex10.m1.1.1.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex10.m1.1.1.1.1.1.1.1.3.cmml" xref="A2.Ex10.m1.1.1.1.1.1.1.1.3">0</cn></apply><apply id="A2.Ex10.m1.1.1.1.1.2.2.2.cmml" xref="A2.Ex10.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex10.m1.1.1.1.1.2.2.2.1.cmml" xref="A2.Ex10.m1.1.1.1.1.2.2.2">superscript</csymbol><apply id="A2.Ex10.m1.1.1.1.1.2.2.2.2.cmml" xref="A2.Ex10.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex10.m1.1.1.1.1.2.2.2.2.1.cmml" xref="A2.Ex10.m1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A2.Ex10.m1.1.1.1.1.2.2.2.2.2.cmml" xref="A2.Ex10.m1.1.1.1.1.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex10.m1.1.1.1.1.2.2.2.2.3.cmml" xref="A2.Ex10.m1.1.1.1.1.2.2.2.2.3">2</cn></apply><cn type="integer" id="A2.Ex10.m1.1.1.1.1.2.2.2.3.cmml" xref="A2.Ex10.m1.1.1.1.1.2.2.2.3">0</cn></apply><apply id="A2.Ex10.m1.1.1.1.1.3.3.3.cmml" xref="A2.Ex10.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex10.m1.1.1.1.1.3.3.3.1.cmml" xref="A2.Ex10.m1.1.1.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex10.m1.1.1.1.1.3.3.3.2.cmml" xref="A2.Ex10.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex10.m1.1.1.1.1.3.3.3.2.1.cmml" xref="A2.Ex10.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex10.m1.1.1.1.1.3.3.3.2.2.cmml" xref="A2.Ex10.m1.1.1.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex10.m1.1.1.1.1.3.3.3.2.3.cmml" xref="A2.Ex10.m1.1.1.1.1.3.3.3.2.3">3</cn></apply><cn type="integer" id="A2.Ex10.m1.1.1.1.1.3.3.3.3.cmml" xref="A2.Ex10.m1.1.1.1.1.3.3.3.3">0</cn></apply></list><csymbol cd="latexml" id="A2.Ex10.m1.1.1.1.1.5.cmml" xref="A2.Ex10.m1.1.1.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex10.m1.1c">\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{0}]^{\top}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A2.SS6.SSS0.Px2.p1.2" class="ltx_p"><span id="A2.SS6.SSS0.Px2.p1.2.1" class="ltx_text" style="font-size:90%;">Notably, Diffusion Forcing can also perform this sampling scheme at sampling time.</span></p>
</div>
</section>
<section id="A2.SS6.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Diffusion Forcing with causal uncertainty</h5>

<div id="A2.SS6.SSS0.Px3.p1" class="ltx_para">
<p id="A2.SS6.SSS0.Px3.p1.1" class="ltx_p"><span id="A2.SS6.SSS0.Px3.p1.1.1" class="ltx_text" style="font-size:90%;">As shown in Figure </span><a href="#S2.F2" title="Figure 2 ‣ Bayesian Filtering. ‣ 2 Related Work and Preliminaries ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">2</span></a><span id="A2.SS6.SSS0.Px3.p1.1.2" class="ltx_text" style="font-size:90%;">, to model causal uncertainty, Diffusion Forcing keeps the far future more uncertain than the near future by having larger noise level </span><math id="A2.SS6.SSS0.Px3.p1.1.m1.1" class="ltx_Math" alttext="k" display="inline"><semantics id="A2.SS6.SSS0.Px3.p1.1.m1.1a"><mi mathsize="90%" id="A2.SS6.SSS0.Px3.p1.1.m1.1.1" xref="A2.SS6.SSS0.Px3.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A2.SS6.SSS0.Px3.p1.1.m1.1b"><ci id="A2.SS6.SSS0.Px3.p1.1.m1.1.1.cmml" xref="A2.SS6.SSS0.Px3.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS6.SSS0.Px3.p1.1.m1.1c">k</annotation></semantics></math><span id="A2.SS6.SSS0.Px3.p1.1.3" class="ltx_text" style="font-size:90%;">, at any time of diffusion. An example pattern looks like this:</span></p>
<table id="A6.EGx39" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A2.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex11.m1.3" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{K},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}" display="inline"><semantics id="A2.Ex11.m1.3a"><msup id="A2.Ex11.m1.3.3" xref="A2.Ex11.m1.3.3.cmml"><mrow id="A2.Ex11.m1.3.3.3.3" xref="A2.Ex11.m1.3.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex11.m1.3.3.3.3.4" xref="A2.Ex11.m1.3.3.3.4.cmml">[</mo><msubsup id="A2.Ex11.m1.1.1.1.1.1" xref="A2.Ex11.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex11.m1.1.1.1.1.1.2.2" xref="A2.Ex11.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex11.m1.1.1.1.1.1.2.3" xref="A2.Ex11.m1.1.1.1.1.1.2.3.cmml">1</mn><mi mathsize="90%" id="A2.Ex11.m1.1.1.1.1.1.3" xref="A2.Ex11.m1.1.1.1.1.1.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex11.m1.3.3.3.3.5" xref="A2.Ex11.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex11.m1.2.2.2.2.2" xref="A2.Ex11.m1.2.2.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex11.m1.2.2.2.2.2.2.2" xref="A2.Ex11.m1.2.2.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex11.m1.2.2.2.2.2.2.3" xref="A2.Ex11.m1.2.2.2.2.2.2.3.cmml">2</mn><mi mathsize="90%" id="A2.Ex11.m1.2.2.2.2.2.3" xref="A2.Ex11.m1.2.2.2.2.2.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex11.m1.3.3.3.3.6" xref="A2.Ex11.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex11.m1.3.3.3.3.3" xref="A2.Ex11.m1.3.3.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex11.m1.3.3.3.3.3.2.2" xref="A2.Ex11.m1.3.3.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex11.m1.3.3.3.3.3.2.3" xref="A2.Ex11.m1.3.3.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex11.m1.3.3.3.3.3.3" xref="A2.Ex11.m1.3.3.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex11.m1.3.3.3.3.7" xref="A2.Ex11.m1.3.3.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex11.m1.3.3.5" xref="A2.Ex11.m1.3.3.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="A2.Ex11.m1.3b"><apply id="A2.Ex11.m1.3.3.cmml" xref="A2.Ex11.m1.3.3"><csymbol cd="ambiguous" id="A2.Ex11.m1.3.3.4.cmml" xref="A2.Ex11.m1.3.3">superscript</csymbol><list id="A2.Ex11.m1.3.3.3.4.cmml" xref="A2.Ex11.m1.3.3.3.3"><apply id="A2.Ex11.m1.1.1.1.1.1.cmml" xref="A2.Ex11.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex11.m1.1.1.1.1.1.1.cmml" xref="A2.Ex11.m1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex11.m1.1.1.1.1.1.2.cmml" xref="A2.Ex11.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex11.m1.1.1.1.1.1.2.1.cmml" xref="A2.Ex11.m1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex11.m1.1.1.1.1.1.2.2.cmml" xref="A2.Ex11.m1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex11.m1.1.1.1.1.1.2.3.cmml" xref="A2.Ex11.m1.1.1.1.1.1.2.3">1</cn></apply><ci id="A2.Ex11.m1.1.1.1.1.1.3.cmml" xref="A2.Ex11.m1.1.1.1.1.1.3">𝐾</ci></apply><apply id="A2.Ex11.m1.2.2.2.2.2.cmml" xref="A2.Ex11.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex11.m1.2.2.2.2.2.1.cmml" xref="A2.Ex11.m1.2.2.2.2.2">superscript</csymbol><apply id="A2.Ex11.m1.2.2.2.2.2.2.cmml" xref="A2.Ex11.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex11.m1.2.2.2.2.2.2.1.cmml" xref="A2.Ex11.m1.2.2.2.2.2">subscript</csymbol><ci id="A2.Ex11.m1.2.2.2.2.2.2.2.cmml" xref="A2.Ex11.m1.2.2.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex11.m1.2.2.2.2.2.2.3.cmml" xref="A2.Ex11.m1.2.2.2.2.2.2.3">2</cn></apply><ci id="A2.Ex11.m1.2.2.2.2.2.3.cmml" xref="A2.Ex11.m1.2.2.2.2.2.3">𝐾</ci></apply><apply id="A2.Ex11.m1.3.3.3.3.3.cmml" xref="A2.Ex11.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex11.m1.3.3.3.3.3.1.cmml" xref="A2.Ex11.m1.3.3.3.3.3">superscript</csymbol><apply id="A2.Ex11.m1.3.3.3.3.3.2.cmml" xref="A2.Ex11.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex11.m1.3.3.3.3.3.2.1.cmml" xref="A2.Ex11.m1.3.3.3.3.3">subscript</csymbol><ci id="A2.Ex11.m1.3.3.3.3.3.2.2.cmml" xref="A2.Ex11.m1.3.3.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex11.m1.3.3.3.3.3.2.3.cmml" xref="A2.Ex11.m1.3.3.3.3.3.2.3">3</cn></apply><ci id="A2.Ex11.m1.3.3.3.3.3.3.cmml" xref="A2.Ex11.m1.3.3.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex11.m1.3.3.5.cmml" xref="A2.Ex11.m1.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex11.m1.3c">\displaystyle[\mathbf{x}_{1}^{K},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex12.m1.4" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{K//2},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}," display="inline"><semantics id="A2.Ex12.m1.4a"><mrow id="A2.Ex12.m1.4.4.1" xref="A2.Ex12.m1.4.4.1.1.cmml"><msup id="A2.Ex12.m1.4.4.1.1" xref="A2.Ex12.m1.4.4.1.1.cmml"><mrow id="A2.Ex12.m1.4.4.1.1.3.3" xref="A2.Ex12.m1.4.4.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex12.m1.4.4.1.1.3.3.4" xref="A2.Ex12.m1.4.4.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex12.m1.4.4.1.1.1.1.1" xref="A2.Ex12.m1.4.4.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex12.m1.4.4.1.1.1.1.1.2.2" xref="A2.Ex12.m1.4.4.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex12.m1.4.4.1.1.1.1.1.2.3" xref="A2.Ex12.m1.4.4.1.1.1.1.1.2.3.cmml">1</mn><mrow id="A2.Ex12.m1.3.3.3.3" xref="A2.Ex12.m1.3.3.3.4.cmml"><mi mathsize="90%" id="A2.Ex12.m1.1.1.1.1" xref="A2.Ex12.m1.1.1.1.1.cmml">K</mi><mo id="A2.Ex12.m1.3.3.3.3.2" xref="A2.Ex12.m1.3.3.3.4.cmml">⁣</mo><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex12.m1.2.2.2.2" xref="A2.Ex12.m1.2.2.2.2.cmml">/</mo><mo id="A2.Ex12.m1.3.3.3.3.3" xref="A2.Ex12.m1.3.3.3.4.cmml">⁣</mo><mrow id="A2.Ex12.m1.3.3.3.3.1" xref="A2.Ex12.m1.3.3.3.3.1.cmml"><mi id="A2.Ex12.m1.3.3.3.3.1.2" xref="A2.Ex12.m1.3.3.3.3.1.2.cmml"></mi><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex12.m1.3.3.3.3.1.1" xref="A2.Ex12.m1.3.3.3.3.1.1.cmml">/</mo><mn mathsize="90%" id="A2.Ex12.m1.3.3.3.3.1.3" xref="A2.Ex12.m1.3.3.3.3.1.3.cmml">2</mn></mrow></mrow></msubsup><mo mathsize="90%" id="A2.Ex12.m1.4.4.1.1.3.3.5" xref="A2.Ex12.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex12.m1.4.4.1.1.2.2.2" xref="A2.Ex12.m1.4.4.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex12.m1.4.4.1.1.2.2.2.2.2" xref="A2.Ex12.m1.4.4.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex12.m1.4.4.1.1.2.2.2.2.3" xref="A2.Ex12.m1.4.4.1.1.2.2.2.2.3.cmml">2</mn><mi mathsize="90%" id="A2.Ex12.m1.4.4.1.1.2.2.2.3" xref="A2.Ex12.m1.4.4.1.1.2.2.2.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex12.m1.4.4.1.1.3.3.6" xref="A2.Ex12.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex12.m1.4.4.1.1.3.3.3" xref="A2.Ex12.m1.4.4.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex12.m1.4.4.1.1.3.3.3.2.2" xref="A2.Ex12.m1.4.4.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex12.m1.4.4.1.1.3.3.3.2.3" xref="A2.Ex12.m1.4.4.1.1.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex12.m1.4.4.1.1.3.3.3.3" xref="A2.Ex12.m1.4.4.1.1.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex12.m1.4.4.1.1.3.3.7" xref="A2.Ex12.m1.4.4.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex12.m1.4.4.1.1.5" xref="A2.Ex12.m1.4.4.1.1.5.cmml">⊤</mo></msup><mo mathsize="90%" id="A2.Ex12.m1.4.4.1.2" xref="A2.Ex12.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex12.m1.4b"><apply id="A2.Ex12.m1.4.4.1.1.cmml" xref="A2.Ex12.m1.4.4.1"><csymbol cd="ambiguous" id="A2.Ex12.m1.4.4.1.1.4.cmml" xref="A2.Ex12.m1.4.4.1">superscript</csymbol><list id="A2.Ex12.m1.4.4.1.1.3.4.cmml" xref="A2.Ex12.m1.4.4.1.1.3.3"><apply id="A2.Ex12.m1.4.4.1.1.1.1.1.cmml" xref="A2.Ex12.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex12.m1.4.4.1.1.1.1.1.1.cmml" xref="A2.Ex12.m1.4.4.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex12.m1.4.4.1.1.1.1.1.2.cmml" xref="A2.Ex12.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex12.m1.4.4.1.1.1.1.1.2.1.cmml" xref="A2.Ex12.m1.4.4.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex12.m1.4.4.1.1.1.1.1.2.2.cmml" xref="A2.Ex12.m1.4.4.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex12.m1.4.4.1.1.1.1.1.2.3.cmml" xref="A2.Ex12.m1.4.4.1.1.1.1.1.2.3">1</cn></apply><list id="A2.Ex12.m1.3.3.3.4.cmml" xref="A2.Ex12.m1.3.3.3.3"><ci id="A2.Ex12.m1.1.1.1.1.cmml" xref="A2.Ex12.m1.1.1.1.1">𝐾</ci><divide id="A2.Ex12.m1.2.2.2.2.cmml" xref="A2.Ex12.m1.2.2.2.2"></divide><apply id="A2.Ex12.m1.3.3.3.3.1.cmml" xref="A2.Ex12.m1.3.3.3.3.1"><divide id="A2.Ex12.m1.3.3.3.3.1.1.cmml" xref="A2.Ex12.m1.3.3.3.3.1.1"></divide><csymbol cd="latexml" id="A2.Ex12.m1.3.3.3.3.1.2.cmml" xref="A2.Ex12.m1.3.3.3.3.1.2">absent</csymbol><cn type="integer" id="A2.Ex12.m1.3.3.3.3.1.3.cmml" xref="A2.Ex12.m1.3.3.3.3.1.3">2</cn></apply></list></apply><apply id="A2.Ex12.m1.4.4.1.1.2.2.2.cmml" xref="A2.Ex12.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex12.m1.4.4.1.1.2.2.2.1.cmml" xref="A2.Ex12.m1.4.4.1.1.2.2.2">superscript</csymbol><apply id="A2.Ex12.m1.4.4.1.1.2.2.2.2.cmml" xref="A2.Ex12.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex12.m1.4.4.1.1.2.2.2.2.1.cmml" xref="A2.Ex12.m1.4.4.1.1.2.2.2">subscript</csymbol><ci id="A2.Ex12.m1.4.4.1.1.2.2.2.2.2.cmml" xref="A2.Ex12.m1.4.4.1.1.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex12.m1.4.4.1.1.2.2.2.2.3.cmml" xref="A2.Ex12.m1.4.4.1.1.2.2.2.2.3">2</cn></apply><ci id="A2.Ex12.m1.4.4.1.1.2.2.2.3.cmml" xref="A2.Ex12.m1.4.4.1.1.2.2.2.3">𝐾</ci></apply><apply id="A2.Ex12.m1.4.4.1.1.3.3.3.cmml" xref="A2.Ex12.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex12.m1.4.4.1.1.3.3.3.1.cmml" xref="A2.Ex12.m1.4.4.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex12.m1.4.4.1.1.3.3.3.2.cmml" xref="A2.Ex12.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex12.m1.4.4.1.1.3.3.3.2.1.cmml" xref="A2.Ex12.m1.4.4.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex12.m1.4.4.1.1.3.3.3.2.2.cmml" xref="A2.Ex12.m1.4.4.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex12.m1.4.4.1.1.3.3.3.2.3.cmml" xref="A2.Ex12.m1.4.4.1.1.3.3.3.2.3">3</cn></apply><ci id="A2.Ex12.m1.4.4.1.1.3.3.3.3.cmml" xref="A2.Ex12.m1.4.4.1.1.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex12.m1.4.4.1.1.5.cmml" xref="A2.Ex12.m1.4.4.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex12.m1.4c">\displaystyle[\mathbf{x}_{1}^{K//2},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex13.m1.4" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{K//2},\mathbf{x}_{3}^{K}]^{\top}," display="inline"><semantics id="A2.Ex13.m1.4a"><mrow id="A2.Ex13.m1.4.4.1" xref="A2.Ex13.m1.4.4.1.1.cmml"><msup id="A2.Ex13.m1.4.4.1.1" xref="A2.Ex13.m1.4.4.1.1.cmml"><mrow id="A2.Ex13.m1.4.4.1.1.3.3" xref="A2.Ex13.m1.4.4.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex13.m1.4.4.1.1.3.3.4" xref="A2.Ex13.m1.4.4.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex13.m1.4.4.1.1.1.1.1" xref="A2.Ex13.m1.4.4.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex13.m1.4.4.1.1.1.1.1.2.2" xref="A2.Ex13.m1.4.4.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex13.m1.4.4.1.1.1.1.1.2.3" xref="A2.Ex13.m1.4.4.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex13.m1.4.4.1.1.1.1.1.3" xref="A2.Ex13.m1.4.4.1.1.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex13.m1.4.4.1.1.3.3.5" xref="A2.Ex13.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex13.m1.4.4.1.1.2.2.2" xref="A2.Ex13.m1.4.4.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex13.m1.4.4.1.1.2.2.2.2.2" xref="A2.Ex13.m1.4.4.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex13.m1.4.4.1.1.2.2.2.2.3" xref="A2.Ex13.m1.4.4.1.1.2.2.2.2.3.cmml">2</mn><mrow id="A2.Ex13.m1.3.3.3.3" xref="A2.Ex13.m1.3.3.3.4.cmml"><mi mathsize="90%" id="A2.Ex13.m1.1.1.1.1" xref="A2.Ex13.m1.1.1.1.1.cmml">K</mi><mo id="A2.Ex13.m1.3.3.3.3.2" xref="A2.Ex13.m1.3.3.3.4.cmml">⁣</mo><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex13.m1.2.2.2.2" xref="A2.Ex13.m1.2.2.2.2.cmml">/</mo><mo id="A2.Ex13.m1.3.3.3.3.3" xref="A2.Ex13.m1.3.3.3.4.cmml">⁣</mo><mrow id="A2.Ex13.m1.3.3.3.3.1" xref="A2.Ex13.m1.3.3.3.3.1.cmml"><mi id="A2.Ex13.m1.3.3.3.3.1.2" xref="A2.Ex13.m1.3.3.3.3.1.2.cmml"></mi><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex13.m1.3.3.3.3.1.1" xref="A2.Ex13.m1.3.3.3.3.1.1.cmml">/</mo><mn mathsize="90%" id="A2.Ex13.m1.3.3.3.3.1.3" xref="A2.Ex13.m1.3.3.3.3.1.3.cmml">2</mn></mrow></mrow></msubsup><mo mathsize="90%" id="A2.Ex13.m1.4.4.1.1.3.3.6" xref="A2.Ex13.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex13.m1.4.4.1.1.3.3.3" xref="A2.Ex13.m1.4.4.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex13.m1.4.4.1.1.3.3.3.2.2" xref="A2.Ex13.m1.4.4.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex13.m1.4.4.1.1.3.3.3.2.3" xref="A2.Ex13.m1.4.4.1.1.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex13.m1.4.4.1.1.3.3.3.3" xref="A2.Ex13.m1.4.4.1.1.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex13.m1.4.4.1.1.3.3.7" xref="A2.Ex13.m1.4.4.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex13.m1.4.4.1.1.5" xref="A2.Ex13.m1.4.4.1.1.5.cmml">⊤</mo></msup><mo mathsize="90%" id="A2.Ex13.m1.4.4.1.2" xref="A2.Ex13.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex13.m1.4b"><apply id="A2.Ex13.m1.4.4.1.1.cmml" xref="A2.Ex13.m1.4.4.1"><csymbol cd="ambiguous" id="A2.Ex13.m1.4.4.1.1.4.cmml" xref="A2.Ex13.m1.4.4.1">superscript</csymbol><list id="A2.Ex13.m1.4.4.1.1.3.4.cmml" xref="A2.Ex13.m1.4.4.1.1.3.3"><apply id="A2.Ex13.m1.4.4.1.1.1.1.1.cmml" xref="A2.Ex13.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex13.m1.4.4.1.1.1.1.1.1.cmml" xref="A2.Ex13.m1.4.4.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex13.m1.4.4.1.1.1.1.1.2.cmml" xref="A2.Ex13.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex13.m1.4.4.1.1.1.1.1.2.1.cmml" xref="A2.Ex13.m1.4.4.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex13.m1.4.4.1.1.1.1.1.2.2.cmml" xref="A2.Ex13.m1.4.4.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex13.m1.4.4.1.1.1.1.1.2.3.cmml" xref="A2.Ex13.m1.4.4.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex13.m1.4.4.1.1.1.1.1.3.cmml" xref="A2.Ex13.m1.4.4.1.1.1.1.1.3">0</cn></apply><apply id="A2.Ex13.m1.4.4.1.1.2.2.2.cmml" xref="A2.Ex13.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex13.m1.4.4.1.1.2.2.2.1.cmml" xref="A2.Ex13.m1.4.4.1.1.2.2.2">superscript</csymbol><apply id="A2.Ex13.m1.4.4.1.1.2.2.2.2.cmml" xref="A2.Ex13.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex13.m1.4.4.1.1.2.2.2.2.1.cmml" xref="A2.Ex13.m1.4.4.1.1.2.2.2">subscript</csymbol><ci id="A2.Ex13.m1.4.4.1.1.2.2.2.2.2.cmml" xref="A2.Ex13.m1.4.4.1.1.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex13.m1.4.4.1.1.2.2.2.2.3.cmml" xref="A2.Ex13.m1.4.4.1.1.2.2.2.2.3">2</cn></apply><list id="A2.Ex13.m1.3.3.3.4.cmml" xref="A2.Ex13.m1.3.3.3.3"><ci id="A2.Ex13.m1.1.1.1.1.cmml" xref="A2.Ex13.m1.1.1.1.1">𝐾</ci><divide id="A2.Ex13.m1.2.2.2.2.cmml" xref="A2.Ex13.m1.2.2.2.2"></divide><apply id="A2.Ex13.m1.3.3.3.3.1.cmml" xref="A2.Ex13.m1.3.3.3.3.1"><divide id="A2.Ex13.m1.3.3.3.3.1.1.cmml" xref="A2.Ex13.m1.3.3.3.3.1.1"></divide><csymbol cd="latexml" id="A2.Ex13.m1.3.3.3.3.1.2.cmml" xref="A2.Ex13.m1.3.3.3.3.1.2">absent</csymbol><cn type="integer" id="A2.Ex13.m1.3.3.3.3.1.3.cmml" xref="A2.Ex13.m1.3.3.3.3.1.3">2</cn></apply></list></apply><apply id="A2.Ex13.m1.4.4.1.1.3.3.3.cmml" xref="A2.Ex13.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex13.m1.4.4.1.1.3.3.3.1.cmml" xref="A2.Ex13.m1.4.4.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex13.m1.4.4.1.1.3.3.3.2.cmml" xref="A2.Ex13.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex13.m1.4.4.1.1.3.3.3.2.1.cmml" xref="A2.Ex13.m1.4.4.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex13.m1.4.4.1.1.3.3.3.2.2.cmml" xref="A2.Ex13.m1.4.4.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex13.m1.4.4.1.1.3.3.3.2.3.cmml" xref="A2.Ex13.m1.4.4.1.1.3.3.3.2.3">3</cn></apply><ci id="A2.Ex13.m1.4.4.1.1.3.3.3.3.cmml" xref="A2.Ex13.m1.4.4.1.1.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex13.m1.4.4.1.1.5.cmml" xref="A2.Ex13.m1.4.4.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex13.m1.4c">\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{K//2},\mathbf{x}_{3}^{K}]^{\top},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex14.m1.6" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{K//2}]^{\top}" display="inline"><semantics id="A2.Ex14.m1.6a"><msup id="A2.Ex14.m1.6.6" xref="A2.Ex14.m1.6.6.cmml"><mrow id="A2.Ex14.m1.6.6.3.3" xref="A2.Ex14.m1.6.6.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex14.m1.6.6.3.3.4" xref="A2.Ex14.m1.6.6.3.4.cmml">[</mo><msubsup id="A2.Ex14.m1.4.4.1.1.1" xref="A2.Ex14.m1.4.4.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex14.m1.4.4.1.1.1.2.2" xref="A2.Ex14.m1.4.4.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex14.m1.4.4.1.1.1.2.3" xref="A2.Ex14.m1.4.4.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex14.m1.4.4.1.1.1.3" xref="A2.Ex14.m1.4.4.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex14.m1.6.6.3.3.5" xref="A2.Ex14.m1.6.6.3.4.cmml">,</mo><msubsup id="A2.Ex14.m1.5.5.2.2.2" xref="A2.Ex14.m1.5.5.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex14.m1.5.5.2.2.2.2.2" xref="A2.Ex14.m1.5.5.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex14.m1.5.5.2.2.2.2.3" xref="A2.Ex14.m1.5.5.2.2.2.2.3.cmml">2</mn><mn mathsize="90%" id="A2.Ex14.m1.5.5.2.2.2.3" xref="A2.Ex14.m1.5.5.2.2.2.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex14.m1.6.6.3.3.6" xref="A2.Ex14.m1.6.6.3.4.cmml">,</mo><msubsup id="A2.Ex14.m1.6.6.3.3.3" xref="A2.Ex14.m1.6.6.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex14.m1.6.6.3.3.3.2.2" xref="A2.Ex14.m1.6.6.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex14.m1.6.6.3.3.3.2.3" xref="A2.Ex14.m1.6.6.3.3.3.2.3.cmml">3</mn><mrow id="A2.Ex14.m1.3.3.3.3" xref="A2.Ex14.m1.3.3.3.4.cmml"><mi mathsize="90%" id="A2.Ex14.m1.1.1.1.1" xref="A2.Ex14.m1.1.1.1.1.cmml">K</mi><mo id="A2.Ex14.m1.3.3.3.3.2" xref="A2.Ex14.m1.3.3.3.4.cmml">⁣</mo><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex14.m1.2.2.2.2" xref="A2.Ex14.m1.2.2.2.2.cmml">/</mo><mo id="A2.Ex14.m1.3.3.3.3.3" xref="A2.Ex14.m1.3.3.3.4.cmml">⁣</mo><mrow id="A2.Ex14.m1.3.3.3.3.1" xref="A2.Ex14.m1.3.3.3.3.1.cmml"><mi id="A2.Ex14.m1.3.3.3.3.1.2" xref="A2.Ex14.m1.3.3.3.3.1.2.cmml"></mi><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex14.m1.3.3.3.3.1.1" xref="A2.Ex14.m1.3.3.3.3.1.1.cmml">/</mo><mn mathsize="90%" id="A2.Ex14.m1.3.3.3.3.1.3" xref="A2.Ex14.m1.3.3.3.3.1.3.cmml">2</mn></mrow></mrow></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex14.m1.6.6.3.3.7" xref="A2.Ex14.m1.6.6.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex14.m1.6.6.5" xref="A2.Ex14.m1.6.6.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="A2.Ex14.m1.6b"><apply id="A2.Ex14.m1.6.6.cmml" xref="A2.Ex14.m1.6.6"><csymbol cd="ambiguous" id="A2.Ex14.m1.6.6.4.cmml" xref="A2.Ex14.m1.6.6">superscript</csymbol><list id="A2.Ex14.m1.6.6.3.4.cmml" xref="A2.Ex14.m1.6.6.3.3"><apply id="A2.Ex14.m1.4.4.1.1.1.cmml" xref="A2.Ex14.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="A2.Ex14.m1.4.4.1.1.1.1.cmml" xref="A2.Ex14.m1.4.4.1.1.1">superscript</csymbol><apply id="A2.Ex14.m1.4.4.1.1.1.2.cmml" xref="A2.Ex14.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="A2.Ex14.m1.4.4.1.1.1.2.1.cmml" xref="A2.Ex14.m1.4.4.1.1.1">subscript</csymbol><ci id="A2.Ex14.m1.4.4.1.1.1.2.2.cmml" xref="A2.Ex14.m1.4.4.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex14.m1.4.4.1.1.1.2.3.cmml" xref="A2.Ex14.m1.4.4.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex14.m1.4.4.1.1.1.3.cmml" xref="A2.Ex14.m1.4.4.1.1.1.3">0</cn></apply><apply id="A2.Ex14.m1.5.5.2.2.2.cmml" xref="A2.Ex14.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="A2.Ex14.m1.5.5.2.2.2.1.cmml" xref="A2.Ex14.m1.5.5.2.2.2">superscript</csymbol><apply id="A2.Ex14.m1.5.5.2.2.2.2.cmml" xref="A2.Ex14.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="A2.Ex14.m1.5.5.2.2.2.2.1.cmml" xref="A2.Ex14.m1.5.5.2.2.2">subscript</csymbol><ci id="A2.Ex14.m1.5.5.2.2.2.2.2.cmml" xref="A2.Ex14.m1.5.5.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex14.m1.5.5.2.2.2.2.3.cmml" xref="A2.Ex14.m1.5.5.2.2.2.2.3">2</cn></apply><cn type="integer" id="A2.Ex14.m1.5.5.2.2.2.3.cmml" xref="A2.Ex14.m1.5.5.2.2.2.3">0</cn></apply><apply id="A2.Ex14.m1.6.6.3.3.3.cmml" xref="A2.Ex14.m1.6.6.3.3.3"><csymbol cd="ambiguous" id="A2.Ex14.m1.6.6.3.3.3.1.cmml" xref="A2.Ex14.m1.6.6.3.3.3">superscript</csymbol><apply id="A2.Ex14.m1.6.6.3.3.3.2.cmml" xref="A2.Ex14.m1.6.6.3.3.3"><csymbol cd="ambiguous" id="A2.Ex14.m1.6.6.3.3.3.2.1.cmml" xref="A2.Ex14.m1.6.6.3.3.3">subscript</csymbol><ci id="A2.Ex14.m1.6.6.3.3.3.2.2.cmml" xref="A2.Ex14.m1.6.6.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex14.m1.6.6.3.3.3.2.3.cmml" xref="A2.Ex14.m1.6.6.3.3.3.2.3">3</cn></apply><list id="A2.Ex14.m1.3.3.3.4.cmml" xref="A2.Ex14.m1.3.3.3.3"><ci id="A2.Ex14.m1.1.1.1.1.cmml" xref="A2.Ex14.m1.1.1.1.1">𝐾</ci><divide id="A2.Ex14.m1.2.2.2.2.cmml" xref="A2.Ex14.m1.2.2.2.2"></divide><apply id="A2.Ex14.m1.3.3.3.3.1.cmml" xref="A2.Ex14.m1.3.3.3.3.1"><divide id="A2.Ex14.m1.3.3.3.3.1.1.cmml" xref="A2.Ex14.m1.3.3.3.3.1.1"></divide><csymbol cd="latexml" id="A2.Ex14.m1.3.3.3.3.1.2.cmml" xref="A2.Ex14.m1.3.3.3.3.1.2">absent</csymbol><cn type="integer" id="A2.Ex14.m1.3.3.3.3.1.3.cmml" xref="A2.Ex14.m1.3.3.3.3.1.3">2</cn></apply></list></apply></list><csymbol cd="latexml" id="A2.Ex14.m1.6.6.5.cmml" xref="A2.Ex14.m1.6.6.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex14.m1.6c">\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{K//2}]^{\top}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex15.m1.3" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{0}]^{\top}" display="inline"><semantics id="A2.Ex15.m1.3a"><msup id="A2.Ex15.m1.3.3" xref="A2.Ex15.m1.3.3.cmml"><mrow id="A2.Ex15.m1.3.3.3.3" xref="A2.Ex15.m1.3.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex15.m1.3.3.3.3.4" xref="A2.Ex15.m1.3.3.3.4.cmml">[</mo><msubsup id="A2.Ex15.m1.1.1.1.1.1" xref="A2.Ex15.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex15.m1.1.1.1.1.1.2.2" xref="A2.Ex15.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex15.m1.1.1.1.1.1.2.3" xref="A2.Ex15.m1.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex15.m1.1.1.1.1.1.3" xref="A2.Ex15.m1.1.1.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex15.m1.3.3.3.3.5" xref="A2.Ex15.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex15.m1.2.2.2.2.2" xref="A2.Ex15.m1.2.2.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex15.m1.2.2.2.2.2.2.2" xref="A2.Ex15.m1.2.2.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex15.m1.2.2.2.2.2.2.3" xref="A2.Ex15.m1.2.2.2.2.2.2.3.cmml">2</mn><mn mathsize="90%" id="A2.Ex15.m1.2.2.2.2.2.3" xref="A2.Ex15.m1.2.2.2.2.2.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex15.m1.3.3.3.3.6" xref="A2.Ex15.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex15.m1.3.3.3.3.3" xref="A2.Ex15.m1.3.3.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex15.m1.3.3.3.3.3.2.2" xref="A2.Ex15.m1.3.3.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex15.m1.3.3.3.3.3.2.3" xref="A2.Ex15.m1.3.3.3.3.3.2.3.cmml">3</mn><mn mathsize="90%" id="A2.Ex15.m1.3.3.3.3.3.3" xref="A2.Ex15.m1.3.3.3.3.3.3.cmml">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex15.m1.3.3.3.3.7" xref="A2.Ex15.m1.3.3.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex15.m1.3.3.5" xref="A2.Ex15.m1.3.3.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="A2.Ex15.m1.3b"><apply id="A2.Ex15.m1.3.3.cmml" xref="A2.Ex15.m1.3.3"><csymbol cd="ambiguous" id="A2.Ex15.m1.3.3.4.cmml" xref="A2.Ex15.m1.3.3">superscript</csymbol><list id="A2.Ex15.m1.3.3.3.4.cmml" xref="A2.Ex15.m1.3.3.3.3"><apply id="A2.Ex15.m1.1.1.1.1.1.cmml" xref="A2.Ex15.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex15.m1.1.1.1.1.1.1.cmml" xref="A2.Ex15.m1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex15.m1.1.1.1.1.1.2.cmml" xref="A2.Ex15.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex15.m1.1.1.1.1.1.2.1.cmml" xref="A2.Ex15.m1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex15.m1.1.1.1.1.1.2.2.cmml" xref="A2.Ex15.m1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex15.m1.1.1.1.1.1.2.3.cmml" xref="A2.Ex15.m1.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex15.m1.1.1.1.1.1.3.cmml" xref="A2.Ex15.m1.1.1.1.1.1.3">0</cn></apply><apply id="A2.Ex15.m1.2.2.2.2.2.cmml" xref="A2.Ex15.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex15.m1.2.2.2.2.2.1.cmml" xref="A2.Ex15.m1.2.2.2.2.2">superscript</csymbol><apply id="A2.Ex15.m1.2.2.2.2.2.2.cmml" xref="A2.Ex15.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex15.m1.2.2.2.2.2.2.1.cmml" xref="A2.Ex15.m1.2.2.2.2.2">subscript</csymbol><ci id="A2.Ex15.m1.2.2.2.2.2.2.2.cmml" xref="A2.Ex15.m1.2.2.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex15.m1.2.2.2.2.2.2.3.cmml" xref="A2.Ex15.m1.2.2.2.2.2.2.3">2</cn></apply><cn type="integer" id="A2.Ex15.m1.2.2.2.2.2.3.cmml" xref="A2.Ex15.m1.2.2.2.2.2.3">0</cn></apply><apply id="A2.Ex15.m1.3.3.3.3.3.cmml" xref="A2.Ex15.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex15.m1.3.3.3.3.3.1.cmml" xref="A2.Ex15.m1.3.3.3.3.3">superscript</csymbol><apply id="A2.Ex15.m1.3.3.3.3.3.2.cmml" xref="A2.Ex15.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex15.m1.3.3.3.3.3.2.1.cmml" xref="A2.Ex15.m1.3.3.3.3.3">subscript</csymbol><ci id="A2.Ex15.m1.3.3.3.3.3.2.2.cmml" xref="A2.Ex15.m1.3.3.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex15.m1.3.3.3.3.3.2.3.cmml" xref="A2.Ex15.m1.3.3.3.3.3.2.3">3</cn></apply><cn type="integer" id="A2.Ex15.m1.3.3.3.3.3.3.cmml" xref="A2.Ex15.m1.3.3.3.3.3.3">0</cn></apply></list><csymbol cd="latexml" id="A2.Ex15.m1.3.3.5.cmml" xref="A2.Ex15.m1.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex15.m1.3c">\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{0}]^{\top}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A2.SS6.SSS0.Px3.p1.2" class="ltx_p"><span id="A2.SS6.SSS0.Px3.p1.2.1" class="ltx_text" style="font-size:90%;">Notable,  </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A2.SS6.SSS0.Px3.p1.2.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib63" title="" class="ltx_ref">63</a><span id="A2.SS6.SSS0.Px3.p1.2.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A2.SS6.SSS0.Px3.p1.2.4" class="ltx_text" style="font-size:90%;"> is the first one to propose such a linear uncertainty sampling scheme for causal diffusion models, although Diffusion Forcing provides a generalization of such scheme in combination of other abilities.</span></p>
</div>
</section>
<section id="A2.SS6.SSS0.Px4" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Diffusion Forcing with stablization</h5>

<div id="A2.SS6.SSS0.Px4.p1" class="ltx_para">
<p id="A2.SS6.SSS0.Px4.p1.1" class="ltx_p"><span id="A2.SS6.SSS0.Px4.p1.1.1" class="ltx_text" style="font-size:90%;">Previously we introduced the autoregressive sampling scheme that Diffusion Forcing can also do. However, such a scheme can accumulate single-step errors because it treats predicted </span><math id="A2.SS6.SSS0.Px4.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}" display="inline"><semantics id="A2.SS6.SSS0.Px4.p1.1.m1.1a"><mi mathsize="90%" id="A2.SS6.SSS0.Px4.p1.1.m1.1.1" xref="A2.SS6.SSS0.Px4.p1.1.m1.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="A2.SS6.SSS0.Px4.p1.1.m1.1b"><ci id="A2.SS6.SSS0.Px4.p1.1.m1.1.1.cmml" xref="A2.SS6.SSS0.Px4.p1.1.m1.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.SS6.SSS0.Px4.p1.1.m1.1c">\mathbf{x}</annotation></semantics></math><span id="A2.SS6.SSS0.Px4.p1.1.2" class="ltx_text" style="font-size:90%;"> as ground truth observation. Diffusion Forcing addresses this problem by telling the model that generated images should be treated as noisy ground truth, as shown in </span><a href="#S2.F2" title="Figure 2 ‣ Bayesian Filtering. ‣ 2 Related Work and Preliminaries ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">2</span></a><span id="A2.SS6.SSS0.Px4.p1.1.3" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
<div id="A2.SS6.SSS0.Px4.p2" class="ltx_para">
<p id="A2.SS6.SSS0.Px4.p2.6" class="ltx_p"><span id="A2.SS6.SSS0.Px4.p2.6.1" class="ltx_text" style="font-size:90%;">It first fully diffuse the first token,</span></p>
<table id="A6.EGx40" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A2.Ex16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex16.m1.3" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{K},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}" display="inline"><semantics id="A2.Ex16.m1.3a"><msup id="A2.Ex16.m1.3.3" xref="A2.Ex16.m1.3.3.cmml"><mrow id="A2.Ex16.m1.3.3.3.3" xref="A2.Ex16.m1.3.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex16.m1.3.3.3.3.4" xref="A2.Ex16.m1.3.3.3.4.cmml">[</mo><msubsup id="A2.Ex16.m1.1.1.1.1.1" xref="A2.Ex16.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex16.m1.1.1.1.1.1.2.2" xref="A2.Ex16.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex16.m1.1.1.1.1.1.2.3" xref="A2.Ex16.m1.1.1.1.1.1.2.3.cmml">1</mn><mi mathsize="90%" id="A2.Ex16.m1.1.1.1.1.1.3" xref="A2.Ex16.m1.1.1.1.1.1.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex16.m1.3.3.3.3.5" xref="A2.Ex16.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex16.m1.2.2.2.2.2" xref="A2.Ex16.m1.2.2.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex16.m1.2.2.2.2.2.2.2" xref="A2.Ex16.m1.2.2.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex16.m1.2.2.2.2.2.2.3" xref="A2.Ex16.m1.2.2.2.2.2.2.3.cmml">2</mn><mi mathsize="90%" id="A2.Ex16.m1.2.2.2.2.2.3" xref="A2.Ex16.m1.2.2.2.2.2.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex16.m1.3.3.3.3.6" xref="A2.Ex16.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex16.m1.3.3.3.3.3" xref="A2.Ex16.m1.3.3.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex16.m1.3.3.3.3.3.2.2" xref="A2.Ex16.m1.3.3.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex16.m1.3.3.3.3.3.2.3" xref="A2.Ex16.m1.3.3.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex16.m1.3.3.3.3.3.3" xref="A2.Ex16.m1.3.3.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex16.m1.3.3.3.3.7" xref="A2.Ex16.m1.3.3.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex16.m1.3.3.5" xref="A2.Ex16.m1.3.3.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="A2.Ex16.m1.3b"><apply id="A2.Ex16.m1.3.3.cmml" xref="A2.Ex16.m1.3.3"><csymbol cd="ambiguous" id="A2.Ex16.m1.3.3.4.cmml" xref="A2.Ex16.m1.3.3">superscript</csymbol><list id="A2.Ex16.m1.3.3.3.4.cmml" xref="A2.Ex16.m1.3.3.3.3"><apply id="A2.Ex16.m1.1.1.1.1.1.cmml" xref="A2.Ex16.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex16.m1.1.1.1.1.1.1.cmml" xref="A2.Ex16.m1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex16.m1.1.1.1.1.1.2.cmml" xref="A2.Ex16.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex16.m1.1.1.1.1.1.2.1.cmml" xref="A2.Ex16.m1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex16.m1.1.1.1.1.1.2.2.cmml" xref="A2.Ex16.m1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex16.m1.1.1.1.1.1.2.3.cmml" xref="A2.Ex16.m1.1.1.1.1.1.2.3">1</cn></apply><ci id="A2.Ex16.m1.1.1.1.1.1.3.cmml" xref="A2.Ex16.m1.1.1.1.1.1.3">𝐾</ci></apply><apply id="A2.Ex16.m1.2.2.2.2.2.cmml" xref="A2.Ex16.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex16.m1.2.2.2.2.2.1.cmml" xref="A2.Ex16.m1.2.2.2.2.2">superscript</csymbol><apply id="A2.Ex16.m1.2.2.2.2.2.2.cmml" xref="A2.Ex16.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex16.m1.2.2.2.2.2.2.1.cmml" xref="A2.Ex16.m1.2.2.2.2.2">subscript</csymbol><ci id="A2.Ex16.m1.2.2.2.2.2.2.2.cmml" xref="A2.Ex16.m1.2.2.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex16.m1.2.2.2.2.2.2.3.cmml" xref="A2.Ex16.m1.2.2.2.2.2.2.3">2</cn></apply><ci id="A2.Ex16.m1.2.2.2.2.2.3.cmml" xref="A2.Ex16.m1.2.2.2.2.2.3">𝐾</ci></apply><apply id="A2.Ex16.m1.3.3.3.3.3.cmml" xref="A2.Ex16.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex16.m1.3.3.3.3.3.1.cmml" xref="A2.Ex16.m1.3.3.3.3.3">superscript</csymbol><apply id="A2.Ex16.m1.3.3.3.3.3.2.cmml" xref="A2.Ex16.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex16.m1.3.3.3.3.3.2.1.cmml" xref="A2.Ex16.m1.3.3.3.3.3">subscript</csymbol><ci id="A2.Ex16.m1.3.3.3.3.3.2.2.cmml" xref="A2.Ex16.m1.3.3.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex16.m1.3.3.3.3.3.2.3.cmml" xref="A2.Ex16.m1.3.3.3.3.3.2.3">3</cn></apply><ci id="A2.Ex16.m1.3.3.3.3.3.3.cmml" xref="A2.Ex16.m1.3.3.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex16.m1.3.3.5.cmml" xref="A2.Ex16.m1.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex16.m1.3c">\displaystyle[\mathbf{x}_{1}^{K},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex17.m1.4" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{K//2},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}," display="inline"><semantics id="A2.Ex17.m1.4a"><mrow id="A2.Ex17.m1.4.4.1" xref="A2.Ex17.m1.4.4.1.1.cmml"><msup id="A2.Ex17.m1.4.4.1.1" xref="A2.Ex17.m1.4.4.1.1.cmml"><mrow id="A2.Ex17.m1.4.4.1.1.3.3" xref="A2.Ex17.m1.4.4.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex17.m1.4.4.1.1.3.3.4" xref="A2.Ex17.m1.4.4.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex17.m1.4.4.1.1.1.1.1" xref="A2.Ex17.m1.4.4.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex17.m1.4.4.1.1.1.1.1.2.2" xref="A2.Ex17.m1.4.4.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex17.m1.4.4.1.1.1.1.1.2.3" xref="A2.Ex17.m1.4.4.1.1.1.1.1.2.3.cmml">1</mn><mrow id="A2.Ex17.m1.3.3.3.3" xref="A2.Ex17.m1.3.3.3.4.cmml"><mi mathsize="90%" id="A2.Ex17.m1.1.1.1.1" xref="A2.Ex17.m1.1.1.1.1.cmml">K</mi><mo id="A2.Ex17.m1.3.3.3.3.2" xref="A2.Ex17.m1.3.3.3.4.cmml">⁣</mo><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex17.m1.2.2.2.2" xref="A2.Ex17.m1.2.2.2.2.cmml">/</mo><mo id="A2.Ex17.m1.3.3.3.3.3" xref="A2.Ex17.m1.3.3.3.4.cmml">⁣</mo><mrow id="A2.Ex17.m1.3.3.3.3.1" xref="A2.Ex17.m1.3.3.3.3.1.cmml"><mi id="A2.Ex17.m1.3.3.3.3.1.2" xref="A2.Ex17.m1.3.3.3.3.1.2.cmml"></mi><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex17.m1.3.3.3.3.1.1" xref="A2.Ex17.m1.3.3.3.3.1.1.cmml">/</mo><mn mathsize="90%" id="A2.Ex17.m1.3.3.3.3.1.3" xref="A2.Ex17.m1.3.3.3.3.1.3.cmml">2</mn></mrow></mrow></msubsup><mo mathsize="90%" id="A2.Ex17.m1.4.4.1.1.3.3.5" xref="A2.Ex17.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex17.m1.4.4.1.1.2.2.2" xref="A2.Ex17.m1.4.4.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex17.m1.4.4.1.1.2.2.2.2.2" xref="A2.Ex17.m1.4.4.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex17.m1.4.4.1.1.2.2.2.2.3" xref="A2.Ex17.m1.4.4.1.1.2.2.2.2.3.cmml">2</mn><mi mathsize="90%" id="A2.Ex17.m1.4.4.1.1.2.2.2.3" xref="A2.Ex17.m1.4.4.1.1.2.2.2.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex17.m1.4.4.1.1.3.3.6" xref="A2.Ex17.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex17.m1.4.4.1.1.3.3.3" xref="A2.Ex17.m1.4.4.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex17.m1.4.4.1.1.3.3.3.2.2" xref="A2.Ex17.m1.4.4.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex17.m1.4.4.1.1.3.3.3.2.3" xref="A2.Ex17.m1.4.4.1.1.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex17.m1.4.4.1.1.3.3.3.3" xref="A2.Ex17.m1.4.4.1.1.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex17.m1.4.4.1.1.3.3.7" xref="A2.Ex17.m1.4.4.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex17.m1.4.4.1.1.5" xref="A2.Ex17.m1.4.4.1.1.5.cmml">⊤</mo></msup><mo mathsize="90%" id="A2.Ex17.m1.4.4.1.2" xref="A2.Ex17.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex17.m1.4b"><apply id="A2.Ex17.m1.4.4.1.1.cmml" xref="A2.Ex17.m1.4.4.1"><csymbol cd="ambiguous" id="A2.Ex17.m1.4.4.1.1.4.cmml" xref="A2.Ex17.m1.4.4.1">superscript</csymbol><list id="A2.Ex17.m1.4.4.1.1.3.4.cmml" xref="A2.Ex17.m1.4.4.1.1.3.3"><apply id="A2.Ex17.m1.4.4.1.1.1.1.1.cmml" xref="A2.Ex17.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex17.m1.4.4.1.1.1.1.1.1.cmml" xref="A2.Ex17.m1.4.4.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex17.m1.4.4.1.1.1.1.1.2.cmml" xref="A2.Ex17.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex17.m1.4.4.1.1.1.1.1.2.1.cmml" xref="A2.Ex17.m1.4.4.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex17.m1.4.4.1.1.1.1.1.2.2.cmml" xref="A2.Ex17.m1.4.4.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex17.m1.4.4.1.1.1.1.1.2.3.cmml" xref="A2.Ex17.m1.4.4.1.1.1.1.1.2.3">1</cn></apply><list id="A2.Ex17.m1.3.3.3.4.cmml" xref="A2.Ex17.m1.3.3.3.3"><ci id="A2.Ex17.m1.1.1.1.1.cmml" xref="A2.Ex17.m1.1.1.1.1">𝐾</ci><divide id="A2.Ex17.m1.2.2.2.2.cmml" xref="A2.Ex17.m1.2.2.2.2"></divide><apply id="A2.Ex17.m1.3.3.3.3.1.cmml" xref="A2.Ex17.m1.3.3.3.3.1"><divide 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xref="A2.Ex17.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex17.m1.4.4.1.1.3.3.3.1.cmml" xref="A2.Ex17.m1.4.4.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex17.m1.4.4.1.1.3.3.3.2.cmml" xref="A2.Ex17.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex17.m1.4.4.1.1.3.3.3.2.1.cmml" xref="A2.Ex17.m1.4.4.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex17.m1.4.4.1.1.3.3.3.2.2.cmml" xref="A2.Ex17.m1.4.4.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex17.m1.4.4.1.1.3.3.3.2.3.cmml" xref="A2.Ex17.m1.4.4.1.1.3.3.3.2.3">3</cn></apply><ci id="A2.Ex17.m1.4.4.1.1.3.3.3.3.cmml" xref="A2.Ex17.m1.4.4.1.1.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex17.m1.4.4.1.1.5.cmml" xref="A2.Ex17.m1.4.4.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex17.m1.4c">\displaystyle[\mathbf{x}_{1}^{K//2},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex18.m1.3" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}" display="inline"><semantics id="A2.Ex18.m1.3a"><msup id="A2.Ex18.m1.3.3" xref="A2.Ex18.m1.3.3.cmml"><mrow id="A2.Ex18.m1.3.3.3.3" xref="A2.Ex18.m1.3.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex18.m1.3.3.3.3.4" xref="A2.Ex18.m1.3.3.3.4.cmml">[</mo><msubsup id="A2.Ex18.m1.1.1.1.1.1" xref="A2.Ex18.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex18.m1.1.1.1.1.1.2.2" xref="A2.Ex18.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex18.m1.1.1.1.1.1.2.3" xref="A2.Ex18.m1.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex18.m1.1.1.1.1.1.3" xref="A2.Ex18.m1.1.1.1.1.1.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex18.m1.3.3.3.3.5" xref="A2.Ex18.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex18.m1.2.2.2.2.2" xref="A2.Ex18.m1.2.2.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex18.m1.2.2.2.2.2.2.2" xref="A2.Ex18.m1.2.2.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex18.m1.2.2.2.2.2.2.3" xref="A2.Ex18.m1.2.2.2.2.2.2.3.cmml">2</mn><mi mathsize="90%" id="A2.Ex18.m1.2.2.2.2.2.3" xref="A2.Ex18.m1.2.2.2.2.2.3.cmml">K</mi></msubsup><mo mathsize="90%" id="A2.Ex18.m1.3.3.3.3.6" xref="A2.Ex18.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex18.m1.3.3.3.3.3" xref="A2.Ex18.m1.3.3.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex18.m1.3.3.3.3.3.2.2" xref="A2.Ex18.m1.3.3.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex18.m1.3.3.3.3.3.2.3" xref="A2.Ex18.m1.3.3.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex18.m1.3.3.3.3.3.3" xref="A2.Ex18.m1.3.3.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex18.m1.3.3.3.3.7" xref="A2.Ex18.m1.3.3.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex18.m1.3.3.5" xref="A2.Ex18.m1.3.3.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="A2.Ex18.m1.3b"><apply id="A2.Ex18.m1.3.3.cmml" xref="A2.Ex18.m1.3.3"><csymbol cd="ambiguous" id="A2.Ex18.m1.3.3.4.cmml" xref="A2.Ex18.m1.3.3">superscript</csymbol><list id="A2.Ex18.m1.3.3.3.4.cmml" xref="A2.Ex18.m1.3.3.3.3"><apply id="A2.Ex18.m1.1.1.1.1.1.cmml" xref="A2.Ex18.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex18.m1.1.1.1.1.1.1.cmml" xref="A2.Ex18.m1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex18.m1.1.1.1.1.1.2.cmml" xref="A2.Ex18.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex18.m1.1.1.1.1.1.2.1.cmml" xref="A2.Ex18.m1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex18.m1.1.1.1.1.1.2.2.cmml" xref="A2.Ex18.m1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex18.m1.1.1.1.1.1.2.3.cmml" xref="A2.Ex18.m1.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex18.m1.1.1.1.1.1.3.cmml" xref="A2.Ex18.m1.1.1.1.1.1.3">0</cn></apply><apply id="A2.Ex18.m1.2.2.2.2.2.cmml" xref="A2.Ex18.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex18.m1.2.2.2.2.2.1.cmml" xref="A2.Ex18.m1.2.2.2.2.2">superscript</csymbol><apply id="A2.Ex18.m1.2.2.2.2.2.2.cmml" xref="A2.Ex18.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex18.m1.2.2.2.2.2.2.1.cmml" xref="A2.Ex18.m1.2.2.2.2.2">subscript</csymbol><ci id="A2.Ex18.m1.2.2.2.2.2.2.2.cmml" xref="A2.Ex18.m1.2.2.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex18.m1.2.2.2.2.2.2.3.cmml" xref="A2.Ex18.m1.2.2.2.2.2.2.3">2</cn></apply><ci id="A2.Ex18.m1.2.2.2.2.2.3.cmml" xref="A2.Ex18.m1.2.2.2.2.2.3">𝐾</ci></apply><apply id="A2.Ex18.m1.3.3.3.3.3.cmml" xref="A2.Ex18.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex18.m1.3.3.3.3.3.1.cmml" xref="A2.Ex18.m1.3.3.3.3.3">superscript</csymbol><apply id="A2.Ex18.m1.3.3.3.3.3.2.cmml" xref="A2.Ex18.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex18.m1.3.3.3.3.3.2.1.cmml" xref="A2.Ex18.m1.3.3.3.3.3">subscript</csymbol><ci id="A2.Ex18.m1.3.3.3.3.3.2.2.cmml" xref="A2.Ex18.m1.3.3.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex18.m1.3.3.3.3.3.2.3.cmml" xref="A2.Ex18.m1.3.3.3.3.3.2.3">3</cn></apply><ci id="A2.Ex18.m1.3.3.3.3.3.3.cmml" xref="A2.Ex18.m1.3.3.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex18.m1.3.3.5.cmml" xref="A2.Ex18.m1.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex18.m1.3c">\displaystyle[\mathbf{x}_{1}^{0},\mathbf{x}_{2}^{K},\mathbf{x}_{3}^{K}]^{\top}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A2.SS6.SSS0.Px4.p2.3" class="ltx_p"><span id="A2.SS6.SSS0.Px4.p2.3.1" class="ltx_text" style="font-size:90%;">Then, it feed the diffused </span><math id="A2.SS6.SSS0.Px4.p2.1.m1.1" class="ltx_Math" alttext="\mathbf{x}_{1}^{0}" display="inline"><semantics id="A2.SS6.SSS0.Px4.p2.1.m1.1a"><msubsup id="A2.SS6.SSS0.Px4.p2.1.m1.1.1" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1.cmml"><mi mathsize="90%" id="A2.SS6.SSS0.Px4.p2.1.m1.1.1.2.2" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.SS6.SSS0.Px4.p2.1.m1.1.1.2.3" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.SS6.SSS0.Px4.p2.1.m1.1.1.3" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1.3.cmml">0</mn></msubsup><annotation-xml encoding="MathML-Content" id="A2.SS6.SSS0.Px4.p2.1.m1.1b"><apply id="A2.SS6.SSS0.Px4.p2.1.m1.1.1.cmml" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A2.SS6.SSS0.Px4.p2.1.m1.1.1.1.cmml" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1">superscript</csymbol><apply id="A2.SS6.SSS0.Px4.p2.1.m1.1.1.2.cmml" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A2.SS6.SSS0.Px4.p2.1.m1.1.1.2.1.cmml" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1">subscript</csymbol><ci id="A2.SS6.SSS0.Px4.p2.1.m1.1.1.2.2.cmml" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.SS6.SSS0.Px4.p2.1.m1.1.1.2.3.cmml" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.SS6.SSS0.Px4.p2.1.m1.1.1.3.cmml" xref="A2.SS6.SSS0.Px4.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS6.SSS0.Px4.p2.1.m1.1c">\mathbf{x}_{1}^{0}</annotation></semantics></math><span id="A2.SS6.SSS0.Px4.p2.3.2" class="ltx_text" style="font-size:90%;"> into the model but tell it is of a slightly higher noise level, as </span><math id="A2.SS6.SSS0.Px4.p2.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{1}^{1}" display="inline"><semantics id="A2.SS6.SSS0.Px4.p2.2.m2.1a"><msubsup id="A2.SS6.SSS0.Px4.p2.2.m2.1.1" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1.cmml"><mi mathsize="90%" id="A2.SS6.SSS0.Px4.p2.2.m2.1.1.2.2" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.SS6.SSS0.Px4.p2.2.m2.1.1.2.3" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.SS6.SSS0.Px4.p2.2.m2.1.1.3" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1.3.cmml">1</mn></msubsup><annotation-xml encoding="MathML-Content" id="A2.SS6.SSS0.Px4.p2.2.m2.1b"><apply id="A2.SS6.SSS0.Px4.p2.2.m2.1.1.cmml" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A2.SS6.SSS0.Px4.p2.2.m2.1.1.1.cmml" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1">superscript</csymbol><apply id="A2.SS6.SSS0.Px4.p2.2.m2.1.1.2.cmml" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A2.SS6.SSS0.Px4.p2.2.m2.1.1.2.1.cmml" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1">subscript</csymbol><ci id="A2.SS6.SSS0.Px4.p2.2.m2.1.1.2.2.cmml" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1.2.2">𝐱</ci><cn type="integer" id="A2.SS6.SSS0.Px4.p2.2.m2.1.1.2.3.cmml" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1.2.3">1</cn></apply><cn type="integer" id="A2.SS6.SSS0.Px4.p2.2.m2.1.1.3.cmml" xref="A2.SS6.SSS0.Px4.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS6.SSS0.Px4.p2.2.m2.1c">\mathbf{x}_{1}^{1}</annotation></semantics></math><span id="A2.SS6.SSS0.Px4.p2.3.3" class="ltx_text" style="font-size:90%;"> to diffuse </span><math id="A2.SS6.SSS0.Px4.p2.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{2}" display="inline"><semantics id="A2.SS6.SSS0.Px4.p2.3.m3.1a"><msub id="A2.SS6.SSS0.Px4.p2.3.m3.1.1" xref="A2.SS6.SSS0.Px4.p2.3.m3.1.1.cmml"><mi mathsize="90%" id="A2.SS6.SSS0.Px4.p2.3.m3.1.1.2" xref="A2.SS6.SSS0.Px4.p2.3.m3.1.1.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.SS6.SSS0.Px4.p2.3.m3.1.1.3" xref="A2.SS6.SSS0.Px4.p2.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="A2.SS6.SSS0.Px4.p2.3.m3.1b"><apply id="A2.SS6.SSS0.Px4.p2.3.m3.1.1.cmml" xref="A2.SS6.SSS0.Px4.p2.3.m3.1.1"><csymbol cd="ambiguous" id="A2.SS6.SSS0.Px4.p2.3.m3.1.1.1.cmml" xref="A2.SS6.SSS0.Px4.p2.3.m3.1.1">subscript</csymbol><ci id="A2.SS6.SSS0.Px4.p2.3.m3.1.1.2.cmml" xref="A2.SS6.SSS0.Px4.p2.3.m3.1.1.2">𝐱</ci><cn type="integer" id="A2.SS6.SSS0.Px4.p2.3.m3.1.1.3.cmml" xref="A2.SS6.SSS0.Px4.p2.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS6.SSS0.Px4.p2.3.m3.1c">\mathbf{x}_{2}</annotation></semantics></math><span id="A2.SS6.SSS0.Px4.p2.3.4" class="ltx_text" style="font-size:90%;">.</span></p>
<table id="A6.EGx41" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A2.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex20.m1.6" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{1},\mathbf{x}_{2}^{K//2},\mathbf{x}_{3}^{K}]^{\top}" display="inline"><semantics id="A2.Ex20.m1.6a"><msup id="A2.Ex20.m1.6.6" xref="A2.Ex20.m1.6.6.cmml"><mrow id="A2.Ex20.m1.6.6.3.3" xref="A2.Ex20.m1.6.6.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex20.m1.6.6.3.3.4" xref="A2.Ex20.m1.6.6.3.4.cmml">[</mo><msubsup id="A2.Ex20.m1.4.4.1.1.1" xref="A2.Ex20.m1.4.4.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex20.m1.4.4.1.1.1.2.2" xref="A2.Ex20.m1.4.4.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex20.m1.4.4.1.1.1.2.3" xref="A2.Ex20.m1.4.4.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex20.m1.4.4.1.1.1.3" xref="A2.Ex20.m1.4.4.1.1.1.3.cmml">1</mn></msubsup><mo mathsize="90%" id="A2.Ex20.m1.6.6.3.3.5" xref="A2.Ex20.m1.6.6.3.4.cmml">,</mo><msubsup id="A2.Ex20.m1.5.5.2.2.2" xref="A2.Ex20.m1.5.5.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex20.m1.5.5.2.2.2.2.2" xref="A2.Ex20.m1.5.5.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex20.m1.5.5.2.2.2.2.3" xref="A2.Ex20.m1.5.5.2.2.2.2.3.cmml">2</mn><mrow id="A2.Ex20.m1.3.3.3.3" xref="A2.Ex20.m1.3.3.3.4.cmml"><mi mathsize="90%" id="A2.Ex20.m1.1.1.1.1" xref="A2.Ex20.m1.1.1.1.1.cmml">K</mi><mo id="A2.Ex20.m1.3.3.3.3.2" xref="A2.Ex20.m1.3.3.3.4.cmml">⁣</mo><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex20.m1.2.2.2.2" xref="A2.Ex20.m1.2.2.2.2.cmml">/</mo><mo id="A2.Ex20.m1.3.3.3.3.3" xref="A2.Ex20.m1.3.3.3.4.cmml">⁣</mo><mrow id="A2.Ex20.m1.3.3.3.3.1" xref="A2.Ex20.m1.3.3.3.3.1.cmml"><mi id="A2.Ex20.m1.3.3.3.3.1.2" xref="A2.Ex20.m1.3.3.3.3.1.2.cmml"></mi><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex20.m1.3.3.3.3.1.1" xref="A2.Ex20.m1.3.3.3.3.1.1.cmml">/</mo><mn mathsize="90%" id="A2.Ex20.m1.3.3.3.3.1.3" xref="A2.Ex20.m1.3.3.3.3.1.3.cmml">2</mn></mrow></mrow></msubsup><mo mathsize="90%" id="A2.Ex20.m1.6.6.3.3.6" xref="A2.Ex20.m1.6.6.3.4.cmml">,</mo><msubsup id="A2.Ex20.m1.6.6.3.3.3" xref="A2.Ex20.m1.6.6.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex20.m1.6.6.3.3.3.2.2" xref="A2.Ex20.m1.6.6.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex20.m1.6.6.3.3.3.2.3" xref="A2.Ex20.m1.6.6.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex20.m1.6.6.3.3.3.3" xref="A2.Ex20.m1.6.6.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex20.m1.6.6.3.3.7" xref="A2.Ex20.m1.6.6.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex20.m1.6.6.5" xref="A2.Ex20.m1.6.6.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="A2.Ex20.m1.6b"><apply id="A2.Ex20.m1.6.6.cmml" xref="A2.Ex20.m1.6.6"><csymbol cd="ambiguous" id="A2.Ex20.m1.6.6.4.cmml" xref="A2.Ex20.m1.6.6">superscript</csymbol><list id="A2.Ex20.m1.6.6.3.4.cmml" xref="A2.Ex20.m1.6.6.3.3"><apply id="A2.Ex20.m1.4.4.1.1.1.cmml" xref="A2.Ex20.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="A2.Ex20.m1.4.4.1.1.1.1.cmml" xref="A2.Ex20.m1.4.4.1.1.1">superscript</csymbol><apply id="A2.Ex20.m1.4.4.1.1.1.2.cmml" xref="A2.Ex20.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="A2.Ex20.m1.4.4.1.1.1.2.1.cmml" xref="A2.Ex20.m1.4.4.1.1.1">subscript</csymbol><ci id="A2.Ex20.m1.4.4.1.1.1.2.2.cmml" xref="A2.Ex20.m1.4.4.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex20.m1.4.4.1.1.1.2.3.cmml" xref="A2.Ex20.m1.4.4.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex20.m1.4.4.1.1.1.3.cmml" xref="A2.Ex20.m1.4.4.1.1.1.3">1</cn></apply><apply id="A2.Ex20.m1.5.5.2.2.2.cmml" xref="A2.Ex20.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="A2.Ex20.m1.5.5.2.2.2.1.cmml" xref="A2.Ex20.m1.5.5.2.2.2">superscript</csymbol><apply id="A2.Ex20.m1.5.5.2.2.2.2.cmml" xref="A2.Ex20.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="A2.Ex20.m1.5.5.2.2.2.2.1.cmml" xref="A2.Ex20.m1.5.5.2.2.2">subscript</csymbol><ci id="A2.Ex20.m1.5.5.2.2.2.2.2.cmml" xref="A2.Ex20.m1.5.5.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex20.m1.5.5.2.2.2.2.3.cmml" xref="A2.Ex20.m1.5.5.2.2.2.2.3">2</cn></apply><list id="A2.Ex20.m1.3.3.3.4.cmml" xref="A2.Ex20.m1.3.3.3.3"><ci id="A2.Ex20.m1.1.1.1.1.cmml" xref="A2.Ex20.m1.1.1.1.1">𝐾</ci><divide id="A2.Ex20.m1.2.2.2.2.cmml" xref="A2.Ex20.m1.2.2.2.2"></divide><apply id="A2.Ex20.m1.3.3.3.3.1.cmml" xref="A2.Ex20.m1.3.3.3.3.1"><divide id="A2.Ex20.m1.3.3.3.3.1.1.cmml" xref="A2.Ex20.m1.3.3.3.3.1.1"></divide><csymbol cd="latexml" id="A2.Ex20.m1.3.3.3.3.1.2.cmml" xref="A2.Ex20.m1.3.3.3.3.1.2">absent</csymbol><cn type="integer" id="A2.Ex20.m1.3.3.3.3.1.3.cmml" xref="A2.Ex20.m1.3.3.3.3.1.3">2</cn></apply></list></apply><apply id="A2.Ex20.m1.6.6.3.3.3.cmml" xref="A2.Ex20.m1.6.6.3.3.3"><csymbol cd="ambiguous" id="A2.Ex20.m1.6.6.3.3.3.1.cmml" xref="A2.Ex20.m1.6.6.3.3.3">superscript</csymbol><apply id="A2.Ex20.m1.6.6.3.3.3.2.cmml" xref="A2.Ex20.m1.6.6.3.3.3"><csymbol cd="ambiguous" id="A2.Ex20.m1.6.6.3.3.3.2.1.cmml" xref="A2.Ex20.m1.6.6.3.3.3">subscript</csymbol><ci id="A2.Ex20.m1.6.6.3.3.3.2.2.cmml" xref="A2.Ex20.m1.6.6.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex20.m1.6.6.3.3.3.2.3.cmml" xref="A2.Ex20.m1.6.6.3.3.3.2.3">3</cn></apply><ci id="A2.Ex20.m1.6.6.3.3.3.3.cmml" xref="A2.Ex20.m1.6.6.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex20.m1.6.6.5.cmml" xref="A2.Ex20.m1.6.6.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex20.m1.6c">\displaystyle[\mathbf{x}_{1}^{1},\mathbf{x}_{2}^{K//2},\mathbf{x}_{3}^{K}]^{\top}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex21.m1.3" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{1},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{K}]^{\top}" display="inline"><semantics id="A2.Ex21.m1.3a"><msup id="A2.Ex21.m1.3.3" xref="A2.Ex21.m1.3.3.cmml"><mrow id="A2.Ex21.m1.3.3.3.3" xref="A2.Ex21.m1.3.3.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex21.m1.3.3.3.3.4" xref="A2.Ex21.m1.3.3.3.4.cmml">[</mo><msubsup id="A2.Ex21.m1.1.1.1.1.1" xref="A2.Ex21.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex21.m1.1.1.1.1.1.2.2" xref="A2.Ex21.m1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex21.m1.1.1.1.1.1.2.3" xref="A2.Ex21.m1.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex21.m1.1.1.1.1.1.3" xref="A2.Ex21.m1.1.1.1.1.1.3.cmml">1</mn></msubsup><mo mathsize="90%" id="A2.Ex21.m1.3.3.3.3.5" xref="A2.Ex21.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex21.m1.2.2.2.2.2" xref="A2.Ex21.m1.2.2.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex21.m1.2.2.2.2.2.2.2" xref="A2.Ex21.m1.2.2.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex21.m1.2.2.2.2.2.2.3" xref="A2.Ex21.m1.2.2.2.2.2.2.3.cmml">2</mn><mn mathsize="90%" id="A2.Ex21.m1.2.2.2.2.2.3" xref="A2.Ex21.m1.2.2.2.2.2.3.cmml">0</mn></msubsup><mo mathsize="90%" id="A2.Ex21.m1.3.3.3.3.6" xref="A2.Ex21.m1.3.3.3.4.cmml">,</mo><msubsup id="A2.Ex21.m1.3.3.3.3.3" xref="A2.Ex21.m1.3.3.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex21.m1.3.3.3.3.3.2.2" xref="A2.Ex21.m1.3.3.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex21.m1.3.3.3.3.3.2.3" xref="A2.Ex21.m1.3.3.3.3.3.2.3.cmml">3</mn><mi mathsize="90%" id="A2.Ex21.m1.3.3.3.3.3.3" xref="A2.Ex21.m1.3.3.3.3.3.3.cmml">K</mi></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex21.m1.3.3.3.3.7" xref="A2.Ex21.m1.3.3.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex21.m1.3.3.5" xref="A2.Ex21.m1.3.3.5.cmml">⊤</mo></msup><annotation-xml encoding="MathML-Content" id="A2.Ex21.m1.3b"><apply id="A2.Ex21.m1.3.3.cmml" xref="A2.Ex21.m1.3.3"><csymbol cd="ambiguous" id="A2.Ex21.m1.3.3.4.cmml" xref="A2.Ex21.m1.3.3">superscript</csymbol><list id="A2.Ex21.m1.3.3.3.4.cmml" xref="A2.Ex21.m1.3.3.3.3"><apply id="A2.Ex21.m1.1.1.1.1.1.cmml" xref="A2.Ex21.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex21.m1.1.1.1.1.1.1.cmml" xref="A2.Ex21.m1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex21.m1.1.1.1.1.1.2.cmml" xref="A2.Ex21.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex21.m1.1.1.1.1.1.2.1.cmml" xref="A2.Ex21.m1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex21.m1.1.1.1.1.1.2.2.cmml" xref="A2.Ex21.m1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex21.m1.1.1.1.1.1.2.3.cmml" xref="A2.Ex21.m1.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex21.m1.1.1.1.1.1.3.cmml" xref="A2.Ex21.m1.1.1.1.1.1.3">1</cn></apply><apply id="A2.Ex21.m1.2.2.2.2.2.cmml" xref="A2.Ex21.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex21.m1.2.2.2.2.2.1.cmml" xref="A2.Ex21.m1.2.2.2.2.2">superscript</csymbol><apply id="A2.Ex21.m1.2.2.2.2.2.2.cmml" xref="A2.Ex21.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="A2.Ex21.m1.2.2.2.2.2.2.1.cmml" xref="A2.Ex21.m1.2.2.2.2.2">subscript</csymbol><ci id="A2.Ex21.m1.2.2.2.2.2.2.2.cmml" xref="A2.Ex21.m1.2.2.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex21.m1.2.2.2.2.2.2.3.cmml" xref="A2.Ex21.m1.2.2.2.2.2.2.3">2</cn></apply><cn type="integer" id="A2.Ex21.m1.2.2.2.2.2.3.cmml" xref="A2.Ex21.m1.2.2.2.2.2.3">0</cn></apply><apply id="A2.Ex21.m1.3.3.3.3.3.cmml" xref="A2.Ex21.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex21.m1.3.3.3.3.3.1.cmml" xref="A2.Ex21.m1.3.3.3.3.3">superscript</csymbol><apply id="A2.Ex21.m1.3.3.3.3.3.2.cmml" xref="A2.Ex21.m1.3.3.3.3.3"><csymbol cd="ambiguous" id="A2.Ex21.m1.3.3.3.3.3.2.1.cmml" xref="A2.Ex21.m1.3.3.3.3.3">subscript</csymbol><ci id="A2.Ex21.m1.3.3.3.3.3.2.2.cmml" xref="A2.Ex21.m1.3.3.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex21.m1.3.3.3.3.3.2.3.cmml" xref="A2.Ex21.m1.3.3.3.3.3.2.3">3</cn></apply><ci id="A2.Ex21.m1.3.3.3.3.3.3.cmml" xref="A2.Ex21.m1.3.3.3.3.3.3">𝐾</ci></apply></list><csymbol cd="latexml" id="A2.Ex21.m1.3.3.5.cmml" xref="A2.Ex21.m1.3.3.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex21.m1.3c">\displaystyle[\mathbf{x}_{1}^{1},\mathbf{x}_{2}^{0},\mathbf{x}_{3}^{K}]^{\top}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A2.SS6.SSS0.Px4.p2.5" class="ltx_p"><span id="A2.SS6.SSS0.Px4.p2.5.1" class="ltx_text" style="font-size:90%;">Then, it feed the diffused </span><math id="A2.SS6.SSS0.Px4.p2.4.m1.1" class="ltx_Math" alttext="\mathbf{x}_{2}^{0}" display="inline"><semantics id="A2.SS6.SSS0.Px4.p2.4.m1.1a"><msubsup id="A2.SS6.SSS0.Px4.p2.4.m1.1.1" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1.cmml"><mi mathsize="90%" id="A2.SS6.SSS0.Px4.p2.4.m1.1.1.2.2" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.SS6.SSS0.Px4.p2.4.m1.1.1.2.3" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1.2.3.cmml">2</mn><mn mathsize="90%" id="A2.SS6.SSS0.Px4.p2.4.m1.1.1.3" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1.3.cmml">0</mn></msubsup><annotation-xml encoding="MathML-Content" id="A2.SS6.SSS0.Px4.p2.4.m1.1b"><apply id="A2.SS6.SSS0.Px4.p2.4.m1.1.1.cmml" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1"><csymbol cd="ambiguous" id="A2.SS6.SSS0.Px4.p2.4.m1.1.1.1.cmml" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1">superscript</csymbol><apply id="A2.SS6.SSS0.Px4.p2.4.m1.1.1.2.cmml" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1"><csymbol cd="ambiguous" id="A2.SS6.SSS0.Px4.p2.4.m1.1.1.2.1.cmml" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1">subscript</csymbol><ci id="A2.SS6.SSS0.Px4.p2.4.m1.1.1.2.2.cmml" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.SS6.SSS0.Px4.p2.4.m1.1.1.2.3.cmml" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1.2.3">2</cn></apply><cn type="integer" id="A2.SS6.SSS0.Px4.p2.4.m1.1.1.3.cmml" xref="A2.SS6.SSS0.Px4.p2.4.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS6.SSS0.Px4.p2.4.m1.1c">\mathbf{x}_{2}^{0}</annotation></semantics></math><span id="A2.SS6.SSS0.Px4.p2.5.2" class="ltx_text" style="font-size:90%;"> into the model but tell it is of a higher noise level, as </span><math id="A2.SS6.SSS0.Px4.p2.5.m2.1" class="ltx_Math" alttext="\mathbf{x}_{2}^{1}" display="inline"><semantics id="A2.SS6.SSS0.Px4.p2.5.m2.1a"><msubsup id="A2.SS6.SSS0.Px4.p2.5.m2.1.1" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1.cmml"><mi mathsize="90%" id="A2.SS6.SSS0.Px4.p2.5.m2.1.1.2.2" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.SS6.SSS0.Px4.p2.5.m2.1.1.2.3" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1.2.3.cmml">2</mn><mn mathsize="90%" id="A2.SS6.SSS0.Px4.p2.5.m2.1.1.3" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1.3.cmml">1</mn></msubsup><annotation-xml encoding="MathML-Content" id="A2.SS6.SSS0.Px4.p2.5.m2.1b"><apply id="A2.SS6.SSS0.Px4.p2.5.m2.1.1.cmml" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1"><csymbol cd="ambiguous" id="A2.SS6.SSS0.Px4.p2.5.m2.1.1.1.cmml" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1">superscript</csymbol><apply id="A2.SS6.SSS0.Px4.p2.5.m2.1.1.2.cmml" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1"><csymbol cd="ambiguous" id="A2.SS6.SSS0.Px4.p2.5.m2.1.1.2.1.cmml" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1">subscript</csymbol><ci id="A2.SS6.SSS0.Px4.p2.5.m2.1.1.2.2.cmml" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1.2.2">𝐱</ci><cn type="integer" id="A2.SS6.SSS0.Px4.p2.5.m2.1.1.2.3.cmml" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1.2.3">2</cn></apply><cn type="integer" id="A2.SS6.SSS0.Px4.p2.5.m2.1.1.3.cmml" xref="A2.SS6.SSS0.Px4.p2.5.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.SS6.SSS0.Px4.p2.5.m2.1c">\mathbf{x}_{2}^{1}</annotation></semantics></math><span id="A2.SS6.SSS0.Px4.p2.5.3" class="ltx_text" style="font-size:90%;">.</span></p>
<table id="A6.EGx42" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A2.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex22.m1.4" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{1},\mathbf{x}_{2}^{1},\mathbf{x}_{3}^{K//2}]^{\top}," display="inline"><semantics id="A2.Ex22.m1.4a"><mrow id="A2.Ex22.m1.4.4.1" xref="A2.Ex22.m1.4.4.1.1.cmml"><msup id="A2.Ex22.m1.4.4.1.1" xref="A2.Ex22.m1.4.4.1.1.cmml"><mrow id="A2.Ex22.m1.4.4.1.1.3.3" xref="A2.Ex22.m1.4.4.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex22.m1.4.4.1.1.3.3.4" xref="A2.Ex22.m1.4.4.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex22.m1.4.4.1.1.1.1.1" xref="A2.Ex22.m1.4.4.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex22.m1.4.4.1.1.1.1.1.2.2" xref="A2.Ex22.m1.4.4.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex22.m1.4.4.1.1.1.1.1.2.3" xref="A2.Ex22.m1.4.4.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex22.m1.4.4.1.1.1.1.1.3" xref="A2.Ex22.m1.4.4.1.1.1.1.1.3.cmml">1</mn></msubsup><mo mathsize="90%" id="A2.Ex22.m1.4.4.1.1.3.3.5" xref="A2.Ex22.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex22.m1.4.4.1.1.2.2.2" xref="A2.Ex22.m1.4.4.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex22.m1.4.4.1.1.2.2.2.2.2" xref="A2.Ex22.m1.4.4.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex22.m1.4.4.1.1.2.2.2.2.3" xref="A2.Ex22.m1.4.4.1.1.2.2.2.2.3.cmml">2</mn><mn mathsize="90%" id="A2.Ex22.m1.4.4.1.1.2.2.2.3" xref="A2.Ex22.m1.4.4.1.1.2.2.2.3.cmml">1</mn></msubsup><mo mathsize="90%" id="A2.Ex22.m1.4.4.1.1.3.3.6" xref="A2.Ex22.m1.4.4.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex22.m1.4.4.1.1.3.3.3" xref="A2.Ex22.m1.4.4.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex22.m1.4.4.1.1.3.3.3.2.2" xref="A2.Ex22.m1.4.4.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex22.m1.4.4.1.1.3.3.3.2.3" xref="A2.Ex22.m1.4.4.1.1.3.3.3.2.3.cmml">3</mn><mrow id="A2.Ex22.m1.3.3.3.3" xref="A2.Ex22.m1.3.3.3.4.cmml"><mi mathsize="90%" id="A2.Ex22.m1.1.1.1.1" xref="A2.Ex22.m1.1.1.1.1.cmml">K</mi><mo id="A2.Ex22.m1.3.3.3.3.2" xref="A2.Ex22.m1.3.3.3.4.cmml">⁣</mo><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex22.m1.2.2.2.2" xref="A2.Ex22.m1.2.2.2.2.cmml">/</mo><mo id="A2.Ex22.m1.3.3.3.3.3" xref="A2.Ex22.m1.3.3.3.4.cmml">⁣</mo><mrow id="A2.Ex22.m1.3.3.3.3.1" xref="A2.Ex22.m1.3.3.3.3.1.cmml"><mi id="A2.Ex22.m1.3.3.3.3.1.2" xref="A2.Ex22.m1.3.3.3.3.1.2.cmml"></mi><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.Ex22.m1.3.3.3.3.1.1" xref="A2.Ex22.m1.3.3.3.3.1.1.cmml">/</mo><mn mathsize="90%" id="A2.Ex22.m1.3.3.3.3.1.3" xref="A2.Ex22.m1.3.3.3.3.1.3.cmml">2</mn></mrow></mrow></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex22.m1.4.4.1.1.3.3.7" xref="A2.Ex22.m1.4.4.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex22.m1.4.4.1.1.5" xref="A2.Ex22.m1.4.4.1.1.5.cmml">⊤</mo></msup><mo mathsize="90%" id="A2.Ex22.m1.4.4.1.2" xref="A2.Ex22.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex22.m1.4b"><apply id="A2.Ex22.m1.4.4.1.1.cmml" xref="A2.Ex22.m1.4.4.1"><csymbol cd="ambiguous" id="A2.Ex22.m1.4.4.1.1.4.cmml" xref="A2.Ex22.m1.4.4.1">superscript</csymbol><list id="A2.Ex22.m1.4.4.1.1.3.4.cmml" xref="A2.Ex22.m1.4.4.1.1.3.3"><apply id="A2.Ex22.m1.4.4.1.1.1.1.1.cmml" xref="A2.Ex22.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex22.m1.4.4.1.1.1.1.1.1.cmml" xref="A2.Ex22.m1.4.4.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex22.m1.4.4.1.1.1.1.1.2.cmml" xref="A2.Ex22.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex22.m1.4.4.1.1.1.1.1.2.1.cmml" xref="A2.Ex22.m1.4.4.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex22.m1.4.4.1.1.1.1.1.2.2.cmml" xref="A2.Ex22.m1.4.4.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex22.m1.4.4.1.1.1.1.1.2.3.cmml" xref="A2.Ex22.m1.4.4.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex22.m1.4.4.1.1.1.1.1.3.cmml" xref="A2.Ex22.m1.4.4.1.1.1.1.1.3">1</cn></apply><apply id="A2.Ex22.m1.4.4.1.1.2.2.2.cmml" xref="A2.Ex22.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex22.m1.4.4.1.1.2.2.2.1.cmml" xref="A2.Ex22.m1.4.4.1.1.2.2.2">superscript</csymbol><apply id="A2.Ex22.m1.4.4.1.1.2.2.2.2.cmml" xref="A2.Ex22.m1.4.4.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex22.m1.4.4.1.1.2.2.2.2.1.cmml" xref="A2.Ex22.m1.4.4.1.1.2.2.2">subscript</csymbol><ci id="A2.Ex22.m1.4.4.1.1.2.2.2.2.2.cmml" xref="A2.Ex22.m1.4.4.1.1.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex22.m1.4.4.1.1.2.2.2.2.3.cmml" xref="A2.Ex22.m1.4.4.1.1.2.2.2.2.3">2</cn></apply><cn type="integer" id="A2.Ex22.m1.4.4.1.1.2.2.2.3.cmml" xref="A2.Ex22.m1.4.4.1.1.2.2.2.3">1</cn></apply><apply id="A2.Ex22.m1.4.4.1.1.3.3.3.cmml" xref="A2.Ex22.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex22.m1.4.4.1.1.3.3.3.1.cmml" xref="A2.Ex22.m1.4.4.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex22.m1.4.4.1.1.3.3.3.2.cmml" xref="A2.Ex22.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex22.m1.4.4.1.1.3.3.3.2.1.cmml" xref="A2.Ex22.m1.4.4.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex22.m1.4.4.1.1.3.3.3.2.2.cmml" xref="A2.Ex22.m1.4.4.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex22.m1.4.4.1.1.3.3.3.2.3.cmml" xref="A2.Ex22.m1.4.4.1.1.3.3.3.2.3">3</cn></apply><list id="A2.Ex22.m1.3.3.3.4.cmml" xref="A2.Ex22.m1.3.3.3.3"><ci id="A2.Ex22.m1.1.1.1.1.cmml" xref="A2.Ex22.m1.1.1.1.1">𝐾</ci><divide id="A2.Ex22.m1.2.2.2.2.cmml" xref="A2.Ex22.m1.2.2.2.2"></divide><apply id="A2.Ex22.m1.3.3.3.3.1.cmml" xref="A2.Ex22.m1.3.3.3.3.1"><divide id="A2.Ex22.m1.3.3.3.3.1.1.cmml" xref="A2.Ex22.m1.3.3.3.3.1.1"></divide><csymbol cd="latexml" id="A2.Ex22.m1.3.3.3.3.1.2.cmml" xref="A2.Ex22.m1.3.3.3.3.1.2">absent</csymbol><cn type="integer" id="A2.Ex22.m1.3.3.3.3.1.3.cmml" xref="A2.Ex22.m1.3.3.3.3.1.3">2</cn></apply></list></apply></list><csymbol cd="latexml" id="A2.Ex22.m1.4.4.1.1.5.cmml" xref="A2.Ex22.m1.4.4.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex22.m1.4c">\displaystyle[\mathbf{x}_{1}^{1},\mathbf{x}_{2}^{1},\mathbf{x}_{3}^{K//2}]^{\top},</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="A2.Ex23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="A2.Ex23.m1.1" class="ltx_Math" alttext="\displaystyle[\mathbf{x}_{1}^{1},\mathbf{x}_{2}^{1},\mathbf{x}_{3}^{0}]^{\top}." display="inline"><semantics id="A2.Ex23.m1.1a"><mrow id="A2.Ex23.m1.1.1.1" xref="A2.Ex23.m1.1.1.1.1.cmml"><msup id="A2.Ex23.m1.1.1.1.1" xref="A2.Ex23.m1.1.1.1.1.cmml"><mrow id="A2.Ex23.m1.1.1.1.1.3.3" xref="A2.Ex23.m1.1.1.1.1.3.4.cmml"><mo maxsize="90%" minsize="90%" id="A2.Ex23.m1.1.1.1.1.3.3.4" xref="A2.Ex23.m1.1.1.1.1.3.4.cmml">[</mo><msubsup id="A2.Ex23.m1.1.1.1.1.1.1.1" xref="A2.Ex23.m1.1.1.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.Ex23.m1.1.1.1.1.1.1.1.2.2" xref="A2.Ex23.m1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex23.m1.1.1.1.1.1.1.1.2.3" xref="A2.Ex23.m1.1.1.1.1.1.1.1.2.3.cmml">1</mn><mn mathsize="90%" id="A2.Ex23.m1.1.1.1.1.1.1.1.3" xref="A2.Ex23.m1.1.1.1.1.1.1.1.3.cmml">1</mn></msubsup><mo mathsize="90%" id="A2.Ex23.m1.1.1.1.1.3.3.5" xref="A2.Ex23.m1.1.1.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex23.m1.1.1.1.1.2.2.2" xref="A2.Ex23.m1.1.1.1.1.2.2.2.cmml"><mi mathsize="90%" id="A2.Ex23.m1.1.1.1.1.2.2.2.2.2" xref="A2.Ex23.m1.1.1.1.1.2.2.2.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex23.m1.1.1.1.1.2.2.2.2.3" xref="A2.Ex23.m1.1.1.1.1.2.2.2.2.3.cmml">2</mn><mn mathsize="90%" id="A2.Ex23.m1.1.1.1.1.2.2.2.3" xref="A2.Ex23.m1.1.1.1.1.2.2.2.3.cmml">1</mn></msubsup><mo mathsize="90%" id="A2.Ex23.m1.1.1.1.1.3.3.6" xref="A2.Ex23.m1.1.1.1.1.3.4.cmml">,</mo><msubsup id="A2.Ex23.m1.1.1.1.1.3.3.3" xref="A2.Ex23.m1.1.1.1.1.3.3.3.cmml"><mi mathsize="90%" id="A2.Ex23.m1.1.1.1.1.3.3.3.2.2" xref="A2.Ex23.m1.1.1.1.1.3.3.3.2.2.cmml">𝐱</mi><mn mathsize="90%" id="A2.Ex23.m1.1.1.1.1.3.3.3.2.3" xref="A2.Ex23.m1.1.1.1.1.3.3.3.2.3.cmml">3</mn><mn mathsize="90%" id="A2.Ex23.m1.1.1.1.1.3.3.3.3" xref="A2.Ex23.m1.1.1.1.1.3.3.3.3.cmml">0</mn></msubsup><mo maxsize="90%" minsize="90%" id="A2.Ex23.m1.1.1.1.1.3.3.7" xref="A2.Ex23.m1.1.1.1.1.3.4.cmml">]</mo></mrow><mo mathsize="90%" id="A2.Ex23.m1.1.1.1.1.5" xref="A2.Ex23.m1.1.1.1.1.5.cmml">⊤</mo></msup><mo lspace="0em" mathsize="90%" id="A2.Ex23.m1.1.1.1.2" xref="A2.Ex23.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.Ex23.m1.1b"><apply id="A2.Ex23.m1.1.1.1.1.cmml" xref="A2.Ex23.m1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex23.m1.1.1.1.1.4.cmml" xref="A2.Ex23.m1.1.1.1">superscript</csymbol><list id="A2.Ex23.m1.1.1.1.1.3.4.cmml" xref="A2.Ex23.m1.1.1.1.1.3.3"><apply id="A2.Ex23.m1.1.1.1.1.1.1.1.cmml" xref="A2.Ex23.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex23.m1.1.1.1.1.1.1.1.1.cmml" xref="A2.Ex23.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="A2.Ex23.m1.1.1.1.1.1.1.1.2.cmml" xref="A2.Ex23.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.Ex23.m1.1.1.1.1.1.1.1.2.1.cmml" xref="A2.Ex23.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A2.Ex23.m1.1.1.1.1.1.1.1.2.2.cmml" xref="A2.Ex23.m1.1.1.1.1.1.1.1.2.2">𝐱</ci><cn type="integer" id="A2.Ex23.m1.1.1.1.1.1.1.1.2.3.cmml" xref="A2.Ex23.m1.1.1.1.1.1.1.1.2.3">1</cn></apply><cn type="integer" id="A2.Ex23.m1.1.1.1.1.1.1.1.3.cmml" xref="A2.Ex23.m1.1.1.1.1.1.1.1.3">1</cn></apply><apply id="A2.Ex23.m1.1.1.1.1.2.2.2.cmml" xref="A2.Ex23.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex23.m1.1.1.1.1.2.2.2.1.cmml" xref="A2.Ex23.m1.1.1.1.1.2.2.2">superscript</csymbol><apply id="A2.Ex23.m1.1.1.1.1.2.2.2.2.cmml" xref="A2.Ex23.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.Ex23.m1.1.1.1.1.2.2.2.2.1.cmml" xref="A2.Ex23.m1.1.1.1.1.2.2.2">subscript</csymbol><ci id="A2.Ex23.m1.1.1.1.1.2.2.2.2.2.cmml" xref="A2.Ex23.m1.1.1.1.1.2.2.2.2.2">𝐱</ci><cn type="integer" id="A2.Ex23.m1.1.1.1.1.2.2.2.2.3.cmml" xref="A2.Ex23.m1.1.1.1.1.2.2.2.2.3">2</cn></apply><cn type="integer" id="A2.Ex23.m1.1.1.1.1.2.2.2.3.cmml" xref="A2.Ex23.m1.1.1.1.1.2.2.2.3">1</cn></apply><apply id="A2.Ex23.m1.1.1.1.1.3.3.3.cmml" xref="A2.Ex23.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex23.m1.1.1.1.1.3.3.3.1.cmml" xref="A2.Ex23.m1.1.1.1.1.3.3.3">superscript</csymbol><apply id="A2.Ex23.m1.1.1.1.1.3.3.3.2.cmml" xref="A2.Ex23.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.Ex23.m1.1.1.1.1.3.3.3.2.1.cmml" xref="A2.Ex23.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="A2.Ex23.m1.1.1.1.1.3.3.3.2.2.cmml" xref="A2.Ex23.m1.1.1.1.1.3.3.3.2.2">𝐱</ci><cn type="integer" id="A2.Ex23.m1.1.1.1.1.3.3.3.2.3.cmml" xref="A2.Ex23.m1.1.1.1.1.3.3.3.2.3">3</cn></apply><cn type="integer" id="A2.Ex23.m1.1.1.1.1.3.3.3.3.cmml" xref="A2.Ex23.m1.1.1.1.1.3.3.3.3">0</cn></apply></list><csymbol cd="latexml" id="A2.Ex23.m1.1.1.1.1.5.cmml" xref="A2.Ex23.m1.1.1.1.1.5">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.Ex23.m1.1c">\displaystyle[\mathbf{x}_{1}^{1},\mathbf{x}_{2}^{1},\mathbf{x}_{3}^{0}]^{\top}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
</div>
</section>
</section>
</section>
<section id="A3" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix" style="font-size:90%;">
<span class="ltx_tag ltx_tag_appendix">Appendix C </span>Extended Related Work</h2>

<section id="A3.SS0.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Reconstructing masked tokens.</h5>

<div id="A3.SS0.SSS0.Px1.p1" class="ltx_para">
<p id="A3.SS0.SSS0.Px1.p1.1" class="ltx_p"><span id="A3.SS0.SSS0.Px1.p1.1.1" class="ltx_text" style="font-size:90%;">Masked Autoencoders for images </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A3.SS0.SSS0.Px1.p1.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib26" title="" class="ltx_ref">26</a><span id="A3.SS0.SSS0.Px1.p1.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A3.SS0.SSS0.Px1.p1.1.4" class="ltx_text" style="font-size:90%;"> and videos </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A3.SS0.SSS0.Px1.p1.1.5.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib17" title="" class="ltx_ref">17</a><span id="A3.SS0.SSS0.Px1.p1.1.6.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A3.SS0.SSS0.Px1.p1.1.7" class="ltx_text" style="font-size:90%;"> are a popular method for representation learning in pixel space. They have been extended to perform diffusion to generate masked patches conditioned on unmasked ones </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A3.SS0.SSS0.Px1.p1.1.8.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib60" title="" class="ltx_ref">60</a>, <a href="#bib.bib19" title="" class="ltx_ref">19</a><span id="A3.SS0.SSS0.Px1.p1.1.9.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A3.SS0.SSS0.Px1.p1.1.10" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
</section>
<section id="A3.SS0.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Casting Image Generation as Sequence Generation.</h5>

<div id="A3.SS0.SSS0.Px2.p1" class="ltx_para">
<p id="A3.SS0.SSS0.Px2.p1.1" class="ltx_p"><cite class="ltx_cite ltx_citemacro_cite"><span id="A3.SS0.SSS0.Px2.p1.1.1.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib58" title="" class="ltx_ref">58</a>, <a href="#bib.bib8" title="" class="ltx_ref">8</a><span id="A3.SS0.SSS0.Px2.p1.1.2.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A3.SS0.SSS0.Px2.p1.1.3" class="ltx_text" style="font-size:90%;"> show that even generative modeling of non-sequential data, such as images, can be fruitfully cast as sequence generative modeling.</span></p>
</div>
</section>
<section id="A3.SS0.SSS0.Px3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Non-Diffusion Probabilistic Sequence Models.</h5>

<div id="A3.SS0.SSS0.Px3.p1" class="ltx_para">
<p id="A3.SS0.SSS0.Px3.p1.1" class="ltx_p"><cite class="ltx_cite ltx_citemacro_cite"><span id="A3.SS0.SSS0.Px3.p1.1.1.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib12" title="" class="ltx_ref">12</a><span id="A3.SS0.SSS0.Px3.p1.1.2.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A3.SS0.SSS0.Px3.p1.1.3" class="ltx_text" style="font-size:90%;"> parameterize token-to-token transitions via a variational auto-encoder. This makes them probabilistic, but does not directly maximize the joint probability of sequences, but rather, enables sampling from the distribution of single-step transitions.</span></p>
</div>
</section>
<section id="A3.SS0.SSS0.Px4" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">AR-Diffusion <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib63" title="" class="ltx_ref">63</a>]</cite>.</h5>

<div id="A3.SS0.SSS0.Px4.p1" class="ltx_para">
<p id="A3.SS0.SSS0.Px4.p1.1" class="ltx_p"><span id="A3.SS0.SSS0.Px4.p1.1.1" class="ltx_text" style="font-size:90%;">Most similar to our work is AR-Diffusion </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A3.SS0.SSS0.Px4.p1.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib63" title="" class="ltx_ref">63</a><span id="A3.SS0.SSS0.Px4.p1.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A3.SS0.SSS0.Px4.p1.1.4" class="ltx_text" style="font-size:90%;"> which similarly aim to train next-token prediction models for sequence diffusion. Key differences are that AR-Diffusion proposes a noise level that is </span><em id="A3.SS0.SSS0.Px4.p1.1.5" class="ltx_emph ltx_font_italic" style="font-size:90%;">linearly</em><span id="A3.SS0.SSS0.Px4.p1.1.6" class="ltx_text" style="font-size:90%;"> dependent on the position of each word in the sequence, while our critical contribution is to have each noise level be </span><em id="A3.SS0.SSS0.Px4.p1.1.7" class="ltx_emph ltx_font_italic" style="font-size:90%;">independent</em><span id="A3.SS0.SSS0.Px4.p1.1.8" class="ltx_text" style="font-size:90%;">, as this uniquely enables our proposed sampling schemes, such as stabilizing auto-regressive generation and conditioning on corrupted observations. Further, AR-Diffusion only explores language modeling and does not explore guidance, while we investigate Diffusion Forcing as a broadly applicable sequence generative model with particular applications to sequential decision making. In particular, we introduce Monte-Carlo Tree Guidance as a novel guidance mechanism.</span></p>
</div>
</section>
</section>
<section id="A4" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix" style="font-size:90%;">
<span class="ltx_tag ltx_tag_appendix">Appendix D </span>Additional Method Details</h2>

<section id="A4.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">D.1 </span>Architecture</h3>

<section id="A4.SS1.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Video Diffusion</h5>

<div id="A4.SS1.SSS0.Px1.p1" class="ltx_para">
<p id="A4.SS1.SSS0.Px1.p1.12" class="ltx_p"><span id="A4.SS1.SSS0.Px1.p1.12.1" class="ltx_text" style="font-size:90%;">We choose both the raw image </span><math id="A4.SS1.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.1.m1.1a"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.1.m1.1.1" xref="A4.SS1.SSS0.Px1.p1.1.m1.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.1.m1.1b"><ci id="A4.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.1.m1.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.1.m1.1c">\mathbf{x}</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.2" class="ltx_text" style="font-size:90%;"> and latent state </span><math id="A4.SS1.SSS0.Px1.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{z}" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.2.m2.1a"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.2.m2.1.1" xref="A4.SS1.SSS0.Px1.p1.2.m2.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.2.m2.1b"><ci id="A4.SS1.SSS0.Px1.p1.2.m2.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.2.m2.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.2.m2.1c">\mathbf{z}</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.3" class="ltx_text" style="font-size:90%;"> to be 2D tensors with channel, width, and height. For simplicity, we use the same width and height for </span><math id="A4.SS1.SSS0.Px1.p1.3.m3.1" class="ltx_Math" alttext="\mathbf{x}" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.3.m3.1a"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.3.m3.1.1" xref="A4.SS1.SSS0.Px1.p1.3.m3.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.3.m3.1b"><ci id="A4.SS1.SSS0.Px1.p1.3.m3.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.3.m3.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.3.m3.1c">\mathbf{x}</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.4" class="ltx_text" style="font-size:90%;"> and </span><math id="A4.SS1.SSS0.Px1.p1.4.m4.1" class="ltx_Math" alttext="\mathbf{z}" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.4.m4.1a"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.4.m4.1.1" xref="A4.SS1.SSS0.Px1.p1.4.m4.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.4.m4.1b"><ci id="A4.SS1.SSS0.Px1.p1.4.m4.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.4.m4.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.4.m4.1c">\mathbf{z}</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.5" class="ltx_text" style="font-size:90%;">. 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xref="A4.SS1.SSS0.Px1.p1.5.m5.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.5.m5.1c">p(\mathbf{x}_{t}^{k_{t}}|\mathbf{z}_{t-1})</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.6" class="ltx_text" style="font-size:90%;"> with a typical diffusion U-net </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS1.SSS0.Px1.p1.12.7.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib45" title="" class="ltx_ref">45</a><span id="A4.SS1.SSS0.Px1.p1.12.8.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS1.SSS0.Px1.p1.12.9" class="ltx_text" style="font-size:90%;">. We use the output of the U-net as the input to a gated recurrent unit (GRU) and use </span><math id="A4.SS1.SSS0.Px1.p1.6.m6.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.6.m6.1a"><msub id="A4.SS1.SSS0.Px1.p1.6.m6.1.1" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.2" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.2.cmml">𝐳</mi><mrow id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.2" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.1" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.3" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.6.m6.1b"><apply id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1">subscript</csymbol><ci id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.2">𝐳</ci><apply id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.cmml" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3"><minus id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.1.cmml" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.1"></minus><ci id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.2.cmml" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.2">𝑡</ci><cn type="integer" id="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.3.cmml" xref="A4.SS1.SSS0.Px1.p1.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.6.m6.1c">\mathbf{z}_{t-1}</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.10" class="ltx_text" style="font-size:90%;"> as the hidden state feed into a GRU. The output of GRU is treated as </span><math id="A4.SS1.SSS0.Px1.p1.7.m7.1" class="ltx_Math" alttext="\mathbf{z}_{t}" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.7.m7.1a"><msub id="A4.SS1.SSS0.Px1.p1.7.m7.1.1" xref="A4.SS1.SSS0.Px1.p1.7.m7.1.1.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.7.m7.1.1.2" xref="A4.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml">𝐳</mi><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.7.m7.1.1.3" xref="A4.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.7.m7.1b"><apply id="A4.SS1.SSS0.Px1.p1.7.m7.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS0.Px1.p1.7.m7.1.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.7.m7.1.1">subscript</csymbol><ci id="A4.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml" xref="A4.SS1.SSS0.Px1.p1.7.m7.1.1.2">𝐳</ci><ci id="A4.SS1.SSS0.Px1.p1.7.m7.1.1.3.cmml" xref="A4.SS1.SSS0.Px1.p1.7.m7.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.7.m7.1c">\mathbf{z}_{t}</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.11" class="ltx_text" style="font-size:90%;">. For observation model </span><math id="A4.SS1.SSS0.Px1.p1.8.m8.1" class="ltx_Math" alttext="p(\mathbf{x}_{t}|\mathbf{z}_{t})" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.8.m8.1a"><mrow id="A4.SS1.SSS0.Px1.p1.8.m8.1.1" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.3" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.2" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml">​</mo><mrow id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.2" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml"><msub id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.2" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.3" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.3.cmml">t</mi></msub><mo fence="false" mathsize="90%" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.1" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml">|</mo><msub id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.2" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.2.cmml">𝐳</mi><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.3" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.3.cmml">t</mi></msub></mrow><mo maxsize="90%" minsize="90%" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.3" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.8.m8.1b"><apply id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1"><times id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.2"></times><ci id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.3">𝑝</ci><apply id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1"><csymbol cd="latexml" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.1">conditional</csymbol><apply id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.1.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2">subscript</csymbol><ci id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.2.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.2">𝐱</ci><ci id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.3.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.2.3">𝑡</ci></apply><apply id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.1.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.2.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.2">𝐳</ci><ci id="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.3.cmml" xref="A4.SS1.SSS0.Px1.p1.8.m8.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.8.m8.1c">p(\mathbf{x}_{t}|\mathbf{z}_{t})</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.12" class="ltx_text" style="font-size:90%;">, we use a </span><math id="A4.SS1.SSS0.Px1.p1.9.m9.1" class="ltx_Math" alttext="1" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.9.m9.1a"><mn mathsize="90%" id="A4.SS1.SSS0.Px1.p1.9.m9.1.1" xref="A4.SS1.SSS0.Px1.p1.9.m9.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.9.m9.1b"><cn type="integer" id="A4.SS1.SSS0.Px1.p1.9.m9.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.9.m9.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.9.m9.1c">1</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.13" class="ltx_text" style="font-size:90%;">-layer resnet </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS1.SSS0.Px1.p1.12.14.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib27" title="" class="ltx_ref">27</a><span id="A4.SS1.SSS0.Px1.p1.12.15.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS1.SSS0.Px1.p1.12.16" class="ltx_text" style="font-size:90%;"> followed by a conv layer. We combine these two models to create an RNN layer, where the latent of a particular time step is </span><math id="A4.SS1.SSS0.Px1.p1.10.m10.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.10.m10.1a"><msub id="A4.SS1.SSS0.Px1.p1.10.m10.1.1" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.2" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.2.cmml">𝐳</mi><mrow id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.2" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.1" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.3" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.10.m10.1b"><apply id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1">subscript</csymbol><ci id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.2.cmml" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.2">𝐳</ci><apply id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.cmml" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3"><minus id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.1.cmml" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.1"></minus><ci id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.2.cmml" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.2">𝑡</ci><cn type="integer" id="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.3.cmml" xref="A4.SS1.SSS0.Px1.p1.10.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.10.m10.1c">\mathbf{z}_{t-1}</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.17" class="ltx_text" style="font-size:90%;">, input is </span><math id="A4.SS1.SSS0.Px1.p1.11.m11.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.11.m11.1a"><msubsup id="A4.SS1.SSS0.Px1.p1.11.m11.1.1" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.2.2" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.2.3" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.2.3.cmml">t</mi><msub id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.2" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.3" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.11.m11.1b"><apply id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1">superscript</csymbol><apply id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.2.cmml" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.2.1.cmml" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1">subscript</csymbol><ci id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.2.2.cmml" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.2.2">𝐱</ci><ci id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.2.3.cmml" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.2.3">𝑡</ci></apply><apply id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.cmml" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.1.cmml" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.2.cmml" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.2">𝑘</ci><ci id="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.3.cmml" xref="A4.SS1.SSS0.Px1.p1.11.m11.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.11.m11.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.18" class="ltx_text" style="font-size:90%;"> and output is </span><math id="A4.SS1.SSS0.Px1.p1.12.m12.1" class="ltx_Math" alttext="\hat{\mathbf{x}}" display="inline"><semantics id="A4.SS1.SSS0.Px1.p1.12.m12.1a"><mover accent="true" id="A4.SS1.SSS0.Px1.p1.12.m12.1.1" xref="A4.SS1.SSS0.Px1.p1.12.m12.1.1.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p1.12.m12.1.1.2" xref="A4.SS1.SSS0.Px1.p1.12.m12.1.1.2.cmml">𝐱</mi><mo mathsize="90%" id="A4.SS1.SSS0.Px1.p1.12.m12.1.1.1" xref="A4.SS1.SSS0.Px1.p1.12.m12.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p1.12.m12.1b"><apply id="A4.SS1.SSS0.Px1.p1.12.m12.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.12.m12.1.1"><ci id="A4.SS1.SSS0.Px1.p1.12.m12.1.1.1.cmml" xref="A4.SS1.SSS0.Px1.p1.12.m12.1.1.1">^</ci><ci id="A4.SS1.SSS0.Px1.p1.12.m12.1.1.2.cmml" xref="A4.SS1.SSS0.Px1.p1.12.m12.1.1.2">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p1.12.m12.1c">\hat{\mathbf{x}}</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p1.12.19" class="ltx_text" style="font-size:90%;">. One can potentially obtain better results by training Diffusion Forcing with a causal transformer architecture. However, since RNN is more efficient for online decision-making, we also stick with it for video prediction and it already gives us satisfying results.</span></p>
</div>
<div id="A4.SS1.SSS0.Px1.p2" class="ltx_para">
<p id="A4.SS1.SSS0.Px1.p2.7" class="ltx_p"><span id="A4.SS1.SSS0.Px1.p2.7.1" class="ltx_text" style="font-size:90%;">We choose the number of channels in </span><math id="A4.SS1.SSS0.Px1.p2.1.m1.1" class="ltx_Math" alttext="\mathbf{z}" display="inline"><semantics id="A4.SS1.SSS0.Px1.p2.1.m1.1a"><mi mathsize="90%" id="A4.SS1.SSS0.Px1.p2.1.m1.1.1" xref="A4.SS1.SSS0.Px1.p2.1.m1.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p2.1.m1.1b"><ci id="A4.SS1.SSS0.Px1.p2.1.m1.1.1.cmml" xref="A4.SS1.SSS0.Px1.p2.1.m1.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p2.1.m1.1c">\mathbf{z}</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p2.7.2" class="ltx_text" style="font-size:90%;"> to be </span><math id="A4.SS1.SSS0.Px1.p2.2.m2.1" class="ltx_Math" alttext="16" display="inline"><semantics id="A4.SS1.SSS0.Px1.p2.2.m2.1a"><mn mathsize="90%" id="A4.SS1.SSS0.Px1.p2.2.m2.1.1" xref="A4.SS1.SSS0.Px1.p2.2.m2.1.1.cmml">16</mn><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p2.2.m2.1b"><cn type="integer" id="A4.SS1.SSS0.Px1.p2.2.m2.1.1.cmml" xref="A4.SS1.SSS0.Px1.p2.2.m2.1.1">16</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p2.2.m2.1c">16</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p2.7.3" class="ltx_text" style="font-size:90%;"> for DMlab and </span><math id="A4.SS1.SSS0.Px1.p2.3.m3.1" class="ltx_Math" alttext="32" display="inline"><semantics id="A4.SS1.SSS0.Px1.p2.3.m3.1a"><mn mathsize="90%" id="A4.SS1.SSS0.Px1.p2.3.m3.1.1" xref="A4.SS1.SSS0.Px1.p2.3.m3.1.1.cmml">32</mn><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p2.3.m3.1b"><cn type="integer" id="A4.SS1.SSS0.Px1.p2.3.m3.1.1.cmml" xref="A4.SS1.SSS0.Px1.p2.3.m3.1.1">32</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p2.3.m3.1c">32</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p2.7.4" class="ltx_text" style="font-size:90%;"> for Minecraft. In total, our Minecraft model consists of </span><math id="A4.SS1.SSS0.Px1.p2.4.m4.1" class="ltx_Math" alttext="36" display="inline"><semantics id="A4.SS1.SSS0.Px1.p2.4.m4.1a"><mn mathsize="90%" id="A4.SS1.SSS0.Px1.p2.4.m4.1.1" xref="A4.SS1.SSS0.Px1.p2.4.m4.1.1.cmml">36</mn><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p2.4.m4.1b"><cn type="integer" id="A4.SS1.SSS0.Px1.p2.4.m4.1.1.cmml" xref="A4.SS1.SSS0.Px1.p2.4.m4.1.1">36</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p2.4.m4.1c">36</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p2.7.5" class="ltx_text" style="font-size:90%;"> million parameters and our DMlab model consists of </span><math id="A4.SS1.SSS0.Px1.p2.5.m5.1" class="ltx_Math" alttext="24" display="inline"><semantics id="A4.SS1.SSS0.Px1.p2.5.m5.1a"><mn mathsize="90%" id="A4.SS1.SSS0.Px1.p2.5.m5.1.1" xref="A4.SS1.SSS0.Px1.p2.5.m5.1.1.cmml">24</mn><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p2.5.m5.1b"><cn type="integer" id="A4.SS1.SSS0.Px1.p2.5.m5.1.1.cmml" xref="A4.SS1.SSS0.Px1.p2.5.m5.1.1">24</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p2.5.m5.1c">24</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p2.7.6" class="ltx_text" style="font-size:90%;"> million parameters. We can potentially obtain a better Minecraft video prediction model with more parameters, but we defer that to future works to keep the training duration reasonable (</span><math id="A4.SS1.SSS0.Px1.p2.6.m6.1" class="ltx_Math" alttext="&lt;1" display="inline"><semantics id="A4.SS1.SSS0.Px1.p2.6.m6.1a"><mrow id="A4.SS1.SSS0.Px1.p2.6.m6.1.1" xref="A4.SS1.SSS0.Px1.p2.6.m6.1.1.cmml"><mi id="A4.SS1.SSS0.Px1.p2.6.m6.1.1.2" xref="A4.SS1.SSS0.Px1.p2.6.m6.1.1.2.cmml"></mi><mo mathsize="90%" id="A4.SS1.SSS0.Px1.p2.6.m6.1.1.1" xref="A4.SS1.SSS0.Px1.p2.6.m6.1.1.1.cmml">&lt;</mo><mn mathsize="90%" id="A4.SS1.SSS0.Px1.p2.6.m6.1.1.3" xref="A4.SS1.SSS0.Px1.p2.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p2.6.m6.1b"><apply id="A4.SS1.SSS0.Px1.p2.6.m6.1.1.cmml" xref="A4.SS1.SSS0.Px1.p2.6.m6.1.1"><lt id="A4.SS1.SSS0.Px1.p2.6.m6.1.1.1.cmml" xref="A4.SS1.SSS0.Px1.p2.6.m6.1.1.1"></lt><csymbol cd="latexml" id="A4.SS1.SSS0.Px1.p2.6.m6.1.1.2.cmml" xref="A4.SS1.SSS0.Px1.p2.6.m6.1.1.2">absent</csymbol><cn type="integer" id="A4.SS1.SSS0.Px1.p2.6.m6.1.1.3.cmml" xref="A4.SS1.SSS0.Px1.p2.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p2.6.m6.1c">&lt;1</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p2.7.7" class="ltx_text" style="font-size:90%;"> day). In maze planning, the number of total parameters is </span><math id="A4.SS1.SSS0.Px1.p2.7.m7.1" class="ltx_Math" alttext="4.33" display="inline"><semantics id="A4.SS1.SSS0.Px1.p2.7.m7.1a"><mn mathsize="90%" id="A4.SS1.SSS0.Px1.p2.7.m7.1.1" xref="A4.SS1.SSS0.Px1.p2.7.m7.1.1.cmml">4.33</mn><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px1.p2.7.m7.1b"><cn type="float" id="A4.SS1.SSS0.Px1.p2.7.m7.1.1.cmml" xref="A4.SS1.SSS0.Px1.p2.7.m7.1.1">4.33</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px1.p2.7.m7.1c">4.33</annotation></semantics></math><span id="A4.SS1.SSS0.Px1.p2.7.8" class="ltx_text" style="font-size:90%;"> million.</span></p>
</div>
</section>
<section id="A4.SS1.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Non-Video Diffusion</h5>

<div id="A4.SS1.SSS0.Px2.p1" class="ltx_para">
<p id="A4.SS1.SSS0.Px2.p1.3" class="ltx_p"><span id="A4.SS1.SSS0.Px2.p1.3.1" class="ltx_text" style="font-size:90%;">For non-spatial </span><math id="A4.SS1.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}" display="inline"><semantics id="A4.SS1.SSS0.Px2.p1.1.m1.1a"><mi mathsize="90%" id="A4.SS1.SSS0.Px2.p1.1.m1.1.1" xref="A4.SS1.SSS0.Px2.p1.1.m1.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px2.p1.1.m1.1b"><ci id="A4.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="A4.SS1.SSS0.Px2.p1.1.m1.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px2.p1.1.m1.1c">\mathbf{x}</annotation></semantics></math><span id="A4.SS1.SSS0.Px2.p1.3.2" class="ltx_text" style="font-size:90%;"> that is not video nor images, we use residue MLPs </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS1.SSS0.Px2.p1.3.3.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib57" title="" class="ltx_ref">57</a><span id="A4.SS1.SSS0.Px2.p1.3.4.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS1.SSS0.Px2.p1.3.5" class="ltx_text" style="font-size:90%;"> instead of Unet as the backbone for the dynamics model. Residue MLP is basically the ResNet </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS1.SSS0.Px2.p1.3.6.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib27" title="" class="ltx_ref">27</a><span id="A4.SS1.SSS0.Px2.p1.3.7.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS1.SSS0.Px2.p1.3.8" class="ltx_text" style="font-size:90%;"> equivalent for MLP. Similar to video prediction, we feed the output of resMLP into a GRU along with </span><math id="A4.SS1.SSS0.Px2.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}" display="inline"><semantics id="A4.SS1.SSS0.Px2.p1.2.m2.1a"><msub id="A4.SS1.SSS0.Px2.p1.2.m2.1.1" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.2" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml">𝐳</mi><mrow id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.2" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.1" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.3" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px2.p1.2.m2.1b"><apply id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.cmml" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1">subscript</csymbol><ci id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.2">𝐳</ci><apply id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3"><minus id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.1.cmml" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.1"></minus><ci id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.2.cmml" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.2">𝑡</ci><cn type="integer" id="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.3.cmml" xref="A4.SS1.SSS0.Px2.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px2.p1.2.m2.1c">\mathbf{z}_{t-1}</annotation></semantics></math><span id="A4.SS1.SSS0.Px2.p1.3.9" class="ltx_text" style="font-size:90%;"> to get </span><math id="A4.SS1.SSS0.Px2.p1.3.m3.1" class="ltx_Math" alttext="\mathbf{z}_{t}" display="inline"><semantics id="A4.SS1.SSS0.Px2.p1.3.m3.1a"><msub id="A4.SS1.SSS0.Px2.p1.3.m3.1.1" xref="A4.SS1.SSS0.Px2.p1.3.m3.1.1.cmml"><mi mathsize="90%" id="A4.SS1.SSS0.Px2.p1.3.m3.1.1.2" xref="A4.SS1.SSS0.Px2.p1.3.m3.1.1.2.cmml">𝐳</mi><mi mathsize="90%" id="A4.SS1.SSS0.Px2.p1.3.m3.1.1.3" xref="A4.SS1.SSS0.Px2.p1.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS0.Px2.p1.3.m3.1b"><apply id="A4.SS1.SSS0.Px2.p1.3.m3.1.1.cmml" xref="A4.SS1.SSS0.Px2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="A4.SS1.SSS0.Px2.p1.3.m3.1.1">subscript</csymbol><ci id="A4.SS1.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="A4.SS1.SSS0.Px2.p1.3.m3.1.1.2">𝐳</ci><ci id="A4.SS1.SSS0.Px2.p1.3.m3.1.1.3.cmml" xref="A4.SS1.SSS0.Px2.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS0.Px2.p1.3.m3.1c">\mathbf{z}_{t}</annotation></semantics></math><span id="A4.SS1.SSS0.Px2.p1.3.10" class="ltx_text" style="font-size:90%;">. Another ResMLP serves as the observation model.</span></p>
</div>
</section>
</section>
<section id="A4.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">D.2 </span>Diffusion parameterization</h3>

<div id="A4.SS2.p1" class="ltx_para">
<p id="A4.SS2.p1.5" class="ltx_p"><span id="A4.SS2.p1.5.1" class="ltx_text" style="font-size:90%;">In diffusion models, there are three equivalent prediction objectives, </span><math id="A4.SS2.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}_{0}" display="inline"><semantics id="A4.SS2.p1.1.m1.1a"><msub id="A4.SS2.p1.1.m1.1.1" xref="A4.SS2.p1.1.m1.1.1.cmml"><mi mathsize="90%" id="A4.SS2.p1.1.m1.1.1.2" xref="A4.SS2.p1.1.m1.1.1.2.cmml">𝐱</mi><mn mathsize="90%" id="A4.SS2.p1.1.m1.1.1.3" xref="A4.SS2.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A4.SS2.p1.1.m1.1b"><apply id="A4.SS2.p1.1.m1.1.1.cmml" xref="A4.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS2.p1.1.m1.1.1.1.cmml" xref="A4.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="A4.SS2.p1.1.m1.1.1.2.cmml" xref="A4.SS2.p1.1.m1.1.1.2">𝐱</ci><cn type="integer" id="A4.SS2.p1.1.m1.1.1.3.cmml" xref="A4.SS2.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS2.p1.1.m1.1c">\mathbf{x}_{0}</annotation></semantics></math><span id="A4.SS2.p1.5.2" class="ltx_text" style="font-size:90%;">, </span><math id="A4.SS2.p1.2.m2.1" class="ltx_Math" alttext="\epsilon" display="inline"><semantics id="A4.SS2.p1.2.m2.1a"><mi mathsize="90%" id="A4.SS2.p1.2.m2.1.1" xref="A4.SS2.p1.2.m2.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="A4.SS2.p1.2.m2.1b"><ci id="A4.SS2.p1.2.m2.1.1.cmml" xref="A4.SS2.p1.2.m2.1.1">italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS2.p1.2.m2.1c">\epsilon</annotation></semantics></math><span id="A4.SS2.p1.5.3" class="ltx_text" style="font-size:90%;"> </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS2.p1.5.4.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib28" title="" class="ltx_ref">28</a><span id="A4.SS2.p1.5.5.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS2.p1.5.6" class="ltx_text" style="font-size:90%;">, and </span><math id="A4.SS2.p1.3.m3.1" class="ltx_Math" alttext="v" display="inline"><semantics id="A4.SS2.p1.3.m3.1a"><mi mathsize="90%" id="A4.SS2.p1.3.m3.1.1" xref="A4.SS2.p1.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A4.SS2.p1.3.m3.1b"><ci id="A4.SS2.p1.3.m3.1.1.cmml" xref="A4.SS2.p1.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS2.p1.3.m3.1c">v</annotation></semantics></math><span id="A4.SS2.p1.5.7" class="ltx_text" style="font-size:90%;"> parameterization </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS2.p1.5.8.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib50" title="" class="ltx_ref">50</a><span id="A4.SS2.p1.5.9.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS2.p1.5.10" class="ltx_text" style="font-size:90%;">. Different objectives lead to different reweighting of loss at different noise levels, together with SNR reweighting. For example, </span><math id="A4.SS2.p1.4.m4.1" class="ltx_Math" alttext="\epsilon" display="inline"><semantics id="A4.SS2.p1.4.m4.1a"><mi mathsize="90%" id="A4.SS2.p1.4.m4.1.1" xref="A4.SS2.p1.4.m4.1.1.cmml">ϵ</mi><annotation-xml encoding="MathML-Content" id="A4.SS2.p1.4.m4.1b"><ci id="A4.SS2.p1.4.m4.1.1.cmml" xref="A4.SS2.p1.4.m4.1.1">italic-ϵ</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS2.p1.4.m4.1c">\epsilon</annotation></semantics></math><span id="A4.SS2.p1.5.11" class="ltx_text" style="font-size:90%;"> parameterization and </span><math id="A4.SS2.p1.5.m5.1" class="ltx_Math" alttext="v" display="inline"><semantics id="A4.SS2.p1.5.m5.1a"><mi mathsize="90%" id="A4.SS2.p1.5.m5.1.1" xref="A4.SS2.p1.5.m5.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A4.SS2.p1.5.m5.1b"><ci id="A4.SS2.p1.5.m5.1.1.cmml" xref="A4.SS2.p1.5.m5.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS2.p1.5.m5.1c">v</annotation></semantics></math><span id="A4.SS2.p1.5.12" class="ltx_text" style="font-size:90%;"> parameterization are essential in generating pixel data that favors high-frequency details.</span></p>
</div>
<div id="A4.SS2.p2" class="ltx_para">
<p id="A4.SS2.p2.1" class="ltx_p"><span id="A4.SS2.p2.1.1" class="ltx_text" style="font-size:90%;">In our experiments, we use </span><math id="A4.SS2.p2.1.m1.1" class="ltx_Math" alttext="v" display="inline"><semantics id="A4.SS2.p2.1.m1.1a"><mi mathsize="90%" id="A4.SS2.p2.1.m1.1.1" xref="A4.SS2.p2.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A4.SS2.p2.1.m1.1b"><ci id="A4.SS2.p2.1.m1.1.1.cmml" xref="A4.SS2.p2.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS2.p2.1.m1.1c">v</annotation></semantics></math><span id="A4.SS2.p2.1.2" class="ltx_text" style="font-size:90%;"> parameterization for video prediction and found it essential to both convergence speed and quality.</span></p>
</div>
<div id="A4.SS2.p3" class="ltx_para">
<p id="A4.SS2.p3.1" class="ltx_p"><span id="A4.SS2.p3.1.1" class="ltx_text" style="font-size:90%;">We observe that </span><math id="A4.SS2.p3.1.m1.1" class="ltx_Math" alttext="\mathbf{x}_{0}" display="inline"><semantics id="A4.SS2.p3.1.m1.1a"><msub id="A4.SS2.p3.1.m1.1.1" xref="A4.SS2.p3.1.m1.1.1.cmml"><mi mathsize="90%" id="A4.SS2.p3.1.m1.1.1.2" xref="A4.SS2.p3.1.m1.1.1.2.cmml">𝐱</mi><mn mathsize="90%" id="A4.SS2.p3.1.m1.1.1.3" xref="A4.SS2.p3.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A4.SS2.p3.1.m1.1b"><apply id="A4.SS2.p3.1.m1.1.1.cmml" xref="A4.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS2.p3.1.m1.1.1.1.cmml" xref="A4.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="A4.SS2.p3.1.m1.1.1.2.cmml" xref="A4.SS2.p3.1.m1.1.1.2">𝐱</ci><cn type="integer" id="A4.SS2.p3.1.m1.1.1.3.cmml" xref="A4.SS2.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS2.p3.1.m1.1c">\mathbf{x}_{0}</annotation></semantics></math><span id="A4.SS2.p3.1.2" class="ltx_text" style="font-size:90%;"> parameterization is strongly favorable in planning and imitation learning, likely because they don’t favor an artificial emphasis on high-frequency details. We observe the benefits of v-parameterization in time-series prediction.</span></p>
</div>
</section>
<section id="A4.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">D.3 </span>SNR reweighting</h3>

<div id="A4.SS3.p1" class="ltx_para">
<p id="A4.SS3.p1.9" class="ltx_p"><span id="A4.SS3.p1.9.1" class="ltx_text" style="font-size:90%;">SNR reweighting </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS3.p1.9.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib24" title="" class="ltx_ref">24</a><span id="A4.SS3.p1.9.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS3.p1.9.4" class="ltx_text" style="font-size:90%;"> is a widely used technique to accelerate the convergence of image diffusion models. In short, it reweighs the diffusion loss proportional to the signal-to-noise ratio (SNR) of noisy </span><math id="A4.SS3.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{x}^{k}" display="inline"><semantics id="A4.SS3.p1.1.m1.1a"><msup id="A4.SS3.p1.1.m1.1.1" xref="A4.SS3.p1.1.m1.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p1.1.m1.1.1.2" xref="A4.SS3.p1.1.m1.1.1.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS3.p1.1.m1.1.1.3" xref="A4.SS3.p1.1.m1.1.1.3.cmml">k</mi></msup><annotation-xml encoding="MathML-Content" id="A4.SS3.p1.1.m1.1b"><apply id="A4.SS3.p1.1.m1.1.1.cmml" xref="A4.SS3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.1.m1.1.1.1.cmml" xref="A4.SS3.p1.1.m1.1.1">superscript</csymbol><ci id="A4.SS3.p1.1.m1.1.1.2.cmml" xref="A4.SS3.p1.1.m1.1.1.2">𝐱</ci><ci id="A4.SS3.p1.1.m1.1.1.3.cmml" xref="A4.SS3.p1.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p1.1.m1.1c">\mathbf{x}^{k}</annotation></semantics></math><span id="A4.SS3.p1.9.5" class="ltx_text" style="font-size:90%;">. In Diffusion Forcing, conditioning variable </span><math id="A4.SS3.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}" display="inline"><semantics id="A4.SS3.p1.2.m2.1a"><msub id="A4.SS3.p1.2.m2.1.1" xref="A4.SS3.p1.2.m2.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p1.2.m2.1.1.2" xref="A4.SS3.p1.2.m2.1.1.2.cmml">𝐳</mi><mrow id="A4.SS3.p1.2.m2.1.1.3" xref="A4.SS3.p1.2.m2.1.1.3.cmml"><mi mathsize="90%" id="A4.SS3.p1.2.m2.1.1.3.2" xref="A4.SS3.p1.2.m2.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS3.p1.2.m2.1.1.3.1" xref="A4.SS3.p1.2.m2.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS3.p1.2.m2.1.1.3.3" xref="A4.SS3.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.SS3.p1.2.m2.1b"><apply id="A4.SS3.p1.2.m2.1.1.cmml" xref="A4.SS3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.2.m2.1.1.1.cmml" xref="A4.SS3.p1.2.m2.1.1">subscript</csymbol><ci id="A4.SS3.p1.2.m2.1.1.2.cmml" xref="A4.SS3.p1.2.m2.1.1.2">𝐳</ci><apply id="A4.SS3.p1.2.m2.1.1.3.cmml" xref="A4.SS3.p1.2.m2.1.1.3"><minus id="A4.SS3.p1.2.m2.1.1.3.1.cmml" xref="A4.SS3.p1.2.m2.1.1.3.1"></minus><ci id="A4.SS3.p1.2.m2.1.1.3.2.cmml" xref="A4.SS3.p1.2.m2.1.1.3.2">𝑡</ci><cn type="integer" id="A4.SS3.p1.2.m2.1.1.3.3.cmml" xref="A4.SS3.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p1.2.m2.1c">\mathbf{z}_{t-1}</annotation></semantics></math><span id="A4.SS3.p1.9.6" class="ltx_text" style="font-size:90%;"> can also contain a non-trivial amount of information about </span><math id="A4.SS3.p1.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{t}" display="inline"><semantics id="A4.SS3.p1.3.m3.1a"><msub id="A4.SS3.p1.3.m3.1.1" xref="A4.SS3.p1.3.m3.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p1.3.m3.1.1.2" xref="A4.SS3.p1.3.m3.1.1.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS3.p1.3.m3.1.1.3" xref="A4.SS3.p1.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS3.p1.3.m3.1b"><apply id="A4.SS3.p1.3.m3.1.1.cmml" xref="A4.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.3.m3.1.1.1.cmml" xref="A4.SS3.p1.3.m3.1.1">subscript</csymbol><ci id="A4.SS3.p1.3.m3.1.1.2.cmml" xref="A4.SS3.p1.3.m3.1.1.2">𝐱</ci><ci id="A4.SS3.p1.3.m3.1.1.3.cmml" xref="A4.SS3.p1.3.m3.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p1.3.m3.1c">\mathbf{x}_{t}</annotation></semantics></math><span id="A4.SS3.p1.9.7" class="ltx_text" style="font-size:90%;">, in addition to </span><math id="A4.SS3.p1.4.m4.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="A4.SS3.p1.4.m4.1a"><msubsup id="A4.SS3.p1.4.m4.1.1" xref="A4.SS3.p1.4.m4.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p1.4.m4.1.1.2.2" xref="A4.SS3.p1.4.m4.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS3.p1.4.m4.1.1.2.3" xref="A4.SS3.p1.4.m4.1.1.2.3.cmml">t</mi><msub id="A4.SS3.p1.4.m4.1.1.3" xref="A4.SS3.p1.4.m4.1.1.3.cmml"><mi mathsize="90%" id="A4.SS3.p1.4.m4.1.1.3.2" xref="A4.SS3.p1.4.m4.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A4.SS3.p1.4.m4.1.1.3.3" xref="A4.SS3.p1.4.m4.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS3.p1.4.m4.1b"><apply id="A4.SS3.p1.4.m4.1.1.cmml" xref="A4.SS3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.4.m4.1.1.1.cmml" xref="A4.SS3.p1.4.m4.1.1">superscript</csymbol><apply id="A4.SS3.p1.4.m4.1.1.2.cmml" xref="A4.SS3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.4.m4.1.1.2.1.cmml" xref="A4.SS3.p1.4.m4.1.1">subscript</csymbol><ci id="A4.SS3.p1.4.m4.1.1.2.2.cmml" xref="A4.SS3.p1.4.m4.1.1.2.2">𝐱</ci><ci id="A4.SS3.p1.4.m4.1.1.2.3.cmml" xref="A4.SS3.p1.4.m4.1.1.2.3">𝑡</ci></apply><apply id="A4.SS3.p1.4.m4.1.1.3.cmml" xref="A4.SS3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="A4.SS3.p1.4.m4.1.1.3.1.cmml" xref="A4.SS3.p1.4.m4.1.1.3">subscript</csymbol><ci id="A4.SS3.p1.4.m4.1.1.3.2.cmml" xref="A4.SS3.p1.4.m4.1.1.3.2">𝑘</ci><ci id="A4.SS3.p1.4.m4.1.1.3.3.cmml" xref="A4.SS3.p1.4.m4.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p1.4.m4.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math><span id="A4.SS3.p1.9.8" class="ltx_text" style="font-size:90%;">. For example, in a deterministic markovian system, if </span><math id="A4.SS3.p1.5.m5.1" class="ltx_Math" alttext="\mathbf{x}_{t-1}^{k_{t-1}}" display="inline"><semantics id="A4.SS3.p1.5.m5.1a"><msubsup id="A4.SS3.p1.5.m5.1.1" xref="A4.SS3.p1.5.m5.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p1.5.m5.1.1.2.2" xref="A4.SS3.p1.5.m5.1.1.2.2.cmml">𝐱</mi><mrow id="A4.SS3.p1.5.m5.1.1.2.3" xref="A4.SS3.p1.5.m5.1.1.2.3.cmml"><mi mathsize="90%" id="A4.SS3.p1.5.m5.1.1.2.3.2" xref="A4.SS3.p1.5.m5.1.1.2.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS3.p1.5.m5.1.1.2.3.1" xref="A4.SS3.p1.5.m5.1.1.2.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS3.p1.5.m5.1.1.2.3.3" xref="A4.SS3.p1.5.m5.1.1.2.3.3.cmml">1</mn></mrow><msub id="A4.SS3.p1.5.m5.1.1.3" xref="A4.SS3.p1.5.m5.1.1.3.cmml"><mi mathsize="90%" id="A4.SS3.p1.5.m5.1.1.3.2" xref="A4.SS3.p1.5.m5.1.1.3.2.cmml">k</mi><mrow id="A4.SS3.p1.5.m5.1.1.3.3" xref="A4.SS3.p1.5.m5.1.1.3.3.cmml"><mi mathsize="90%" id="A4.SS3.p1.5.m5.1.1.3.3.2" xref="A4.SS3.p1.5.m5.1.1.3.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS3.p1.5.m5.1.1.3.3.1" xref="A4.SS3.p1.5.m5.1.1.3.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS3.p1.5.m5.1.1.3.3.3" xref="A4.SS3.p1.5.m5.1.1.3.3.3.cmml">1</mn></mrow></msub></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS3.p1.5.m5.1b"><apply id="A4.SS3.p1.5.m5.1.1.cmml" xref="A4.SS3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.5.m5.1.1.1.cmml" xref="A4.SS3.p1.5.m5.1.1">superscript</csymbol><apply id="A4.SS3.p1.5.m5.1.1.2.cmml" xref="A4.SS3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.5.m5.1.1.2.1.cmml" xref="A4.SS3.p1.5.m5.1.1">subscript</csymbol><ci id="A4.SS3.p1.5.m5.1.1.2.2.cmml" xref="A4.SS3.p1.5.m5.1.1.2.2">𝐱</ci><apply id="A4.SS3.p1.5.m5.1.1.2.3.cmml" xref="A4.SS3.p1.5.m5.1.1.2.3"><minus id="A4.SS3.p1.5.m5.1.1.2.3.1.cmml" xref="A4.SS3.p1.5.m5.1.1.2.3.1"></minus><ci id="A4.SS3.p1.5.m5.1.1.2.3.2.cmml" xref="A4.SS3.p1.5.m5.1.1.2.3.2">𝑡</ci><cn type="integer" id="A4.SS3.p1.5.m5.1.1.2.3.3.cmml" xref="A4.SS3.p1.5.m5.1.1.2.3.3">1</cn></apply></apply><apply id="A4.SS3.p1.5.m5.1.1.3.cmml" xref="A4.SS3.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="A4.SS3.p1.5.m5.1.1.3.1.cmml" xref="A4.SS3.p1.5.m5.1.1.3">subscript</csymbol><ci id="A4.SS3.p1.5.m5.1.1.3.2.cmml" xref="A4.SS3.p1.5.m5.1.1.3.2">𝑘</ci><apply id="A4.SS3.p1.5.m5.1.1.3.3.cmml" xref="A4.SS3.p1.5.m5.1.1.3.3"><minus id="A4.SS3.p1.5.m5.1.1.3.3.1.cmml" xref="A4.SS3.p1.5.m5.1.1.3.3.1"></minus><ci id="A4.SS3.p1.5.m5.1.1.3.3.2.cmml" xref="A4.SS3.p1.5.m5.1.1.3.3.2">𝑡</ci><cn type="integer" id="A4.SS3.p1.5.m5.1.1.3.3.3.cmml" xref="A4.SS3.p1.5.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p1.5.m5.1c">\mathbf{x}_{t-1}^{k_{t-1}}</annotation></semantics></math><span id="A4.SS3.p1.9.9" class="ltx_text" style="font-size:90%;"> has its noise level </span><math id="A4.SS3.p1.6.m6.1" class="ltx_Math" alttext="k_{t-1}=0" display="inline"><semantics id="A4.SS3.p1.6.m6.1a"><mrow id="A4.SS3.p1.6.m6.1.1" xref="A4.SS3.p1.6.m6.1.1.cmml"><msub id="A4.SS3.p1.6.m6.1.1.2" xref="A4.SS3.p1.6.m6.1.1.2.cmml"><mi mathsize="90%" id="A4.SS3.p1.6.m6.1.1.2.2" xref="A4.SS3.p1.6.m6.1.1.2.2.cmml">k</mi><mrow id="A4.SS3.p1.6.m6.1.1.2.3" xref="A4.SS3.p1.6.m6.1.1.2.3.cmml"><mi mathsize="90%" id="A4.SS3.p1.6.m6.1.1.2.3.2" xref="A4.SS3.p1.6.m6.1.1.2.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS3.p1.6.m6.1.1.2.3.1" xref="A4.SS3.p1.6.m6.1.1.2.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS3.p1.6.m6.1.1.2.3.3" xref="A4.SS3.p1.6.m6.1.1.2.3.3.cmml">1</mn></mrow></msub><mo mathsize="90%" id="A4.SS3.p1.6.m6.1.1.1" xref="A4.SS3.p1.6.m6.1.1.1.cmml">=</mo><mn mathsize="90%" id="A4.SS3.p1.6.m6.1.1.3" xref="A4.SS3.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS3.p1.6.m6.1b"><apply id="A4.SS3.p1.6.m6.1.1.cmml" xref="A4.SS3.p1.6.m6.1.1"><eq id="A4.SS3.p1.6.m6.1.1.1.cmml" xref="A4.SS3.p1.6.m6.1.1.1"></eq><apply 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id="A4.SS3.p1.7.m7.1a"><msub id="A4.SS3.p1.7.m7.1.1" xref="A4.SS3.p1.7.m7.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p1.7.m7.1.1.2" xref="A4.SS3.p1.7.m7.1.1.2.cmml">𝐳</mi><mrow id="A4.SS3.p1.7.m7.1.1.3" xref="A4.SS3.p1.7.m7.1.1.3.cmml"><mi mathsize="90%" id="A4.SS3.p1.7.m7.1.1.3.2" xref="A4.SS3.p1.7.m7.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS3.p1.7.m7.1.1.3.1" xref="A4.SS3.p1.7.m7.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS3.p1.7.m7.1.1.3.3" xref="A4.SS3.p1.7.m7.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.SS3.p1.7.m7.1b"><apply id="A4.SS3.p1.7.m7.1.1.cmml" xref="A4.SS3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.7.m7.1.1.1.cmml" xref="A4.SS3.p1.7.m7.1.1">subscript</csymbol><ci id="A4.SS3.p1.7.m7.1.1.2.cmml" xref="A4.SS3.p1.7.m7.1.1.2">𝐳</ci><apply id="A4.SS3.p1.7.m7.1.1.3.cmml" xref="A4.SS3.p1.7.m7.1.1.3"><minus id="A4.SS3.p1.7.m7.1.1.3.1.cmml" xref="A4.SS3.p1.7.m7.1.1.3.1"></minus><ci id="A4.SS3.p1.7.m7.1.1.3.2.cmml" xref="A4.SS3.p1.7.m7.1.1.3.2">𝑡</ci><cn type="integer" id="A4.SS3.p1.7.m7.1.1.3.3.cmml" xref="A4.SS3.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p1.7.m7.1c">\mathbf{z}_{t-1}</annotation></semantics></math><span id="A4.SS3.p1.9.11" class="ltx_text" style="font-size:90%;"> contains all the information needed to predict </span><math id="A4.SS3.p1.8.m8.1" class="ltx_Math" alttext="\mathbf{x}^{0}_{t}" display="inline"><semantics id="A4.SS3.p1.8.m8.1a"><msubsup id="A4.SS3.p1.8.m8.1.1" xref="A4.SS3.p1.8.m8.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p1.8.m8.1.1.2.2" xref="A4.SS3.p1.8.m8.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS3.p1.8.m8.1.1.3" xref="A4.SS3.p1.8.m8.1.1.3.cmml">t</mi><mn mathsize="90%" id="A4.SS3.p1.8.m8.1.1.2.3" xref="A4.SS3.p1.8.m8.1.1.2.3.cmml">0</mn></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS3.p1.8.m8.1b"><apply id="A4.SS3.p1.8.m8.1.1.cmml" xref="A4.SS3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.8.m8.1.1.1.cmml" xref="A4.SS3.p1.8.m8.1.1">subscript</csymbol><apply id="A4.SS3.p1.8.m8.1.1.2.cmml" xref="A4.SS3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.8.m8.1.1.2.1.cmml" xref="A4.SS3.p1.8.m8.1.1">superscript</csymbol><ci id="A4.SS3.p1.8.m8.1.1.2.2.cmml" xref="A4.SS3.p1.8.m8.1.1.2.2">𝐱</ci><cn type="integer" id="A4.SS3.p1.8.m8.1.1.2.3.cmml" xref="A4.SS3.p1.8.m8.1.1.2.3">0</cn></apply><ci id="A4.SS3.p1.8.m8.1.1.3.cmml" xref="A4.SS3.p1.8.m8.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p1.8.m8.1c">\mathbf{x}^{0}_{t}</annotation></semantics></math><span id="A4.SS3.p1.9.12" class="ltx_text" style="font-size:90%;"> regardless of the noise level of </span><math id="A4.SS3.p1.9.m9.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="A4.SS3.p1.9.m9.1a"><msubsup id="A4.SS3.p1.9.m9.1.1" xref="A4.SS3.p1.9.m9.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p1.9.m9.1.1.2.2" xref="A4.SS3.p1.9.m9.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS3.p1.9.m9.1.1.2.3" xref="A4.SS3.p1.9.m9.1.1.2.3.cmml">t</mi><msub id="A4.SS3.p1.9.m9.1.1.3" xref="A4.SS3.p1.9.m9.1.1.3.cmml"><mi mathsize="90%" id="A4.SS3.p1.9.m9.1.1.3.2" xref="A4.SS3.p1.9.m9.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A4.SS3.p1.9.m9.1.1.3.3" xref="A4.SS3.p1.9.m9.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS3.p1.9.m9.1b"><apply id="A4.SS3.p1.9.m9.1.1.cmml" xref="A4.SS3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.9.m9.1.1.1.cmml" xref="A4.SS3.p1.9.m9.1.1">superscript</csymbol><apply id="A4.SS3.p1.9.m9.1.1.2.cmml" xref="A4.SS3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="A4.SS3.p1.9.m9.1.1.2.1.cmml" xref="A4.SS3.p1.9.m9.1.1">subscript</csymbol><ci id="A4.SS3.p1.9.m9.1.1.2.2.cmml" xref="A4.SS3.p1.9.m9.1.1.2.2">𝐱</ci><ci id="A4.SS3.p1.9.m9.1.1.2.3.cmml" xref="A4.SS3.p1.9.m9.1.1.2.3">𝑡</ci></apply><apply id="A4.SS3.p1.9.m9.1.1.3.cmml" xref="A4.SS3.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="A4.SS3.p1.9.m9.1.1.3.1.cmml" xref="A4.SS3.p1.9.m9.1.1.3">subscript</csymbol><ci id="A4.SS3.p1.9.m9.1.1.3.2.cmml" xref="A4.SS3.p1.9.m9.1.1.3.2">𝑘</ci><ci id="A4.SS3.p1.9.m9.1.1.3.3.cmml" xref="A4.SS3.p1.9.m9.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p1.9.m9.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math><span id="A4.SS3.p1.9.13" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
<div id="A4.SS3.p2" class="ltx_para">
<p id="A4.SS3.p2.22" class="ltx_p"><span id="A4.SS3.p2.22.1" class="ltx_text" style="font-size:90%;">Therefore we re-derive SNR reweighting to reflect this change in Diffusion Forcing. We follow the intuition of original SNR reweighting to loosely define SNR in a sequence with independent levels of noises at different time steps. Denote </span><math id="A4.SS3.p2.1.m1.1" class="ltx_Math" alttext="S_{t}" display="inline"><semantics id="A4.SS3.p2.1.m1.1a"><msub id="A4.SS3.p2.1.m1.1.1" xref="A4.SS3.p2.1.m1.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p2.1.m1.1.1.2" xref="A4.SS3.p2.1.m1.1.1.2.cmml">S</mi><mi mathsize="90%" id="A4.SS3.p2.1.m1.1.1.3" xref="A4.SS3.p2.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.1.m1.1b"><apply id="A4.SS3.p2.1.m1.1.1.cmml" xref="A4.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.1.m1.1.1.1.cmml" xref="A4.SS3.p2.1.m1.1.1">subscript</csymbol><ci id="A4.SS3.p2.1.m1.1.1.2.cmml" xref="A4.SS3.p2.1.m1.1.1.2">𝑆</ci><ci id="A4.SS3.p2.1.m1.1.1.3.cmml" xref="A4.SS3.p2.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.1.m1.1c">S_{t}</annotation></semantics></math><span id="A4.SS3.p2.22.2" class="ltx_text" style="font-size:90%;"> as the normalized SNR reweighting factor for </span><math id="A4.SS3.p2.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="A4.SS3.p2.2.m2.1a"><msubsup id="A4.SS3.p2.2.m2.1.1" xref="A4.SS3.p2.2.m2.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p2.2.m2.1.1.2.2" xref="A4.SS3.p2.2.m2.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS3.p2.2.m2.1.1.2.3" xref="A4.SS3.p2.2.m2.1.1.2.3.cmml">t</mi><msub id="A4.SS3.p2.2.m2.1.1.3" xref="A4.SS3.p2.2.m2.1.1.3.cmml"><mi mathsize="90%" id="A4.SS3.p2.2.m2.1.1.3.2" xref="A4.SS3.p2.2.m2.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A4.SS3.p2.2.m2.1.1.3.3" xref="A4.SS3.p2.2.m2.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.2.m2.1b"><apply id="A4.SS3.p2.2.m2.1.1.cmml" xref="A4.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.2.m2.1.1.1.cmml" xref="A4.SS3.p2.2.m2.1.1">superscript</csymbol><apply id="A4.SS3.p2.2.m2.1.1.2.cmml" xref="A4.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.2.m2.1.1.2.1.cmml" xref="A4.SS3.p2.2.m2.1.1">subscript</csymbol><ci id="A4.SS3.p2.2.m2.1.1.2.2.cmml" xref="A4.SS3.p2.2.m2.1.1.2.2">𝐱</ci><ci id="A4.SS3.p2.2.m2.1.1.2.3.cmml" xref="A4.SS3.p2.2.m2.1.1.2.3">𝑡</ci></apply><apply id="A4.SS3.p2.2.m2.1.1.3.cmml" xref="A4.SS3.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="A4.SS3.p2.2.m2.1.1.3.1.cmml" xref="A4.SS3.p2.2.m2.1.1.3">subscript</csymbol><ci id="A4.SS3.p2.2.m2.1.1.3.2.cmml" xref="A4.SS3.p2.2.m2.1.1.3.2">𝑘</ci><ci id="A4.SS3.p2.2.m2.1.1.3.3.cmml" xref="A4.SS3.p2.2.m2.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.2.m2.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math><span id="A4.SS3.p2.22.3" class="ltx_text" style="font-size:90%;"> following its normal derivation in diffusion models. For example, if one uses min snr strategy  </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS3.p2.22.4.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib24" title="" class="ltx_ref">24</a><span id="A4.SS3.p2.22.5.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS3.p2.22.6" class="ltx_text" style="font-size:90%;">, its reweighting factor will always fall between </span><math id="A4.SS3.p2.3.m3.2" class="ltx_Math" alttext="[0,C]" display="inline"><semantics id="A4.SS3.p2.3.m3.2a"><mrow id="A4.SS3.p2.3.m3.2.3.2" xref="A4.SS3.p2.3.m3.2.3.1.cmml"><mo maxsize="90%" minsize="90%" id="A4.SS3.p2.3.m3.2.3.2.1" xref="A4.SS3.p2.3.m3.2.3.1.cmml">[</mo><mn mathsize="90%" id="A4.SS3.p2.3.m3.1.1" xref="A4.SS3.p2.3.m3.1.1.cmml">0</mn><mo mathsize="90%" id="A4.SS3.p2.3.m3.2.3.2.2" xref="A4.SS3.p2.3.m3.2.3.1.cmml">,</mo><mi mathsize="90%" id="A4.SS3.p2.3.m3.2.2" xref="A4.SS3.p2.3.m3.2.2.cmml">C</mi><mo maxsize="90%" minsize="90%" id="A4.SS3.p2.3.m3.2.3.2.3" xref="A4.SS3.p2.3.m3.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.3.m3.2b"><interval closure="closed" id="A4.SS3.p2.3.m3.2.3.1.cmml" xref="A4.SS3.p2.3.m3.2.3.2"><cn type="integer" id="A4.SS3.p2.3.m3.1.1.cmml" xref="A4.SS3.p2.3.m3.1.1">0</cn><ci id="A4.SS3.p2.3.m3.2.2.cmml" xref="A4.SS3.p2.3.m3.2.2">𝐶</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.3.m3.2c">[0,C]</annotation></semantics></math><span id="A4.SS3.p2.22.7" class="ltx_text" style="font-size:90%;"> which we divide by </span><math id="A4.SS3.p2.4.m4.1" class="ltx_Math" alttext="C" display="inline"><semantics id="A4.SS3.p2.4.m4.1a"><mi mathsize="90%" id="A4.SS3.p2.4.m4.1.1" xref="A4.SS3.p2.4.m4.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.4.m4.1b"><ci id="A4.SS3.p2.4.m4.1.1.cmml" xref="A4.SS3.p2.4.m4.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.4.m4.1c">C</annotation></semantics></math><span id="A4.SS3.p2.22.8" class="ltx_text" style="font-size:90%;"> to get </span><math id="A4.SS3.p2.5.m5.2" class="ltx_Math" alttext="S_{t}\in[0,1]" display="inline"><semantics id="A4.SS3.p2.5.m5.2a"><mrow id="A4.SS3.p2.5.m5.2.3" xref="A4.SS3.p2.5.m5.2.3.cmml"><msub id="A4.SS3.p2.5.m5.2.3.2" xref="A4.SS3.p2.5.m5.2.3.2.cmml"><mi mathsize="90%" id="A4.SS3.p2.5.m5.2.3.2.2" xref="A4.SS3.p2.5.m5.2.3.2.2.cmml">S</mi><mi mathsize="90%" id="A4.SS3.p2.5.m5.2.3.2.3" xref="A4.SS3.p2.5.m5.2.3.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A4.SS3.p2.5.m5.2.3.1" xref="A4.SS3.p2.5.m5.2.3.1.cmml">∈</mo><mrow id="A4.SS3.p2.5.m5.2.3.3.2" xref="A4.SS3.p2.5.m5.2.3.3.1.cmml"><mo maxsize="90%" minsize="90%" id="A4.SS3.p2.5.m5.2.3.3.2.1" xref="A4.SS3.p2.5.m5.2.3.3.1.cmml">[</mo><mn mathsize="90%" id="A4.SS3.p2.5.m5.1.1" xref="A4.SS3.p2.5.m5.1.1.cmml">0</mn><mo mathsize="90%" id="A4.SS3.p2.5.m5.2.3.3.2.2" xref="A4.SS3.p2.5.m5.2.3.3.1.cmml">,</mo><mn mathsize="90%" id="A4.SS3.p2.5.m5.2.2" xref="A4.SS3.p2.5.m5.2.2.cmml">1</mn><mo maxsize="90%" minsize="90%" id="A4.SS3.p2.5.m5.2.3.3.2.3" xref="A4.SS3.p2.5.m5.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.5.m5.2b"><apply id="A4.SS3.p2.5.m5.2.3.cmml" xref="A4.SS3.p2.5.m5.2.3"><in id="A4.SS3.p2.5.m5.2.3.1.cmml" xref="A4.SS3.p2.5.m5.2.3.1"></in><apply id="A4.SS3.p2.5.m5.2.3.2.cmml" xref="A4.SS3.p2.5.m5.2.3.2"><csymbol cd="ambiguous" id="A4.SS3.p2.5.m5.2.3.2.1.cmml" xref="A4.SS3.p2.5.m5.2.3.2">subscript</csymbol><ci id="A4.SS3.p2.5.m5.2.3.2.2.cmml" xref="A4.SS3.p2.5.m5.2.3.2.2">𝑆</ci><ci id="A4.SS3.p2.5.m5.2.3.2.3.cmml" xref="A4.SS3.p2.5.m5.2.3.2.3">𝑡</ci></apply><interval closure="closed" id="A4.SS3.p2.5.m5.2.3.3.1.cmml" xref="A4.SS3.p2.5.m5.2.3.3.2"><cn type="integer" id="A4.SS3.p2.5.m5.1.1.cmml" xref="A4.SS3.p2.5.m5.1.1">0</cn><cn type="integer" id="A4.SS3.p2.5.m5.2.2.cmml" xref="A4.SS3.p2.5.m5.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.5.m5.2c">S_{t}\in[0,1]</annotation></semantics></math><span id="A4.SS3.p2.22.9" class="ltx_text" style="font-size:90%;">. Define signal decay factor </span><math id="A4.SS3.p2.6.m6.1" class="ltx_Math" alttext="0&lt;\gamma&lt;1" display="inline"><semantics id="A4.SS3.p2.6.m6.1a"><mrow id="A4.SS3.p2.6.m6.1.1" xref="A4.SS3.p2.6.m6.1.1.cmml"><mn mathsize="90%" id="A4.SS3.p2.6.m6.1.1.2" xref="A4.SS3.p2.6.m6.1.1.2.cmml">0</mn><mo mathsize="90%" id="A4.SS3.p2.6.m6.1.1.3" xref="A4.SS3.p2.6.m6.1.1.3.cmml">&lt;</mo><mi mathsize="90%" id="A4.SS3.p2.6.m6.1.1.4" xref="A4.SS3.p2.6.m6.1.1.4.cmml">γ</mi><mo mathsize="90%" id="A4.SS3.p2.6.m6.1.1.5" xref="A4.SS3.p2.6.m6.1.1.5.cmml">&lt;</mo><mn mathsize="90%" id="A4.SS3.p2.6.m6.1.1.6" xref="A4.SS3.p2.6.m6.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.6.m6.1b"><apply id="A4.SS3.p2.6.m6.1.1.cmml" xref="A4.SS3.p2.6.m6.1.1"><and id="A4.SS3.p2.6.m6.1.1a.cmml" xref="A4.SS3.p2.6.m6.1.1"></and><apply id="A4.SS3.p2.6.m6.1.1b.cmml" xref="A4.SS3.p2.6.m6.1.1"><lt id="A4.SS3.p2.6.m6.1.1.3.cmml" xref="A4.SS3.p2.6.m6.1.1.3"></lt><cn type="integer" id="A4.SS3.p2.6.m6.1.1.2.cmml" xref="A4.SS3.p2.6.m6.1.1.2">0</cn><ci id="A4.SS3.p2.6.m6.1.1.4.cmml" xref="A4.SS3.p2.6.m6.1.1.4">𝛾</ci></apply><apply id="A4.SS3.p2.6.m6.1.1c.cmml" xref="A4.SS3.p2.6.m6.1.1"><lt id="A4.SS3.p2.6.m6.1.1.5.cmml" xref="A4.SS3.p2.6.m6.1.1.5"></lt><share href="#A4.SS3.p2.6.m6.1.1.4.cmml" id="A4.SS3.p2.6.m6.1.1d.cmml" xref="A4.SS3.p2.6.m6.1.1"></share><cn type="integer" id="A4.SS3.p2.6.m6.1.1.6.cmml" xref="A4.SS3.p2.6.m6.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.6.m6.1c">0&lt;\gamma&lt;1</annotation></semantics></math><span id="A4.SS3.p2.22.10" class="ltx_text" style="font-size:90%;">, measuring what proportion of signal in </span><math id="A4.SS3.p2.7.m7.1" class="ltx_Math" alttext="\mathbf{x}_{t-1}^{k_{t-1}}" display="inline"><semantics id="A4.SS3.p2.7.m7.1a"><msubsup id="A4.SS3.p2.7.m7.1.1" xref="A4.SS3.p2.7.m7.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p2.7.m7.1.1.2.2" xref="A4.SS3.p2.7.m7.1.1.2.2.cmml">𝐱</mi><mrow id="A4.SS3.p2.7.m7.1.1.2.3" xref="A4.SS3.p2.7.m7.1.1.2.3.cmml"><mi mathsize="90%" id="A4.SS3.p2.7.m7.1.1.2.3.2" xref="A4.SS3.p2.7.m7.1.1.2.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS3.p2.7.m7.1.1.2.3.1" xref="A4.SS3.p2.7.m7.1.1.2.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS3.p2.7.m7.1.1.2.3.3" xref="A4.SS3.p2.7.m7.1.1.2.3.3.cmml">1</mn></mrow><msub id="A4.SS3.p2.7.m7.1.1.3" xref="A4.SS3.p2.7.m7.1.1.3.cmml"><mi mathsize="90%" id="A4.SS3.p2.7.m7.1.1.3.2" xref="A4.SS3.p2.7.m7.1.1.3.2.cmml">k</mi><mrow id="A4.SS3.p2.7.m7.1.1.3.3" xref="A4.SS3.p2.7.m7.1.1.3.3.cmml"><mi mathsize="90%" id="A4.SS3.p2.7.m7.1.1.3.3.2" xref="A4.SS3.p2.7.m7.1.1.3.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS3.p2.7.m7.1.1.3.3.1" xref="A4.SS3.p2.7.m7.1.1.3.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS3.p2.7.m7.1.1.3.3.3" xref="A4.SS3.p2.7.m7.1.1.3.3.3.cmml">1</mn></mrow></msub></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.7.m7.1b"><apply 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id="A4.SS3.p2.7.m7.1.1.3.3.cmml" xref="A4.SS3.p2.7.m7.1.1.3.3"><minus id="A4.SS3.p2.7.m7.1.1.3.3.1.cmml" xref="A4.SS3.p2.7.m7.1.1.3.3.1"></minus><ci id="A4.SS3.p2.7.m7.1.1.3.3.2.cmml" xref="A4.SS3.p2.7.m7.1.1.3.3.2">𝑡</ci><cn type="integer" id="A4.SS3.p2.7.m7.1.1.3.3.3.cmml" xref="A4.SS3.p2.7.m7.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.7.m7.1c">\mathbf{x}_{t-1}^{k_{t-1}}</annotation></semantics></math><span id="A4.SS3.p2.22.11" class="ltx_text" style="font-size:90%;"> contribute to denoising </span><math id="A4.SS3.p2.8.m8.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="A4.SS3.p2.8.m8.1a"><msubsup id="A4.SS3.p2.8.m8.1.1" xref="A4.SS3.p2.8.m8.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p2.8.m8.1.1.2.2" xref="A4.SS3.p2.8.m8.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS3.p2.8.m8.1.1.2.3" xref="A4.SS3.p2.8.m8.1.1.2.3.cmml">t</mi><msub id="A4.SS3.p2.8.m8.1.1.3" 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xref="A4.SS3.p2.8.m8.1.1.3.2">𝑘</ci><ci id="A4.SS3.p2.8.m8.1.1.3.3.cmml" xref="A4.SS3.p2.8.m8.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.8.m8.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math><span id="A4.SS3.p2.22.12" class="ltx_text" style="font-size:90%;">. This is the simple exponential decay model of sequential information. Now, define cumulated SNR recursively as the running mean of </span><math id="A4.SS3.p2.9.m9.1" class="ltx_Math" alttext="S_{t}" display="inline"><semantics id="A4.SS3.p2.9.m9.1a"><msub id="A4.SS3.p2.9.m9.1.1" xref="A4.SS3.p2.9.m9.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p2.9.m9.1.1.2" xref="A4.SS3.p2.9.m9.1.1.2.cmml">S</mi><mi mathsize="90%" id="A4.SS3.p2.9.m9.1.1.3" xref="A4.SS3.p2.9.m9.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.9.m9.1b"><apply id="A4.SS3.p2.9.m9.1.1.cmml" xref="A4.SS3.p2.9.m9.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.9.m9.1.1.1.cmml" xref="A4.SS3.p2.9.m9.1.1">subscript</csymbol><ci id="A4.SS3.p2.9.m9.1.1.2.cmml" xref="A4.SS3.p2.9.m9.1.1.2">𝑆</ci><ci id="A4.SS3.p2.9.m9.1.1.3.cmml" xref="A4.SS3.p2.9.m9.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.9.m9.1c">S_{t}</annotation></semantics></math><span id="A4.SS3.p2.22.13" class="ltx_text" style="font-size:90%;">: </span><math 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xref="A4.SS3.p2.10.m10.1.1.1.1.3">subscript</csymbol><ci id="A4.SS3.p2.10.m10.1.1.1.1.3.2.cmml" xref="A4.SS3.p2.10.m10.1.1.1.1.3.2">𝑆</ci><ci id="A4.SS3.p2.10.m10.1.1.1.1.3.3.cmml" xref="A4.SS3.p2.10.m10.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.10.m10.1c">\bar{S}_{t}=\gamma\bar{S}_{t-1}+(1-\gamma)S_{t}</annotation></semantics></math><span id="A4.SS3.p2.22.14" class="ltx_text" style="font-size:90%;"> to account for signals contributed by the entire noisy history to the denoising at time step </span><math id="A4.SS3.p2.11.m11.1" class="ltx_Math" alttext="t" display="inline"><semantics id="A4.SS3.p2.11.m11.1a"><mi mathsize="90%" id="A4.SS3.p2.11.m11.1.1" xref="A4.SS3.p2.11.m11.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.11.m11.1b"><ci id="A4.SS3.p2.11.m11.1.1.cmml" xref="A4.SS3.p2.11.m11.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.11.m11.1c">t</annotation></semantics></math><span id="A4.SS3.p2.22.15" class="ltx_text" style="font-size:90%;">. The other factor that contributes to the denoising is </span><math id="A4.SS3.p2.12.m12.1" class="ltx_Math" alttext="S_{t}" display="inline"><semantics id="A4.SS3.p2.12.m12.1a"><msub id="A4.SS3.p2.12.m12.1.1" xref="A4.SS3.p2.12.m12.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p2.12.m12.1.1.2" xref="A4.SS3.p2.12.m12.1.1.2.cmml">S</mi><mi mathsize="90%" id="A4.SS3.p2.12.m12.1.1.3" xref="A4.SS3.p2.12.m12.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.12.m12.1b"><apply id="A4.SS3.p2.12.m12.1.1.cmml" xref="A4.SS3.p2.12.m12.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.12.m12.1.1.1.cmml" xref="A4.SS3.p2.12.m12.1.1">subscript</csymbol><ci id="A4.SS3.p2.12.m12.1.1.2.cmml" xref="A4.SS3.p2.12.m12.1.1.2">𝑆</ci><ci id="A4.SS3.p2.12.m12.1.1.3.cmml" xref="A4.SS3.p2.12.m12.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.12.m12.1c">S_{t}</annotation></semantics></math><span id="A4.SS3.p2.22.16" class="ltx_text" style="font-size:90%;"> of noisy observation </span><math id="A4.SS3.p2.13.m13.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="A4.SS3.p2.13.m13.1a"><msubsup id="A4.SS3.p2.13.m13.1.1" xref="A4.SS3.p2.13.m13.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p2.13.m13.1.1.2.2" xref="A4.SS3.p2.13.m13.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS3.p2.13.m13.1.1.2.3" xref="A4.SS3.p2.13.m13.1.1.2.3.cmml">t</mi><msub id="A4.SS3.p2.13.m13.1.1.3" xref="A4.SS3.p2.13.m13.1.1.3.cmml"><mi mathsize="90%" id="A4.SS3.p2.13.m13.1.1.3.2" xref="A4.SS3.p2.13.m13.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A4.SS3.p2.13.m13.1.1.3.3" xref="A4.SS3.p2.13.m13.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.13.m13.1b"><apply id="A4.SS3.p2.13.m13.1.1.cmml" xref="A4.SS3.p2.13.m13.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.13.m13.1.1.1.cmml" xref="A4.SS3.p2.13.m13.1.1">superscript</csymbol><apply id="A4.SS3.p2.13.m13.1.1.2.cmml" xref="A4.SS3.p2.13.m13.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.13.m13.1.1.2.1.cmml" xref="A4.SS3.p2.13.m13.1.1">subscript</csymbol><ci id="A4.SS3.p2.13.m13.1.1.2.2.cmml" xref="A4.SS3.p2.13.m13.1.1.2.2">𝐱</ci><ci id="A4.SS3.p2.13.m13.1.1.2.3.cmml" xref="A4.SS3.p2.13.m13.1.1.2.3">𝑡</ci></apply><apply id="A4.SS3.p2.13.m13.1.1.3.cmml" xref="A4.SS3.p2.13.m13.1.1.3"><csymbol cd="ambiguous" id="A4.SS3.p2.13.m13.1.1.3.1.cmml" xref="A4.SS3.p2.13.m13.1.1.3">subscript</csymbol><ci id="A4.SS3.p2.13.m13.1.1.3.2.cmml" xref="A4.SS3.p2.13.m13.1.1.3.2">𝑘</ci><ci id="A4.SS3.p2.13.m13.1.1.3.3.cmml" xref="A4.SS3.p2.13.m13.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.13.m13.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math><span id="A4.SS3.p2.22.17" class="ltx_text" style="font-size:90%;">. To combine them, we use a simplified model for independent events. Notice </span><math id="A4.SS3.p2.14.m14.1" class="ltx_Math" alttext="S_{t}" display="inline"><semantics id="A4.SS3.p2.14.m14.1a"><msub id="A4.SS3.p2.14.m14.1.1" xref="A4.SS3.p2.14.m14.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p2.14.m14.1.1.2" xref="A4.SS3.p2.14.m14.1.1.2.cmml">S</mi><mi mathsize="90%" id="A4.SS3.p2.14.m14.1.1.3" xref="A4.SS3.p2.14.m14.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.14.m14.1b"><apply id="A4.SS3.p2.14.m14.1.1.cmml" xref="A4.SS3.p2.14.m14.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.14.m14.1.1.1.cmml" xref="A4.SS3.p2.14.m14.1.1">subscript</csymbol><ci id="A4.SS3.p2.14.m14.1.1.2.cmml" xref="A4.SS3.p2.14.m14.1.1.2">𝑆</ci><ci id="A4.SS3.p2.14.m14.1.1.3.cmml" xref="A4.SS3.p2.14.m14.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.14.m14.1c">S_{t}</annotation></semantics></math><span id="A4.SS3.p2.22.18" class="ltx_text" style="font-size:90%;"> and </span><math id="A4.SS3.p2.15.m15.1" class="ltx_Math" alttext="\bar{S}_{t}" display="inline"><semantics id="A4.SS3.p2.15.m15.1a"><msub id="A4.SS3.p2.15.m15.1.1" xref="A4.SS3.p2.15.m15.1.1.cmml"><mover accent="true" id="A4.SS3.p2.15.m15.1.1.2" xref="A4.SS3.p2.15.m15.1.1.2.cmml"><mi mathsize="90%" id="A4.SS3.p2.15.m15.1.1.2.2" xref="A4.SS3.p2.15.m15.1.1.2.2.cmml">S</mi><mo mathsize="90%" id="A4.SS3.p2.15.m15.1.1.2.1" xref="A4.SS3.p2.15.m15.1.1.2.1.cmml">¯</mo></mover><mi mathsize="90%" id="A4.SS3.p2.15.m15.1.1.3" xref="A4.SS3.p2.15.m15.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.15.m15.1b"><apply id="A4.SS3.p2.15.m15.1.1.cmml" xref="A4.SS3.p2.15.m15.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.15.m15.1.1.1.cmml" xref="A4.SS3.p2.15.m15.1.1">subscript</csymbol><apply id="A4.SS3.p2.15.m15.1.1.2.cmml" xref="A4.SS3.p2.15.m15.1.1.2"><ci id="A4.SS3.p2.15.m15.1.1.2.1.cmml" xref="A4.SS3.p2.15.m15.1.1.2.1">¯</ci><ci id="A4.SS3.p2.15.m15.1.1.2.2.cmml" xref="A4.SS3.p2.15.m15.1.1.2.2">𝑆</ci></apply><ci id="A4.SS3.p2.15.m15.1.1.3.cmml" xref="A4.SS3.p2.15.m15.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.15.m15.1c">\bar{S}_{t}</annotation></semantics></math><span id="A4.SS3.p2.22.19" class="ltx_text" style="font-size:90%;"> always falls in range </span><math id="A4.SS3.p2.16.m16.2" class="ltx_Math" alttext="[0,1]" display="inline"><semantics id="A4.SS3.p2.16.m16.2a"><mrow id="A4.SS3.p2.16.m16.2.3.2" xref="A4.SS3.p2.16.m16.2.3.1.cmml"><mo maxsize="90%" minsize="90%" id="A4.SS3.p2.16.m16.2.3.2.1" xref="A4.SS3.p2.16.m16.2.3.1.cmml">[</mo><mn mathsize="90%" id="A4.SS3.p2.16.m16.1.1" xref="A4.SS3.p2.16.m16.1.1.cmml">0</mn><mo mathsize="90%" id="A4.SS3.p2.16.m16.2.3.2.2" xref="A4.SS3.p2.16.m16.2.3.1.cmml">,</mo><mn mathsize="90%" id="A4.SS3.p2.16.m16.2.2" xref="A4.SS3.p2.16.m16.2.2.cmml">1</mn><mo maxsize="90%" minsize="90%" id="A4.SS3.p2.16.m16.2.3.2.3" xref="A4.SS3.p2.16.m16.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.16.m16.2b"><interval closure="closed" id="A4.SS3.p2.16.m16.2.3.1.cmml" xref="A4.SS3.p2.16.m16.2.3.2"><cn type="integer" id="A4.SS3.p2.16.m16.1.1.cmml" xref="A4.SS3.p2.16.m16.1.1">0</cn><cn type="integer" id="A4.SS3.p2.16.m16.2.2.cmml" xref="A4.SS3.p2.16.m16.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.16.m16.2c">[0,1]</annotation></semantics></math><span id="A4.SS3.p2.22.20" class="ltx_text" style="font-size:90%;">, and therefore can be reinterpreted as probabilities of having all the signal one needs to perfect denoise </span><math id="A4.SS3.p2.17.m17.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="A4.SS3.p2.17.m17.1a"><msubsup id="A4.SS3.p2.17.m17.1.1" xref="A4.SS3.p2.17.m17.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p2.17.m17.1.1.2.2" xref="A4.SS3.p2.17.m17.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS3.p2.17.m17.1.1.2.3" xref="A4.SS3.p2.17.m17.1.1.2.3.cmml">t</mi><msub id="A4.SS3.p2.17.m17.1.1.3" xref="A4.SS3.p2.17.m17.1.1.3.cmml"><mi mathsize="90%" id="A4.SS3.p2.17.m17.1.1.3.2" xref="A4.SS3.p2.17.m17.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A4.SS3.p2.17.m17.1.1.3.3" xref="A4.SS3.p2.17.m17.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.17.m17.1b"><apply id="A4.SS3.p2.17.m17.1.1.cmml" xref="A4.SS3.p2.17.m17.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.17.m17.1.1.1.cmml" xref="A4.SS3.p2.17.m17.1.1">superscript</csymbol><apply id="A4.SS3.p2.17.m17.1.1.2.cmml" xref="A4.SS3.p2.17.m17.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.17.m17.1.1.2.1.cmml" xref="A4.SS3.p2.17.m17.1.1">subscript</csymbol><ci id="A4.SS3.p2.17.m17.1.1.2.2.cmml" xref="A4.SS3.p2.17.m17.1.1.2.2">𝐱</ci><ci id="A4.SS3.p2.17.m17.1.1.2.3.cmml" xref="A4.SS3.p2.17.m17.1.1.2.3">𝑡</ci></apply><apply id="A4.SS3.p2.17.m17.1.1.3.cmml" xref="A4.SS3.p2.17.m17.1.1.3"><csymbol cd="ambiguous" id="A4.SS3.p2.17.m17.1.1.3.1.cmml" xref="A4.SS3.p2.17.m17.1.1.3">subscript</csymbol><ci id="A4.SS3.p2.17.m17.1.1.3.2.cmml" xref="A4.SS3.p2.17.m17.1.1.3.2">𝑘</ci><ci id="A4.SS3.p2.17.m17.1.1.3.3.cmml" xref="A4.SS3.p2.17.m17.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.17.m17.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math><span id="A4.SS3.p2.22.21" class="ltx_text" style="font-size:90%;">. Since the noise level at </span><math id="A4.SS3.p2.18.m18.1" class="ltx_Math" alttext="t" display="inline"><semantics id="A4.SS3.p2.18.m18.1a"><mi mathsize="90%" id="A4.SS3.p2.18.m18.1.1" xref="A4.SS3.p2.18.m18.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.18.m18.1b"><ci id="A4.SS3.p2.18.m18.1.1.cmml" xref="A4.SS3.p2.18.m18.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.18.m18.1c">t</annotation></semantics></math><span id="A4.SS3.p2.22.22" class="ltx_text" style="font-size:90%;"> is independent of prior noise levels, we can view </span><math id="A4.SS3.p2.19.m19.1" class="ltx_Math" alttext="S_{t}" display="inline"><semantics id="A4.SS3.p2.19.m19.1a"><msub id="A4.SS3.p2.19.m19.1.1" xref="A4.SS3.p2.19.m19.1.1.cmml"><mi mathsize="90%" id="A4.SS3.p2.19.m19.1.1.2" xref="A4.SS3.p2.19.m19.1.1.2.cmml">S</mi><mi mathsize="90%" id="A4.SS3.p2.19.m19.1.1.3" xref="A4.SS3.p2.19.m19.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.19.m19.1b"><apply id="A4.SS3.p2.19.m19.1.1.cmml" xref="A4.SS3.p2.19.m19.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.19.m19.1.1.1.cmml" xref="A4.SS3.p2.19.m19.1.1">subscript</csymbol><ci id="A4.SS3.p2.19.m19.1.1.2.cmml" xref="A4.SS3.p2.19.m19.1.1.2">𝑆</ci><ci id="A4.SS3.p2.19.m19.1.1.3.cmml" xref="A4.SS3.p2.19.m19.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.19.m19.1c">S_{t}</annotation></semantics></math><span id="A4.SS3.p2.22.23" class="ltx_text" style="font-size:90%;"> and </span><math id="A4.SS3.p2.20.m20.1" class="ltx_Math" alttext="\bar{S}_{t-1}" display="inline"><semantics id="A4.SS3.p2.20.m20.1a"><msub id="A4.SS3.p2.20.m20.1.1" xref="A4.SS3.p2.20.m20.1.1.cmml"><mover accent="true" id="A4.SS3.p2.20.m20.1.1.2" xref="A4.SS3.p2.20.m20.1.1.2.cmml"><mi mathsize="90%" id="A4.SS3.p2.20.m20.1.1.2.2" xref="A4.SS3.p2.20.m20.1.1.2.2.cmml">S</mi><mo mathsize="90%" id="A4.SS3.p2.20.m20.1.1.2.1" xref="A4.SS3.p2.20.m20.1.1.2.1.cmml">¯</mo></mover><mrow id="A4.SS3.p2.20.m20.1.1.3" xref="A4.SS3.p2.20.m20.1.1.3.cmml"><mi mathsize="90%" id="A4.SS3.p2.20.m20.1.1.3.2" xref="A4.SS3.p2.20.m20.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS3.p2.20.m20.1.1.3.1" xref="A4.SS3.p2.20.m20.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS3.p2.20.m20.1.1.3.3" xref="A4.SS3.p2.20.m20.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.20.m20.1b"><apply id="A4.SS3.p2.20.m20.1.1.cmml" xref="A4.SS3.p2.20.m20.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.20.m20.1.1.1.cmml" xref="A4.SS3.p2.20.m20.1.1">subscript</csymbol><apply id="A4.SS3.p2.20.m20.1.1.2.cmml" xref="A4.SS3.p2.20.m20.1.1.2"><ci id="A4.SS3.p2.20.m20.1.1.2.1.cmml" xref="A4.SS3.p2.20.m20.1.1.2.1">¯</ci><ci id="A4.SS3.p2.20.m20.1.1.2.2.cmml" xref="A4.SS3.p2.20.m20.1.1.2.2">𝑆</ci></apply><apply id="A4.SS3.p2.20.m20.1.1.3.cmml" xref="A4.SS3.p2.20.m20.1.1.3"><minus id="A4.SS3.p2.20.m20.1.1.3.1.cmml" xref="A4.SS3.p2.20.m20.1.1.3.1"></minus><ci id="A4.SS3.p2.20.m20.1.1.3.2.cmml" xref="A4.SS3.p2.20.m20.1.1.3.2">𝑡</ci><cn type="integer" id="A4.SS3.p2.20.m20.1.1.3.3.cmml" xref="A4.SS3.p2.20.m20.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.20.m20.1c">\bar{S}_{t-1}</annotation></semantics></math><span id="A4.SS3.p2.22.24" class="ltx_text" style="font-size:90%;"> as probabilities of independent events, and thus can composed to define a joint probability </span><math id="A4.SS3.p2.21.m21.2" class="ltx_Math" alttext="S^{\prime}_{t}=1-(1-S_{t})(1-\bar{S}_{t-1})" display="inline"><semantics id="A4.SS3.p2.21.m21.2a"><mrow id="A4.SS3.p2.21.m21.2.2" xref="A4.SS3.p2.21.m21.2.2.cmml"><msubsup id="A4.SS3.p2.21.m21.2.2.4" xref="A4.SS3.p2.21.m21.2.2.4.cmml"><mi mathsize="90%" id="A4.SS3.p2.21.m21.2.2.4.2.2" xref="A4.SS3.p2.21.m21.2.2.4.2.2.cmml">S</mi><mi mathsize="90%" id="A4.SS3.p2.21.m21.2.2.4.3" xref="A4.SS3.p2.21.m21.2.2.4.3.cmml">t</mi><mo mathsize="90%" id="A4.SS3.p2.21.m21.2.2.4.2.3" xref="A4.SS3.p2.21.m21.2.2.4.2.3.cmml">′</mo></msubsup><mo mathsize="90%" id="A4.SS3.p2.21.m21.2.2.3" xref="A4.SS3.p2.21.m21.2.2.3.cmml">=</mo><mrow id="A4.SS3.p2.21.m21.2.2.2" xref="A4.SS3.p2.21.m21.2.2.2.cmml"><mn mathsize="90%" id="A4.SS3.p2.21.m21.2.2.2.4" xref="A4.SS3.p2.21.m21.2.2.2.4.cmml">1</mn><mo mathsize="90%" id="A4.SS3.p2.21.m21.2.2.2.3" xref="A4.SS3.p2.21.m21.2.2.2.3.cmml">−</mo><mrow id="A4.SS3.p2.21.m21.2.2.2.2" xref="A4.SS3.p2.21.m21.2.2.2.2.cmml"><mrow id="A4.SS3.p2.21.m21.1.1.1.1.1.1" xref="A4.SS3.p2.21.m21.1.1.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A4.SS3.p2.21.m21.1.1.1.1.1.1.2" xref="A4.SS3.p2.21.m21.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.SS3.p2.21.m21.1.1.1.1.1.1.1" xref="A4.SS3.p2.21.m21.1.1.1.1.1.1.1.cmml"><mn mathsize="90%" id="A4.SS3.p2.21.m21.1.1.1.1.1.1.1.2" xref="A4.SS3.p2.21.m21.1.1.1.1.1.1.1.2.cmml">1</mn><mo mathsize="90%" 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id="A4.SS3.p2.22.m22.1.1.2.2" xref="A4.SS3.p2.22.m22.1.1.2.2.cmml">S</mi><mi mathsize="90%" id="A4.SS3.p2.22.m22.1.1.3" xref="A4.SS3.p2.22.m22.1.1.3.cmml">t</mi><mo mathsize="90%" id="A4.SS3.p2.22.m22.1.1.2.3" xref="A4.SS3.p2.22.m22.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS3.p2.22.m22.1b"><apply id="A4.SS3.p2.22.m22.1.1.cmml" xref="A4.SS3.p2.22.m22.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.22.m22.1.1.1.cmml" xref="A4.SS3.p2.22.m22.1.1">subscript</csymbol><apply id="A4.SS3.p2.22.m22.1.1.2.cmml" xref="A4.SS3.p2.22.m22.1.1"><csymbol cd="ambiguous" id="A4.SS3.p2.22.m22.1.1.2.1.cmml" xref="A4.SS3.p2.22.m22.1.1">superscript</csymbol><ci id="A4.SS3.p2.22.m22.1.1.2.2.cmml" xref="A4.SS3.p2.22.m22.1.1.2.2">𝑆</ci><ci id="A4.SS3.p2.22.m22.1.1.2.3.cmml" xref="A4.SS3.p2.22.m22.1.1.2.3">′</ci></apply><ci id="A4.SS3.p2.22.m22.1.1.3.cmml" xref="A4.SS3.p2.22.m22.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p2.22.m22.1c">S^{\prime}_{t}</annotation></semantics></math><span id="A4.SS3.p2.22.26" class="ltx_text" style="font-size:90%;"> as our fused SNR reweighting factor for diffusion training.</span></p>
</div>
<div id="A4.SS3.p3" class="ltx_para">
<p id="A4.SS3.p3.1" class="ltx_p"><span id="A4.SS3.p3.1.1" class="ltx_text" style="font-size:90%;">In our experiments, we choose to follow the min-SNR reweighting strategy  </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS3.p3.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib24" title="" class="ltx_ref">24</a><span id="A4.SS3.p3.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS3.p3.1.4" class="ltx_text" style="font-size:90%;"> to derive the </span><math id="A4.SS3.p3.1.m1.1" class="ltx_Math" alttext="S" display="inline"><semantics id="A4.SS3.p3.1.m1.1a"><mi mathsize="90%" id="A4.SS3.p3.1.m1.1.1" xref="A4.SS3.p3.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="A4.SS3.p3.1.m1.1b"><ci id="A4.SS3.p3.1.m1.1.1.cmml" xref="A4.SS3.p3.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS3.p3.1.m1.1c">S</annotation></semantics></math><span id="A4.SS3.p3.1.5" class="ltx_text" style="font-size:90%;">. Our new SNR reweighting proves extremely useful to accelerate the convergence of video prediction, while we didn’t observe a boost on non-image domains so we didn’t use it there.</span></p>
</div>
</section>
<section id="A4.SS4" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">D.4 </span>Noise schedule</h3>

<div id="A4.SS4.p1" class="ltx_para">
<p id="A4.SS4.p1.1" class="ltx_p"><span id="A4.SS4.p1.1.1" class="ltx_text" style="font-size:90%;">We use sigmoid noise schedule </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS4.p1.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib9" title="" class="ltx_ref">9</a><span id="A4.SS4.p1.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS4.p1.1.4" class="ltx_text" style="font-size:90%;"> for video prediction, linear noise schedule for maze planning, and cosine schedule for everything else.</span></p>
</div>
</section>
<section id="A4.SS5" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">D.5 </span>Implementation Details of Sampling with Guidance</h3>

<section id="A4.SS5.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Corner case of sampling noise</h5>

<div id="A4.SS5.SSS0.Px1.p1" class="ltx_para">
<p id="A4.SS5.SSS0.Px1.p1.6" class="ltx_p"><span id="A4.SS5.SSS0.Px1.p1.6.1" class="ltx_text" style="font-size:90%;">In our sampling algorithm, due to the flexibility of the scheduling matrix </span><math id="A4.SS5.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="\mathcal{K}" display="inline"><semantics id="A4.SS5.SSS0.Px1.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A4.SS5.SSS0.Px1.p1.1.m1.1.1" xref="A4.SS5.SSS0.Px1.p1.1.m1.1.1.cmml">𝒦</mi><annotation-xml encoding="MathML-Content" id="A4.SS5.SSS0.Px1.p1.1.m1.1b"><ci id="A4.SS5.SSS0.Px1.p1.1.m1.1.1.cmml" xref="A4.SS5.SSS0.Px1.p1.1.m1.1.1">𝒦</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS5.SSS0.Px1.p1.1.m1.1c">\mathcal{K}</annotation></semantics></math><span id="A4.SS5.SSS0.Px1.p1.6.2" class="ltx_text" style="font-size:90%;">, there are corner cases when </span><math id="A4.SS5.SSS0.Px1.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="A4.SS5.SSS0.Px1.p1.2.m2.1a"><msubsup id="A4.SS5.SSS0.Px1.p1.2.m2.1.1" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.cmml"><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.2.2" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.2.3" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.2.3.cmml">t</mi><msub id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.cmml"><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.2" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.3" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS5.SSS0.Px1.p1.2.m2.1b"><apply id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.cmml" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1">superscript</csymbol><apply id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.2.1.cmml" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1">subscript</csymbol><ci id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.2.2.cmml" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.2.2">𝐱</ci><ci id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.2.3.cmml" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.2.3">𝑡</ci></apply><apply id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.1.cmml" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3">subscript</csymbol><ci id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.2.cmml" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.2">𝑘</ci><ci id="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.3.cmml" xref="A4.SS5.SSS0.Px1.p1.2.m2.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS5.SSS0.Px1.p1.2.m2.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math><span id="A4.SS5.SSS0.Px1.p1.6.3" class="ltx_text" style="font-size:90%;"> is required to stay at its same noise level during a sampling step. The core question of this corner case is whether we should update</span><math id="A4.SS5.SSS0.Px1.p1.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{k_{t}}" display="inline"><semantics id="A4.SS5.SSS0.Px1.p1.3.m3.1a"><msubsup id="A4.SS5.SSS0.Px1.p1.3.m3.1.1" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.cmml"><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.2.2" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.2.3" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.2.3.cmml">t</mi><msub id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.cmml"><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.2" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.3" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.3.cmml">t</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS5.SSS0.Px1.p1.3.m3.1b"><apply id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.cmml" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.1.cmml" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1">superscript</csymbol><apply id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.2.1.cmml" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1">subscript</csymbol><ci id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.2.2.cmml" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.2.2">𝐱</ci><ci id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.2.3.cmml" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.2.3">𝑡</ci></apply><apply id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.cmml" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.1.cmml" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3">subscript</csymbol><ci id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.2.cmml" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.2">𝑘</ci><ci id="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.3.cmml" xref="A4.SS5.SSS0.Px1.p1.3.m3.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS5.SSS0.Px1.p1.3.m3.1c">\mathbf{x}_{t}^{k_{t}}</annotation></semantics></math><span id="A4.SS5.SSS0.Px1.p1.6.4" class="ltx_text" style="font-size:90%;"> at all. One option is just copying over the old value. The other option is to run a backward diffusion followed by a forward diffusion back to its old noise level to resample under the diffusion process. While we conclude this can be an open question, we prefer the later approach, resampling, and use it in Monte Carlo Guidance to generate multiple samples. We note that even if one take the first approach, the guidance gradient can still flow back in the time steps before </span><math id="A4.SS5.SSS0.Px1.p1.4.m4.1" class="ltx_Math" alttext="t" display="inline"><semantics id="A4.SS5.SSS0.Px1.p1.4.m4.1a"><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.4.m4.1.1" xref="A4.SS5.SSS0.Px1.p1.4.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A4.SS5.SSS0.Px1.p1.4.m4.1b"><ci id="A4.SS5.SSS0.Px1.p1.4.m4.1.1.cmml" xref="A4.SS5.SSS0.Px1.p1.4.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS5.SSS0.Px1.p1.4.m4.1c">t</annotation></semantics></math><span id="A4.SS5.SSS0.Px1.p1.6.5" class="ltx_text" style="font-size:90%;"> as the dynamics model </span><math id="A4.SS5.SSS0.Px1.p1.5.m5.1" class="ltx_Math" alttext="p(\mathbf{z}_{t}|\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1})" display="inline"><semantics id="A4.SS5.SSS0.Px1.p1.5.m5.1a"><mrow id="A4.SS5.SSS0.Px1.p1.5.m5.1.1" xref="A4.SS5.SSS0.Px1.p1.5.m5.1.1.cmml"><mi 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xref="A4.SS5.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.2">𝑡</ci><cn type="integer" id="A4.SS5.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.3.cmml" xref="A4.SS5.SSS0.Px1.p1.5.m5.1.1.1.1.1.2.2.2.3.3">1</cn></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS5.SSS0.Px1.p1.5.m5.1c">p(\mathbf{z}_{t}|\mathbf{x}_{t}^{k_{t}},\mathbf{z}_{t-1})</annotation></semantics></math><span id="A4.SS5.SSS0.Px1.p1.6.6" class="ltx_text" style="font-size:90%;"> can still propagate the guidance gradient to </span><math id="A4.SS5.SSS0.Px1.p1.6.m6.1" class="ltx_Math" alttext="\mathbf{z}_{t-1}" display="inline"><semantics id="A4.SS5.SSS0.Px1.p1.6.m6.1a"><msub id="A4.SS5.SSS0.Px1.p1.6.m6.1.1" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.cmml"><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.2" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.2.cmml">𝐳</mi><mrow id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.cmml"><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.2" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.2.cmml">t</mi><mo mathsize="90%" id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.1" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.1.cmml">−</mo><mn mathsize="90%" id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.3" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.SS5.SSS0.Px1.p1.6.m6.1b"><apply id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.cmml" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.1.cmml" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1">subscript</csymbol><ci id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.2.cmml" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.2">𝐳</ci><apply id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.cmml" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3"><minus id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.1.cmml" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.1"></minus><ci id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.2.cmml" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.2">𝑡</ci><cn type="integer" id="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.3.cmml" xref="A4.SS5.SSS0.Px1.p1.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS5.SSS0.Px1.p1.6.m6.1c">\mathbf{z}_{t-1}</annotation></semantics></math><span id="A4.SS5.SSS0.Px1.p1.6.7" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
<div id="A4.SS5.SSS0.Px1.p2" class="ltx_para">
<p id="A4.SS5.SSS0.Px1.p2.6" class="ltx_p"><span id="A4.SS5.SSS0.Px1.p2.6.1" class="ltx_text" style="font-size:90%;">Other than Monte Carlo Guidance, this corner case only happens when </span><math id="A4.SS5.SSS0.Px1.p2.1.m1.1" class="ltx_Math" alttext="k_{t}=0" display="inline"><semantics id="A4.SS5.SSS0.Px1.p2.1.m1.1a"><mrow id="A4.SS5.SSS0.Px1.p2.1.m1.1.1" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.cmml"><msub id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.cmml"><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.2" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.2.cmml">k</mi><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.3" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.1" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.1.cmml">=</mo><mn mathsize="90%" id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.3" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS5.SSS0.Px1.p2.1.m1.1b"><apply id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.cmml" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1"><eq id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.1.cmml" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.1"></eq><apply id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.cmml" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.1.cmml" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2">subscript</csymbol><ci id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.2.cmml" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.2">𝑘</ci><ci id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.3.cmml" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.2.3">𝑡</ci></apply><cn type="integer" id="A4.SS5.SSS0.Px1.p2.1.m1.1.1.3.cmml" xref="A4.SS5.SSS0.Px1.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS5.SSS0.Px1.p2.1.m1.1c">k_{t}=0</annotation></semantics></math><span id="A4.SS5.SSS0.Px1.p2.6.2" class="ltx_text" style="font-size:90%;"> or </span><math id="A4.SS5.SSS0.Px1.p2.2.m2.1" class="ltx_Math" alttext="k_{t}=K" display="inline"><semantics id="A4.SS5.SSS0.Px1.p2.2.m2.1a"><mrow id="A4.SS5.SSS0.Px1.p2.2.m2.1.1" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.cmml"><msub id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.cmml"><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.2" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.2.cmml">k</mi><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.3" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.1" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.1.cmml">=</mo><mi mathsize="90%" id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.3" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS5.SSS0.Px1.p2.2.m2.1b"><apply id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.cmml" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1"><eq id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.1.cmml" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.1"></eq><apply id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.cmml" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.1.cmml" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2">subscript</csymbol><ci id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.2.cmml" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.2">𝑘</ci><ci id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.3.cmml" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.2.3">𝑡</ci></apply><ci id="A4.SS5.SSS0.Px1.p2.2.m2.1.1.3.cmml" xref="A4.SS5.SSS0.Px1.p2.2.m2.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS5.SSS0.Px1.p2.2.m2.1c">k_{t}=K</annotation></semantics></math><span id="A4.SS5.SSS0.Px1.p2.6.3" class="ltx_text" style="font-size:90%;"> throughout our experiments. That is, we chose our </span><math id="A4.SS5.SSS0.Px1.p2.3.m3.1" class="ltx_Math" alttext="\mathcal{K}" display="inline"><semantics id="A4.SS5.SSS0.Px1.p2.3.m3.1a"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A4.SS5.SSS0.Px1.p2.3.m3.1.1" xref="A4.SS5.SSS0.Px1.p2.3.m3.1.1.cmml">𝒦</mi><annotation-xml encoding="MathML-Content" id="A4.SS5.SSS0.Px1.p2.3.m3.1b"><ci id="A4.SS5.SSS0.Px1.p2.3.m3.1.1.cmml" xref="A4.SS5.SSS0.Px1.p2.3.m3.1.1">𝒦</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS5.SSS0.Px1.p2.3.m3.1c">\mathcal{K}</annotation></semantics></math><span id="A4.SS5.SSS0.Px1.p2.6.4" class="ltx_text" style="font-size:90%;"> such that once any token gets diffused slightly, it will keep diffusing. 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</div>
</section>
<section id="A4.SS5.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Guidance for maze planning</h5>

<div id="A4.SS5.SSS0.Px2.p1" class="ltx_para">
<p id="A4.SS5.SSS0.Px2.p1.3" class="ltx_p"><span id="A4.SS5.SSS0.Px2.p1.3.1" class="ltx_text" style="font-size:90%;">In maze planning, our main baseline Diffuer </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS5.SSS0.Px2.p1.3.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib36" title="" class="ltx_ref">36</a><span id="A4.SS5.SSS0.Px2.p1.3.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS5.SSS0.Px2.p1.3.4" class="ltx_text" style="font-size:90%;"> discards the reward from the dataset and directly plans with the goal position and velocity. We adopt the same convention for Diffusion Forcing. 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This objective is challenging due to the non-convex nature of 2D maze, but we found Diffusion Forcing can still reliably find plans without bumping into walls. However, we also observe that the agent tend to leave the goal location due to the nature of the provided dataset - the goal location is just one possible waypoint for the robot to pass through, and there are no trajectories that simply stay at the goal. We also tried this reward for guidance with Diffuser, but it didn’t work even with a good amount of tuning.</span></p>
</div>
</section>
</section>
<section id="A4.SS6" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">D.6 </span>Performance Optimization</h3>

<div id="A4.SS6.p1" class="ltx_para">
<p id="A4.SS6.p1.3" class="ltx_p"><span id="A4.SS6.p1.3.1" class="ltx_text" style="font-size:90%;">Accelerating the diffusion sampling of Diffusion Forcing is similar to that of normal diffusion models. We adopt DDIM </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A4.SS6.p1.3.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib55" title="" class="ltx_ref">55</a><span id="A4.SS6.p1.3.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A4.SS6.p1.3.4" class="ltx_text" style="font-size:90%;"> sampling for the diffusion of each token. While we use </span><math id="A4.SS6.p1.1.m1.1" class="ltx_Math" alttext="K=1000" display="inline"><semantics id="A4.SS6.p1.1.m1.1a"><mrow id="A4.SS6.p1.1.m1.1.1" xref="A4.SS6.p1.1.m1.1.1.cmml"><mi mathsize="90%" id="A4.SS6.p1.1.m1.1.1.2" xref="A4.SS6.p1.1.m1.1.1.2.cmml">K</mi><mo mathsize="90%" id="A4.SS6.p1.1.m1.1.1.1" xref="A4.SS6.p1.1.m1.1.1.1.cmml">=</mo><mn mathsize="90%" id="A4.SS6.p1.1.m1.1.1.3" xref="A4.SS6.p1.1.m1.1.1.3.cmml">1000</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS6.p1.1.m1.1b"><apply id="A4.SS6.p1.1.m1.1.1.cmml" xref="A4.SS6.p1.1.m1.1.1"><eq id="A4.SS6.p1.1.m1.1.1.1.cmml" xref="A4.SS6.p1.1.m1.1.1.1"></eq><ci id="A4.SS6.p1.1.m1.1.1.2.cmml" xref="A4.SS6.p1.1.m1.1.1.2">𝐾</ci><cn type="integer" id="A4.SS6.p1.1.m1.1.1.3.cmml" xref="A4.SS6.p1.1.m1.1.1.3">1000</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p1.1.m1.1c">K=1000</annotation></semantics></math><span id="A4.SS6.p1.3.5" class="ltx_text" style="font-size:90%;"> steps of diffusion, we sample with only </span><math id="A4.SS6.p1.2.m2.1" class="ltx_Math" alttext="100" display="inline"><semantics id="A4.SS6.p1.2.m2.1a"><mn mathsize="90%" id="A4.SS6.p1.2.m2.1.1" xref="A4.SS6.p1.2.m2.1.1.cmml">100</mn><annotation-xml encoding="MathML-Content" id="A4.SS6.p1.2.m2.1b"><cn type="integer" id="A4.SS6.p1.2.m2.1.1.cmml" xref="A4.SS6.p1.2.m2.1.1">100</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p1.2.m2.1c">100</annotation></semantics></math><span id="A4.SS6.p1.3.6" class="ltx_text" style="font-size:90%;"> DDIM for video prediction and </span><math id="A4.SS6.p1.3.m3.1" class="ltx_Math" alttext="50" display="inline"><semantics id="A4.SS6.p1.3.m3.1a"><mn mathsize="90%" id="A4.SS6.p1.3.m3.1.1" xref="A4.SS6.p1.3.m3.1.1.cmml">50</mn><annotation-xml encoding="MathML-Content" id="A4.SS6.p1.3.m3.1b"><cn type="integer" id="A4.SS6.p1.3.m3.1.1.cmml" xref="A4.SS6.p1.3.m3.1.1">50</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p1.3.m3.1c">50</annotation></semantics></math><span id="A4.SS6.p1.3.7" class="ltx_text" style="font-size:90%;"> for non-video domains.</span></p>
</div>
<div id="A4.SS6.p2" class="ltx_para">
<p id="A4.SS6.p2.5" class="ltx_p"><span id="A4.SS6.p2.5.1" class="ltx_text" style="font-size:90%;">While Diffusion Forcing can be implemented with transformers, we use an RNN as the backbone for Diffusion Forcing experiments it’s widely used in decision-making for its flexibility and efficiency in online decision-making systems. To further reduce training time and GPU memory usage, we use frame-stacking to stack multiple observed images as a single </span><math id="A4.SS6.p2.1.m1.1" class="ltx_Math" alttext="\mathbf{x}" display="inline"><semantics id="A4.SS6.p2.1.m1.1a"><mi mathsize="90%" id="A4.SS6.p2.1.m1.1.1" xref="A4.SS6.p2.1.m1.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="A4.SS6.p2.1.m1.1b"><ci id="A4.SS6.p2.1.m1.1.1.cmml" xref="A4.SS6.p2.1.m1.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p2.1.m1.1c">\mathbf{x}</annotation></semantics></math><span id="A4.SS6.p2.5.2" class="ltx_text" style="font-size:90%;">. This is due to the fact that adjacent tokens can be very similar - e.g. recording the same motion at higher fps can lead to this. We deem that it’s wasteful if we rollout the dynamics model multiple times to generate almost identical tokens. For video datasets, we manually examine how many time steps it takes to require a minimal level of prediction power instead of copying frames over. There is another reason why we use frame stacking - many diffusion model techniques such as different noise schedules are designed to model </span><math id="A4.SS6.p2.2.m2.1" class="ltx_Math" alttext="\mathbf{x}" display="inline"><semantics id="A4.SS6.p2.2.m2.1a"><mi mathsize="90%" id="A4.SS6.p2.2.m2.1.1" xref="A4.SS6.p2.2.m2.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="A4.SS6.p2.2.m2.1b"><ci id="A4.SS6.p2.2.m2.1.1.cmml" xref="A4.SS6.p2.2.m2.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p2.2.m2.1c">\mathbf{x}</annotation></semantics></math><span id="A4.SS6.p2.5.3" class="ltx_text" style="font-size:90%;"> with correlated elements or redundancy. Low-dimensional systems may need drastically different hyperparameters when they lack the data redundancy these techniques are tested on. Frame stacking is thus also helpful for our non-image experiments so we can start with canonical hyperparameters of diffusion models. We use a frame stack of </span><math id="A4.SS6.p2.3.m3.1" class="ltx_Math" alttext="4" display="inline"><semantics id="A4.SS6.p2.3.m3.1a"><mn mathsize="90%" id="A4.SS6.p2.3.m3.1.1" xref="A4.SS6.p2.3.m3.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="A4.SS6.p2.3.m3.1b"><cn type="integer" id="A4.SS6.p2.3.m3.1.1.cmml" xref="A4.SS6.p2.3.m3.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p2.3.m3.1c">4</annotation></semantics></math><span id="A4.SS6.p2.5.4" class="ltx_text" style="font-size:90%;"> for DMlab video prediction, </span><math id="A4.SS6.p2.4.m4.1" class="ltx_Math" alttext="8" display="inline"><semantics id="A4.SS6.p2.4.m4.1a"><mn mathsize="90%" id="A4.SS6.p2.4.m4.1.1" xref="A4.SS6.p2.4.m4.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="A4.SS6.p2.4.m4.1b"><cn type="integer" id="A4.SS6.p2.4.m4.1.1.cmml" xref="A4.SS6.p2.4.m4.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p2.4.m4.1c">8</annotation></semantics></math><span id="A4.SS6.p2.5.5" class="ltx_text" style="font-size:90%;"> for Minecraft and </span><math id="A4.SS6.p2.5.m5.1" class="ltx_Math" alttext="10" display="inline"><semantics id="A4.SS6.p2.5.m5.1a"><mn mathsize="90%" id="A4.SS6.p2.5.m5.1.1" xref="A4.SS6.p2.5.m5.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="A4.SS6.p2.5.m5.1b"><cn type="integer" id="A4.SS6.p2.5.m5.1.1.cmml" xref="A4.SS6.p2.5.m5.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p2.5.m5.1c">10</annotation></semantics></math><span id="A4.SS6.p2.5.6" class="ltx_text" style="font-size:90%;"> for maze planning.</span></p>
</div>
<div id="A4.SS6.p3" class="ltx_para">
<p id="A4.SS6.p3.10" class="ltx_p"><span id="A4.SS6.p3.10.1" class="ltx_text" style="font-size:90%;">At sampling time, we also have a design choice to reduce compute usage, as reflected in line 8 of Algorithm </span><a href="#alg2" title="Algorithm 2 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">2</span></a><span id="A4.SS6.p3.10.2" class="ltx_text" style="font-size:90%;">. 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id="A4.SS6.p3.2.m2.1.1.1.cmml" xref="A4.SS6.p3.2.m2.1.1">subscript</csymbol><ci id="A4.SS6.p3.2.m2.1.1.2.cmml" xref="A4.SS6.p3.2.m2.1.1.2">𝐳</ci><ci id="A4.SS6.p3.2.m2.1.1.3.cmml" xref="A4.SS6.p3.2.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p3.2.m2.1c">\mathbf{z}_{t}</annotation></semantics></math><span id="A4.SS6.p3.10.4" class="ltx_text" style="font-size:90%;">, instead of recalculating </span><math id="A4.SS6.p3.3.m3.1" class="ltx_Math" alttext="\mathbf{z}_{t}" display="inline"><semantics id="A4.SS6.p3.3.m3.1a"><msub id="A4.SS6.p3.3.m3.1.1" xref="A4.SS6.p3.3.m3.1.1.cmml"><mi mathsize="90%" id="A4.SS6.p3.3.m3.1.1.2" xref="A4.SS6.p3.3.m3.1.1.2.cmml">𝐳</mi><mi mathsize="90%" id="A4.SS6.p3.3.m3.1.1.3" xref="A4.SS6.p3.3.m3.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS6.p3.3.m3.1b"><apply id="A4.SS6.p3.3.m3.1.1.cmml" xref="A4.SS6.p3.3.m3.1.1"><csymbol cd="ambiguous" id="A4.SS6.p3.3.m3.1.1.1.cmml" 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Since the model is trained to condition on </span><math id="A4.SS6.p3.5.m5.1" class="ltx_Math" alttext="\mathbf{z}_{t}" display="inline"><semantics id="A4.SS6.p3.5.m5.1a"><msub id="A4.SS6.p3.5.m5.1.1" xref="A4.SS6.p3.5.m5.1.1.cmml"><mi mathsize="90%" id="A4.SS6.p3.5.m5.1.1.2" xref="A4.SS6.p3.5.m5.1.1.2.cmml">𝐳</mi><mi mathsize="90%" id="A4.SS6.p3.5.m5.1.1.3" xref="A4.SS6.p3.5.m5.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS6.p3.5.m5.1b"><apply id="A4.SS6.p3.5.m5.1.1.cmml" xref="A4.SS6.p3.5.m5.1.1"><csymbol cd="ambiguous" id="A4.SS6.p3.5.m5.1.1.1.cmml" xref="A4.SS6.p3.5.m5.1.1">subscript</csymbol><ci id="A4.SS6.p3.5.m5.1.1.2.cmml" xref="A4.SS6.p3.5.m5.1.1.2">𝐳</ci><ci id="A4.SS6.p3.5.m5.1.1.3.cmml" xref="A4.SS6.p3.5.m5.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p3.5.m5.1c">\mathbf{z}_{t}</annotation></semantics></math><span id="A4.SS6.p3.10.7" class="ltx_text" style="font-size:90%;"> estimated from arbitrary noisy history, we recognize that both are valid approaches. The reason why the choose line </span><math id="A4.SS6.p3.6.m6.1" class="ltx_Math" alttext="8" display="inline"><semantics id="A4.SS6.p3.6.m6.1a"><mn mathsize="90%" id="A4.SS6.p3.6.m6.1.1" xref="A4.SS6.p3.6.m6.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="A4.SS6.p3.6.m6.1b"><cn type="integer" id="A4.SS6.p3.6.m6.1.1.cmml" xref="A4.SS6.p3.6.m6.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p3.6.m6.1c">8</annotation></semantics></math><span id="A4.SS6.p3.10.8" class="ltx_text" style="font-size:90%;"> is two fold. First, it cuts the compute by half, avoiding computing posterior every step. Second, this happens to be what we want for stabilization - </span><math id="A4.SS6.p3.7.m7.1" class="ltx_Math" alttext="\mathbf{z}_{t}^{\text{new}}" display="inline"><semantics id="A4.SS6.p3.7.m7.1a"><msubsup id="A4.SS6.p3.7.m7.1.1" xref="A4.SS6.p3.7.m7.1.1.cmml"><mi mathsize="90%" id="A4.SS6.p3.7.m7.1.1.2.2" xref="A4.SS6.p3.7.m7.1.1.2.2.cmml">𝐳</mi><mi mathsize="90%" id="A4.SS6.p3.7.m7.1.1.2.3" xref="A4.SS6.p3.7.m7.1.1.2.3.cmml">t</mi><mtext mathsize="90%" id="A4.SS6.p3.7.m7.1.1.3" xref="A4.SS6.p3.7.m7.1.1.3a.cmml">new</mtext></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS6.p3.7.m7.1b"><apply id="A4.SS6.p3.7.m7.1.1.cmml" xref="A4.SS6.p3.7.m7.1.1"><csymbol cd="ambiguous" id="A4.SS6.p3.7.m7.1.1.1.cmml" xref="A4.SS6.p3.7.m7.1.1">superscript</csymbol><apply id="A4.SS6.p3.7.m7.1.1.2.cmml" xref="A4.SS6.p3.7.m7.1.1"><csymbol cd="ambiguous" id="A4.SS6.p3.7.m7.1.1.2.1.cmml" xref="A4.SS6.p3.7.m7.1.1">subscript</csymbol><ci id="A4.SS6.p3.7.m7.1.1.2.2.cmml" xref="A4.SS6.p3.7.m7.1.1.2.2">𝐳</ci><ci id="A4.SS6.p3.7.m7.1.1.2.3.cmml" xref="A4.SS6.p3.7.m7.1.1.2.3">𝑡</ci></apply><ci id="A4.SS6.p3.7.m7.1.1.3a.cmml" xref="A4.SS6.p3.7.m7.1.1.3"><mtext mathsize="63%" id="A4.SS6.p3.7.m7.1.1.3.cmml" xref="A4.SS6.p3.7.m7.1.1.3">new</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p3.7.m7.1c">\mathbf{z}_{t}^{\text{new}}</annotation></semantics></math><span id="A4.SS6.p3.10.9" class="ltx_text" style="font-size:90%;"> already contains the information of the clean </span><math id="A4.SS6.p3.8.m8.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{\text{new}}" display="inline"><semantics id="A4.SS6.p3.8.m8.1a"><msubsup id="A4.SS6.p3.8.m8.1.1" xref="A4.SS6.p3.8.m8.1.1.cmml"><mi mathsize="90%" id="A4.SS6.p3.8.m8.1.1.2.2" xref="A4.SS6.p3.8.m8.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS6.p3.8.m8.1.1.2.3" xref="A4.SS6.p3.8.m8.1.1.2.3.cmml">t</mi><mtext mathsize="90%" id="A4.SS6.p3.8.m8.1.1.3" xref="A4.SS6.p3.8.m8.1.1.3a.cmml">new</mtext></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS6.p3.8.m8.1b"><apply id="A4.SS6.p3.8.m8.1.1.cmml" xref="A4.SS6.p3.8.m8.1.1"><csymbol cd="ambiguous" id="A4.SS6.p3.8.m8.1.1.1.cmml" xref="A4.SS6.p3.8.m8.1.1">superscript</csymbol><apply id="A4.SS6.p3.8.m8.1.1.2.cmml" xref="A4.SS6.p3.8.m8.1.1"><csymbol cd="ambiguous" id="A4.SS6.p3.8.m8.1.1.2.1.cmml" xref="A4.SS6.p3.8.m8.1.1">subscript</csymbol><ci id="A4.SS6.p3.8.m8.1.1.2.2.cmml" xref="A4.SS6.p3.8.m8.1.1.2.2">𝐱</ci><ci id="A4.SS6.p3.8.m8.1.1.2.3.cmml" xref="A4.SS6.p3.8.m8.1.1.2.3">𝑡</ci></apply><ci id="A4.SS6.p3.8.m8.1.1.3a.cmml" xref="A4.SS6.p3.8.m8.1.1.3"><mtext mathsize="63%" id="A4.SS6.p3.8.m8.1.1.3.cmml" xref="A4.SS6.p3.8.m8.1.1.3">new</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p3.8.m8.1c">\mathbf{x}_{t}^{\text{new}}</annotation></semantics></math><span id="A4.SS6.p3.10.10" class="ltx_text" style="font-size:90%;"> under our simplified observation model, and happens to be estimated with </span><math id="A4.SS6.p3.9.m9.1" class="ltx_Math" alttext="k=k_{t}" display="inline"><semantics id="A4.SS6.p3.9.m9.1a"><mrow id="A4.SS6.p3.9.m9.1.1" xref="A4.SS6.p3.9.m9.1.1.cmml"><mi mathsize="90%" id="A4.SS6.p3.9.m9.1.1.2" xref="A4.SS6.p3.9.m9.1.1.2.cmml">k</mi><mo mathsize="90%" id="A4.SS6.p3.9.m9.1.1.1" xref="A4.SS6.p3.9.m9.1.1.1.cmml">=</mo><msub id="A4.SS6.p3.9.m9.1.1.3" xref="A4.SS6.p3.9.m9.1.1.3.cmml"><mi mathsize="90%" id="A4.SS6.p3.9.m9.1.1.3.2" xref="A4.SS6.p3.9.m9.1.1.3.2.cmml">k</mi><mi mathsize="90%" id="A4.SS6.p3.9.m9.1.1.3.3" xref="A4.SS6.p3.9.m9.1.1.3.3.cmml">t</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS6.p3.9.m9.1b"><apply id="A4.SS6.p3.9.m9.1.1.cmml" xref="A4.SS6.p3.9.m9.1.1"><eq id="A4.SS6.p3.9.m9.1.1.1.cmml" xref="A4.SS6.p3.9.m9.1.1.1"></eq><ci id="A4.SS6.p3.9.m9.1.1.2.cmml" xref="A4.SS6.p3.9.m9.1.1.2">𝑘</ci><apply id="A4.SS6.p3.9.m9.1.1.3.cmml" xref="A4.SS6.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="A4.SS6.p3.9.m9.1.1.3.1.cmml" xref="A4.SS6.p3.9.m9.1.1.3">subscript</csymbol><ci id="A4.SS6.p3.9.m9.1.1.3.2.cmml" xref="A4.SS6.p3.9.m9.1.1.3.2">𝑘</ci><ci id="A4.SS6.p3.9.m9.1.1.3.3.cmml" xref="A4.SS6.p3.9.m9.1.1.3.3">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p3.9.m9.1c">k=k_{t}</annotation></semantics></math><span id="A4.SS6.p3.10.11" class="ltx_text" style="font-size:90%;">, a noise level higher than that of </span><math id="A4.SS6.p3.10.m10.1" class="ltx_Math" alttext="\mathbf{x}_{t}^{\text{new}}" display="inline"><semantics id="A4.SS6.p3.10.m10.1a"><msubsup id="A4.SS6.p3.10.m10.1.1" xref="A4.SS6.p3.10.m10.1.1.cmml"><mi mathsize="90%" id="A4.SS6.p3.10.m10.1.1.2.2" xref="A4.SS6.p3.10.m10.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A4.SS6.p3.10.m10.1.1.2.3" xref="A4.SS6.p3.10.m10.1.1.2.3.cmml">t</mi><mtext mathsize="90%" id="A4.SS6.p3.10.m10.1.1.3" xref="A4.SS6.p3.10.m10.1.1.3a.cmml">new</mtext></msubsup><annotation-xml encoding="MathML-Content" id="A4.SS6.p3.10.m10.1b"><apply id="A4.SS6.p3.10.m10.1.1.cmml" xref="A4.SS6.p3.10.m10.1.1"><csymbol cd="ambiguous" id="A4.SS6.p3.10.m10.1.1.1.cmml" xref="A4.SS6.p3.10.m10.1.1">superscript</csymbol><apply id="A4.SS6.p3.10.m10.1.1.2.cmml" xref="A4.SS6.p3.10.m10.1.1"><csymbol cd="ambiguous" id="A4.SS6.p3.10.m10.1.1.2.1.cmml" xref="A4.SS6.p3.10.m10.1.1">subscript</csymbol><ci id="A4.SS6.p3.10.m10.1.1.2.2.cmml" xref="A4.SS6.p3.10.m10.1.1.2.2">𝐱</ci><ci id="A4.SS6.p3.10.m10.1.1.2.3.cmml" xref="A4.SS6.p3.10.m10.1.1.2.3">𝑡</ci></apply><ci id="A4.SS6.p3.10.m10.1.1.3a.cmml" xref="A4.SS6.p3.10.m10.1.1.3"><mtext mathsize="63%" id="A4.SS6.p3.10.m10.1.1.3.cmml" xref="A4.SS6.p3.10.m10.1.1.3">new</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS6.p3.10.m10.1c">\mathbf{x}_{t}^{\text{new}}</annotation></semantics></math><span id="A4.SS6.p3.10.12" class="ltx_text" style="font-size:90%;">. This happens to implement the behavior we want for stabilization.</span></p>
</div>
</section>
<section id="A4.SS7" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">D.7 </span>Sampling schedule for causal uncertainty</h3>

<div id="A4.SS7.p1" class="ltx_para">
<p id="A4.SS7.p1.3" class="ltx_p"><span id="A4.SS7.p1.3.1" class="ltx_text" style="font-size:90%;">Inference is depicted in </span><a href="#alg2" title="In Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">Algorithm</span> <span class="ltx_text ltx_ref_tag">2</span></a><span id="A4.SS7.p1.3.2" class="ltx_text" style="font-size:90%;"> and Figure </span><a href="#S2.F2" title="Figure 2 ‣ Bayesian Filtering. ‣ 2 Related Work and Preliminaries ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">2</span></a><span id="A4.SS7.p1.3.3" class="ltx_text" style="font-size:90%;">. In Equation </span><a href="#A4.E1" title="Equation D.1 ‣ D.7 Sampling schedule for causal uncertainty ‣ Appendix D Additional Method Details ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">D.1</span></a><span id="A4.SS7.p1.3.4" class="ltx_text" style="font-size:90%;">, we illustrate a specific instantiation of the </span><math id="A4.SS7.p1.1.m1.1" class="ltx_Math" alttext="\mathcal{K}" display="inline"><semantics id="A4.SS7.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A4.SS7.p1.1.m1.1.1" xref="A4.SS7.p1.1.m1.1.1.cmml">𝒦</mi><annotation-xml encoding="MathML-Content" id="A4.SS7.p1.1.m1.1b"><ci id="A4.SS7.p1.1.m1.1.1.cmml" xref="A4.SS7.p1.1.m1.1.1">𝒦</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS7.p1.1.m1.1c">\mathcal{K}</annotation></semantics></math><span id="A4.SS7.p1.3.5" class="ltx_text" style="font-size:90%;"> matrix we used for causal planning. For simplicity, we denote the case where a latent </span><math id="A4.SS7.p1.2.m2.1" class="ltx_Math" alttext="\mathbf{z}_{0}" display="inline"><semantics id="A4.SS7.p1.2.m2.1a"><msub id="A4.SS7.p1.2.m2.1.1" xref="A4.SS7.p1.2.m2.1.1.cmml"><mi mathsize="90%" id="A4.SS7.p1.2.m2.1.1.2" xref="A4.SS7.p1.2.m2.1.1.2.cmml">𝐳</mi><mn mathsize="90%" id="A4.SS7.p1.2.m2.1.1.3" xref="A4.SS7.p1.2.m2.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A4.SS7.p1.2.m2.1b"><apply id="A4.SS7.p1.2.m2.1.1.cmml" xref="A4.SS7.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A4.SS7.p1.2.m2.1.1.1.cmml" xref="A4.SS7.p1.2.m2.1.1">subscript</csymbol><ci id="A4.SS7.p1.2.m2.1.1.2.cmml" xref="A4.SS7.p1.2.m2.1.1.2">𝐳</ci><cn type="integer" id="A4.SS7.p1.2.m2.1.1.3.cmml" xref="A4.SS7.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS7.p1.2.m2.1c">\mathbf{z}_{0}</annotation></semantics></math><span id="A4.SS7.p1.3.6" class="ltx_text" style="font-size:90%;"> is given and aim to generate </span><math id="A4.SS7.p1.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{1:H+1}" display="inline"><semantics id="A4.SS7.p1.3.m3.1a"><msub id="A4.SS7.p1.3.m3.1.1" xref="A4.SS7.p1.3.m3.1.1.cmml"><mi mathsize="90%" id="A4.SS7.p1.3.m3.1.1.2" xref="A4.SS7.p1.3.m3.1.1.2.cmml">𝐱</mi><mrow id="A4.SS7.p1.3.m3.1.1.3" xref="A4.SS7.p1.3.m3.1.1.3.cmml"><mn mathsize="90%" id="A4.SS7.p1.3.m3.1.1.3.2" xref="A4.SS7.p1.3.m3.1.1.3.2.cmml">1</mn><mo lspace="0.278em" mathsize="90%" rspace="0.278em" id="A4.SS7.p1.3.m3.1.1.3.1" xref="A4.SS7.p1.3.m3.1.1.3.1.cmml">:</mo><mrow id="A4.SS7.p1.3.m3.1.1.3.3" xref="A4.SS7.p1.3.m3.1.1.3.3.cmml"><mi mathsize="90%" id="A4.SS7.p1.3.m3.1.1.3.3.2" xref="A4.SS7.p1.3.m3.1.1.3.3.2.cmml">H</mi><mo mathsize="90%" id="A4.SS7.p1.3.m3.1.1.3.3.1" xref="A4.SS7.p1.3.m3.1.1.3.3.1.cmml">+</mo><mn mathsize="90%" id="A4.SS7.p1.3.m3.1.1.3.3.3" xref="A4.SS7.p1.3.m3.1.1.3.3.3.cmml">1</mn></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.SS7.p1.3.m3.1b"><apply id="A4.SS7.p1.3.m3.1.1.cmml" xref="A4.SS7.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A4.SS7.p1.3.m3.1.1.1.cmml" xref="A4.SS7.p1.3.m3.1.1">subscript</csymbol><ci id="A4.SS7.p1.3.m3.1.1.2.cmml" xref="A4.SS7.p1.3.m3.1.1.2">𝐱</ci><apply id="A4.SS7.p1.3.m3.1.1.3.cmml" xref="A4.SS7.p1.3.m3.1.1.3"><ci id="A4.SS7.p1.3.m3.1.1.3.1.cmml" xref="A4.SS7.p1.3.m3.1.1.3.1">:</ci><cn type="integer" id="A4.SS7.p1.3.m3.1.1.3.2.cmml" xref="A4.SS7.p1.3.m3.1.1.3.2">1</cn><apply id="A4.SS7.p1.3.m3.1.1.3.3.cmml" xref="A4.SS7.p1.3.m3.1.1.3.3"><plus id="A4.SS7.p1.3.m3.1.1.3.3.1.cmml" xref="A4.SS7.p1.3.m3.1.1.3.3.1"></plus><ci id="A4.SS7.p1.3.m3.1.1.3.3.2.cmml" xref="A4.SS7.p1.3.m3.1.1.3.3.2">𝐻</ci><cn type="integer" id="A4.SS7.p1.3.m3.1.1.3.3.3.cmml" xref="A4.SS7.p1.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS7.p1.3.m3.1c">\mathbf{x}_{1:H+1}</annotation></semantics></math><span id="A4.SS7.p1.3.7" class="ltx_text" style="font-size:90%;">.</span></p>
<table id="A6.EGx43" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A4.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A4.E1.m1.1" class="ltx_Math" alttext="\displaystyle\mathcal{K}^{\mathrm{pyramid}}=\begin{bmatrix}K&amp;K&amp;K&amp;...&amp;K\\
K-1&amp;K&amp;K&amp;...&amp;K\\
K-2&amp;K-1&amp;K&amp;...&amp;K\\
\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots\\
1&amp;2&amp;3&amp;...&amp;H\\
0&amp;1&amp;2&amp;...&amp;H-1\\
\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots\\
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id="A4.E1.m1.1.1.1.1.8.3.1.cmml" xref="A4.E1.m1.1.1.1.1.8.3.1">0</cn><ci id="A4.E1.m1.1.1.1.1.8.4.1.cmml" xref="A4.E1.m1.1.1.1.1.8.4.1">…</ci><cn type="integer" id="A4.E1.m1.1.1.1.1.8.5.1.cmml" xref="A4.E1.m1.1.1.1.1.8.5.1">1</cn></matrixrow><matrixrow id="A4.E1.m1.1.1.1.1i.cmml" xref="A4.E1.m1.1.1.1.1"><cn type="integer" id="A4.E1.m1.1.1.1.1.9.1.1.cmml" xref="A4.E1.m1.1.1.1.1.9.1.1">0</cn><cn type="integer" id="A4.E1.m1.1.1.1.1.9.2.1.cmml" xref="A4.E1.m1.1.1.1.1.9.2.1">0</cn><cn type="integer" id="A4.E1.m1.1.1.1.1.9.3.1.cmml" xref="A4.E1.m1.1.1.1.1.9.3.1">0</cn><ci id="A4.E1.m1.1.1.1.1.9.4.1.cmml" xref="A4.E1.m1.1.1.1.1.9.4.1">…</ci><cn type="integer" id="A4.E1.m1.1.1.1.1.9.5.1.cmml" xref="A4.E1.m1.1.1.1.1.9.5.1">0</cn></matrixrow></matrix></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.E1.m1.1c">\displaystyle\mathcal{K}^{\mathrm{pyramid}}=\begin{bmatrix}K&amp;K&amp;K&amp;...&amp;K\\
K-1&amp;K&amp;K&amp;...&amp;K\\
K-2&amp;K-1&amp;K&amp;...&amp;K\\
\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots\\
1&amp;2&amp;3&amp;...&amp;H\\
0&amp;1&amp;2&amp;...&amp;H-1\\
\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots\\
0&amp;0&amp;0&amp;...&amp;1\\
0&amp;0&amp;0&amp;...&amp;0\end{bmatrix}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(D.1)</span></td>
</tr></tbody>
</table>
</div>
<div id="A4.SS7.p2" class="ltx_para">
<p id="A4.SS7.p2.7" class="ltx_p"><span id="A4.SS7.p2.7.1" class="ltx_text" style="font-size:90%;">Diffusion Forcing begins by sampling our sequences as white noise with noise level </span><math id="A4.SS7.p2.1.m1.1" class="ltx_Math" alttext="K" display="inline"><semantics id="A4.SS7.p2.1.m1.1a"><mi mathsize="90%" id="A4.SS7.p2.1.m1.1.1" xref="A4.SS7.p2.1.m1.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="A4.SS7.p2.1.m1.1b"><ci id="A4.SS7.p2.1.m1.1.1.cmml" xref="A4.SS7.p2.1.m1.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS7.p2.1.m1.1c">K</annotation></semantics></math><span id="A4.SS7.p2.7.2" class="ltx_text" style="font-size:90%;">. It then denoises along each row </span><math id="A4.SS7.p2.2.m2.3" class="ltx_Math" alttext="m=1,\dots,M" display="inline"><semantics id="A4.SS7.p2.2.m2.3a"><mrow id="A4.SS7.p2.2.m2.3.4" xref="A4.SS7.p2.2.m2.3.4.cmml"><mi mathsize="90%" id="A4.SS7.p2.2.m2.3.4.2" xref="A4.SS7.p2.2.m2.3.4.2.cmml">m</mi><mo mathsize="90%" id="A4.SS7.p2.2.m2.3.4.1" xref="A4.SS7.p2.2.m2.3.4.1.cmml">=</mo><mrow id="A4.SS7.p2.2.m2.3.4.3.2" xref="A4.SS7.p2.2.m2.3.4.3.1.cmml"><mn mathsize="90%" id="A4.SS7.p2.2.m2.1.1" xref="A4.SS7.p2.2.m2.1.1.cmml">1</mn><mo mathsize="90%" id="A4.SS7.p2.2.m2.3.4.3.2.1" xref="A4.SS7.p2.2.m2.3.4.3.1.cmml">,</mo><mi mathsize="90%" mathvariant="normal" id="A4.SS7.p2.2.m2.2.2" xref="A4.SS7.p2.2.m2.2.2.cmml">…</mi><mo mathsize="90%" id="A4.SS7.p2.2.m2.3.4.3.2.2" xref="A4.SS7.p2.2.m2.3.4.3.1.cmml">,</mo><mi mathsize="90%" id="A4.SS7.p2.2.m2.3.3" xref="A4.SS7.p2.2.m2.3.3.cmml">M</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS7.p2.2.m2.3b"><apply id="A4.SS7.p2.2.m2.3.4.cmml" xref="A4.SS7.p2.2.m2.3.4"><eq id="A4.SS7.p2.2.m2.3.4.1.cmml" xref="A4.SS7.p2.2.m2.3.4.1"></eq><ci id="A4.SS7.p2.2.m2.3.4.2.cmml" xref="A4.SS7.p2.2.m2.3.4.2">𝑚</ci><list id="A4.SS7.p2.2.m2.3.4.3.1.cmml" xref="A4.SS7.p2.2.m2.3.4.3.2"><cn type="integer" id="A4.SS7.p2.2.m2.1.1.cmml" xref="A4.SS7.p2.2.m2.1.1">1</cn><ci id="A4.SS7.p2.2.m2.2.2.cmml" xref="A4.SS7.p2.2.m2.2.2">…</ci><ci id="A4.SS7.p2.2.m2.3.3.cmml" xref="A4.SS7.p2.2.m2.3.3">𝑀</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS7.p2.2.m2.3c">m=1,\dots,M</annotation></semantics></math><span id="A4.SS7.p2.7.3" class="ltx_text" style="font-size:90%;"> of </span><math id="A4.SS7.p2.3.m3.1" class="ltx_Math" alttext="\mathcal{K}" display="inline"><semantics id="A4.SS7.p2.3.m3.1a"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A4.SS7.p2.3.m3.1.1" xref="A4.SS7.p2.3.m3.1.1.cmml">𝒦</mi><annotation-xml encoding="MathML-Content" id="A4.SS7.p2.3.m3.1b"><ci id="A4.SS7.p2.3.m3.1.1.cmml" xref="A4.SS7.p2.3.m3.1.1">𝒦</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS7.p2.3.m3.1c">\mathcal{K}</annotation></semantics></math><span id="A4.SS7.p2.7.4" class="ltx_text" style="font-size:90%;"> in decreasing order. It does so by proceeding sequentially through frames </span><math id="A4.SS7.p2.4.m4.3" class="ltx_Math" alttext="t=1,\dots,T" display="inline"><semantics id="A4.SS7.p2.4.m4.3a"><mrow id="A4.SS7.p2.4.m4.3.4" xref="A4.SS7.p2.4.m4.3.4.cmml"><mi mathsize="90%" id="A4.SS7.p2.4.m4.3.4.2" xref="A4.SS7.p2.4.m4.3.4.2.cmml">t</mi><mo mathsize="90%" id="A4.SS7.p2.4.m4.3.4.1" xref="A4.SS7.p2.4.m4.3.4.1.cmml">=</mo><mrow id="A4.SS7.p2.4.m4.3.4.3.2" xref="A4.SS7.p2.4.m4.3.4.3.1.cmml"><mn mathsize="90%" id="A4.SS7.p2.4.m4.1.1" xref="A4.SS7.p2.4.m4.1.1.cmml">1</mn><mo mathsize="90%" id="A4.SS7.p2.4.m4.3.4.3.2.1" xref="A4.SS7.p2.4.m4.3.4.3.1.cmml">,</mo><mi mathsize="90%" mathvariant="normal" id="A4.SS7.p2.4.m4.2.2" xref="A4.SS7.p2.4.m4.2.2.cmml">…</mi><mo mathsize="90%" id="A4.SS7.p2.4.m4.3.4.3.2.2" xref="A4.SS7.p2.4.m4.3.4.3.1.cmml">,</mo><mi mathsize="90%" id="A4.SS7.p2.4.m4.3.3" xref="A4.SS7.p2.4.m4.3.3.cmml">T</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS7.p2.4.m4.3b"><apply id="A4.SS7.p2.4.m4.3.4.cmml" xref="A4.SS7.p2.4.m4.3.4"><eq id="A4.SS7.p2.4.m4.3.4.1.cmml" xref="A4.SS7.p2.4.m4.3.4.1"></eq><ci id="A4.SS7.p2.4.m4.3.4.2.cmml" xref="A4.SS7.p2.4.m4.3.4.2">𝑡</ci><list id="A4.SS7.p2.4.m4.3.4.3.1.cmml" xref="A4.SS7.p2.4.m4.3.4.3.2"><cn type="integer" id="A4.SS7.p2.4.m4.1.1.cmml" xref="A4.SS7.p2.4.m4.1.1">1</cn><ci id="A4.SS7.p2.4.m4.2.2.cmml" xref="A4.SS7.p2.4.m4.2.2">…</ci><ci id="A4.SS7.p2.4.m4.3.3.cmml" xref="A4.SS7.p2.4.m4.3.3">𝑇</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS7.p2.4.m4.3c">t=1,\dots,T</annotation></semantics></math><span id="A4.SS7.p2.7.5" class="ltx_text" style="font-size:90%;">, updating the latent (Line 5 of Algorithm </span><a href="#alg2" title="Algorithm 2 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">2</span></a><span id="A4.SS7.p2.7.6" class="ltx_text" style="font-size:90%;">), and then partially applying the backward process to noise level </span><math id="A4.SS7.p2.5.m5.2" class="ltx_Math" alttext="k=\mathcal{K}_{m,t}" display="inline"><semantics id="A4.SS7.p2.5.m5.2a"><mrow id="A4.SS7.p2.5.m5.2.3" xref="A4.SS7.p2.5.m5.2.3.cmml"><mi mathsize="90%" id="A4.SS7.p2.5.m5.2.3.2" xref="A4.SS7.p2.5.m5.2.3.2.cmml">k</mi><mo mathsize="90%" id="A4.SS7.p2.5.m5.2.3.1" xref="A4.SS7.p2.5.m5.2.3.1.cmml">=</mo><msub id="A4.SS7.p2.5.m5.2.3.3" xref="A4.SS7.p2.5.m5.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A4.SS7.p2.5.m5.2.3.3.2" xref="A4.SS7.p2.5.m5.2.3.3.2.cmml">𝒦</mi><mrow id="A4.SS7.p2.5.m5.2.2.2.4" xref="A4.SS7.p2.5.m5.2.2.2.3.cmml"><mi mathsize="90%" id="A4.SS7.p2.5.m5.1.1.1.1" xref="A4.SS7.p2.5.m5.1.1.1.1.cmml">m</mi><mo mathsize="90%" id="A4.SS7.p2.5.m5.2.2.2.4.1" xref="A4.SS7.p2.5.m5.2.2.2.3.cmml">,</mo><mi mathsize="90%" id="A4.SS7.p2.5.m5.2.2.2.2" xref="A4.SS7.p2.5.m5.2.2.2.2.cmml">t</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS7.p2.5.m5.2b"><apply id="A4.SS7.p2.5.m5.2.3.cmml" xref="A4.SS7.p2.5.m5.2.3"><eq id="A4.SS7.p2.5.m5.2.3.1.cmml" xref="A4.SS7.p2.5.m5.2.3.1"></eq><ci id="A4.SS7.p2.5.m5.2.3.2.cmml" xref="A4.SS7.p2.5.m5.2.3.2">𝑘</ci><apply id="A4.SS7.p2.5.m5.2.3.3.cmml" xref="A4.SS7.p2.5.m5.2.3.3"><csymbol cd="ambiguous" id="A4.SS7.p2.5.m5.2.3.3.1.cmml" xref="A4.SS7.p2.5.m5.2.3.3">subscript</csymbol><ci id="A4.SS7.p2.5.m5.2.3.3.2.cmml" xref="A4.SS7.p2.5.m5.2.3.3.2">𝒦</ci><list id="A4.SS7.p2.5.m5.2.2.2.3.cmml" xref="A4.SS7.p2.5.m5.2.2.2.4"><ci id="A4.SS7.p2.5.m5.1.1.1.1.cmml" xref="A4.SS7.p2.5.m5.1.1.1.1">𝑚</ci><ci id="A4.SS7.p2.5.m5.2.2.2.2.cmml" xref="A4.SS7.p2.5.m5.2.2.2.2">𝑡</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS7.p2.5.m5.2c">k=\mathcal{K}_{m,t}</annotation></semantics></math><span id="A4.SS7.p2.7.7" class="ltx_text" style="font-size:90%;"> dictated by the scheduling matrix </span><math id="A4.SS7.p2.6.m6.1" class="ltx_Math" alttext="\mathcal{K}" display="inline"><semantics id="A4.SS7.p2.6.m6.1a"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A4.SS7.p2.6.m6.1.1" xref="A4.SS7.p2.6.m6.1.1.cmml">𝒦</mi><annotation-xml encoding="MathML-Content" id="A4.SS7.p2.6.m6.1b"><ci id="A4.SS7.p2.6.m6.1.1.cmml" xref="A4.SS7.p2.6.m6.1.1">𝒦</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS7.p2.6.m6.1c">\mathcal{K}</annotation></semantics></math><span id="A4.SS7.p2.7.8" class="ltx_text" style="font-size:90%;"> (Line 6-7 of Algorithm </span><a href="#alg2" title="Algorithm 2 ‣ Diffusion Forcing: Different noise nevels for different tokens. ‣ 3 Diffusion Forcing ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">2</span></a><span id="A4.SS7.p2.7.9" class="ltx_text" style="font-size:90%;">). We call a </span><math id="A4.SS7.p2.7.m7.1" class="ltx_Math" alttext="\mathcal{K}" display="inline"><semantics id="A4.SS7.p2.7.m7.1a"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="A4.SS7.p2.7.m7.1.1" xref="A4.SS7.p2.7.m7.1.1.cmml">𝒦</mi><annotation-xml encoding="MathML-Content" id="A4.SS7.p2.7.m7.1b"><ci id="A4.SS7.p2.7.m7.1.1.cmml" xref="A4.SS7.p2.7.m7.1.1">𝒦</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS7.p2.7.m7.1c">\mathcal{K}</annotation></semantics></math><span id="A4.SS7.p2.7.10" class="ltx_text" style="font-size:90%;"> like this pyramid scheduling, as the tokens in the far future are kept at higher noise level than near future.</span></p>
</div>
</section>
<section id="A4.SS8" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">D.8 </span>Implementation Details of Timeseries Regression</h3>

<div id="A4.SS8.p1" class="ltx_para">
<p id="A4.SS8.p1.1" class="ltx_p"><span id="A4.SS8.p1.1.1" class="ltx_text" style="font-size:90%;">We follow the implementation of pytorch-ts, where the validation set is a random subset of the training set with the same number of sequences as the test set. We use early stopping when validation crpssum hasn’t increased for 6 epochs. We leverage the same architecture (1 mlp and 4 grus) as well as a batch size of 32.</span></p>
</div>
</section>
<section id="A4.SS9" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">D.9 </span>Compute Resources</h3>

<div id="A4.SS9.p1" class="ltx_para">
<p id="A4.SS9.p1.8" class="ltx_p"><span id="A4.SS9.p1.8.1" class="ltx_text" style="font-size:90%;">All of our experiments use </span><math id="A4.SS9.p1.1.m1.1" class="ltx_Math" alttext="fp16" display="inline"><semantics id="A4.SS9.p1.1.m1.1a"><mrow id="A4.SS9.p1.1.m1.1.1" xref="A4.SS9.p1.1.m1.1.1.cmml"><mi mathsize="90%" id="A4.SS9.p1.1.m1.1.1.2" xref="A4.SS9.p1.1.m1.1.1.2.cmml">f</mi><mo lspace="0em" rspace="0em" id="A4.SS9.p1.1.m1.1.1.1" xref="A4.SS9.p1.1.m1.1.1.1.cmml">​</mo><mi mathsize="90%" id="A4.SS9.p1.1.m1.1.1.3" xref="A4.SS9.p1.1.m1.1.1.3.cmml">p</mi><mo lspace="0em" rspace="0em" id="A4.SS9.p1.1.m1.1.1.1a" xref="A4.SS9.p1.1.m1.1.1.1.cmml">​</mo><mn mathsize="90%" id="A4.SS9.p1.1.m1.1.1.4" xref="A4.SS9.p1.1.m1.1.1.4.cmml">16</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS9.p1.1.m1.1b"><apply id="A4.SS9.p1.1.m1.1.1.cmml" xref="A4.SS9.p1.1.m1.1.1"><times id="A4.SS9.p1.1.m1.1.1.1.cmml" xref="A4.SS9.p1.1.m1.1.1.1"></times><ci id="A4.SS9.p1.1.m1.1.1.2.cmml" xref="A4.SS9.p1.1.m1.1.1.2">𝑓</ci><ci id="A4.SS9.p1.1.m1.1.1.3.cmml" xref="A4.SS9.p1.1.m1.1.1.3">𝑝</ci><cn type="integer" id="A4.SS9.p1.1.m1.1.1.4.cmml" xref="A4.SS9.p1.1.m1.1.1.4">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p1.1.m1.1c">fp16</annotation></semantics></math><span id="A4.SS9.p1.8.2" class="ltx_text" style="font-size:90%;"> mixed precision training. Time series, maze planning, compositionally, visual imitation experiments can be trained with a single </span><math id="A4.SS9.p1.2.m2.1" class="ltx_Math" alttext="2080Ti" display="inline"><semantics id="A4.SS9.p1.2.m2.1a"><mrow id="A4.SS9.p1.2.m2.1.1" xref="A4.SS9.p1.2.m2.1.1.cmml"><mn mathsize="90%" id="A4.SS9.p1.2.m2.1.1.2" xref="A4.SS9.p1.2.m2.1.1.2.cmml">2080</mn><mo lspace="0em" rspace="0em" id="A4.SS9.p1.2.m2.1.1.1" xref="A4.SS9.p1.2.m2.1.1.1.cmml">​</mo><mi mathsize="90%" id="A4.SS9.p1.2.m2.1.1.3" xref="A4.SS9.p1.2.m2.1.1.3.cmml">T</mi><mo lspace="0em" rspace="0em" id="A4.SS9.p1.2.m2.1.1.1a" xref="A4.SS9.p1.2.m2.1.1.1.cmml">​</mo><mi mathsize="90%" id="A4.SS9.p1.2.m2.1.1.4" xref="A4.SS9.p1.2.m2.1.1.4.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS9.p1.2.m2.1b"><apply id="A4.SS9.p1.2.m2.1.1.cmml" xref="A4.SS9.p1.2.m2.1.1"><times id="A4.SS9.p1.2.m2.1.1.1.cmml" xref="A4.SS9.p1.2.m2.1.1.1"></times><cn type="integer" id="A4.SS9.p1.2.m2.1.1.2.cmml" xref="A4.SS9.p1.2.m2.1.1.2">2080</cn><ci id="A4.SS9.p1.2.m2.1.1.3.cmml" xref="A4.SS9.p1.2.m2.1.1.3">𝑇</ci><ci id="A4.SS9.p1.2.m2.1.1.4.cmml" xref="A4.SS9.p1.2.m2.1.1.4">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p1.2.m2.1c">2080Ti</annotation></semantics></math><span id="A4.SS9.p1.8.3" class="ltx_text" style="font-size:90%;"> with </span><math id="A4.SS9.p1.3.m3.1" class="ltx_Math" alttext="11" display="inline"><semantics id="A4.SS9.p1.3.m3.1a"><mn mathsize="90%" id="A4.SS9.p1.3.m3.1.1" xref="A4.SS9.p1.3.m3.1.1.cmml">11</mn><annotation-xml encoding="MathML-Content" id="A4.SS9.p1.3.m3.1b"><cn type="integer" id="A4.SS9.p1.3.m3.1.1.cmml" xref="A4.SS9.p1.3.m3.1.1">11</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p1.3.m3.1c">11</annotation></semantics></math><span id="A4.SS9.p1.8.4" class="ltx_text" style="font-size:90%;">GB of memory. We tune the batch size such that we fully use the memory of GPUs. This translates to a batch size of </span><math id="A4.SS9.p1.4.m4.1" class="ltx_Math" alttext="2048" display="inline"><semantics id="A4.SS9.p1.4.m4.1a"><mn mathsize="90%" id="A4.SS9.p1.4.m4.1.1" xref="A4.SS9.p1.4.m4.1.1.cmml">2048</mn><annotation-xml encoding="MathML-Content" id="A4.SS9.p1.4.m4.1b"><cn type="integer" id="A4.SS9.p1.4.m4.1.1.cmml" xref="A4.SS9.p1.4.m4.1.1">2048</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p1.4.m4.1c">2048</annotation></semantics></math><span id="A4.SS9.p1.8.5" class="ltx_text" style="font-size:90%;"> for maze planning and compositional experiments, and </span><math id="A4.SS9.p1.5.m5.1" class="ltx_Math" alttext="32" display="inline"><semantics id="A4.SS9.p1.5.m5.1a"><mn mathsize="90%" id="A4.SS9.p1.5.m5.1.1" xref="A4.SS9.p1.5.m5.1.1.cmml">32</mn><annotation-xml encoding="MathML-Content" id="A4.SS9.p1.5.m5.1b"><cn type="integer" id="A4.SS9.p1.5.m5.1.1.cmml" xref="A4.SS9.p1.5.m5.1.1">32</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p1.5.m5.1c">32</annotation></semantics></math><span id="A4.SS9.p1.8.6" class="ltx_text" style="font-size:90%;"> for visual imitation learning. While we use early stopping on the validation set for time series experiments, we did not carefully search for the minimal number of training steps required, though the model usually converges between </span><math id="A4.SS9.p1.6.m6.1" class="ltx_Math" alttext="50" display="inline"><semantics id="A4.SS9.p1.6.m6.1a"><mn mathsize="90%" id="A4.SS9.p1.6.m6.1.1" xref="A4.SS9.p1.6.m6.1.1.cmml">50</mn><annotation-xml encoding="MathML-Content" id="A4.SS9.p1.6.m6.1b"><cn type="integer" id="A4.SS9.p1.6.m6.1.1.cmml" xref="A4.SS9.p1.6.m6.1.1">50</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p1.6.m6.1c">50</annotation></semantics></math><span id="A4.SS9.p1.8.7" class="ltx_text" style="font-size:90%;">k to </span><math id="A4.SS9.p1.7.m7.1" class="ltx_Math" alttext="100k" display="inline"><semantics id="A4.SS9.p1.7.m7.1a"><mrow id="A4.SS9.p1.7.m7.1.1" xref="A4.SS9.p1.7.m7.1.1.cmml"><mn mathsize="90%" id="A4.SS9.p1.7.m7.1.1.2" xref="A4.SS9.p1.7.m7.1.1.2.cmml">100</mn><mo lspace="0em" rspace="0em" id="A4.SS9.p1.7.m7.1.1.1" xref="A4.SS9.p1.7.m7.1.1.1.cmml">​</mo><mi mathsize="90%" id="A4.SS9.p1.7.m7.1.1.3" xref="A4.SS9.p1.7.m7.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS9.p1.7.m7.1b"><apply id="A4.SS9.p1.7.m7.1.1.cmml" xref="A4.SS9.p1.7.m7.1.1"><times id="A4.SS9.p1.7.m7.1.1.1.cmml" xref="A4.SS9.p1.7.m7.1.1.1"></times><cn type="integer" id="A4.SS9.p1.7.m7.1.1.2.cmml" xref="A4.SS9.p1.7.m7.1.1.2">100</cn><ci id="A4.SS9.p1.7.m7.1.1.3.cmml" xref="A4.SS9.p1.7.m7.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p1.7.m7.1c">100k</annotation></semantics></math><span id="A4.SS9.p1.8.8" class="ltx_text" style="font-size:90%;"> steps. The above environments thus usually take </span><math id="A4.SS9.p1.8.m8.1" class="ltx_Math" alttext="4-8" display="inline"><semantics id="A4.SS9.p1.8.m8.1a"><mrow id="A4.SS9.p1.8.m8.1.1" xref="A4.SS9.p1.8.m8.1.1.cmml"><mn mathsize="90%" id="A4.SS9.p1.8.m8.1.1.2" xref="A4.SS9.p1.8.m8.1.1.2.cmml">4</mn><mo mathsize="90%" id="A4.SS9.p1.8.m8.1.1.1" xref="A4.SS9.p1.8.m8.1.1.1.cmml">−</mo><mn mathsize="90%" id="A4.SS9.p1.8.m8.1.1.3" xref="A4.SS9.p1.8.m8.1.1.3.cmml">8</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS9.p1.8.m8.1b"><apply id="A4.SS9.p1.8.m8.1.1.cmml" xref="A4.SS9.p1.8.m8.1.1"><minus id="A4.SS9.p1.8.m8.1.1.1.cmml" xref="A4.SS9.p1.8.m8.1.1.1"></minus><cn type="integer" id="A4.SS9.p1.8.m8.1.1.2.cmml" xref="A4.SS9.p1.8.m8.1.1.2">4</cn><cn type="integer" id="A4.SS9.p1.8.m8.1.1.3.cmml" xref="A4.SS9.p1.8.m8.1.1.3">8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p1.8.m8.1c">4-8</annotation></semantics></math><span id="A4.SS9.p1.8.9" class="ltx_text" style="font-size:90%;"> hours to train although there is without doubt a significant potential for speed up.</span></p>
</div>
<div id="A4.SS9.p2" class="ltx_para">
<p id="A4.SS9.p2.5" class="ltx_p"><span id="A4.SS9.p2.5.1" class="ltx_text" style="font-size:90%;">Video prediction is GPU intensive. We use </span><math id="A4.SS9.p2.1.m1.1" class="ltx_Math" alttext="8" display="inline"><semantics id="A4.SS9.p2.1.m1.1a"><mn mathsize="90%" id="A4.SS9.p2.1.m1.1.1" xref="A4.SS9.p2.1.m1.1.1.cmml">8</mn><annotation-xml encoding="MathML-Content" id="A4.SS9.p2.1.m1.1b"><cn type="integer" id="A4.SS9.p2.1.m1.1.1.cmml" xref="A4.SS9.p2.1.m1.1.1">8</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p2.1.m1.1c">8</annotation></semantics></math><span id="A4.SS9.p2.5.2" class="ltx_text" style="font-size:90%;"> A100 GPUs for both video prediction datasets. We train for </span><math id="A4.SS9.p2.2.m2.1" class="ltx_Math" alttext="50K" display="inline"><semantics id="A4.SS9.p2.2.m2.1a"><mrow id="A4.SS9.p2.2.m2.1.1" xref="A4.SS9.p2.2.m2.1.1.cmml"><mn mathsize="90%" id="A4.SS9.p2.2.m2.1.1.2" xref="A4.SS9.p2.2.m2.1.1.2.cmml">50</mn><mo lspace="0em" rspace="0em" id="A4.SS9.p2.2.m2.1.1.1" xref="A4.SS9.p2.2.m2.1.1.1.cmml">​</mo><mi mathsize="90%" id="A4.SS9.p2.2.m2.1.1.3" xref="A4.SS9.p2.2.m2.1.1.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS9.p2.2.m2.1b"><apply id="A4.SS9.p2.2.m2.1.1.cmml" xref="A4.SS9.p2.2.m2.1.1"><times id="A4.SS9.p2.2.m2.1.1.1.cmml" xref="A4.SS9.p2.2.m2.1.1.1"></times><cn type="integer" id="A4.SS9.p2.2.m2.1.1.2.cmml" xref="A4.SS9.p2.2.m2.1.1.2">50</cn><ci id="A4.SS9.p2.2.m2.1.1.3.cmml" xref="A4.SS9.p2.2.m2.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p2.2.m2.1c">50K</annotation></semantics></math><span id="A4.SS9.p2.5.3" class="ltx_text" style="font-size:90%;"> steps with a batch size of </span><math id="A4.SS9.p2.3.m3.1" class="ltx_Math" alttext="8\times 16" display="inline"><semantics id="A4.SS9.p2.3.m3.1a"><mrow id="A4.SS9.p2.3.m3.1.1" xref="A4.SS9.p2.3.m3.1.1.cmml"><mn mathsize="90%" id="A4.SS9.p2.3.m3.1.1.2" xref="A4.SS9.p2.3.m3.1.1.2.cmml">8</mn><mo lspace="0.222em" mathsize="90%" rspace="0.222em" id="A4.SS9.p2.3.m3.1.1.1" xref="A4.SS9.p2.3.m3.1.1.1.cmml">×</mo><mn mathsize="90%" id="A4.SS9.p2.3.m3.1.1.3" xref="A4.SS9.p2.3.m3.1.1.3.cmml">16</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS9.p2.3.m3.1b"><apply id="A4.SS9.p2.3.m3.1.1.cmml" xref="A4.SS9.p2.3.m3.1.1"><times id="A4.SS9.p2.3.m3.1.1.1.cmml" xref="A4.SS9.p2.3.m3.1.1.1"></times><cn type="integer" id="A4.SS9.p2.3.m3.1.1.2.cmml" xref="A4.SS9.p2.3.m3.1.1.2">8</cn><cn type="integer" id="A4.SS9.p2.3.m3.1.1.3.cmml" xref="A4.SS9.p2.3.m3.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p2.3.m3.1c">8\times 16</annotation></semantics></math><span id="A4.SS9.p2.5.4" class="ltx_text" style="font-size:90%;">. It usually take </span><math id="A4.SS9.p2.4.m4.1" class="ltx_Math" alttext="12" display="inline"><semantics id="A4.SS9.p2.4.m4.1a"><mn mathsize="90%" id="A4.SS9.p2.4.m4.1.1" xref="A4.SS9.p2.4.m4.1.1.cmml">12</mn><annotation-xml encoding="MathML-Content" id="A4.SS9.p2.4.m4.1b"><cn type="integer" id="A4.SS9.p2.4.m4.1.1.cmml" xref="A4.SS9.p2.4.m4.1.1">12</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p2.4.m4.1c">12</annotation></semantics></math><span id="A4.SS9.p2.5.5" class="ltx_text" style="font-size:90%;"> hours to converge at </span><math id="A4.SS9.p2.5.m5.1" class="ltx_Math" alttext="40K" display="inline"><semantics id="A4.SS9.p2.5.m5.1a"><mrow id="A4.SS9.p2.5.m5.1.1" xref="A4.SS9.p2.5.m5.1.1.cmml"><mn mathsize="90%" id="A4.SS9.p2.5.m5.1.1.2" xref="A4.SS9.p2.5.m5.1.1.2.cmml">40</mn><mo lspace="0em" rspace="0em" id="A4.SS9.p2.5.m5.1.1.1" xref="A4.SS9.p2.5.m5.1.1.1.cmml">​</mo><mi mathsize="90%" id="A4.SS9.p2.5.m5.1.1.3" xref="A4.SS9.p2.5.m5.1.1.3.cmml">K</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS9.p2.5.m5.1b"><apply id="A4.SS9.p2.5.m5.1.1.cmml" xref="A4.SS9.p2.5.m5.1.1"><times id="A4.SS9.p2.5.m5.1.1.1.cmml" xref="A4.SS9.p2.5.m5.1.1.1"></times><cn type="integer" id="A4.SS9.p2.5.m5.1.1.2.cmml" xref="A4.SS9.p2.5.m5.1.1.2">40</cn><ci id="A4.SS9.p2.5.m5.1.1.3.cmml" xref="A4.SS9.p2.5.m5.1.1.3">𝐾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS9.p2.5.m5.1c">40K</annotation></semantics></math><span id="A4.SS9.p2.5.6" class="ltx_text" style="font-size:90%;"> steps of training (occasional validation time also included).</span></p>
</div>
</section>
</section>
<section id="A5" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix" style="font-size:90%;">
<span class="ltx_tag ltx_tag_appendix">Appendix E </span>Additional Experiment Results </h2>

<section id="A5.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:90%;">
<span class="ltx_tag ltx_tag_subsection">E.1 </span>Multivariate Probabilistic Time Series Forecasting</h3>

<figure id="A5.F6" class="ltx_figure"><img src="/html/2407.01392/assets/figures/ts_electricity.png" id="A5.F6.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="598" height="359" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:90%;"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>Prediction intervals of Diffusion Forcing for the first prediction window of the test set in the Electricity time series dataset. Only the first 16 features out of 370 are plotted.</figcaption>
</figure>
<div id="A5.SS1.p1" class="ltx_para">
<p id="A5.SS1.p1.1" class="ltx_p"><span id="A5.SS1.p1.1.1" class="ltx_text" style="font-size:90%;">To illustrate Diffusion Forcing’s new training objective does not degrade it as a generic sequence model, we evaluate Diffusion Forcing on high-dimensional and long-horizon sequence prediction tasks in time series prediction. We adopt multiple time series datasets with real-world applications from GluonTS </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A5.SS1.p1.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib2" title="" class="ltx_ref">2</a><span id="A5.SS1.p1.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A5.SS1.p1.1.4" class="ltx_text" style="font-size:90%;"> and evaluate Diffusion Forcing with strong baselines with standard metrics in this domain. In this section, we mainly focus on the results and analysis. For a detailed description of datasets and the metric, we refer the reader to Appendix </span><a href="#A6.SS4" title="F.4 Dataset for time series ‣ Appendix F Additional details about datasets ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">F.4</span></a><span id="A5.SS1.p1.1.5" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
<section id="A5.SS1.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Problem Formulation</h5>

<div id="A5.SS1.SSS0.Px1.p1" class="ltx_para">
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style="font-size:90%;"> be a sequence (multivariate time series) of </span><math id="A5.SS1.SSS0.Px1.p1.2.m2.1" class="ltx_Math" alttext="D" display="inline"><semantics id="A5.SS1.SSS0.Px1.p1.2.m2.1a"><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.2.m2.1.1" xref="A5.SS1.SSS0.Px1.p1.2.m2.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px1.p1.2.m2.1b"><ci id="A5.SS1.SSS0.Px1.p1.2.m2.1.1.cmml" xref="A5.SS1.SSS0.Px1.p1.2.m2.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px1.p1.2.m2.1c">D</annotation></semantics></math><span id="A5.SS1.SSS0.Px1.p1.12.3" class="ltx_text" style="font-size:90%;">-dimensional observations </span><math id="A5.SS1.SSS0.Px1.p1.3.m3.1" class="ltx_Math" alttext="\mathbf{x}_{t}\in\mathbb{R}^{D}" display="inline"><semantics id="A5.SS1.SSS0.Px1.p1.3.m3.1a"><mrow id="A5.SS1.SSS0.Px1.p1.3.m3.1.1" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.cmml"><msub id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.cmml"><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.2" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.2.cmml">𝐱</mi><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.3" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.3.cmml">t</mi></msub><mo mathsize="90%" id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.1" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.1.cmml">∈</mo><msup id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.cmml"><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.2" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.3" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.3.cmml">D</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px1.p1.3.m3.1b"><apply id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.cmml" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1"><in id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.1.cmml" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.1"></in><apply id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.1.cmml" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2">subscript</csymbol><ci id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.2.cmml" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.2">𝐱</ci><ci id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.3.cmml" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.2.3">𝑡</ci></apply><apply id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.cmml" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.1.cmml" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3">superscript</csymbol><ci id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.2.cmml" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.2">ℝ</ci><ci id="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.3.cmml" xref="A5.SS1.SSS0.Px1.p1.3.m3.1.1.3.3">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px1.p1.3.m3.1c">\mathbf{x}_{t}\in\mathbb{R}^{D}</annotation></semantics></math><span id="A5.SS1.SSS0.Px1.p1.12.4" class="ltx_text" style="font-size:90%;"> of some underlying dynamical process, sampled in discrete time steps </span><math id="A5.SS1.SSS0.Px1.p1.4.m4.3" class="ltx_Math" alttext="t\in\left\{1,\dots,T\right\}" display="inline"><semantics id="A5.SS1.SSS0.Px1.p1.4.m4.3a"><mrow id="A5.SS1.SSS0.Px1.p1.4.m4.3.4" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.cmml"><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.2" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.2.cmml">t</mi><mo mathsize="90%" id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.1" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.1.cmml">∈</mo><mrow id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.2" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.1.cmml"><mo id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.2.1" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.1.cmml">{</mo><mn mathsize="90%" id="A5.SS1.SSS0.Px1.p1.4.m4.1.1" xref="A5.SS1.SSS0.Px1.p1.4.m4.1.1.cmml">1</mn><mo mathsize="90%" id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.2.2" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.1.cmml">,</mo><mi mathsize="90%" mathvariant="normal" id="A5.SS1.SSS0.Px1.p1.4.m4.2.2" xref="A5.SS1.SSS0.Px1.p1.4.m4.2.2.cmml">…</mi><mo mathsize="90%" id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.2.3" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.1.cmml">,</mo><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.4.m4.3.3" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.3.cmml">T</mi><mo id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.2.4" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px1.p1.4.m4.3b"><apply id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.cmml" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4"><in id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.1.cmml" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.1"></in><ci id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.2.cmml" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.2">𝑡</ci><set id="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.1.cmml" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.4.3.2"><cn type="integer" id="A5.SS1.SSS0.Px1.p1.4.m4.1.1.cmml" xref="A5.SS1.SSS0.Px1.p1.4.m4.1.1">1</cn><ci id="A5.SS1.SSS0.Px1.p1.4.m4.2.2.cmml" xref="A5.SS1.SSS0.Px1.p1.4.m4.2.2">…</ci><ci id="A5.SS1.SSS0.Px1.p1.4.m4.3.3.cmml" xref="A5.SS1.SSS0.Px1.p1.4.m4.3.3">𝑇</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px1.p1.4.m4.3c">t\in\left\{1,\dots,T\right\}</annotation></semantics></math><span id="A5.SS1.SSS0.Px1.p1.12.5" class="ltx_text" style="font-size:90%;">, where </span><math id="A5.SS1.SSS0.Px1.p1.5.m5.1" class="ltx_Math" alttext="T\in\mathbb{N}" display="inline"><semantics id="A5.SS1.SSS0.Px1.p1.5.m5.1a"><mrow id="A5.SS1.SSS0.Px1.p1.5.m5.1.1" xref="A5.SS1.SSS0.Px1.p1.5.m5.1.1.cmml"><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.5.m5.1.1.2" xref="A5.SS1.SSS0.Px1.p1.5.m5.1.1.2.cmml">T</mi><mo mathsize="90%" id="A5.SS1.SSS0.Px1.p1.5.m5.1.1.1" xref="A5.SS1.SSS0.Px1.p1.5.m5.1.1.1.cmml">∈</mo><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.5.m5.1.1.3" xref="A5.SS1.SSS0.Px1.p1.5.m5.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px1.p1.5.m5.1b"><apply id="A5.SS1.SSS0.Px1.p1.5.m5.1.1.cmml" xref="A5.SS1.SSS0.Px1.p1.5.m5.1.1"><in id="A5.SS1.SSS0.Px1.p1.5.m5.1.1.1.cmml" xref="A5.SS1.SSS0.Px1.p1.5.m5.1.1.1"></in><ci id="A5.SS1.SSS0.Px1.p1.5.m5.1.1.2.cmml" xref="A5.SS1.SSS0.Px1.p1.5.m5.1.1.2">𝑇</ci><ci id="A5.SS1.SSS0.Px1.p1.5.m5.1.1.3.cmml" xref="A5.SS1.SSS0.Px1.p1.5.m5.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px1.p1.5.m5.1c">T\in\mathbb{N}</annotation></semantics></math><span id="A5.SS1.SSS0.Px1.p1.12.6" class="ltx_text" style="font-size:90%;">. In the problem setting of probabilistic time series forecasting, the sequence </span><math id="A5.SS1.SSS0.Px1.p1.6.m6.2" class="ltx_Math" alttext="\boldsymbol{X}=\left\{\boldsymbol{X}_{c},\boldsymbol{X}_{p}\right\}" display="inline"><semantics id="A5.SS1.SSS0.Px1.p1.6.m6.2a"><mrow id="A5.SS1.SSS0.Px1.p1.6.m6.2.2" xref="A5.SS1.SSS0.Px1.p1.6.m6.2.2.cmml"><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.6.m6.2.2.4" xref="A5.SS1.SSS0.Px1.p1.6.m6.2.2.4.cmml">𝑿</mi><mo mathsize="90%" id="A5.SS1.SSS0.Px1.p1.6.m6.2.2.3" xref="A5.SS1.SSS0.Px1.p1.6.m6.2.2.3.cmml">=</mo><mrow id="A5.SS1.SSS0.Px1.p1.6.m6.2.2.2.2" xref="A5.SS1.SSS0.Px1.p1.6.m6.2.2.2.3.cmml"><mo id="A5.SS1.SSS0.Px1.p1.6.m6.2.2.2.2.3" xref="A5.SS1.SSS0.Px1.p1.6.m6.2.2.2.3.cmml">{</mo><msub id="A5.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1" xref="A5.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.cmml"><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.2" xref="A5.SS1.SSS0.Px1.p1.6.m6.1.1.1.1.1.2.cmml">𝑿</mi><mi mathsize="90%" 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encoding="application/x-tex" id="A5.SS1.SSS0.Px1.p1.6.m6.2c">\boldsymbol{X}=\left\{\boldsymbol{X}_{c},\boldsymbol{X}_{p}\right\}</annotation></semantics></math><span id="A5.SS1.SSS0.Px1.p1.12.7" class="ltx_text" style="font-size:90%;"> is split into two subsequences at time step </span><math id="A5.SS1.SSS0.Px1.p1.7.m7.1" class="ltx_Math" alttext="t_{0}\in\mathbb{N}" display="inline"><semantics id="A5.SS1.SSS0.Px1.p1.7.m7.1a"><mrow id="A5.SS1.SSS0.Px1.p1.7.m7.1.1" xref="A5.SS1.SSS0.Px1.p1.7.m7.1.1.cmml"><msub id="A5.SS1.SSS0.Px1.p1.7.m7.1.1.2" xref="A5.SS1.SSS0.Px1.p1.7.m7.1.1.2.cmml"><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.7.m7.1.1.2.2" xref="A5.SS1.SSS0.Px1.p1.7.m7.1.1.2.2.cmml">t</mi><mn mathsize="90%" id="A5.SS1.SSS0.Px1.p1.7.m7.1.1.2.3" xref="A5.SS1.SSS0.Px1.p1.7.m7.1.1.2.3.cmml">0</mn></msub><mo mathsize="90%" id="A5.SS1.SSS0.Px1.p1.7.m7.1.1.1" xref="A5.SS1.SSS0.Px1.p1.7.m7.1.1.1.cmml">∈</mo><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.7.m7.1.1.3" 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style="font-size:90%;"> with </span><math id="A5.SS1.SSS0.Px1.p1.8.m8.1" class="ltx_Math" alttext="1&lt;t_{0}\leq T" display="inline"><semantics id="A5.SS1.SSS0.Px1.p1.8.m8.1a"><mrow id="A5.SS1.SSS0.Px1.p1.8.m8.1.1" xref="A5.SS1.SSS0.Px1.p1.8.m8.1.1.cmml"><mn mathsize="90%" id="A5.SS1.SSS0.Px1.p1.8.m8.1.1.2" xref="A5.SS1.SSS0.Px1.p1.8.m8.1.1.2.cmml">1</mn><mo mathsize="90%" id="A5.SS1.SSS0.Px1.p1.8.m8.1.1.3" xref="A5.SS1.SSS0.Px1.p1.8.m8.1.1.3.cmml">&lt;</mo><msub id="A5.SS1.SSS0.Px1.p1.8.m8.1.1.4" xref="A5.SS1.SSS0.Px1.p1.8.m8.1.1.4.cmml"><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.8.m8.1.1.4.2" xref="A5.SS1.SSS0.Px1.p1.8.m8.1.1.4.2.cmml">t</mi><mn mathsize="90%" id="A5.SS1.SSS0.Px1.p1.8.m8.1.1.4.3" xref="A5.SS1.SSS0.Px1.p1.8.m8.1.1.4.3.cmml">0</mn></msub><mo mathsize="90%" id="A5.SS1.SSS0.Px1.p1.8.m8.1.1.5" xref="A5.SS1.SSS0.Px1.p1.8.m8.1.1.5.cmml">≤</mo><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.8.m8.1.1.6" xref="A5.SS1.SSS0.Px1.p1.8.m8.1.1.6.cmml">T</mi></mrow><annotation-xml 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id="A5.SS1.SSS0.Px1.p1.11.m11.1.1.1.1.3.3.3.cmml" xref="A5.SS1.SSS0.Px1.p1.11.m11.1.1.1.1.3.3.3">0</cn></apply></apply></apply><ci id="A5.SS1.SSS0.Px1.p1.11.m11.1.1.1.3.cmml" xref="A5.SS1.SSS0.Px1.p1.11.m11.1.1.1.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px1.p1.11.m11.1c">\boldsymbol{X}_{p}:=\left\{\mathbf{x}_{t}\right\}_{t=t_{0}}^{T}</annotation></semantics></math><span id="A5.SS1.SSS0.Px1.p1.12.12" class="ltx_text" style="font-size:90%;"> of length </span><math id="A5.SS1.SSS0.Px1.p1.12.m12.1" class="ltx_Math" alttext="T-t_{0}+1" display="inline"><semantics id="A5.SS1.SSS0.Px1.p1.12.m12.1a"><mrow id="A5.SS1.SSS0.Px1.p1.12.m12.1.1" xref="A5.SS1.SSS0.Px1.p1.12.m12.1.1.cmml"><mrow id="A5.SS1.SSS0.Px1.p1.12.m12.1.1.2" xref="A5.SS1.SSS0.Px1.p1.12.m12.1.1.2.cmml"><mi mathsize="90%" id="A5.SS1.SSS0.Px1.p1.12.m12.1.1.2.2" xref="A5.SS1.SSS0.Px1.p1.12.m12.1.1.2.2.cmml">T</mi><mo mathsize="90%" id="A5.SS1.SSS0.Px1.p1.12.m12.1.1.2.1" 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Then, the task is to model the conditional joint probability distribution</span></p>
<table id="A5.E1" class="ltx_equation ltx_eqn_table">

<tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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id="A5.E1.m1.2.2.2.1.1.1.1.3.3.3.3.cmml" xref="A5.E1.m1.2.2.2.1.1.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.E1.m1.2c">q{\left(\mathbf{x}_{t_{0}:T}\mid\mathbf{x}_{1:t_{0}-1}\right)}:=\prod_{t=t_{0}}^{T}{q{\left(\mathbf{x}_{t}\mid\mathbf{x}_{1:t-1}\right)}}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(E.1)</span></td>
</tr></tbody>
</table>
<p id="A5.SS1.SSS0.Px1.p1.13" class="ltx_p"><span id="A5.SS1.SSS0.Px1.p1.13.1" class="ltx_text" style="font-size:90%;">over the samples in the prediction window. If we know the distribution in (</span><a href="#A5.E1" title="Equation E.1 ‣ Problem Formulation ‣ E.1 Multivariate Probabilistic Time Series Forecasting ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">E.1</span></a><span id="A5.SS1.SSS0.Px1.p1.13.2" class="ltx_text" style="font-size:90%;">), we can sample forecast prediction sequences given some initial context from the evidence sequence. 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encoding="application/x-tex" id="A5.SS1.SSS0.Px1.p1.13.m1.1c">q{\left(\mathbf{x}_{t_{0}:T}\mid\mathbf{x}_{1:t_{0}-1}\right)}</annotation></semantics></math><span id="A5.SS1.SSS0.Px1.p1.13.3" class="ltx_text" style="font-size:90%;">. Instead, we construct a statistical model that approximates the generative process in (</span><a href="#A5.E1" title="Equation E.1 ‣ Problem Formulation ‣ E.1 Multivariate Probabilistic Time Series Forecasting ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:90%;"><span class="ltx_text ltx_ref_tag">E.1</span></a><span id="A5.SS1.SSS0.Px1.p1.13.4" class="ltx_text" style="font-size:90%;">) and estimates quantiles via Monte Carlo sampling of simulated trajectories. In this way, confidence levels or uncertainty measures can be calculated, and point forecasts can be produced as the mean or median trajectory </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A5.SS1.SSS0.Px1.p1.13.5.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib35" title="" class="ltx_ref">35</a><span id="A5.SS1.SSS0.Px1.p1.13.6.2" class="ltx_text" style="font-size:90%;">]</span></cite><span id="A5.SS1.SSS0.Px1.p1.13.7" class="ltx_text" style="font-size:90%;">.</span></p>
</div>
</section>
<section id="A5.SS1.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:90%;">Results.</h5>

<div id="A5.SS1.SSS0.Px2.p1" class="ltx_para">
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caption = Results for time series forecasting. We report the test set </span><math id="A5.SS1.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="\text{CRPS}_{\textbf{sum}}" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.1.m1.1a"><msub id="A5.SS1.SSS0.Px2.p1.1.m1.1.1" xref="A5.SS1.SSS0.Px2.p1.1.m1.1.1.cmml"><mtext mathsize="90%" id="A5.SS1.SSS0.Px2.p1.1.m1.1.1.2" xref="A5.SS1.SSS0.Px2.p1.1.m1.1.1.2a.cmml">CRPS</mtext><mtext class="ltx_mathvariant_bold" mathsize="90%" id="A5.SS1.SSS0.Px2.p1.1.m1.1.1.3" xref="A5.SS1.SSS0.Px2.p1.1.m1.1.1.3a.cmml">sum</mtext></msub><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.1.m1.1b"><apply id="A5.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A5.SS1.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.1.m1.1.1">subscript</csymbol><ci id="A5.SS1.SSS0.Px2.p1.1.m1.1.1.2a.cmml" xref="A5.SS1.SSS0.Px2.p1.1.m1.1.1.2"><mtext mathsize="90%" id="A5.SS1.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="A5.SS1.SSS0.Px2.p1.1.m1.1.1.2">CRPS</mtext></ci><ci id="A5.SS1.SSS0.Px2.p1.1.m1.1.1.3a.cmml" xref="A5.SS1.SSS0.Px2.p1.1.m1.1.1.3"><mtext class="ltx_mathvariant_bold" mathsize="63%" id="A5.SS1.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="A5.SS1.SSS0.Px2.p1.1.m1.1.1.3">sum</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.1.m1.1c">\text{CRPS}_{\textbf{sum}}</annotation></semantics></math><span id="A5.SS1.SSS0.Px2.p1.71.72" class="ltx_text" style="font-size:90%;"> (the lower, the better) of comparable methods on six time series datasets. We measure the mean and standard deviation of our method from five runs trained with different seeds.,
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encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.3.2.m2.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.3.2.m2.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.3.2.m2.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.3.2.m2.1c">\pm</annotation></semantics></math> 0.003  0.880 <math id="A5.SS1.SSS0.Px2.p1.4.3.m3.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.4.3.m3.1a"><mo id="A5.SS1.SSS0.Px2.p1.4.3.m3.1.1" xref="A5.SS1.SSS0.Px2.p1.4.3.m3.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.4.3.m3.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.4.3.m3.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.4.3.m3.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.4.3.m3.1c">\pm</annotation></semantics></math> 0.004  0.350 <math id="A5.SS1.SSS0.Px2.p1.5.4.m4.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.5.4.m4.1a"><mo id="A5.SS1.SSS0.Px2.p1.5.4.m4.1.1" xref="A5.SS1.SSS0.Px2.p1.5.4.m4.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.5.4.m4.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.5.4.m4.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.5.4.m4.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.5.4.m4.1c">\pm</annotation></semantics></math> 0.002  -  - <span id="A5.SS1.SSS0.Px2.p1.71.70.3" class="ltx_ERROR undefined">\NN</span>VAR <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib43" title="" class="ltx_ref">43</a>]</cite>  0.005 <math id="A5.SS1.SSS0.Px2.p1.6.5.m5.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.6.5.m5.1a"><mo id="A5.SS1.SSS0.Px2.p1.6.5.m5.1.1" xref="A5.SS1.SSS0.Px2.p1.6.5.m5.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.6.5.m5.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.6.5.m5.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.6.5.m5.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.6.5.m5.1c">\pm</annotation></semantics></math> 0.000  0.830 <math id="A5.SS1.SSS0.Px2.p1.7.6.m6.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.7.6.m6.1a"><mo id="A5.SS1.SSS0.Px2.p1.7.6.m6.1.1" xref="A5.SS1.SSS0.Px2.p1.7.6.m6.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.7.6.m6.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.7.6.m6.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.7.6.m6.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.7.6.m6.1c">\pm</annotation></semantics></math> 0.006  0.039 <math id="A5.SS1.SSS0.Px2.p1.8.7.m7.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.8.7.m7.1a"><mo id="A5.SS1.SSS0.Px2.p1.8.7.m7.1.1" xref="A5.SS1.SSS0.Px2.p1.8.7.m7.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.8.7.m7.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.8.7.m7.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.8.7.m7.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.8.7.m7.1c">\pm</annotation></semantics></math> 0.001  0.290 <math id="A5.SS1.SSS0.Px2.p1.9.8.m8.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.9.8.m8.1a"><mo id="A5.SS1.SSS0.Px2.p1.9.8.m8.1.1" xref="A5.SS1.SSS0.Px2.p1.9.8.m8.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.9.8.m8.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.9.8.m8.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.9.8.m8.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.9.8.m8.1c">\pm</annotation></semantics></math> 0.001  -  - <span id="A5.SS1.SSS0.Px2.p1.71.70.4" class="ltx_ERROR undefined">\NN</span>VAR-Lasso <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib43" title="" class="ltx_ref">43</a>]</cite>  0.012 <math id="A5.SS1.SSS0.Px2.p1.10.9.m9.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.10.9.m9.1a"><mo id="A5.SS1.SSS0.Px2.p1.10.9.m9.1.1" xref="A5.SS1.SSS0.Px2.p1.10.9.m9.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.10.9.m9.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.10.9.m9.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.10.9.m9.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.10.9.m9.1c">\pm</annotation></semantics></math> 0.000  0.510 <math id="A5.SS1.SSS0.Px2.p1.11.10.m10.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.11.10.m10.1a"><mo id="A5.SS1.SSS0.Px2.p1.11.10.m10.1.1" xref="A5.SS1.SSS0.Px2.p1.11.10.m10.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.11.10.m10.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.11.10.m10.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.11.10.m10.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.11.10.m10.1c">\pm</annotation></semantics></math> 0.006  0.025 <math id="A5.SS1.SSS0.Px2.p1.12.11.m11.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.12.11.m11.1a"><mo id="A5.SS1.SSS0.Px2.p1.12.11.m11.1.1" xref="A5.SS1.SSS0.Px2.p1.12.11.m11.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.12.11.m11.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.12.11.m11.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.12.11.m11.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.12.11.m11.1c">\pm</annotation></semantics></math> 0.000  0.150 <math id="A5.SS1.SSS0.Px2.p1.13.12.m12.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.13.12.m12.1a"><mo id="A5.SS1.SSS0.Px2.p1.13.12.m12.1.1" xref="A5.SS1.SSS0.Px2.p1.13.12.m12.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.13.12.m12.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.13.12.m12.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.13.12.m12.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.13.12.m12.1c">\pm</annotation></semantics></math> 0.002  -  3.100 <math id="A5.SS1.SSS0.Px2.p1.14.13.m13.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.14.13.m13.1a"><mo id="A5.SS1.SSS0.Px2.p1.14.13.m13.1.1" xref="A5.SS1.SSS0.Px2.p1.14.13.m13.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.14.13.m13.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.14.13.m13.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.14.13.m13.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.14.13.m13.1c">\pm</annotation></semantics></math> 0.004 <span id="A5.SS1.SSS0.Px2.p1.71.70.5" class="ltx_ERROR undefined">\NN</span>GARCH <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib59" title="" class="ltx_ref">59</a>]</cite>  0.023 <math id="A5.SS1.SSS0.Px2.p1.15.14.m14.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.15.14.m14.1a"><mo id="A5.SS1.SSS0.Px2.p1.15.14.m14.1.1" xref="A5.SS1.SSS0.Px2.p1.15.14.m14.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.15.14.m14.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.15.14.m14.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.15.14.m14.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.15.14.m14.1c">\pm</annotation></semantics></math> 0.000  0.880 <math id="A5.SS1.SSS0.Px2.p1.16.15.m15.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.16.15.m15.1a"><mo id="A5.SS1.SSS0.Px2.p1.16.15.m15.1.1" xref="A5.SS1.SSS0.Px2.p1.16.15.m15.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.16.15.m15.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.16.15.m15.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.16.15.m15.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.16.15.m15.1c">\pm</annotation></semantics></math> 0.002  0.190 <math id="A5.SS1.SSS0.Px2.p1.17.16.m16.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.17.16.m16.1a"><mo id="A5.SS1.SSS0.Px2.p1.17.16.m16.1.1" xref="A5.SS1.SSS0.Px2.p1.17.16.m16.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.17.16.m16.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.17.16.m16.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.17.16.m16.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.17.16.m16.1c">\pm</annotation></semantics></math> 0.001  0.370 <math id="A5.SS1.SSS0.Px2.p1.18.17.m17.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.18.17.m17.1a"><mo id="A5.SS1.SSS0.Px2.p1.18.17.m17.1.1" xref="A5.SS1.SSS0.Px2.p1.18.17.m17.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.18.17.m17.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.18.17.m17.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.18.17.m17.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.18.17.m17.1c">\pm</annotation></semantics></math> 0.001  -  - <span id="A5.SS1.SSS0.Px2.p1.71.70.6" class="ltx_ERROR undefined">\NN</span>DeepAR <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib52" title="" class="ltx_ref">52</a>]</cite>  -  0.336 <math id="A5.SS1.SSS0.Px2.p1.19.18.m18.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.19.18.m18.1a"><mo id="A5.SS1.SSS0.Px2.p1.19.18.m18.1.1" xref="A5.SS1.SSS0.Px2.p1.19.18.m18.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.19.18.m18.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.19.18.m18.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.19.18.m18.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.19.18.m18.1c">\pm</annotation></semantics></math> 0.014  0.023 <math id="A5.SS1.SSS0.Px2.p1.20.19.m19.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.20.19.m19.1a"><mo id="A5.SS1.SSS0.Px2.p1.20.19.m19.1.1" xref="A5.SS1.SSS0.Px2.p1.20.19.m19.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.20.19.m19.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.20.19.m19.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.20.19.m19.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.20.19.m19.1c">\pm</annotation></semantics></math> 0.001  0.055 <math id="A5.SS1.SSS0.Px2.p1.21.20.m20.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.21.20.m20.1a"><mo id="A5.SS1.SSS0.Px2.p1.21.20.m20.1.1" xref="A5.SS1.SSS0.Px2.p1.21.20.m20.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.21.20.m20.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.21.20.m20.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.21.20.m20.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.21.20.m20.1c">\pm</annotation></semantics></math> 0.003  -  0.127 <math id="A5.SS1.SSS0.Px2.p1.22.21.m21.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.22.21.m21.1a"><mo id="A5.SS1.SSS0.Px2.p1.22.21.m21.1.1" xref="A5.SS1.SSS0.Px2.p1.22.21.m21.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.22.21.m21.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.22.21.m21.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.22.21.m21.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.22.21.m21.1c">\pm</annotation></semantics></math> 0.042 <span id="A5.SS1.SSS0.Px2.p1.71.70.7" class="ltx_ERROR undefined">\NN</span>LSTM-Copula <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib51" title="" class="ltx_ref">51</a>]</cite>  0.007 <math id="A5.SS1.SSS0.Px2.p1.23.22.m22.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.23.22.m22.1a"><mo id="A5.SS1.SSS0.Px2.p1.23.22.m22.1.1" xref="A5.SS1.SSS0.Px2.p1.23.22.m22.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.23.22.m22.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.23.22.m22.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.23.22.m22.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.23.22.m22.1c">\pm</annotation></semantics></math> 0.000  0.319 <math id="A5.SS1.SSS0.Px2.p1.24.23.m23.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.24.23.m23.1a"><mo id="A5.SS1.SSS0.Px2.p1.24.23.m23.1.1" xref="A5.SS1.SSS0.Px2.p1.24.23.m23.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.24.23.m23.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.24.23.m23.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.24.23.m23.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.24.23.m23.1c">\pm</annotation></semantics></math> 0.011  0.064 <math id="A5.SS1.SSS0.Px2.p1.25.24.m24.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.25.24.m24.1a"><mo id="A5.SS1.SSS0.Px2.p1.25.24.m24.1.1" xref="A5.SS1.SSS0.Px2.p1.25.24.m24.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.25.24.m24.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.25.24.m24.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.25.24.m24.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.25.24.m24.1c">\pm</annotation></semantics></math> 0.008  0.103 <math id="A5.SS1.SSS0.Px2.p1.26.25.m25.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.26.25.m25.1a"><mo id="A5.SS1.SSS0.Px2.p1.26.25.m25.1.1" xref="A5.SS1.SSS0.Px2.p1.26.25.m25.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.26.25.m25.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.26.25.m25.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.26.25.m25.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.26.25.m25.1c">\pm</annotation></semantics></math> 0.006  0.326 <math id="A5.SS1.SSS0.Px2.p1.27.26.m26.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.27.26.m26.1a"><mo id="A5.SS1.SSS0.Px2.p1.27.26.m26.1.1" xref="A5.SS1.SSS0.Px2.p1.27.26.m26.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.27.26.m26.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.27.26.m26.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.27.26.m26.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.27.26.m26.1c">\pm</annotation></semantics></math> 0.007  0.241 <math id="A5.SS1.SSS0.Px2.p1.28.27.m27.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.28.27.m27.1a"><mo id="A5.SS1.SSS0.Px2.p1.28.27.m27.1.1" xref="A5.SS1.SSS0.Px2.p1.28.27.m27.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.28.27.m27.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.28.27.m27.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.28.27.m27.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.28.27.m27.1c">\pm</annotation></semantics></math> 0.033 <span id="A5.SS1.SSS0.Px2.p1.71.70.8" class="ltx_ERROR undefined">\NN</span>GP-Copula <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib51" title="" class="ltx_ref">51</a>]</cite>  0.007 <math id="A5.SS1.SSS0.Px2.p1.29.28.m28.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.29.28.m28.1a"><mo id="A5.SS1.SSS0.Px2.p1.29.28.m28.1.1" xref="A5.SS1.SSS0.Px2.p1.29.28.m28.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.29.28.m28.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.29.28.m28.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.29.28.m28.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.29.28.m28.1c">\pm</annotation></semantics></math> 0.000  0.337 <math id="A5.SS1.SSS0.Px2.p1.30.29.m29.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.30.29.m29.1a"><mo id="A5.SS1.SSS0.Px2.p1.30.29.m29.1.1" xref="A5.SS1.SSS0.Px2.p1.30.29.m29.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.30.29.m29.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.30.29.m29.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.30.29.m29.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.30.29.m29.1c">\pm</annotation></semantics></math> 0.024  0.025 <math id="A5.SS1.SSS0.Px2.p1.31.30.m30.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.31.30.m30.1a"><mo id="A5.SS1.SSS0.Px2.p1.31.30.m30.1.1" xref="A5.SS1.SSS0.Px2.p1.31.30.m30.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.31.30.m30.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.31.30.m30.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.31.30.m30.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.31.30.m30.1c">\pm</annotation></semantics></math> 0.002  0.078 <math id="A5.SS1.SSS0.Px2.p1.32.31.m31.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.32.31.m31.1a"><mo id="A5.SS1.SSS0.Px2.p1.32.31.m31.1.1" xref="A5.SS1.SSS0.Px2.p1.32.31.m31.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.32.31.m31.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.32.31.m31.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.32.31.m31.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.32.31.m31.1c">\pm</annotation></semantics></math> 0.002  0.208 <math id="A5.SS1.SSS0.Px2.p1.33.32.m32.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.33.32.m32.1a"><mo id="A5.SS1.SSS0.Px2.p1.33.32.m32.1.1" xref="A5.SS1.SSS0.Px2.p1.33.32.m32.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.33.32.m32.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.33.32.m32.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.33.32.m32.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.33.32.m32.1c">\pm</annotation></semantics></math> 0.183  0.086 <math id="A5.SS1.SSS0.Px2.p1.34.33.m33.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.34.33.m33.1a"><mo id="A5.SS1.SSS0.Px2.p1.34.33.m33.1.1" xref="A5.SS1.SSS0.Px2.p1.34.33.m33.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.34.33.m33.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.34.33.m33.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.34.33.m33.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.34.33.m33.1c">\pm</annotation></semantics></math> 0.004 <span id="A5.SS1.SSS0.Px2.p1.71.70.9" class="ltx_ERROR undefined">\NN</span>KVAE <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib39" title="" class="ltx_ref">39</a>]</cite>  0.014 <math id="A5.SS1.SSS0.Px2.p1.35.34.m34.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.35.34.m34.1a"><mo id="A5.SS1.SSS0.Px2.p1.35.34.m34.1.1" xref="A5.SS1.SSS0.Px2.p1.35.34.m34.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.35.34.m34.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.35.34.m34.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.35.34.m34.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.35.34.m34.1c">\pm</annotation></semantics></math> 0.002  0.340 <math id="A5.SS1.SSS0.Px2.p1.36.35.m35.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.36.35.m35.1a"><mo id="A5.SS1.SSS0.Px2.p1.36.35.m35.1.1" xref="A5.SS1.SSS0.Px2.p1.36.35.m35.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.36.35.m35.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.36.35.m35.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.36.35.m35.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.36.35.m35.1c">\pm</annotation></semantics></math> 0.025  0.051 <math id="A5.SS1.SSS0.Px2.p1.37.36.m36.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.37.36.m36.1a"><mo id="A5.SS1.SSS0.Px2.p1.37.36.m36.1.1" xref="A5.SS1.SSS0.Px2.p1.37.36.m36.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.37.36.m36.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.37.36.m36.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.37.36.m36.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.37.36.m36.1c">\pm</annotation></semantics></math> 0.019  0.100 <math id="A5.SS1.SSS0.Px2.p1.38.37.m37.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.38.37.m37.1a"><mo id="A5.SS1.SSS0.Px2.p1.38.37.m37.1.1" xref="A5.SS1.SSS0.Px2.p1.38.37.m37.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.38.37.m37.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.38.37.m37.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.38.37.m37.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.38.37.m37.1c">\pm</annotation></semantics></math> 0.005  -  0.095 <math id="A5.SS1.SSS0.Px2.p1.39.38.m38.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.39.38.m38.1a"><mo id="A5.SS1.SSS0.Px2.p1.39.38.m38.1.1" xref="A5.SS1.SSS0.Px2.p1.39.38.m38.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.39.38.m38.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.39.38.m38.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.39.38.m38.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.39.38.m38.1c">\pm</annotation></semantics></math> 0.012 <span id="A5.SS1.SSS0.Px2.p1.71.70.10" class="ltx_ERROR undefined">\NN</span>NKF <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib14" title="" class="ltx_ref">14</a>]</cite>  -  0.320 <math id="A5.SS1.SSS0.Px2.p1.40.39.m39.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.40.39.m39.1a"><mo id="A5.SS1.SSS0.Px2.p1.40.39.m39.1.1" xref="A5.SS1.SSS0.Px2.p1.40.39.m39.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.40.39.m39.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.40.39.m39.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.40.39.m39.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.40.39.m39.1c">\pm</annotation></semantics></math> 0.020  0.016 <math id="A5.SS1.SSS0.Px2.p1.41.40.m40.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.41.40.m40.1a"><mo id="A5.SS1.SSS0.Px2.p1.41.40.m40.1.1" xref="A5.SS1.SSS0.Px2.p1.41.40.m40.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.41.40.m40.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.41.40.m40.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.41.40.m40.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.41.40.m40.1c">\pm</annotation></semantics></math> 0.001  0.100 <math id="A5.SS1.SSS0.Px2.p1.42.41.m41.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.42.41.m41.1a"><mo id="A5.SS1.SSS0.Px2.p1.42.41.m41.1.1" xref="A5.SS1.SSS0.Px2.p1.42.41.m41.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.42.41.m41.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.42.41.m41.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.42.41.m41.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.42.41.m41.1c">\pm</annotation></semantics></math> 0.002  -  0.071 <math id="A5.SS1.SSS0.Px2.p1.43.42.m42.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.43.42.m42.1a"><mo id="A5.SS1.SSS0.Px2.p1.43.42.m42.1.1" xref="A5.SS1.SSS0.Px2.p1.43.42.m42.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.43.42.m42.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.43.42.m42.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.43.42.m42.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.43.42.m42.1c">\pm</annotation></semantics></math> 0.002 <span id="A5.SS1.SSS0.Px2.p1.71.70.11" class="ltx_ERROR undefined">\NN</span>Transformer-MAF <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib49" title="" class="ltx_ref">49</a>]</cite>  0.005 <math id="A5.SS1.SSS0.Px2.p1.44.43.m43.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.44.43.m43.1a"><mo id="A5.SS1.SSS0.Px2.p1.44.43.m43.1.1" xref="A5.SS1.SSS0.Px2.p1.44.43.m43.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.44.43.m43.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.44.43.m43.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.44.43.m43.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.44.43.m43.1c">\pm</annotation></semantics></math> 0.003  0.301 <math id="A5.SS1.SSS0.Px2.p1.45.44.m44.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.45.44.m44.1a"><mo id="A5.SS1.SSS0.Px2.p1.45.44.m44.1.1" xref="A5.SS1.SSS0.Px2.p1.45.44.m44.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.45.44.m44.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.45.44.m44.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.45.44.m44.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.45.44.m44.1c">\pm</annotation></semantics></math> 0.014  0.021 <math id="A5.SS1.SSS0.Px2.p1.46.45.m45.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.46.45.m45.1a"><mo id="A5.SS1.SSS0.Px2.p1.46.45.m45.1.1" xref="A5.SS1.SSS0.Px2.p1.46.45.m45.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.46.45.m45.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.46.45.m45.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.46.45.m45.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.46.45.m45.1c">\pm</annotation></semantics></math> 0.000  0.056 <math id="A5.SS1.SSS0.Px2.p1.47.46.m46.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.47.46.m46.1a"><mo id="A5.SS1.SSS0.Px2.p1.47.46.m46.1.1" xref="A5.SS1.SSS0.Px2.p1.47.46.m46.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.47.46.m46.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.47.46.m46.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.47.46.m46.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.47.46.m46.1c">\pm</annotation></semantics></math> 0.001  0.179 <math id="A5.SS1.SSS0.Px2.p1.48.47.m47.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.48.47.m47.1a"><mo id="A5.SS1.SSS0.Px2.p1.48.47.m47.1.1" xref="A5.SS1.SSS0.Px2.p1.48.47.m47.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.48.47.m47.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.48.47.m47.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.48.47.m47.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.48.47.m47.1c">\pm</annotation></semantics></math> 0.002  0.063 <math id="A5.SS1.SSS0.Px2.p1.49.48.m48.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.49.48.m48.1a"><mo id="A5.SS1.SSS0.Px2.p1.49.48.m48.1.1" xref="A5.SS1.SSS0.Px2.p1.49.48.m48.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.49.48.m48.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.49.48.m48.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.49.48.m48.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.49.48.m48.1c">\pm</annotation></semantics></math> 0.003 <span id="A5.SS1.SSS0.Px2.p1.71.70.12" class="ltx_ERROR undefined">\NN</span>TimeGrad <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib48" title="" class="ltx_ref">48</a>]</cite>  0.006 <math id="A5.SS1.SSS0.Px2.p1.50.49.m49.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.50.49.m49.1a"><mo id="A5.SS1.SSS0.Px2.p1.50.49.m49.1.1" xref="A5.SS1.SSS0.Px2.p1.50.49.m49.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.50.49.m49.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.50.49.m49.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.50.49.m49.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.50.49.m49.1c">\pm</annotation></semantics></math> 0.001  0.287 <math id="A5.SS1.SSS0.Px2.p1.51.50.m50.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.51.50.m50.1a"><mo id="A5.SS1.SSS0.Px2.p1.51.50.m50.1.1" xref="A5.SS1.SSS0.Px2.p1.51.50.m50.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.51.50.m50.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.51.50.m50.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.51.50.m50.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.51.50.m50.1c">\pm</annotation></semantics></math> 0.020  0.021 <math id="A5.SS1.SSS0.Px2.p1.52.51.m51.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.52.51.m51.1a"><mo id="A5.SS1.SSS0.Px2.p1.52.51.m51.1.1" xref="A5.SS1.SSS0.Px2.p1.52.51.m51.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.52.51.m51.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.52.51.m51.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.52.51.m51.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.52.51.m51.1c">\pm</annotation></semantics></math> 0.001  0.044 <math id="A5.SS1.SSS0.Px2.p1.53.52.m52.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.53.52.m52.1a"><mo id="A5.SS1.SSS0.Px2.p1.53.52.m52.1.1" xref="A5.SS1.SSS0.Px2.p1.53.52.m52.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.53.52.m52.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.53.52.m52.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.53.52.m52.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.53.52.m52.1c">\pm</annotation></semantics></math> 0.006  0.114 <math id="A5.SS1.SSS0.Px2.p1.54.53.m53.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.54.53.m53.1a"><mo id="A5.SS1.SSS0.Px2.p1.54.53.m53.1.1" xref="A5.SS1.SSS0.Px2.p1.54.53.m53.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.54.53.m53.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.54.53.m53.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.54.53.m53.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.54.53.m53.1c">\pm</annotation></semantics></math> 0.020  0.049 <math id="A5.SS1.SSS0.Px2.p1.55.54.m54.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.55.54.m54.1a"><mo id="A5.SS1.SSS0.Px2.p1.55.54.m54.1.1" xref="A5.SS1.SSS0.Px2.p1.55.54.m54.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.55.54.m54.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.55.54.m54.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.55.54.m54.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.55.54.m54.1c">\pm</annotation></semantics></math> 0.002 <span id="A5.SS1.SSS0.Px2.p1.71.70.13" class="ltx_ERROR undefined">\NN</span>ScoreGrad sub-VP SDE <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib64" title="" class="ltx_ref">64</a>]</cite>  0.006 <math id="A5.SS1.SSS0.Px2.p1.56.55.m55.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.56.55.m55.1a"><mo id="A5.SS1.SSS0.Px2.p1.56.55.m55.1.1" xref="A5.SS1.SSS0.Px2.p1.56.55.m55.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.56.55.m55.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.56.55.m55.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.56.55.m55.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.56.55.m55.1c">\pm</annotation></semantics></math> 0.001  <span id="A5.SS1.SSS0.Px2.p1.71.70.14" class="ltx_text ltx_font_bold">0.256</span> <math id="A5.SS1.SSS0.Px2.p1.57.56.m56.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.57.56.m56.1a"><mo id="A5.SS1.SSS0.Px2.p1.57.56.m56.1.1" xref="A5.SS1.SSS0.Px2.p1.57.56.m56.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.57.56.m56.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.57.56.m56.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.57.56.m56.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.57.56.m56.1c">\pm</annotation></semantics></math> <span id="A5.SS1.SSS0.Px2.p1.71.70.15" class="ltx_text ltx_font_bold">0.015</span>  <span id="A5.SS1.SSS0.Px2.p1.71.70.16" class="ltx_text ltx_font_bold">0.019</span> <math id="A5.SS1.SSS0.Px2.p1.58.57.m57.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.58.57.m57.1a"><mo id="A5.SS1.SSS0.Px2.p1.58.57.m57.1.1" xref="A5.SS1.SSS0.Px2.p1.58.57.m57.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.58.57.m57.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.58.57.m57.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.58.57.m57.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.58.57.m57.1c">\pm</annotation></semantics></math> <span id="A5.SS1.SSS0.Px2.p1.71.70.17" class="ltx_text ltx_font_bold">0.001</span>  0.041 <math id="A5.SS1.SSS0.Px2.p1.59.58.m58.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.59.58.m58.1a"><mo id="A5.SS1.SSS0.Px2.p1.59.58.m58.1.1" xref="A5.SS1.SSS0.Px2.p1.59.58.m58.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.59.58.m58.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.59.58.m58.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.59.58.m58.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.59.58.m58.1c">\pm</annotation></semantics></math> 0.004  0.101 <math id="A5.SS1.SSS0.Px2.p1.60.59.m59.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.60.59.m59.1a"><mo id="A5.SS1.SSS0.Px2.p1.60.59.m59.1.1" xref="A5.SS1.SSS0.Px2.p1.60.59.m59.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.60.59.m59.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.60.59.m59.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.60.59.m59.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.60.59.m59.1c">\pm</annotation></semantics></math> 0.004  <span id="A5.SS1.SSS0.Px2.p1.71.70.18" class="ltx_text ltx_font_bold">0.043</span> <math id="A5.SS1.SSS0.Px2.p1.61.60.m60.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.61.60.m60.1a"><mo id="A5.SS1.SSS0.Px2.p1.61.60.m60.1.1" xref="A5.SS1.SSS0.Px2.p1.61.60.m60.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.61.60.m60.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.61.60.m60.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.61.60.m60.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.61.60.m60.1c">\pm</annotation></semantics></math> <span id="A5.SS1.SSS0.Px2.p1.71.70.19" class="ltx_text ltx_font_bold">0.002</span> <span id="A5.SS1.SSS0.Px2.p1.71.70.20" class="ltx_ERROR undefined">\NN</span><span id="A5.SS1.SSS0.Px2.p1.71.70.21" class="ltx_text ltx_font_bold">Ours</span>  <span id="A5.SS1.SSS0.Px2.p1.71.70.22" class="ltx_text ltx_font_bold">0.003</span> <math id="A5.SS1.SSS0.Px2.p1.62.61.m61.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.62.61.m61.1a"><mo id="A5.SS1.SSS0.Px2.p1.62.61.m61.1.1" xref="A5.SS1.SSS0.Px2.p1.62.61.m61.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.62.61.m61.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.62.61.m61.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.62.61.m61.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.62.61.m61.1c">\pm</annotation></semantics></math> <span id="A5.SS1.SSS0.Px2.p1.71.70.23" class="ltx_text ltx_font_bold">0.001</span>  0.289 <math id="A5.SS1.SSS0.Px2.p1.63.62.m62.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.63.62.m62.1a"><mo id="A5.SS1.SSS0.Px2.p1.63.62.m62.1.1" xref="A5.SS1.SSS0.Px2.p1.63.62.m62.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.63.62.m62.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.63.62.m62.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.63.62.m62.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.63.62.m62.1c">\pm</annotation></semantics></math> 0.002  0.023 <math id="A5.SS1.SSS0.Px2.p1.64.63.m63.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.64.63.m63.1a"><mo id="A5.SS1.SSS0.Px2.p1.64.63.m63.1.1" xref="A5.SS1.SSS0.Px2.p1.64.63.m63.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.64.63.m63.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.64.63.m63.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.64.63.m63.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.64.63.m63.1c">\pm</annotation></semantics></math> 0.001  <span id="A5.SS1.SSS0.Px2.p1.71.70.24" class="ltx_text ltx_font_bold">0.040</span> <math id="A5.SS1.SSS0.Px2.p1.65.64.m64.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.65.64.m64.1a"><mo id="A5.SS1.SSS0.Px2.p1.65.64.m64.1.1" xref="A5.SS1.SSS0.Px2.p1.65.64.m64.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.65.64.m64.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.65.64.m64.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.65.64.m64.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.65.64.m64.1c">\pm</annotation></semantics></math> <span id="A5.SS1.SSS0.Px2.p1.71.70.25" class="ltx_text ltx_font_bold">0.004</span>  <span id="A5.SS1.SSS0.Px2.p1.71.70.26" class="ltx_text ltx_font_bold">0.075</span> <math id="A5.SS1.SSS0.Px2.p1.66.65.m65.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.66.65.m65.1a"><mo id="A5.SS1.SSS0.Px2.p1.66.65.m65.1.1" xref="A5.SS1.SSS0.Px2.p1.66.65.m65.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.66.65.m65.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.66.65.m65.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.66.65.m65.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.66.65.m65.1c">\pm</annotation></semantics></math> <span id="A5.SS1.SSS0.Px2.p1.71.70.27" class="ltx_text ltx_font_bold">0.002</span>  0.085 <math id="A5.SS1.SSS0.Px2.p1.67.66.m66.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.67.66.m66.1a"><mo id="A5.SS1.SSS0.Px2.p1.67.66.m66.1.1" xref="A5.SS1.SSS0.Px2.p1.67.66.m66.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.67.66.m66.1b"><csymbol cd="latexml" id="A5.SS1.SSS0.Px2.p1.67.66.m66.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.67.66.m66.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.67.66.m66.1c">\pm</annotation></semantics></math> 0.007
<span id="A5.SS1.SSS0.Px2.p1.71.70.28" class="ltx_ERROR undefined">\LL</span>

We evaluate the effectiveness of Diffusion Forcing as a sequence model on the canonical task of multivariate time series forecasting by following the experiment setup of  <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib51" title="" class="ltx_ref">51</a>, <a href="#bib.bib49" title="" class="ltx_ref">49</a>, <a href="#bib.bib48" title="" class="ltx_ref">48</a>, <a href="#bib.bib56" title="" class="ltx_ref">56</a>, <a href="#bib.bib64" title="" class="ltx_ref">64</a>]</cite>
Concretely, we benchmark Diffusion Forcing on the datasets Solar, Electricity, Traffic, Taxi, and Wikipedia. These datasets have different dimensionality, domains, sampling frequencies, and capture seasonal patterns of different lengths. The features of each dataset are detailed in Table <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:tab:ts_data</span>. We access the datasets from GluonTS <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib2" title="" class="ltx_ref">2</a>]</cite>, and set the context and prediction windows to the same length for each dataset. Additionally, we employ the same covariates as <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib48" title="" class="ltx_ref">48</a>]</cite>.
We evaluate the performance of the model quantitatively by estimating the Summed Continuous Ranked Probability Score <math id="A5.SS1.SSS0.Px2.p1.68.67.m67.1" class="ltx_Math" alttext="\operatorname{CRPS}_{\text{sum}}" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.68.67.m67.1a"><msub id="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1" xref="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.cmml"><mi id="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.2" xref="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.2.cmml">CRPS</mi><mtext id="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.3" xref="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.3a.cmml">sum</mtext></msub><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.68.67.m67.1b"><apply id="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1"><csymbol cd="ambiguous" id="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1">subscript</csymbol><ci id="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.2.cmml" xref="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.2">CRPS</ci><ci id="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.3a.cmml" xref="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.3"><mtext id="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.3.cmml" xref="A5.SS1.SSS0.Px2.p1.68.67.m67.1.1.3">sum</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.68.67.m67.1c">\operatorname{CRPS}_{\text{sum}}</annotation></semantics></math> via quantiles. As a metric, <math id="A5.SS1.SSS0.Px2.p1.69.68.m68.1" class="ltx_Math" alttext="\operatorname{CRPS}_{\text{sum}}" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.69.68.m68.1a"><msub id="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1" xref="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.cmml"><mi id="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.2" xref="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.2.cmml">CRPS</mi><mtext id="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.3" xref="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.3a.cmml">sum</mtext></msub><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.69.68.m68.1b"><apply id="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1"><csymbol cd="ambiguous" id="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1">subscript</csymbol><ci id="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.2.cmml" xref="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.2">CRPS</ci><ci id="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.3a.cmml" xref="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.3"><mtext id="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.3.cmml" xref="A5.SS1.SSS0.Px2.p1.69.68.m68.1.1.3">sum</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.69.68.m68.1c">\operatorname{CRPS}_{\text{sum}}</annotation></semantics></math> measures how well a forecast distribution matches the ground truth distribution. We provide detailed descriptions of the metric in Appendix <a href="#A6.SS4" title="F.4 Dataset for time series ‣ Appendix F Additional details about datasets ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">F.4</span></a>.
We benchmark with other diffusion-based methods in time series forecasting, such as TimeGrad <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib48" title="" class="ltx_ref">48</a>]</cite> and the transformer-based Transformer-MAF <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib49" title="" class="ltx_ref">49</a>]</cite>. In particular, the main baseline of interest, TimeGrad <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib48" title="" class="ltx_ref">48</a>]</cite>, is a next-token diffusion sequence model trained with teacher forcing.
We track the <math id="A5.SS1.SSS0.Px2.p1.70.69.m69.1" class="ltx_Math" alttext="\operatorname{CRPS}_{\text{sum}}" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.70.69.m69.1a"><msub id="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1" xref="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.cmml"><mi id="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.2" xref="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.2.cmml">CRPS</mi><mtext id="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.3" xref="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.3a.cmml">sum</mtext></msub><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.70.69.m69.1b"><apply id="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1"><csymbol cd="ambiguous" id="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1">subscript</csymbol><ci id="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.2.cmml" xref="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.2">CRPS</ci><ci id="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.3a.cmml" xref="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.3"><mtext id="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.3.cmml" xref="A5.SS1.SSS0.Px2.p1.70.69.m69.1.1.3">sum</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.70.69.m69.1c">\operatorname{CRPS}_{\text{sum}}</annotation></semantics></math> metric on the validation set and use early stopping when the metric has not improved for 6 consecutive epochs, while all epochs are fixed to 100 batches across datasets. We then measure the <math id="A5.SS1.SSS0.Px2.p1.71.70.m70.1" class="ltx_Math" alttext="\operatorname{CRPS}_{\text{sum}}" display="inline"><semantics id="A5.SS1.SSS0.Px2.p1.71.70.m70.1a"><msub id="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1" xref="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.cmml"><mi id="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.2" xref="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.2.cmml">CRPS</mi><mtext id="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.3" xref="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.3a.cmml">sum</mtext></msub><annotation-xml encoding="MathML-Content" id="A5.SS1.SSS0.Px2.p1.71.70.m70.1b"><apply id="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1"><csymbol cd="ambiguous" id="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.1.cmml" xref="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1">subscript</csymbol><ci id="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.2.cmml" xref="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.2">CRPS</ci><ci id="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.3a.cmml" xref="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.3"><mtext id="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.3.cmml" xref="A5.SS1.SSS0.Px2.p1.71.70.m70.1.1.3">sum</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.SS1.SSS0.Px2.p1.71.70.m70.1c">\operatorname{CRPS}_{\text{sum}}</annotation></semantics></math> on the test set at the end of training, which we report in Table <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:tab:results_ts</span>. We use the exact same architecture and hyperparameters for all time series datasets and experiments.
Diffusion Forcing outperforms all prior methods except for  <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib64" title="" class="ltx_ref">64</a>]</cite> with which Diffusion Forcing is overall tied, except for the Wikipedia dataset, on which Diffusion Forcing takes fourth place.
Note that time series is not the core application of Diffusion Forcing, and that we merely seek to demonstrate that the Diffusion Forcing objective is applicable to diverse domains with no apparent trade-off in performance over baseline objectives.</span></p>
</div>
</section>
</section>
<section id="A5.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:70%;">
<span class="ltx_tag ltx_tag_subsection">E.2 </span>Additional results in compositional generation</h3>

<div id="A5.SS2.p1" class="ltx_para">
<p id="A5.SS2.p1.1" class="ltx_p"><span id="A5.SS2.p1.1.1" class="ltx_text" style="font-size:70%;">Since Diffusion Forcing models the joint distribution of any subset of a sequence, we can leverage this unique property to achieve compositional behavior - i.e., Diffusion Forcing can sample from the distribution of </span><em id="A5.SS2.p1.1.2" class="ltx_emph ltx_font_italic" style="font-size:70%;">subsets</em><span id="A5.SS2.p1.1.3" class="ltx_text" style="font-size:70%;"> of the trajectory and compose these sub-trajectories into new trajectories.</span></p>
</div>
<div id="A5.SS2.p2" class="ltx_para">
<p id="A5.SS2.p2.1" class="ltx_p"><span id="A5.SS2.p2.1.1" class="ltx_text" style="font-size:70%;">In particular, we show that we can also have flexible control over how compositional Diffusion Forcing is. As shown in </span><a href="#A5.F7" title="Figure 7 ‣ E.2 Additional results in compositional generation ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">7</span></a><span id="A5.SS2.p2.1.2" class="ltx_text" style="font-size:70%;">, consider a dataset of trajectories on a 2D, square plane, where all trajectories start from one corner and end up in the opposite corner, forming a cross shape. When no compositional behavior is desired, one can let the models replicate the cross-shaped distribution by allowing full memory of the HMM model.
When one desires compositional such as generating a V-shaped trajectory, which stitches two sub trajectories together, one can let the model generate shorter plans with no-memory context using MPC. (Add figures).</span></p>
</div>
<figure id="A5.F7" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_3">
<figure id="A5.F7.sf1" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2407.01392/assets/figures/dataset_diagonal2d_thick_v3.png" id="A5.F7.sf1.g1" class="ltx_graphics ltx_centering ltx_img_square" width="598" height="598" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F7.sf1.4.1.1" class="ltx_text" style="font-size:129%;">(a)</span> </span><span id="A5.F7.sf1.5.2" class="ltx_text" style="font-size:129%;">Dataset</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_3">
<figure id="A5.F7.sf2" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2407.01392/assets/figures/w_memory.png" id="A5.F7.sf2.g1" class="ltx_graphics ltx_centering ltx_img_square" width="598" height="598" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F7.sf2.4.1.1" class="ltx_text" style="font-size:129%;">(b)</span> </span><span id="A5.F7.sf2.5.2" class="ltx_text" style="font-size:129%;">W/ memory</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_3">
<figure id="A5.F7.sf3" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2407.01392/assets/figures/wo_memory_v2.png" id="A5.F7.sf3.g1" class="ltx_graphics ltx_centering ltx_img_square" width="598" height="598" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F7.sf3.4.1.1" class="ltx_text" style="font-size:129%;">(c)</span> </span><span id="A5.F7.sf3.5.2" class="ltx_text" style="font-size:129%;">W/o memory</span></figcaption>
</figure>
</div>
</div>
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F7.7.1.1" class="ltx_text" style="font-size:129%;">Figure 7</span>: </span><span id="A5.F7.8.2" class="ltx_text" style="font-size:129%;">Given a dataset of trajectories (<a href="#A5.F7.sf1" title="Figure 7(a) ‣ Figure 7 ‣ E.2 Additional results in compositional generation ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7(a)</span></a>), Diffusion Forcing models the joint distribution of all subsequences of arbitrary length. At sampling time, we can sample from the trajectory distribution by sampling Diffusion Forcing with full horizon (<a href="#A5.F7.sf2" title="Figure 7(b) ‣ Figure 7 ‣ E.2 Additional results in compositional generation ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7(b)</span></a>) or recover Markovian dynamics by disregarding previous states (<a href="#A5.F7.sf3" title="Figure 7(c) ‣ Figure 7 ‣ E.2 Additional results in compositional generation ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7(c)</span></a>).</span></figcaption>
</figure>
</section>
<section id="A5.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:70%;">
<span class="ltx_tag ltx_tag_subsection">E.3 </span>Additional results in video prediction (wo/ cherry picking)</h3>

<section id="A5.SS3.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:70%;">Infinite Rollout without sliding window</h5>

<div id="A5.SS3.SSS0.Px1.p1" class="ltx_para">
<p id="A5.SS3.SSS0.Px1.p1.5" class="ltx_p"><span id="A5.SS3.SSS0.Px1.p1.5.1" class="ltx_text" style="font-size:70%;">Diffusion Forcing can rollout longer than maximum training horizon</span><em id="A5.SS3.SSS0.Px1.p1.5.2" class="ltx_emph ltx_font_italic" style="font-size:70%;">without sliding window</em><span id="A5.SS3.SSS0.Px1.p1.5.3" class="ltx_text" style="font-size:70%;">. That is, we run Diffusion Forcing’s RNN continuously without ever reinitializing </span><math id="A5.SS3.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="\mathbf{z}_{0}" display="inline"><semantics id="A5.SS3.SSS0.Px1.p1.1.m1.1a"><msub id="A5.SS3.SSS0.Px1.p1.1.m1.1.1" xref="A5.SS3.SSS0.Px1.p1.1.m1.1.1.cmml"><mi mathsize="70%" id="A5.SS3.SSS0.Px1.p1.1.m1.1.1.2" xref="A5.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml">𝐳</mi><mn mathsize="70%" id="A5.SS3.SSS0.Px1.p1.1.m1.1.1.3" xref="A5.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A5.SS3.SSS0.Px1.p1.1.m1.1b"><apply id="A5.SS3.SSS0.Px1.p1.1.m1.1.1.cmml" xref="A5.SS3.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A5.SS3.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="A5.SS3.SSS0.Px1.p1.1.m1.1.1">subscript</csymbol><ci id="A5.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="A5.SS3.SSS0.Px1.p1.1.m1.1.1.2">𝐳</ci><cn type="integer" id="A5.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="A5.SS3.SSS0.Px1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.SS3.SSS0.Px1.p1.1.m1.1c">\mathbf{z}_{0}</annotation></semantics></math><span id="A5.SS3.SSS0.Px1.p1.5.4" class="ltx_text" style="font-size:70%;">. This is a surprising effect we observed from the rollout stabilization property of Diffusion Forcing. In Figure </span><a href="#A5.F8" title="Figure 8 ‣ Infinite Rollout without sliding window ‣ E.3 Additional results in video prediction (wo/ cherry picking) ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">8</span></a><span id="A5.SS3.SSS0.Px1.p1.5.5" class="ltx_text" style="font-size:70%;">, </span><a href="#A5.F10" title="Figure 10 ‣ Infinite Rollout without sliding window ‣ E.3 Additional results in video prediction (wo/ cherry picking) ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">10</span></a><span id="A5.SS3.SSS0.Px1.p1.5.6" class="ltx_text" style="font-size:70%;">, we use Diffusion Forcing to generate video sequences of length </span><math id="A5.SS3.SSS0.Px1.p1.2.m2.1" class="ltx_Math" alttext="180" display="inline"><semantics id="A5.SS3.SSS0.Px1.p1.2.m2.1a"><mn mathsize="70%" id="A5.SS3.SSS0.Px1.p1.2.m2.1.1" xref="A5.SS3.SSS0.Px1.p1.2.m2.1.1.cmml">180</mn><annotation-xml encoding="MathML-Content" id="A5.SS3.SSS0.Px1.p1.2.m2.1b"><cn type="integer" id="A5.SS3.SSS0.Px1.p1.2.m2.1.1.cmml" xref="A5.SS3.SSS0.Px1.p1.2.m2.1.1">180</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.SS3.SSS0.Px1.p1.2.m2.1c">180</annotation></semantics></math><span id="A5.SS3.SSS0.Px1.p1.5.7" class="ltx_text" style="font-size:70%;"> and visualize subsampled sequences. Notably, Diffusion Forcing used in these visualizations is trained with a maximum length of </span><math id="A5.SS3.SSS0.Px1.p1.3.m3.1" class="ltx_Math" alttext="72" display="inline"><semantics id="A5.SS3.SSS0.Px1.p1.3.m3.1a"><mn mathsize="70%" id="A5.SS3.SSS0.Px1.p1.3.m3.1.1" xref="A5.SS3.SSS0.Px1.p1.3.m3.1.1.cmml">72</mn><annotation-xml encoding="MathML-Content" id="A5.SS3.SSS0.Px1.p1.3.m3.1b"><cn type="integer" id="A5.SS3.SSS0.Px1.p1.3.m3.1.1.cmml" xref="A5.SS3.SSS0.Px1.p1.3.m3.1.1">72</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.SS3.SSS0.Px1.p1.3.m3.1c">72</annotation></semantics></math><span id="A5.SS3.SSS0.Px1.p1.5.8" class="ltx_text" style="font-size:70%;"> frames for Minecraft and </span><math id="A5.SS3.SSS0.Px1.p1.4.m4.1" class="ltx_Math" alttext="36" display="inline"><semantics id="A5.SS3.SSS0.Px1.p1.4.m4.1a"><mn mathsize="70%" id="A5.SS3.SSS0.Px1.p1.4.m4.1.1" xref="A5.SS3.SSS0.Px1.p1.4.m4.1.1.cmml">36</mn><annotation-xml encoding="MathML-Content" id="A5.SS3.SSS0.Px1.p1.4.m4.1b"><cn type="integer" id="A5.SS3.SSS0.Px1.p1.4.m4.1.1.cmml" xref="A5.SS3.SSS0.Px1.p1.4.m4.1.1">36</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.SS3.SSS0.Px1.p1.4.m4.1c">36</annotation></semantics></math><span id="A5.SS3.SSS0.Px1.p1.5.9" class="ltx_text" style="font-size:70%;"> frames for DMLab, illustrating it can rollout 2x-5x times longer than it’s trained on </span><em id="A5.SS3.SSS0.Px1.p1.5.10" class="ltx_emph ltx_font_italic" style="font-size:70%;">without sliding window</em><span id="A5.SS3.SSS0.Px1.p1.5.11" class="ltx_text" style="font-size:70%;">. In addition, we also tried rolling these models out for </span><math id="A5.SS3.SSS0.Px1.p1.5.m5.1" class="ltx_Math" alttext="2000" display="inline"><semantics id="A5.SS3.SSS0.Px1.p1.5.m5.1a"><mn mathsize="70%" id="A5.SS3.SSS0.Px1.p1.5.m5.1.1" xref="A5.SS3.SSS0.Px1.p1.5.m5.1.1.cmml">2000</mn><annotation-xml encoding="MathML-Content" id="A5.SS3.SSS0.Px1.p1.5.m5.1b"><cn type="integer" id="A5.SS3.SSS0.Px1.p1.5.m5.1.1.cmml" xref="A5.SS3.SSS0.Px1.p1.5.m5.1.1">2000</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.SS3.SSS0.Px1.p1.5.m5.1c">2000</annotation></semantics></math><span id="A5.SS3.SSS0.Px1.p1.5.12" class="ltx_text" style="font-size:70%;"> frames and without seeing the model blowing up on both datasets. There are occasional cases where the Minecraft agent gets stuck and the entire screen is the “dirt” block, but this is more of a dataset issue </span><a href="#A5.F16" title="Figure 16 ‣ E.5 Real robot experiment setup ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">16</span></a><span id="A5.SS3.SSS0.Px1.p1.5.13" class="ltx_text" style="font-size:70%;"> and the agent is able to recover after it turns around.</span></p>
</div>
<figure id="A5.F8" class="ltx_figure"><img src="/html/2407.01392/assets/figures/appendix_vis/df_minecraft_long_1.png" id="A5.F8.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="598" height="333" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F8.15.4.1" class="ltx_text" style="font-size:129%;">Figure 8</span>: </span><span id="A5.F8.6.3" class="ltx_text" style="font-size:129%;">Visualization shows Diffusion Forcing trained on <math id="A5.F8.4.1.m1.1" class="ltx_Math" alttext="72" display="inline"><semantics id="A5.F8.4.1.m1.1b"><mn id="A5.F8.4.1.m1.1.1" xref="A5.F8.4.1.m1.1.1.cmml">72</mn><annotation-xml encoding="MathML-Content" id="A5.F8.4.1.m1.1c"><cn type="integer" id="A5.F8.4.1.m1.1.1.cmml" xref="A5.F8.4.1.m1.1.1">72</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F8.4.1.m1.1d">72</annotation></semantics></math> frames is able to rollout <math id="A5.F8.5.2.m2.1" class="ltx_Math" alttext="180" display="inline"><semantics id="A5.F8.5.2.m2.1b"><mn id="A5.F8.5.2.m2.1.1" xref="A5.F8.5.2.m2.1.1.cmml">180</mn><annotation-xml encoding="MathML-Content" id="A5.F8.5.2.m2.1c"><cn type="integer" id="A5.F8.5.2.m2.1.1.cmml" xref="A5.F8.5.2.m2.1.1">180</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F8.5.2.m2.1d">180</annotation></semantics></math> frames on Minecraft dataset <em id="A5.F8.6.3.1" class="ltx_emph ltx_font_italic">without sliding window</em>. The visualization shows a non-cherry-picked subsampling of these <math id="A5.F8.6.3.m3.1" class="ltx_Math" alttext="180" display="inline"><semantics id="A5.F8.6.3.m3.1b"><mn id="A5.F8.6.3.m3.1.1" xref="A5.F8.6.3.m3.1.1.cmml">180</mn><annotation-xml encoding="MathML-Content" id="A5.F8.6.3.m3.1c"><cn type="integer" id="A5.F8.6.3.m3.1.1.cmml" xref="A5.F8.6.3.m3.1.1">180</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F8.6.3.m3.1d">180</annotation></semantics></math> frames, although Diffusion Forcing can roll out much longer (such as 2000 frames) on this dataset.</span></figcaption>
</figure>
<figure id="A5.F9" class="ltx_figure"><img src="/html/2407.01392/assets/figures/appendix_vis/df_minecraft_long_1.png" id="A5.F9.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="598" height="333" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F9.15.4.1" class="ltx_text" style="font-size:129%;">Figure 9</span>: </span><span id="A5.F9.6.3" class="ltx_text" style="font-size:129%;">Diffusion Forcing trained on <math id="A5.F9.4.1.m1.1" class="ltx_Math" alttext="72" display="inline"><semantics id="A5.F9.4.1.m1.1b"><mn id="A5.F9.4.1.m1.1.1" xref="A5.F9.4.1.m1.1.1.cmml">72</mn><annotation-xml encoding="MathML-Content" id="A5.F9.4.1.m1.1c"><cn type="integer" id="A5.F9.4.1.m1.1.1.cmml" xref="A5.F9.4.1.m1.1.1">72</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F9.4.1.m1.1d">72</annotation></semantics></math> frames is able to rollout <math id="A5.F9.5.2.m2.1" class="ltx_Math" alttext="180" display="inline"><semantics id="A5.F9.5.2.m2.1b"><mn id="A5.F9.5.2.m2.1.1" xref="A5.F9.5.2.m2.1.1.cmml">180</mn><annotation-xml encoding="MathML-Content" id="A5.F9.5.2.m2.1c"><cn type="integer" id="A5.F9.5.2.m2.1.1.cmml" xref="A5.F9.5.2.m2.1.1">180</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F9.5.2.m2.1d">180</annotation></semantics></math> frames on Minecraft dataset <em id="A5.F9.6.3.1" class="ltx_emph ltx_font_italic">without sliding window</em>. The visualization shows a non-cherry-picked subsampling of these <math id="A5.F9.6.3.m3.1" class="ltx_Math" alttext="180" display="inline"><semantics id="A5.F9.6.3.m3.1b"><mn id="A5.F9.6.3.m3.1.1" xref="A5.F9.6.3.m3.1.1.cmml">180</mn><annotation-xml encoding="MathML-Content" id="A5.F9.6.3.m3.1c"><cn type="integer" id="A5.F9.6.3.m3.1.1.cmml" xref="A5.F9.6.3.m3.1.1">180</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F9.6.3.m3.1d">180</annotation></semantics></math> frames, although Diffusion Forcing can roll out much longer (such as 2000 frames) on this dataset. The first few frames marked in red are the ground truth images of the dataset used for conditioning.</span></figcaption>
</figure>
<figure id="A5.F10" class="ltx_figure"><img src="/html/2407.01392/assets/figures/appendix_vis/df_dmlab_long_0.png" id="A5.F10.g1" class="ltx_graphics ltx_centering ltx_img_square" width="598" height="665" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F10.15.4.1" class="ltx_text" style="font-size:129%;">Figure 10</span>: </span><span id="A5.F10.6.3" class="ltx_text" style="font-size:129%;">Visualization shows Diffusion Forcing trained on <math id="A5.F10.4.1.m1.1" class="ltx_Math" alttext="36" display="inline"><semantics id="A5.F10.4.1.m1.1b"><mn id="A5.F10.4.1.m1.1.1" xref="A5.F10.4.1.m1.1.1.cmml">36</mn><annotation-xml encoding="MathML-Content" id="A5.F10.4.1.m1.1c"><cn type="integer" id="A5.F10.4.1.m1.1.1.cmml" xref="A5.F10.4.1.m1.1.1">36</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F10.4.1.m1.1d">36</annotation></semantics></math> frames is able to rollout <math id="A5.F10.5.2.m2.1" class="ltx_Math" alttext="180" display="inline"><semantics id="A5.F10.5.2.m2.1b"><mn id="A5.F10.5.2.m2.1.1" xref="A5.F10.5.2.m2.1.1.cmml">180</mn><annotation-xml encoding="MathML-Content" id="A5.F10.5.2.m2.1c"><cn type="integer" id="A5.F10.5.2.m2.1.1.cmml" xref="A5.F10.5.2.m2.1.1">180</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F10.5.2.m2.1d">180</annotation></semantics></math> frames on DMLab dataset <em id="A5.F10.6.3.1" class="ltx_emph ltx_font_italic">without sliding window</em>. The visualization shows a non-cherry-picked subsampling of these <math id="A5.F10.6.3.m3.1" class="ltx_Math" alttext="180" display="inline"><semantics id="A5.F10.6.3.m3.1b"><mn id="A5.F10.6.3.m3.1.1" xref="A5.F10.6.3.m3.1.1.cmml">180</mn><annotation-xml encoding="MathML-Content" id="A5.F10.6.3.m3.1c"><cn type="integer" id="A5.F10.6.3.m3.1.1.cmml" xref="A5.F10.6.3.m3.1.1">180</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F10.6.3.m3.1d">180</annotation></semantics></math> frames, although Diffusion Forcing can roll out almost infinitely on this dataset. The first few frames marked in red are the ground truth images of the dataset used for conditioning.</span></figcaption>
</figure>
<figure id="A5.F11" class="ltx_figure"><img src="/html/2407.01392/assets/figures/appendix_vis/df_dmlab_long_1.png" id="A5.F11.g1" class="ltx_graphics ltx_centering ltx_img_square" width="598" height="665" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F11.15.4.1" class="ltx_text" style="font-size:129%;">Figure 11</span>: </span><span id="A5.F11.6.3" class="ltx_text" style="font-size:129%;">Visualization shows Diffusion Forcing trained on <math id="A5.F11.4.1.m1.1" class="ltx_Math" alttext="36" display="inline"><semantics id="A5.F11.4.1.m1.1b"><mn id="A5.F11.4.1.m1.1.1" xref="A5.F11.4.1.m1.1.1.cmml">36</mn><annotation-xml encoding="MathML-Content" id="A5.F11.4.1.m1.1c"><cn type="integer" id="A5.F11.4.1.m1.1.1.cmml" xref="A5.F11.4.1.m1.1.1">36</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F11.4.1.m1.1d">36</annotation></semantics></math> frames is able to rollout <math id="A5.F11.5.2.m2.1" class="ltx_Math" alttext="180" display="inline"><semantics id="A5.F11.5.2.m2.1b"><mn id="A5.F11.5.2.m2.1.1" xref="A5.F11.5.2.m2.1.1.cmml">180</mn><annotation-xml encoding="MathML-Content" id="A5.F11.5.2.m2.1c"><cn type="integer" id="A5.F11.5.2.m2.1.1.cmml" xref="A5.F11.5.2.m2.1.1">180</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F11.5.2.m2.1d">180</annotation></semantics></math> frames on DMLab dataset <em id="A5.F11.6.3.1" class="ltx_emph ltx_font_italic">without sliding window</em>. The visualization shows a non-cherry-picked subsampling of these <math id="A5.F11.6.3.m3.1" class="ltx_Math" alttext="180" display="inline"><semantics id="A5.F11.6.3.m3.1b"><mn id="A5.F11.6.3.m3.1.1" xref="A5.F11.6.3.m3.1.1.cmml">180</mn><annotation-xml encoding="MathML-Content" id="A5.F11.6.3.m3.1c"><cn type="integer" id="A5.F11.6.3.m3.1.1.cmml" xref="A5.F11.6.3.m3.1.1">180</cn></annotation-xml><annotation encoding="application/x-tex" id="A5.F11.6.3.m3.1d">180</annotation></semantics></math> frames, although Diffusion Forcing can roll out almost infinitely on this dataset. The first few frames marked in red are the ground truth images of the dataset used for conditioning.</span></figcaption>
</figure>
</section>
<section id="A5.SS3.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:70%;">Consistency</h5>

<div id="A5.SS3.SSS0.Px2.p1" class="ltx_para">
<p id="A5.SS3.SSS0.Px2.p1.1" class="ltx_p"><span id="A5.SS3.SSS0.Px2.p1.1.1" class="ltx_text" style="font-size:70%;">We also present additional results where we only generate within our maximum training length. As shown in figure </span><a href="#A5.F13" title="Figure 13 ‣ Consistency ‣ E.3 Additional results in video prediction (wo/ cherry picking) ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">13</span></a><span id="A5.SS3.SSS0.Px2.p1.1.2" class="ltx_text" style="font-size:70%;"> </span><a href="#A5.F12" title="Figure 12 ‣ Consistency ‣ E.3 Additional results in video prediction (wo/ cherry picking) ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">12</span></a><span id="A5.SS3.SSS0.Px2.p1.1.3" class="ltx_text" style="font-size:70%;">, Diffusion Forcing can generate consistent videos. Results are not cherry-picked.</span></p>
</div>
<figure id="A5.F12" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2407.01392/assets/figures/appendix_vis/dmlab_df_0.png" id="A5.F12.g1" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="598" height="266" alt="Refer to caption"></div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2407.01392/assets/figures/appendix_vis/dmlab_df_1.png" id="A5.F12.g2" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="598" height="266" alt="Refer to caption"></div>
</div>
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F12.4.1.1" class="ltx_text" style="font-size:129%;">Figure 12</span>: </span><span id="A5.F12.5.2" class="ltx_text" style="font-size:129%;">Additional non-cherry-picked video prediction results on DMLab dataset, generated within maximum training length. The first few frames marked in red are the ground truth images of the dataset used for conditioning.</span></figcaption>
</figure>
<figure id="A5.F13" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2407.01392/assets/figures/appendix_vis/minecraft_df_0.png" id="A5.F13.g1" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="598" height="133" alt="Refer to caption"></div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2407.01392/assets/figures/appendix_vis/minecraft_df_0.png" id="A5.F13.g2" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="598" height="133" alt="Refer to caption"></div>
</div>
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F13.4.1.1" class="ltx_text" style="font-size:129%;">Figure 13</span>: </span><span id="A5.F13.5.2" class="ltx_text" style="font-size:129%;">Additional non-cherry-picked video prediction results on Minecraft dataset, generated within maximum training length. The first few frames marked in red are the ground truth images of the dataset used for conditioning.</span></figcaption>
</figure>
</section>
</section>
<section id="A5.SS4" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:70%;">
<span class="ltx_tag ltx_tag_subsection">E.4 </span>Additional results in planning</h3>

<div id="A5.SS4.p1" class="ltx_para">
<p id="A5.SS4.p1.1" class="ltx_p"><span id="A5.SS4.p1.1.1" class="ltx_text" style="font-size:70%;">We provide some additional visualizations of causal planning in  </span><a href="#A5.F15" title="Figure 15 ‣ E.4 Additional results in planning ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">15</span></a><span id="A5.SS4.p1.1.2" class="ltx_text" style="font-size:70%;">. We also present additional visualization of Diffusion Forcing performing model predictive control in action. As shown in figure </span><a href="#A5.F14" title="Figure 14 ‣ E.4 Additional results in planning ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">14</span></a><span id="A5.SS4.p1.1.3" class="ltx_text" style="font-size:70%;">, Diffusion Forcing can generate plans of shorter horizons since it’s flexible horizon.</span></p>
</div>
<figure id="A5.F14" class="ltx_figure"><img src="/html/2407.01392/assets/figures/appendix_vis/df_medium_close_mpc.png" id="A5.F14.g1" class="ltx_graphics ltx_centering ltx_img_square" width="598" height="524" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F14.4.1.1" class="ltx_text" style="font-size:129%;">Figure 14</span>: </span><span id="A5.F14.5.2" class="ltx_text" style="font-size:129%;">Example MPC planning for maze medium environment. Blue indicated trajectories actually executed already. Red is the plan.</span></figcaption>
</figure>
<figure id="A5.F15" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2407.01392/assets/figures/appendix_vis/df_medium_close.png" id="A5.F15.g1" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="598" height="75" alt="Refer to caption"></div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2407.01392/assets/figures/appendix_vis/df_large_close.png" id="A5.F15.g2" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="598" height="107" alt="Refer to caption"></div>
</div>
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F15.4.1.1" class="ltx_text" style="font-size:129%;">Figure 15</span>: </span><span id="A5.F15.5.2" class="ltx_text" style="font-size:129%;">Example plans generated for maze medium (above) and maze large (below) environments.</span></figcaption>
</figure>
</section>
<section id="A5.SS5" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:70%;">
<span class="ltx_tag ltx_tag_subsection">E.5 </span>Real robot experiment setup</h3>

<div id="A5.SS5.p1" class="ltx_para">
<p id="A5.SS5.p1.1" class="ltx_p"><span id="A5.SS5.p1.1.1" class="ltx_text" style="font-size:70%;">In Figure </span><a href="#A5.F16" title="Figure 16 ‣ E.5 Real robot experiment setup ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">16</span></a><span id="A5.SS5.p1.1.2" class="ltx_text" style="font-size:70%;"> we visualize our robot experiment setup with corruption on observation. The dataset is collected when the target bag isn’t present, while we test with such a bag in the scene zero-shot for the imitation learning experiment with observation corruption. The typical failure mode is when the robot no longer reacts to the visual clues of the randomized location of objects. We didn’t observe the robot to act wildly due to visual distractors.</span></p>
</div>
<figure id="A5.F16" class="ltx_figure"><img src="/html/2407.01392/assets/figures/appendix_vis/robot_corrupt.jpg" id="A5.F16.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="299" height="225" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering" style="font-size:70%;"><span class="ltx_tag ltx_tag_figure"><span id="A5.F16.4.1.1" class="ltx_text" style="font-size:129%;">Figure 16</span>: </span><span id="A5.F16.5.2" class="ltx_text" style="font-size:129%;">We randomly throw a target bag on the table as a strong visual distractor. Diffusion Forcing can be prompted to treat observation as corrupted than ground truth.</span></figcaption>
</figure>
</section>
</section>
<section id="A6" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix" style="font-size:70%;">
<span class="ltx_tag ltx_tag_appendix">Appendix F </span>Additional details about datasets</h2>

<section id="A6.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:70%;">
<span class="ltx_tag ltx_tag_subsection">F.1 </span>Dataset for video diffusion</h3>

<div id="A6.SS1.p1" class="ltx_para">
<p id="A6.SS1.p1.1" class="ltx_p"><span id="A6.SS1.p1.1.1" class="ltx_text" style="font-size:70%;">We adopt the video prediction dataset Minecraft and DMlab used by TECO</span><cite class="ltx_cite ltx_citemacro_cite"><span id="A6.SS1.p1.1.2.1" class="ltx_text" style="font-size:70%;">[</span><a href="#bib.bib65" title="" class="ltx_ref">65</a><span id="A6.SS1.p1.1.3.2" class="ltx_text" style="font-size:70%;">]</span></cite><span id="A6.SS1.p1.1.4" class="ltx_text" style="font-size:70%;">.</span></p>
</div>
<section id="A6.SS1.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:70%;">Minecraft Navigation</h5>

<div id="A6.SS1.SSS0.Px1.p1" class="ltx_para">
<p id="A6.SS1.SSS0.Px1.p1.3" class="ltx_p"><span id="A6.SS1.SSS0.Px1.p1.3.1" class="ltx_text" style="font-size:70%;">The Minecraft navigation dataset consists of first-person-view videos of random walks in the Minecraft ‘swamp‘ biome. The agent walks via a technique called ‘sprint jump‘ which allows it to jump across blocks without getting stuck at 1 block obstacles. The agent walks straight most of the time, with small chances of turning left or right. The height and width of the video is </span><math id="A6.SS1.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="128" display="inline"><semantics id="A6.SS1.SSS0.Px1.p1.1.m1.1a"><mn mathsize="70%" id="A6.SS1.SSS0.Px1.p1.1.m1.1.1" xref="A6.SS1.SSS0.Px1.p1.1.m1.1.1.cmml">128</mn><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px1.p1.1.m1.1b"><cn type="integer" id="A6.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px1.p1.1.m1.1.1">128</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px1.p1.1.m1.1c">128</annotation></semantics></math><span id="A6.SS1.SSS0.Px1.p1.3.2" class="ltx_text" style="font-size:70%;"> pixels and we trim long videos to subsequences of </span><math id="A6.SS1.SSS0.Px1.p1.2.m2.1" class="ltx_Math" alttext="72" display="inline"><semantics id="A6.SS1.SSS0.Px1.p1.2.m2.1a"><mn mathsize="70%" id="A6.SS1.SSS0.Px1.p1.2.m2.1.1" xref="A6.SS1.SSS0.Px1.p1.2.m2.1.1.cmml">72</mn><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px1.p1.2.m2.1b"><cn type="integer" id="A6.SS1.SSS0.Px1.p1.2.m2.1.1.cmml" xref="A6.SS1.SSS0.Px1.p1.2.m2.1.1">72</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px1.p1.2.m2.1c">72</annotation></semantics></math><span id="A6.SS1.SSS0.Px1.p1.3.3" class="ltx_text" style="font-size:70%;"> frames. The dataset comes with paired action data but we discard them to bring more stochasticity to the prediction task. Due to limited compute, we only train on about </span><math id="A6.SS1.SSS0.Px1.p1.3.m3.1" class="ltx_Math" alttext="10\%" display="inline"><semantics id="A6.SS1.SSS0.Px1.p1.3.m3.1a"><mrow id="A6.SS1.SSS0.Px1.p1.3.m3.1.1" xref="A6.SS1.SSS0.Px1.p1.3.m3.1.1.cmml"><mn mathsize="70%" id="A6.SS1.SSS0.Px1.p1.3.m3.1.1.2" xref="A6.SS1.SSS0.Px1.p1.3.m3.1.1.2.cmml">10</mn><mo mathsize="70%" id="A6.SS1.SSS0.Px1.p1.3.m3.1.1.1" xref="A6.SS1.SSS0.Px1.p1.3.m3.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px1.p1.3.m3.1b"><apply id="A6.SS1.SSS0.Px1.p1.3.m3.1.1.cmml" xref="A6.SS1.SSS0.Px1.p1.3.m3.1.1"><csymbol cd="latexml" id="A6.SS1.SSS0.Px1.p1.3.m3.1.1.1.cmml" xref="A6.SS1.SSS0.Px1.p1.3.m3.1.1.1">percent</csymbol><cn type="integer" id="A6.SS1.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="A6.SS1.SSS0.Px1.p1.3.m3.1.1.2">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px1.p1.3.m3.1c">10\%</annotation></semantics></math><span id="A6.SS1.SSS0.Px1.p1.3.4" class="ltx_text" style="font-size:70%;"> of the total subsequences.</span></p>
</div>
<div id="A6.SS1.SSS0.Px1.p2" class="ltx_para">
<p id="A6.SS1.SSS0.Px1.p2.1" class="ltx_p"><span id="A6.SS1.SSS0.Px1.p2.1.1" class="ltx_text" style="font-size:70%;">One problem we noticed about the dataset is when the agent runs into obstacles with a height of 2 blocks or more. In this case, the agent will get stuck and the entire video sequence will consist of grey granite patterns or brown dirty patterns. This leads to a huge amount of frames with these patterns, making video models predict meaningless frames. Yet, we deem this as a problem of this dataset itself.</span></p>
</div>
</section>
<section id="A6.SS1.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:70%;">DMLab Navigation</h5>

<div id="A6.SS1.SSS0.Px2.p1" class="ltx_para">
<p id="A6.SS1.SSS0.Px2.p1.2" class="ltx_p"><span id="A6.SS1.SSS0.Px2.p1.2.1" class="ltx_text" style="font-size:70%;">Deepmind Lab navigation dataset consists of random walks in a 3D maze environment. For DMLab, the resolution is </span><math id="A6.SS1.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="64" display="inline"><semantics id="A6.SS1.SSS0.Px2.p1.1.m1.1a"><mn mathsize="70%" id="A6.SS1.SSS0.Px2.p1.1.m1.1.1" xref="A6.SS1.SSS0.Px2.p1.1.m1.1.1.cmml">64</mn><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px2.p1.1.m1.1b"><cn type="integer" id="A6.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p1.1.m1.1.1">64</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px2.p1.1.m1.1c">64</annotation></semantics></math><span id="A6.SS1.SSS0.Px2.p1.2.2" class="ltx_text" style="font-size:70%;"> pixels and we use subsequences of </span><math id="A6.SS1.SSS0.Px2.p1.2.m2.1" class="ltx_Math" alttext="48" display="inline"><semantics id="A6.SS1.SSS0.Px2.p1.2.m2.1a"><mn mathsize="70%" id="A6.SS1.SSS0.Px2.p1.2.m2.1.1" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.cmml">48</mn><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px2.p1.2.m2.1b"><cn type="integer" id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.cmml" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1">48</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px2.p1.2.m2.1c">48</annotation></semantics></math><span id="A6.SS1.SSS0.Px2.p1.2.3" class="ltx_text" style="font-size:70%;"> frames. We also disregard the provided actions due to training.</span></p>
</div>
<div id="A6.SS1.SSS0.Px2.p2" class="ltx_para">
<p id="A6.SS1.SSS0.Px2.p2.2" class="ltx_p"><span id="A6.SS1.SSS0.Px2.p2.2.1" class="ltx_text" style="font-size:70%;">We note that the VQ-VAE latent that stable video diffusion </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A6.SS1.SSS0.Px2.p2.2.2.1" class="ltx_text" style="font-size:70%;">[</span><a href="#bib.bib4" title="" class="ltx_ref">4</a><span id="A6.SS1.SSS0.Px2.p2.2.3.2" class="ltx_text" style="font-size:70%;">]</span></cite><span id="A6.SS1.SSS0.Px2.p2.2.4" class="ltx_text" style="font-size:70%;"> diffuses is also only </span><math id="A6.SS1.SSS0.Px2.p2.1.m1.1" class="ltx_Math" alttext="128\times 128\times 3" display="inline"><semantics id="A6.SS1.SSS0.Px2.p2.1.m1.1a"><mrow id="A6.SS1.SSS0.Px2.p2.1.m1.1.1" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.cmml"><mn mathsize="70%" id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.2" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.2.cmml">128</mn><mo lspace="0.222em" mathsize="70%" rspace="0.222em" id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.cmml">×</mo><mn mathsize="70%" id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.3" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.3.cmml">128</mn><mo lspace="0.222em" mathsize="70%" rspace="0.222em" id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1a" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.cmml">×</mo><mn mathsize="70%" id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.4" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.4.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px2.p2.1.m1.1b"><apply id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1"><times id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1"></times><cn type="integer" id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.2.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.2">128</cn><cn type="integer" id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.3.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.3">128</cn><cn type="integer" id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.4.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.4">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px2.p2.1.m1.1c">128\times 128\times 3</annotation></semantics></math><span id="A6.SS1.SSS0.Px2.p2.2.5" class="ltx_text" style="font-size:70%;">, indicating Diffusion Forcing has the potential to scale up to higher resolution images with pre-trained image encoder and decoders. Due to the sheer size of the datasets, we only use about </span><math id="A6.SS1.SSS0.Px2.p2.2.m2.1" class="ltx_Math" alttext="10\%" display="inline"><semantics id="A6.SS1.SSS0.Px2.p2.2.m2.1a"><mrow id="A6.SS1.SSS0.Px2.p2.2.m2.1.1" xref="A6.SS1.SSS0.Px2.p2.2.m2.1.1.cmml"><mn mathsize="70%" id="A6.SS1.SSS0.Px2.p2.2.m2.1.1.2" xref="A6.SS1.SSS0.Px2.p2.2.m2.1.1.2.cmml">10</mn><mo mathsize="70%" id="A6.SS1.SSS0.Px2.p2.2.m2.1.1.1" xref="A6.SS1.SSS0.Px2.p2.2.m2.1.1.1.cmml">%</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px2.p2.2.m2.1b"><apply id="A6.SS1.SSS0.Px2.p2.2.m2.1.1.cmml" xref="A6.SS1.SSS0.Px2.p2.2.m2.1.1"><csymbol cd="latexml" id="A6.SS1.SSS0.Px2.p2.2.m2.1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p2.2.m2.1.1.1">percent</csymbol><cn type="integer" id="A6.SS1.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="A6.SS1.SSS0.Px2.p2.2.m2.1.1.2">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px2.p2.2.m2.1c">10\%</annotation></semantics></math><span id="A6.SS1.SSS0.Px2.p2.2.6" class="ltx_text" style="font-size:70%;"> of the total data sequences for training due to limited computing, as we observe that doing so already allows us to make good generations from initial frames from the test set.</span></p>
</div>
</section>
</section>
<section id="A6.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:70%;">
<span class="ltx_tag ltx_tag_subsection">F.2 </span>Dataset for planning</h3>

<div id="A6.SS2.p1" class="ltx_para">
<p id="A6.SS2.p1.1" class="ltx_p"><span id="A6.SS2.p1.1.1" class="ltx_text" style="font-size:70%;">D4RL </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A6.SS2.p1.1.2.1" class="ltx_text" style="font-size:70%;">[</span><a href="#bib.bib18" title="" class="ltx_ref">18</a><span id="A6.SS2.p1.1.3.2" class="ltx_text" style="font-size:70%;">]</span></cite><span id="A6.SS2.p1.1.4" class="ltx_text" style="font-size:70%;"> is a standard offline RL benchmark featuring a wide range of reinforcement learning environments. Each environment is associated with a provided dataset of offline interactions with the environment featuring state, action and reward trajectories.</span></p>
</div>
<div id="A6.SS2.p2" class="ltx_para">
<p id="A6.SS2.p2.1" class="ltx_p"><span id="A6.SS2.p2.1.1" class="ltx_text" style="font-size:70%;">Like Diffuer </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A6.SS2.p2.1.2.1" class="ltx_text" style="font-size:70%;">[</span><a href="#bib.bib36" title="" class="ltx_ref">36</a><span id="A6.SS2.p2.1.3.2" class="ltx_text" style="font-size:70%;">]</span></cite><span id="A6.SS2.p2.1.4" class="ltx_text" style="font-size:70%;">, we choose the 3 maze environments as they are challenging long-horizon, multi-modal, sparse reward problems uniquely suited for visualization and evaluating planning algorithms. The ID for the 3 used environments are “maze2d-medium-v1”, “maze2d-large-v1”, “maze2d-umaze-v1”. In each environment, one controls the acceleration of a robot to walk it towards a goal. The observation space is </span><math id="A6.SS2.p2.1.m1.1" class="ltx_Math" alttext="4" display="inline"><semantics id="A6.SS2.p2.1.m1.1a"><mn mathsize="70%" id="A6.SS2.p2.1.m1.1.1" xref="A6.SS2.p2.1.m1.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="A6.SS2.p2.1.m1.1b"><cn type="integer" id="A6.SS2.p2.1.m1.1.1.cmml" xref="A6.SS2.p2.1.m1.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS2.p2.1.m1.1c">4</annotation></semantics></math><span id="A6.SS2.p2.1.5" class="ltx_text" style="font-size:70%;"> dimensional, featuring 2D location and velocity. The action space is 2D acceleration. The agent always receives a random start location and the goal is to reach a fixed goal position for each maze. The agent receives a reward of 1 if it is within a circle of radius 0.5 centered at the goal state, and 0 otherwise.</span></p>
</div>
<div id="A6.SS2.p3" class="ltx_para">
<p id="A6.SS2.p3.1" class="ltx_p"><span id="A6.SS2.p3.1.1" class="ltx_text" style="font-size:70%;">The offline RL dataset for the maze environments consists of random walks in the maze. Specifically, the authors first designate all intersections and turns in the maze as waypoints and code an agent to navigate between waypoints with some randomization. As a result, the random walks are generated in a way that the path is collision-free with the walls. The random walks introduces stochasticity to the dataset, as trajectories in the dataset are never towards a specific goal.</span></p>
</div>
<div id="A6.SS2.p4" class="ltx_para">
<p id="A6.SS2.p4.1" class="ltx_p"><span id="A6.SS2.p4.1.1" class="ltx_text" style="font-size:70%;">There are a few choices adopted from our main baseline Diffuser </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A6.SS2.p4.1.2.1" class="ltx_text" style="font-size:70%;">[</span><a href="#bib.bib36" title="" class="ltx_ref">36</a><span id="A6.SS2.p4.1.3.2" class="ltx_text" style="font-size:70%;">]</span></cite><span id="A6.SS2.p4.1.4" class="ltx_text" style="font-size:70%;">: we disregard the reward in the dataset and plan with goals only. We also evaluate a multi-gaol variant of each environment (labeled as “multi” in Table </span><a href="#S4.SS1" title="4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">4.1</span></a><span id="A6.SS2.p4.1.5" class="ltx_text" style="font-size:70%;">), where the goal is randomized just like the starting position.</span></p>
</div>
</section>
<section id="A6.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:70%;">
<span class="ltx_tag ltx_tag_subsection">F.3 </span>Dataset for robot learning</h3>

<div id="A6.SS3.p1" class="ltx_para">
<p id="A6.SS3.p1.1" class="ltx_p"><span id="A6.SS3.p1.1.1" class="ltx_text" style="font-size:70%;">We choose a long horizon robotic manipulation task as described in Section </span><a href="#S4.SS4" title="4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">4.4</span></a><span id="A6.SS3.p1.1.2" class="ltx_text" style="font-size:70%;">: Consider a tabletop with three slots where we can place objects. One places an apple at slot A or slot B randomly, and then places an orange at the other slot between A and B. A robot is challenged to swap the position of two fruits using the third slot C. That is, it can only move a fruit to an empty slot at a time. For example, when the apple is at slot A and the orange is at slot B, it may move the apple to slot C, leaving slot A empty. Then move the orange to slot A and finally move the apple from slot C to slot B. In figure </span><a href="#S4.F4" title="Figure 4 ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">4</span></a><span id="A6.SS3.p1.1.3" class="ltx_text" style="font-size:70%;">, we illustrate the non-markovian property of the task: When the apple is at slot B and the orange is at slot C, one cannot tell what the immediate action is without knowing the initial positions of objects.</span></p>
</div>
<div id="A6.SS3.p2" class="ltx_para">
<p id="A6.SS3.p2.2" class="ltx_p"><span id="A6.SS3.p2.2.1" class="ltx_text" style="font-size:70%;">We put stickers on the table indicating a circular region occupied by any slot. Each circular region is designed to be about double the diameter of a fruit. To make sure the task requires visual feedback, we also randomize the location of a fruit inside the slot. We collected </span><math id="A6.SS3.p2.1.m1.1" class="ltx_Math" alttext="150" display="inline"><semantics id="A6.SS3.p2.1.m1.1a"><mn mathsize="70%" id="A6.SS3.p2.1.m1.1.1" xref="A6.SS3.p2.1.m1.1.1.cmml">150</mn><annotation-xml encoding="MathML-Content" id="A6.SS3.p2.1.m1.1b"><cn type="integer" id="A6.SS3.p2.1.m1.1.1.cmml" xref="A6.SS3.p2.1.m1.1.1">150</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS3.p2.1.m1.1c">150</annotation></semantics></math><span id="A6.SS3.p2.2.2" class="ltx_text" style="font-size:70%;"> expert demonstrations of a franka robot performing the task using VR teleoperation and impedance control. Among them, each initial slot configuration makes up half of the dataset. We record videos from two camera views, one from hand camera and one in the front capturing all three slots. Each demonstration also comes with </span><math id="A6.SS3.p2.2.m2.1" class="ltx_Math" alttext="6" display="inline"><semantics id="A6.SS3.p2.2.m2.1a"><mn mathsize="70%" id="A6.SS3.p2.2.m2.1.1" xref="A6.SS3.p2.2.m2.1.1.cmml">6</mn><annotation-xml encoding="MathML-Content" id="A6.SS3.p2.2.m2.1b"><cn type="integer" id="A6.SS3.p2.2.m2.1.1.cmml" xref="A6.SS3.p2.2.m2.1.1">6</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS3.p2.2.m2.1c">6</annotation></semantics></math><span id="A6.SS3.p2.2.3" class="ltx_text" style="font-size:70%;"> dof actions of the robot hand. During the data collection, since one successful demonstration will swap the position of two objects, its end configuration will naturally serve as the starting configuration of the other randomized location, which we leverage to save time.</span></p>
</div>
<div id="A6.SS3.p3" class="ltx_para">
<p id="A6.SS3.p3.3" class="ltx_p"><span id="A6.SS3.p3.3.1" class="ltx_text" style="font-size:70%;">Each demonstration comprises </span><math id="A6.SS3.p3.1.m1.1" class="ltx_Math" alttext="500-600" display="inline"><semantics id="A6.SS3.p3.1.m1.1a"><mrow id="A6.SS3.p3.1.m1.1.1" xref="A6.SS3.p3.1.m1.1.1.cmml"><mn mathsize="70%" id="A6.SS3.p3.1.m1.1.1.2" xref="A6.SS3.p3.1.m1.1.1.2.cmml">500</mn><mo mathsize="70%" id="A6.SS3.p3.1.m1.1.1.1" xref="A6.SS3.p3.1.m1.1.1.1.cmml">−</mo><mn mathsize="70%" id="A6.SS3.p3.1.m1.1.1.3" xref="A6.SS3.p3.1.m1.1.1.3.cmml">600</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS3.p3.1.m1.1b"><apply id="A6.SS3.p3.1.m1.1.1.cmml" xref="A6.SS3.p3.1.m1.1.1"><minus id="A6.SS3.p3.1.m1.1.1.1.cmml" xref="A6.SS3.p3.1.m1.1.1.1"></minus><cn type="integer" id="A6.SS3.p3.1.m1.1.1.2.cmml" xref="A6.SS3.p3.1.m1.1.1.2">500</cn><cn type="integer" id="A6.SS3.p3.1.m1.1.1.3.cmml" xref="A6.SS3.p3.1.m1.1.1.3">600</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS3.p3.1.m1.1c">500-600</annotation></semantics></math><span id="A6.SS3.p3.3.2" class="ltx_text" style="font-size:70%;"> frames and actions. We train Diffusion Forcing on the entire sequence. However, since adjacent frames are visually close, we pad and downsample the videos to </span><math id="A6.SS3.p3.2.m2.1" class="ltx_Math" alttext="40" display="inline"><semantics id="A6.SS3.p3.2.m2.1a"><mn mathsize="70%" id="A6.SS3.p3.2.m2.1.1" xref="A6.SS3.p3.2.m2.1.1.cmml">40</mn><annotation-xml encoding="MathML-Content" id="A6.SS3.p3.2.m2.1b"><cn type="integer" id="A6.SS3.p3.2.m2.1.1.cmml" xref="A6.SS3.p3.2.m2.1.1">40</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS3.p3.2.m2.1c">40</annotation></semantics></math><span id="A6.SS3.p3.3.3" class="ltx_text" style="font-size:70%;"> frames where each frame is bundled with </span><math id="A6.SS3.p3.3.m3.1" class="ltx_Math" alttext="15" display="inline"><semantics id="A6.SS3.p3.3.m3.1a"><mn mathsize="70%" id="A6.SS3.p3.3.m3.1.1" xref="A6.SS3.p3.3.m3.1.1.cmml">15</mn><annotation-xml encoding="MathML-Content" id="A6.SS3.p3.3.m3.1b"><cn type="integer" id="A6.SS3.p3.3.m3.1.1.cmml" xref="A6.SS3.p3.3.m3.1.1">15</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS3.p3.3.m3.1c">15</annotation></semantics></math><span id="A6.SS3.p3.3.4" class="ltx_text" style="font-size:70%;"> actions.</span></p>
</div>
</section>
<section id="A6.SS4" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection" style="font-size:70%;">
<span class="ltx_tag ltx_tag_subsection">F.4 </span>Dataset for time series </h3>

<div id="A6.SS4.p1" class="ltx_para">
<span id="A6.SS4.p1.6" class="ltx_ERROR undefined">\ctable</span>
<p id="A6.SS4.p1.5" class="ltx_p"><span id="A6.SS4.p1.5.1" class="ltx_text" style="font-size:70%;">[
caption = Characteristics of the GluonTS datasets used to benchmark Diffusion Forcing in the domain of time series forecasting.,
label = tab:ts_data,
pos = ht,
doinside = ,
center,
star
]lccccc
</span><span id="A6.SS4.p1.5.2" class="ltx_ERROR undefined">\FL</span><span id="A6.SS4.p1.5.3" class="ltx_text" style="font-size:70%;">Dataset &amp; Dimension  Domain  Frequency  Steps  Prediction length
</span><span id="A6.SS4.p1.5.4" class="ltx_ERROR undefined">\ML</span><span id="A6.SS4.p1.5.5" class="ltx_text" style="font-size:70%;">Exchange  8  </span><math id="A6.SS4.p1.1.m1.1" class="ltx_Math" alttext="\mathbb{R}^{+}" display="inline"><semantics id="A6.SS4.p1.1.m1.1a"><msup id="A6.SS4.p1.1.m1.1.1" xref="A6.SS4.p1.1.m1.1.1.cmml"><mi mathsize="70%" id="A6.SS4.p1.1.m1.1.1.2" xref="A6.SS4.p1.1.m1.1.1.2.cmml">ℝ</mi><mo mathsize="70%" id="A6.SS4.p1.1.m1.1.1.3" xref="A6.SS4.p1.1.m1.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.1.m1.1b"><apply id="A6.SS4.p1.1.m1.1.1.cmml" xref="A6.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A6.SS4.p1.1.m1.1.1.1.cmml" xref="A6.SS4.p1.1.m1.1.1">superscript</csymbol><ci id="A6.SS4.p1.1.m1.1.1.2.cmml" xref="A6.SS4.p1.1.m1.1.1.2">ℝ</ci><plus id="A6.SS4.p1.1.m1.1.1.3.cmml" xref="A6.SS4.p1.1.m1.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.1.m1.1c">\mathbb{R}^{+}</annotation></semantics></math><span id="A6.SS4.p1.5.6" class="ltx_text" style="font-size:70%;">  BUSINESS DAY  6,071  30 </span><span id="A6.SS4.p1.5.7" class="ltx_ERROR undefined">\NN</span><span id="A6.SS4.p1.5.8" class="ltx_text" style="font-size:70%;">Solar  137  </span><math id="A6.SS4.p1.2.m2.1" class="ltx_Math" alttext="\mathbb{R}^{+}" display="inline"><semantics id="A6.SS4.p1.2.m2.1a"><msup id="A6.SS4.p1.2.m2.1.1" xref="A6.SS4.p1.2.m2.1.1.cmml"><mi mathsize="70%" id="A6.SS4.p1.2.m2.1.1.2" xref="A6.SS4.p1.2.m2.1.1.2.cmml">ℝ</mi><mo mathsize="70%" id="A6.SS4.p1.2.m2.1.1.3" xref="A6.SS4.p1.2.m2.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.2.m2.1b"><apply id="A6.SS4.p1.2.m2.1.1.cmml" xref="A6.SS4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A6.SS4.p1.2.m2.1.1.1.cmml" xref="A6.SS4.p1.2.m2.1.1">superscript</csymbol><ci id="A6.SS4.p1.2.m2.1.1.2.cmml" xref="A6.SS4.p1.2.m2.1.1.2">ℝ</ci><plus id="A6.SS4.p1.2.m2.1.1.3.cmml" xref="A6.SS4.p1.2.m2.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.2.m2.1c">\mathbb{R}^{+}</annotation></semantics></math><span id="A6.SS4.p1.5.9" class="ltx_text" style="font-size:70%;">  HOUR  7,009  24 </span><span id="A6.SS4.p1.5.10" class="ltx_ERROR undefined">\NN</span><span id="A6.SS4.p1.5.11" class="ltx_text" style="font-size:70%;">Electricity  370  </span><math id="A6.SS4.p1.3.m3.1" class="ltx_Math" alttext="\mathbb{R}^{+}" display="inline"><semantics id="A6.SS4.p1.3.m3.1a"><msup id="A6.SS4.p1.3.m3.1.1" xref="A6.SS4.p1.3.m3.1.1.cmml"><mi mathsize="70%" id="A6.SS4.p1.3.m3.1.1.2" xref="A6.SS4.p1.3.m3.1.1.2.cmml">ℝ</mi><mo mathsize="70%" id="A6.SS4.p1.3.m3.1.1.3" xref="A6.SS4.p1.3.m3.1.1.3.cmml">+</mo></msup><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.3.m3.1b"><apply id="A6.SS4.p1.3.m3.1.1.cmml" xref="A6.SS4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A6.SS4.p1.3.m3.1.1.1.cmml" xref="A6.SS4.p1.3.m3.1.1">superscript</csymbol><ci id="A6.SS4.p1.3.m3.1.1.2.cmml" xref="A6.SS4.p1.3.m3.1.1.2">ℝ</ci><plus id="A6.SS4.p1.3.m3.1.1.3.cmml" xref="A6.SS4.p1.3.m3.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.3.m3.1c">\mathbb{R}^{+}</annotation></semantics></math><span id="A6.SS4.p1.5.12" class="ltx_text" style="font-size:70%;">  HOUR  5,833  24 </span><span id="A6.SS4.p1.5.13" class="ltx_ERROR undefined">\NN</span><span id="A6.SS4.p1.5.14" class="ltx_text" style="font-size:70%;">Traffic  963  (0,1)  HOUR  4,001  24 </span><span id="A6.SS4.p1.5.15" class="ltx_ERROR undefined">\NN</span><span id="A6.SS4.p1.5.16" class="ltx_text" style="font-size:70%;">Taxi  1,214  </span><math id="A6.SS4.p1.4.m4.1" class="ltx_Math" alttext="\mathbb{N}" display="inline"><semantics id="A6.SS4.p1.4.m4.1a"><mi mathsize="70%" id="A6.SS4.p1.4.m4.1.1" xref="A6.SS4.p1.4.m4.1.1.cmml">ℕ</mi><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.4.m4.1b"><ci id="A6.SS4.p1.4.m4.1.1.cmml" xref="A6.SS4.p1.4.m4.1.1">ℕ</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.4.m4.1c">\mathbb{N}</annotation></semantics></math><span id="A6.SS4.p1.5.17" class="ltx_text" style="font-size:70%;">  30-MIN  1,488  24 </span><span id="A6.SS4.p1.5.18" class="ltx_ERROR undefined">\NN</span><span id="A6.SS4.p1.5.19" class="ltx_text" style="font-size:70%;">Wikipedia  2,000  </span><math id="A6.SS4.p1.5.m5.1" class="ltx_Math" alttext="\mathbb{N}" display="inline"><semantics id="A6.SS4.p1.5.m5.1a"><mi mathsize="70%" id="A6.SS4.p1.5.m5.1.1" xref="A6.SS4.p1.5.m5.1.1.cmml">ℕ</mi><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.5.m5.1b"><ci id="A6.SS4.p1.5.m5.1.1.cmml" xref="A6.SS4.p1.5.m5.1.1">ℕ</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.5.m5.1c">\mathbb{N}</annotation></semantics></math><span id="A6.SS4.p1.5.20" class="ltx_text" style="font-size:70%;">  DAY  792  30
</span><span id="A6.SS4.p1.5.21" class="ltx_ERROR undefined">\LL</span><span id="A6.SS4.p1.5.22" class="ltx_text" style="font-size:70%;">

We use a set of time series datasets accessible via GluonTS </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A6.SS4.p1.5.23.1" class="ltx_text" style="font-size:70%;">[</span><a href="#bib.bib2" title="" class="ltx_ref">2</a><span id="A6.SS4.p1.5.24.2" class="ltx_text" style="font-size:70%;">]</span></cite><span id="A6.SS4.p1.5.25" class="ltx_text" style="font-size:70%;">, which are adopted from prior works like </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A6.SS4.p1.5.26.1" class="ltx_text" style="font-size:70%;">[</span><a href="#bib.bib67" title="" class="ltx_ref">67</a>, <a href="#bib.bib40" title="" class="ltx_ref">40</a>, <a href="#bib.bib53" title="" class="ltx_ref">53</a><span id="A6.SS4.p1.5.27.2" class="ltx_text" style="font-size:70%;">]</span></cite><span id="A6.SS4.p1.5.28" class="ltx_text" style="font-size:70%;">. These datasets capture real-world data of high-dimensional dynamics like monetary exchange rates or the electricity grid. In Table </span><span class="ltx_ref ltx_missing_label ltx_ref_self" style="font-size:70%;">LABEL:tab:ts_data</span><span id="A6.SS4.p1.5.29" class="ltx_text" style="font-size:70%;">, we provide a summary of the features of these datasets, such as the dimensionality, the domains, the sampling frequency, the length of the mulitvariate sequence in the training set, and the prediction length. We access the datasets in Table </span><span class="ltx_ref ltx_missing_label ltx_ref_self" style="font-size:70%;">LABEL:tab:ts_data</span><span id="A6.SS4.p1.5.30" class="ltx_text" style="font-size:70%;"> via GluonTS and wrap the data processing functions implemented in GluonTS in our own dataloaders. Each dataset consists of one long multivariate sequence, which is the training split, and a set of short sequences that make up the test split. We construct a validation set of the same cardinality as the held-out test set as a randomly sampled subset of subsequences from the training set. All splits are normalized by the mean and the standard deviation of the features in the training split.</span></p>
</div>
<section id="A6.SS4.SSS0.Px1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:70%;">Covariates</h5>

<div id="A6.SS4.SSS0.Px1.p1" class="ltx_para">
<p id="A6.SS4.SSS0.Px1.p1.2" class="ltx_p"><span id="A6.SS4.SSS0.Px1.p1.2.1" class="ltx_text" style="font-size:70%;">Often, statistical models that approximate (</span><a href="#A5.E1" title="Equation E.1 ‣ Problem Formulation ‣ E.1 Multivariate Probabilistic Time Series Forecasting ‣ Appendix E Additional Experiment Results ‣ Acknowledgements. ‣ 5 Discussion ‣ 4.5 Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence Model ‣ Potential for pre-training with video. ‣ 4.4 Robotics: Long horizon imitation learning and robust visuomotor control ‣ 4.3 Controllable Sequential Compositional Generation ‣ Benefit of Flexible Horizon. ‣ 4.2 Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control. ‣ 4.1 Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout. ‣ 4 Experiments ‣ Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion" class="ltx_ref" style="font-size:70%;"><span class="ltx_text ltx_ref_tag">E.1</span></a><span id="A6.SS4.SSS0.Px1.p1.2.2" class="ltx_text" style="font-size:70%;">) benefit from manually curated features as additional input to the observations. A sequence of covariates </span><math id="A6.SS4.SSS0.Px1.p1.1.m1.1" class="ltx_Math" alttext="\boldsymbol{C}=\left\{\mathbf{c}_{t}\right\}_{t=1}^{T}" display="inline"><semantics id="A6.SS4.SSS0.Px1.p1.1.m1.1a"><mrow id="A6.SS4.SSS0.Px1.p1.1.m1.1.1" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.3" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.3.cmml">𝑪</mi><mo mathsize="70%" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.2" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.2.cmml">=</mo><msubsup id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.cmml"><mrow id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.cmml"><mo id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.2" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.cmml">{</mo><msub id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.2" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.2.cmml">𝐜</mi><mi mathsize="70%" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.3" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.3.cmml">t</mi></msub><mo id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.3" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.cmml">}</mo></mrow><mrow id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.2" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.2.cmml">t</mi><mo mathsize="70%" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.1" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.1.cmml">=</mo><mn mathsize="70%" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.3" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.3.cmml">1</mn></mrow><mi mathsize="70%" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.3" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.3.cmml">T</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px1.p1.1.m1.1b"><apply id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1"><eq id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.2"></eq><ci id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.3">𝑪</ci><apply id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.2.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1">superscript</csymbol><apply id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.2.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1">subscript</csymbol><set id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.2.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1"><apply id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.2">𝐜</ci><ci id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.1.1.1.3">𝑡</ci></apply></set><apply id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3"><eq id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.1.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.1"></eq><ci id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.2.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.2">𝑡</ci><cn type="integer" id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.3.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.1.3.3">1</cn></apply></apply><ci id="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.3.cmml" xref="A6.SS4.SSS0.Px1.p1.1.m1.1.1.1.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px1.p1.1.m1.1c">\boldsymbol{C}=\left\{\mathbf{c}_{t}\right\}_{t=1}^{T}</annotation></semantics></math><span id="A6.SS4.SSS0.Px1.p1.2.3" class="ltx_text" style="font-size:70%;"> can be constructed to help the model recognize seasonal patterns and other temporal dependencies. We follow the implementation in </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A6.SS4.SSS0.Px1.p1.2.4.1" class="ltx_text" style="font-size:70%;">[</span><a href="#bib.bib49" title="" class="ltx_ref">49</a><span id="A6.SS4.SSS0.Px1.p1.2.5.2" class="ltx_text" style="font-size:70%;">]</span></cite><span id="A6.SS4.SSS0.Px1.p1.2.6" class="ltx_text" style="font-size:70%;"> to construct the covariate sequence as a function of the frequency of each dataset in Table </span><span class="ltx_ref ltx_missing_label ltx_ref_self" style="font-size:70%;">LABEL:tab:ts_data</span><span id="A6.SS4.SSS0.Px1.p1.2.7" class="ltx_text" style="font-size:70%;">. As such, our covariates are composed of lagged inputs, as well as learned embeddings and handcrafted temporal features that encode information such as the hour of the day or the day of the month, depending on the sampling rate of the particular time series that is being modeled. Therefore, covariates are known for the entire interval </span><math id="A6.SS4.SSS0.Px1.p1.2.m2.2" class="ltx_Math" alttext="\left[1,T\right]" display="inline"><semantics id="A6.SS4.SSS0.Px1.p1.2.m2.2a"><mrow id="A6.SS4.SSS0.Px1.p1.2.m2.2.3.2" xref="A6.SS4.SSS0.Px1.p1.2.m2.2.3.1.cmml"><mo id="A6.SS4.SSS0.Px1.p1.2.m2.2.3.2.1" xref="A6.SS4.SSS0.Px1.p1.2.m2.2.3.1.cmml">[</mo><mn mathsize="70%" id="A6.SS4.SSS0.Px1.p1.2.m2.1.1" xref="A6.SS4.SSS0.Px1.p1.2.m2.1.1.cmml">1</mn><mo mathsize="70%" id="A6.SS4.SSS0.Px1.p1.2.m2.2.3.2.2" xref="A6.SS4.SSS0.Px1.p1.2.m2.2.3.1.cmml">,</mo><mi mathsize="70%" id="A6.SS4.SSS0.Px1.p1.2.m2.2.2" xref="A6.SS4.SSS0.Px1.p1.2.m2.2.2.cmml">T</mi><mo id="A6.SS4.SSS0.Px1.p1.2.m2.2.3.2.3" xref="A6.SS4.SSS0.Px1.p1.2.m2.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px1.p1.2.m2.2b"><interval closure="closed" id="A6.SS4.SSS0.Px1.p1.2.m2.2.3.1.cmml" xref="A6.SS4.SSS0.Px1.p1.2.m2.2.3.2"><cn type="integer" id="A6.SS4.SSS0.Px1.p1.2.m2.1.1.cmml" xref="A6.SS4.SSS0.Px1.p1.2.m2.1.1">1</cn><ci id="A6.SS4.SSS0.Px1.p1.2.m2.2.2.cmml" xref="A6.SS4.SSS0.Px1.p1.2.m2.2.2">𝑇</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px1.p1.2.m2.2c">\left[1,T\right]</annotation></semantics></math><span id="A6.SS4.SSS0.Px1.p1.2.8" class="ltx_text" style="font-size:70%;">, even at inference. We can easily incorporate covariates into the probabilistic framework as</span></p>
<table id="A6.E1" class="ltx_equation ltx_eqn_table">

<tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(F.1)</span></td>
</tr></tbody>
</table>
<p id="A6.SS4.SSS0.Px1.p1.3" class="ltx_p"><span id="A6.SS4.SSS0.Px1.p1.3.1" class="ltx_text" style="font-size:70%;">The benefit obtained from covariates is highly dependent on the characteristics of both the dataset and the model used, as well as the feature engineering practices followed.</span></p>
</div>
</section>
<section id="A6.SS4.SSS0.Px2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph" style="font-size:70%;">Metric</h5>

<div id="A6.SS4.SSS0.Px2.p1" class="ltx_para">
<p id="A6.SS4.SSS0.Px2.p1.10" class="ltx_p"><span id="A6.SS4.SSS0.Px2.p1.10.1" class="ltx_text" style="font-size:70%;">The Continuous Ranked Probability Score (CRPS) </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A6.SS4.SSS0.Px2.p1.10.2.1" class="ltx_text" style="font-size:70%;">[</span><a href="#bib.bib44" title="" class="ltx_ref">44</a><span id="A6.SS4.SSS0.Px2.p1.10.3.2" class="ltx_text" style="font-size:70%;">]</span></cite><span id="A6.SS4.SSS0.Px2.p1.10.4" class="ltx_text" style="font-size:70%;"> is a scoring function that measures how good the forecast distribution matches the ground truth distribution:</span></p>
<table id="A6.Ex1" class="ltx_equation ltx_eqn_table">

<tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="A6.Ex1.m1.6.6.1.1.1.1.1.1.1"><times id="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.2.cmml" xref="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.2"></times><ci id="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.3.cmml" xref="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.3">𝕀</ci><set id="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.1.2.cmml" xref="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.1.1"><apply id="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml" xref="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.1.1.1"><leq id="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.1.1.1.1"></leq><ci id="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A6.Ex1.m1.6.6.1.1.1.1.1.1.1.1.1.1.3">𝑧</ci></apply></set></apply></apply><cn type="integer" id="A6.Ex1.m1.6.6.1.1.1.3.cmml" xref="A6.Ex1.m1.6.6.1.1.1.3">2</cn></apply><apply id="A6.Ex1.m1.1.1.cmml" xref="A6.Ex1.m1.1.1"><diff id="A6.Ex1.m1.1.1.2.cmml" xref="A6.Ex1.m1.1.1.2"></diff><ci id="A6.Ex1.m1.1.1.1.1.1.cmml" xref="A6.Ex1.m1.1.1.1.1.1">𝑧</ci></apply><ci id="A6.Ex1.m1.6.6.1.1.3a.cmml" xref="A6.Ex1.m1.6.6.1.1.3"><mtext mathsize="70%" id="A6.Ex1.m1.6.6.1.1.3.cmml" xref="A6.Ex1.m1.6.6.1.1.3">,</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.Ex1.m1.6c">\operatorname{CRPS}(F,x)=\int_{\mathbb{R}}\left(F(z)-\mathbb{I}\left\{x\leq z\right\}\right)^{2}\differential{z}\text{,}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A6.SS4.SSS0.Px2.p1.8" class="ltx_p"><span id="A6.SS4.SSS0.Px2.p1.8.1" class="ltx_text" style="font-size:70%;">where </span><math id="A6.SS4.SSS0.Px2.p1.1.m1.1" class="ltx_Math" alttext="F(z)" display="inline"><semantics id="A6.SS4.SSS0.Px2.p1.1.m1.1a"><mrow id="A6.SS4.SSS0.Px2.p1.1.m1.1.2" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.2.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.1.m1.1.2.2" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.2.2.cmml">F</mi><mo lspace="0em" rspace="0em" id="A6.SS4.SSS0.Px2.p1.1.m1.1.2.1" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.2.1.cmml">​</mo><mrow id="A6.SS4.SSS0.Px2.p1.1.m1.1.2.3.2" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.2.cmml"><mo maxsize="70%" minsize="70%" id="A6.SS4.SSS0.Px2.p1.1.m1.1.2.3.2.1" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.2.cmml">(</mo><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.1.m1.1.1" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.1.cmml">z</mi><mo maxsize="70%" minsize="70%" id="A6.SS4.SSS0.Px2.p1.1.m1.1.2.3.2.2" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p1.1.m1.1b"><apply id="A6.SS4.SSS0.Px2.p1.1.m1.1.2.cmml" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.2"><times id="A6.SS4.SSS0.Px2.p1.1.m1.1.2.1.cmml" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.2.1"></times><ci id="A6.SS4.SSS0.Px2.p1.1.m1.1.2.2.cmml" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.2.2">𝐹</ci><ci id="A6.SS4.SSS0.Px2.p1.1.m1.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.1.m1.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p1.1.m1.1c">F(z)</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p1.8.2" class="ltx_text" style="font-size:70%;"> is the univariate cumulative distribution function (CDF) over the predicted value, </span><math id="A6.SS4.SSS0.Px2.p1.2.m2.1" class="ltx_Math" alttext="x" display="inline"><semantics id="A6.SS4.SSS0.Px2.p1.2.m2.1a"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.2.m2.1.1" xref="A6.SS4.SSS0.Px2.p1.2.m2.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p1.2.m2.1b"><ci 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xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.2" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.2.cmml">x</mi><mo mathsize="70%" id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.1" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.cmml">≤</mo><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.3" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.3.cmml">z</mi></mrow><mo id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.3" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p1.3.m3.1b"><apply id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1"><times id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.2"></times><ci id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.3.cmml" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.3">𝕀</ci><set id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.2.cmml" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1"><apply id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1"><leq id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.1"></leq><ci id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.2.cmml" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.2">𝑥</ci><ci id="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.3.cmml" xref="A6.SS4.SSS0.Px2.p1.3.m3.1.1.1.1.1.3">𝑧</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p1.3.m3.1c">\mathbb{I}\left\{x\leq z\right\}</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p1.8.4" class="ltx_text" style="font-size:70%;"> is the indicator function that is one if </span><math id="A6.SS4.SSS0.Px2.p1.4.m4.1" class="ltx_Math" alttext="x\leq z" display="inline"><semantics id="A6.SS4.SSS0.Px2.p1.4.m4.1a"><mrow id="A6.SS4.SSS0.Px2.p1.4.m4.1.1" xref="A6.SS4.SSS0.Px2.p1.4.m4.1.1.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.4.m4.1.1.2" xref="A6.SS4.SSS0.Px2.p1.4.m4.1.1.2.cmml">x</mi><mo mathsize="70%" id="A6.SS4.SSS0.Px2.p1.4.m4.1.1.1" xref="A6.SS4.SSS0.Px2.p1.4.m4.1.1.1.cmml">≤</mo><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.4.m4.1.1.3" xref="A6.SS4.SSS0.Px2.p1.4.m4.1.1.3.cmml">z</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p1.4.m4.1b"><apply id="A6.SS4.SSS0.Px2.p1.4.m4.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.4.m4.1.1"><leq id="A6.SS4.SSS0.Px2.p1.4.m4.1.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.4.m4.1.1.1"></leq><ci id="A6.SS4.SSS0.Px2.p1.4.m4.1.1.2.cmml" xref="A6.SS4.SSS0.Px2.p1.4.m4.1.1.2">𝑥</ci><ci id="A6.SS4.SSS0.Px2.p1.4.m4.1.1.3.cmml" xref="A6.SS4.SSS0.Px2.p1.4.m4.1.1.3">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p1.4.m4.1c">x\leq z</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p1.8.5" class="ltx_text" style="font-size:70%;"> and zero otherwise. By summing the </span><math id="A6.SS4.SSS0.Px2.p1.5.m5.1" class="ltx_Math" alttext="D" display="inline"><semantics id="A6.SS4.SSS0.Px2.p1.5.m5.1a"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.5.m5.1.1" xref="A6.SS4.SSS0.Px2.p1.5.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p1.5.m5.1b"><ci id="A6.SS4.SSS0.Px2.p1.5.m5.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.5.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p1.5.m5.1c">D</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p1.8.6" class="ltx_text" style="font-size:70%;">-dimensional time series along the feature dimension for simulated samples (resulting in </span><math id="A6.SS4.SSS0.Px2.p1.6.m6.1" class="ltx_Math" alttext="\hat{F}_{\text{sum}}(t)" display="inline"><semantics id="A6.SS4.SSS0.Px2.p1.6.m6.1a"><mrow id="A6.SS4.SSS0.Px2.p1.6.m6.1.2" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.cmml"><msub id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.cmml"><mover accent="true" id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2.2" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2.2.cmml">F</mi><mo mathsize="70%" id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2.1" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2.1.cmml">^</mo></mover><mtext mathsize="70%" id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.3" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.3a.cmml">sum</mtext></msub><mo lspace="0em" rspace="0em" id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.1" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.1.cmml">​</mo><mrow id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.3.2" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.cmml"><mo maxsize="70%" minsize="70%" id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.3.2.1" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.cmml">(</mo><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.6.m6.1.1" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.1.cmml">t</mi><mo maxsize="70%" minsize="70%" id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.3.2.2" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p1.6.m6.1b"><apply id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.cmml" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2"><times id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.1.cmml" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.1"></times><apply id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.cmml" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.1.cmml" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2">subscript</csymbol><apply id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2.cmml" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2"><ci id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2.1.cmml" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2.1">^</ci><ci id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2.2.cmml" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.2.2">𝐹</ci></apply><ci id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.3a.cmml" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.3"><mtext mathsize="49%" id="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.3.cmml" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.2.2.3">sum</mtext></ci></apply><ci id="A6.SS4.SSS0.Px2.p1.6.m6.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.6.m6.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p1.6.m6.1c">\hat{F}_{\text{sum}}(t)</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p1.8.7" class="ltx_text" style="font-size:70%;">) and ground truth data (as </span><math id="A6.SS4.SSS0.Px2.p1.7.m7.2" class="ltx_Math" alttext="\sum_{i}x^{0}_{i,t}" display="inline"><semantics id="A6.SS4.SSS0.Px2.p1.7.m7.2a"><mrow id="A6.SS4.SSS0.Px2.p1.7.m7.2.3" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.cmml"><msub id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1.cmml"><mo maxsize="70%" minsize="70%" stretchy="true" id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1.2" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1.2.cmml">∑</mo><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1.3" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1.3.cmml">i</mi></msub><msubsup 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id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1">subscript</csymbol><sum id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1.2.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1.2"></sum><ci id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1.3.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.1.3">𝑖</ci></apply><apply id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2"><csymbol cd="ambiguous" id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2.1.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2">subscript</csymbol><apply id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2.2.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2"><csymbol cd="ambiguous" id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2.2.1.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2">superscript</csymbol><ci id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2.2.2.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2.2.2">𝑥</ci><cn type="integer" id="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2.2.3.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.3.2.2.3">0</cn></apply><list id="A6.SS4.SSS0.Px2.p1.7.m7.2.2.2.3.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.2.2.4"><ci id="A6.SS4.SSS0.Px2.p1.7.m7.1.1.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.1.1.1.1">𝑖</ci><ci id="A6.SS4.SSS0.Px2.p1.7.m7.2.2.2.2.cmml" xref="A6.SS4.SSS0.Px2.p1.7.m7.2.2.2.2">𝑡</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p1.7.m7.2c">\sum_{i}x^{0}_{i,t}</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p1.8.8" class="ltx_text" style="font-size:70%;">), we can report the </span><math id="A6.SS4.SSS0.Px2.p1.8.m8.1" class="ltx_Math" alttext="\operatorname{CRPS}_{\text{sum}}" display="inline"><semantics id="A6.SS4.SSS0.Px2.p1.8.m8.1a"><msub id="A6.SS4.SSS0.Px2.p1.8.m8.1.1" xref="A6.SS4.SSS0.Px2.p1.8.m8.1.1.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.8.m8.1.1.2" xref="A6.SS4.SSS0.Px2.p1.8.m8.1.1.2.cmml">CRPS</mi><mtext mathsize="70%" id="A6.SS4.SSS0.Px2.p1.8.m8.1.1.3" xref="A6.SS4.SSS0.Px2.p1.8.m8.1.1.3a.cmml">sum</mtext></msub><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p1.8.m8.1b"><apply id="A6.SS4.SSS0.Px2.p1.8.m8.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A6.SS4.SSS0.Px2.p1.8.m8.1.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.8.m8.1.1">subscript</csymbol><ci id="A6.SS4.SSS0.Px2.p1.8.m8.1.1.2.cmml" xref="A6.SS4.SSS0.Px2.p1.8.m8.1.1.2">CRPS</ci><ci id="A6.SS4.SSS0.Px2.p1.8.m8.1.1.3a.cmml" xref="A6.SS4.SSS0.Px2.p1.8.m8.1.1.3"><mtext mathsize="49%" id="A6.SS4.SSS0.Px2.p1.8.m8.1.1.3.cmml" xref="A6.SS4.SSS0.Px2.p1.8.m8.1.1.3">sum</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p1.8.m8.1c">\operatorname{CRPS}_{\text{sum}}</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p1.8.9" class="ltx_text" style="font-size:70%;"></span></p>
<table id="A6.Ex2" class="ltx_equation ltx_eqn_table">

<tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math id="A6.Ex2.m1.7" class="ltx_Math" alttext="\operatorname{CRPS}_{\text{sum}}=\mathbb{E}_{t\sim\mathcal{U}\left(t_{0},T\right)}\left[\operatorname{CRPS}{\left(\hat{F}_{\text{sum}}(t),\sum_{i}x^{0}_{i,t}\right)}\right]" display="block"><semantics id="A6.Ex2.m1.7a"><mrow id="A6.Ex2.m1.7.7" xref="A6.Ex2.m1.7.7.cmml"><msub id="A6.Ex2.m1.7.7.3" xref="A6.Ex2.m1.7.7.3.cmml"><mi mathsize="70%" id="A6.Ex2.m1.7.7.3.2" xref="A6.Ex2.m1.7.7.3.2.cmml">CRPS</mi><mtext mathsize="70%" id="A6.Ex2.m1.7.7.3.3" xref="A6.Ex2.m1.7.7.3.3a.cmml">sum</mtext></msub><mo mathsize="70%" id="A6.Ex2.m1.7.7.2" xref="A6.Ex2.m1.7.7.2.cmml">=</mo><mrow id="A6.Ex2.m1.7.7.1" xref="A6.Ex2.m1.7.7.1.cmml"><msub id="A6.Ex2.m1.7.7.1.3" xref="A6.Ex2.m1.7.7.1.3.cmml"><mi mathsize="70%" id="A6.Ex2.m1.7.7.1.3.2" xref="A6.Ex2.m1.7.7.1.3.2.cmml">𝔼</mi><mrow id="A6.Ex2.m1.2.2.2" xref="A6.Ex2.m1.2.2.2.cmml"><mi mathsize="70%" id="A6.Ex2.m1.2.2.2.4" xref="A6.Ex2.m1.2.2.2.4.cmml">t</mi><mo 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xref="A6.Ex2.m1.2.2.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></msub><mo lspace="0em" rspace="0em" id="A6.Ex2.m1.7.7.1.2" xref="A6.Ex2.m1.7.7.1.2.cmml">​</mo><mrow id="A6.Ex2.m1.7.7.1.1.1" xref="A6.Ex2.m1.7.7.1.1.2.cmml"><mo id="A6.Ex2.m1.7.7.1.1.1.2" xref="A6.Ex2.m1.7.7.1.1.2.1.cmml">[</mo><mrow id="A6.Ex2.m1.7.7.1.1.1.1.2" xref="A6.Ex2.m1.7.7.1.1.1.1.3.cmml"><mi mathsize="70%" id="A6.Ex2.m1.6.6" xref="A6.Ex2.m1.6.6.cmml">CRPS</mi><mo id="A6.Ex2.m1.7.7.1.1.1.1.2a" xref="A6.Ex2.m1.7.7.1.1.1.1.3.cmml">⁡</mo><mrow id="A6.Ex2.m1.7.7.1.1.1.1.2.2" xref="A6.Ex2.m1.7.7.1.1.1.1.3.cmml"><mo id="A6.Ex2.m1.7.7.1.1.1.1.2.2.3" xref="A6.Ex2.m1.7.7.1.1.1.1.3.cmml">(</mo><mrow id="A6.Ex2.m1.7.7.1.1.1.1.1.1.1" xref="A6.Ex2.m1.7.7.1.1.1.1.1.1.1.cmml"><msub id="A6.Ex2.m1.7.7.1.1.1.1.1.1.1.2" xref="A6.Ex2.m1.7.7.1.1.1.1.1.1.1.2.cmml"><mover accent="true" id="A6.Ex2.m1.7.7.1.1.1.1.1.1.1.2.2" xref="A6.Ex2.m1.7.7.1.1.1.1.1.1.1.2.2.cmml"><mi mathsize="70%" id="A6.Ex2.m1.7.7.1.1.1.1.1.1.1.2.2.2" 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id="A6.Ex2.m1.7c">\operatorname{CRPS}_{\text{sum}}=\mathbb{E}_{t\sim\mathcal{U}\left(t_{0},T\right)}\left[\operatorname{CRPS}{\left(\hat{F}_{\text{sum}}(t),\sum_{i}x^{0}_{i,t}\right)}\right]</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
</table>
<p id="A6.SS4.SSS0.Px2.p1.9" class="ltx_p"><span id="A6.SS4.SSS0.Px2.p1.9.1" class="ltx_text" style="font-size:70%;">as the average over the prediction window. The lower the </span><math id="A6.SS4.SSS0.Px2.p1.9.m1.1" class="ltx_Math" alttext="\operatorname{CRPS}_{\text{sum}}" display="inline"><semantics id="A6.SS4.SSS0.Px2.p1.9.m1.1a"><msub id="A6.SS4.SSS0.Px2.p1.9.m1.1.1" xref="A6.SS4.SSS0.Px2.p1.9.m1.1.1.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p1.9.m1.1.1.2" xref="A6.SS4.SSS0.Px2.p1.9.m1.1.1.2.cmml">CRPS</mi><mtext mathsize="70%" id="A6.SS4.SSS0.Px2.p1.9.m1.1.1.3" xref="A6.SS4.SSS0.Px2.p1.9.m1.1.1.3a.cmml">sum</mtext></msub><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p1.9.m1.1b"><apply id="A6.SS4.SSS0.Px2.p1.9.m1.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.9.m1.1.1"><csymbol cd="ambiguous" id="A6.SS4.SSS0.Px2.p1.9.m1.1.1.1.cmml" xref="A6.SS4.SSS0.Px2.p1.9.m1.1.1">subscript</csymbol><ci id="A6.SS4.SSS0.Px2.p1.9.m1.1.1.2.cmml" xref="A6.SS4.SSS0.Px2.p1.9.m1.1.1.2">CRPS</ci><ci id="A6.SS4.SSS0.Px2.p1.9.m1.1.1.3a.cmml" xref="A6.SS4.SSS0.Px2.p1.9.m1.1.1.3"><mtext mathsize="49%" id="A6.SS4.SSS0.Px2.p1.9.m1.1.1.3.cmml" xref="A6.SS4.SSS0.Px2.p1.9.m1.1.1.3">sum</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p1.9.m1.1c">\operatorname{CRPS}_{\text{sum}}</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p1.9.2" class="ltx_text" style="font-size:70%;"> value, the better does the predicted distribution match the data distribution.</span></p>
</div>
<div id="A6.SS4.SSS0.Px2.p2" class="ltx_para">
<p id="A6.SS4.SSS0.Px2.p2.4" class="ltx_p"><span id="A6.SS4.SSS0.Px2.p2.4.1" class="ltx_text" style="font-size:70%;">First, we manually sum the time series along the feature dimension and estimate the CDF </span><math id="A6.SS4.SSS0.Px2.p2.1.m1.1" class="ltx_Math" alttext="\hat{F}_{\text{sum}}(t)" display="inline"><semantics id="A6.SS4.SSS0.Px2.p2.1.m1.1a"><mrow id="A6.SS4.SSS0.Px2.p2.1.m1.1.2" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.cmml"><msub id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.cmml"><mover accent="true" id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2.2" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2.2.cmml">F</mi><mo mathsize="70%" id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2.1" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2.1.cmml">^</mo></mover><mtext mathsize="70%" id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.3" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.3a.cmml">sum</mtext></msub><mo lspace="0em" rspace="0em" id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.1" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.1.cmml">​</mo><mrow id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.3.2" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.cmml"><mo maxsize="70%" minsize="70%" id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.3.2.1" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.cmml">(</mo><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p2.1.m1.1.1" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.1.cmml">t</mi><mo maxsize="70%" minsize="70%" id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.3.2.2" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p2.1.m1.1b"><apply id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.cmml" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2"><times id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.1.cmml" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.1"></times><apply id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.cmml" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2"><csymbol cd="ambiguous" id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.1.cmml" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2">subscript</csymbol><apply id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2.cmml" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2"><ci id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2.1.cmml" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2.1">^</ci><ci id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2.2.cmml" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.2.2">𝐹</ci></apply><ci id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.3a.cmml" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.3"><mtext mathsize="49%" id="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.3.cmml" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.2.2.3">sum</mtext></ci></apply><ci id="A6.SS4.SSS0.Px2.p2.1.m1.1.1.cmml" xref="A6.SS4.SSS0.Px2.p2.1.m1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p2.1.m1.1c">\hat{F}_{\text{sum}}(t)</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p2.4.2" class="ltx_text" style="font-size:70%;"> via 19 quantile levels at each time step </span><math id="A6.SS4.SSS0.Px2.p2.2.m2.1" class="ltx_Math" alttext="t" display="inline"><semantics id="A6.SS4.SSS0.Px2.p2.2.m2.1a"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p2.2.m2.1.1" xref="A6.SS4.SSS0.Px2.p2.2.m2.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p2.2.m2.1b"><ci id="A6.SS4.SSS0.Px2.p2.2.m2.1.1.cmml" xref="A6.SS4.SSS0.Px2.p2.2.m2.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p2.2.m2.1c">t</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p2.4.3" class="ltx_text" style="font-size:70%;"> from 100 sampled trajectories. We then use the implementation in GluonTs </span><cite class="ltx_cite ltx_citemacro_cite"><span id="A6.SS4.SSS0.Px2.p2.4.4.1" class="ltx_text" style="font-size:70%;">[</span><a href="#bib.bib2" title="" class="ltx_ref">2</a><span id="A6.SS4.SSS0.Px2.p2.4.5.2" class="ltx_text" style="font-size:70%;">]</span></cite><span id="A6.SS4.SSS0.Px2.p2.4.6" class="ltx_text" style="font-size:70%;"> to compute the </span><math id="A6.SS4.SSS0.Px2.p2.3.m3.1" class="ltx_Math" alttext="\operatorname{CRPS}" display="inline"><semantics id="A6.SS4.SSS0.Px2.p2.3.m3.1a"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p2.3.m3.1.1" xref="A6.SS4.SSS0.Px2.p2.3.m3.1.1.cmml">CRPS</mi><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p2.3.m3.1b"><ci id="A6.SS4.SSS0.Px2.p2.3.m3.1.1.cmml" xref="A6.SS4.SSS0.Px2.p2.3.m3.1.1">CRPS</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p2.3.m3.1c">\operatorname{CRPS}</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p2.4.7" class="ltx_text" style="font-size:70%;">, which we report as </span><math id="A6.SS4.SSS0.Px2.p2.4.m4.1" class="ltx_Math" alttext="\operatorname{CRPS}_{\text{sum}}" display="inline"><semantics id="A6.SS4.SSS0.Px2.p2.4.m4.1a"><msub id="A6.SS4.SSS0.Px2.p2.4.m4.1.1" xref="A6.SS4.SSS0.Px2.p2.4.m4.1.1.cmml"><mi mathsize="70%" id="A6.SS4.SSS0.Px2.p2.4.m4.1.1.2" xref="A6.SS4.SSS0.Px2.p2.4.m4.1.1.2.cmml">CRPS</mi><mtext mathsize="70%" id="A6.SS4.SSS0.Px2.p2.4.m4.1.1.3" xref="A6.SS4.SSS0.Px2.p2.4.m4.1.1.3a.cmml">sum</mtext></msub><annotation-xml encoding="MathML-Content" id="A6.SS4.SSS0.Px2.p2.4.m4.1b"><apply id="A6.SS4.SSS0.Px2.p2.4.m4.1.1.cmml" xref="A6.SS4.SSS0.Px2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="A6.SS4.SSS0.Px2.p2.4.m4.1.1.1.cmml" xref="A6.SS4.SSS0.Px2.p2.4.m4.1.1">subscript</csymbol><ci id="A6.SS4.SSS0.Px2.p2.4.m4.1.1.2.cmml" xref="A6.SS4.SSS0.Px2.p2.4.m4.1.1.2">CRPS</ci><ci id="A6.SS4.SSS0.Px2.p2.4.m4.1.1.3a.cmml" xref="A6.SS4.SSS0.Px2.p2.4.m4.1.1.3"><mtext mathsize="49%" id="A6.SS4.SSS0.Px2.p2.4.m4.1.1.3.cmml" xref="A6.SS4.SSS0.Px2.p2.4.m4.1.1.3">sum</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.SSS0.Px2.p2.4.m4.1c">\operatorname{CRPS}_{\text{sum}}</annotation></semantics></math><span id="A6.SS4.SSS0.Px2.p2.4.8" class="ltx_text" style="font-size:70%;"> in Table </span><span class="ltx_ref ltx_missing_label ltx_ref_self" style="font-size:70%;">LABEL:tab:results_ts</span><span id="A6.SS4.SSS0.Px2.p2.4.9" class="ltx_text" style="font-size:70%;">. While we aggregate the data manually, we verify that the numerical error relative to the GluonTS implementation remains orders of magnitude below the precision threshold of the reported metric.</span></p>
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2407.01392
This paper presents Diffusion Forcing, a new training paradigm where a diffusion model is trained to denoise a set of tokens withindependentper-token noise levels.
We apply Diffusion Forcing to sequence generative modeling by training a causal next-token prediction model to generate one or several future tokens without fully diffusing past ones. Our approach is shown to combine the strengths of next-token prediction models, such as variable-length generation, with the strengths of full-sequence diffusion models, such as the ability to guide sampling to desirable trajectories. Our method offers a range of additional capabilities, such as (1)  rolling-out sequences of continuous tokens, such as video, with lengths past the training horizon, where baselines diverge and (2)  new sampling and guiding schemes that uniquely profit from Diffusion Forcing’s variable-horizon and causal architecture, and which lead to marked performance gains in decision-making and planning tasks. In addition to its empirical success, our method is proven to optimize a variational lower bound on the likelihoods of all subsequences of tokens drawn from the true joint distribution. Project website:https://boyuan.space/diffusion-forcing
abstract
Diffusion Forcing: Next-token PredictionMeets Full-Sequence Diffusion
Abstract
2407.01392
1IntroductionProbabilistic sequence modeling plays a crucial role in diverse machine learning applications including natural language processing, video predictionand decision making.
Next-token prediction models in particular have a number of desirable properties. They enable the generation of sequences with varying length(generating only a single token or an “infinite” number of tokens via auto-regressive sampling) , can be conditioned on varying amounts of history, support efficient tree search, and can be used for online feedback control.Current next-token prediction models are trained viateacher forcing, where the model predicts the immediate next token based on a ground truth history of previous tokens.
This results in two limitations: (1)  there is no mechanism by which one can guide the sampling of a sequence to minimize a certain objective, and
(2)  current next-token models easily becomeunstableon continuous data. For example, when attempting to auto-regressively generate a video (as opposed to textor vector-quantized latents)  past the training horizon, slight errors in frame-to-frame predictions accumulate and the model diverges.Full-sequence diffusionseemingly offers a solution.
Commonly used in video generation and long-horizon planning, one directly models the joint distribution of a fixed number of tokens by diffusing their concatenation, where the noise level is identical across all tokens.
They offerdiffusion guidanceto guide sampling to a desirable sequence, invaluable in decision-making (planning)  applications.
They further excel at generating continuous signals such as video.
However, full-sequence diffusion is universally parameterized via non-causal, unmasked architectures.
In addition to restricting sampling to full sequences, as opposed to variable length generation, we show that this limits the possibilities for both guidance and subsequence generation (Figure1) .
Further, we demonstrate that a naive attempt at combining the best of both worlds by training a next-token prediction model for full-sequence diffusion leads to poor generations, intuitively because it does not model the fact that small uncertainty in an early token necessitates high uncertainty in a later one.In this paper, we introduceDiffusion Forcing(DF) , a training and sampling paradigm where each token is associated with arandom, independentnoise level, and where tokens can be denoised according to arbitrary, independent, per-token schedules through a shared next-or-next-few-token prediction model. Our approach is motivated by the observation that noising tokens is a form ofpartial masking—zero noise means a token is unmasked, and complete noise fully masks out a token. Thus, DF forces the model to learn to “unmask” any collection of variably noised tokens (Figure2) . Simultaneously, by parameterizing predictions as a composition of next-token prediction models, our system can flexibly generate varying length sequences as well as compositionally generalize to new trajectories (Figure1) .We implement DF for sequence generation asCausal Diffusion Forcing(CDF) , in which future tokens depend on past ones via a causal architecture.
We train the model to denoise all tokens of a sequence at once, with an independent noise level per token.
During sampling, CDF gradually denoises a sequence of Gaussian noise frames into clean samples where different frames may have different noise levels at each denoising step.
Like next-token prediction models, CDF can generate variable-length sequences; unlike next-token prediction, it does so stabily from the immediate next token to thousands of tokens in the future – even for continuous tokens. Moreover, like full-sequence diffusion it accepts guidance towards high-reward generations. Synergistically leveraging causality, flexible horizon, and variable noise schedules, CDF enables a new capability,
Monte Carlo Tree Guidance (MCTG) , that dramatically improves the sampling of high-reward generations compared to non-causal full-sequence diffusion models. Fig.1overviews these capabilities.Figure 1:Diffusion Forcing capabilities.Today, different applications such as language modeling, planning, or video generationrely oneitherauto-regressive next-token predictionorfull-sequence diffusion, according to their respective unique capabilities. The proposed Diffusion Forcing is a novel sequence generative model that enjoys key strengths of both model types.In summary, our contributions are:
(1)  We propose Diffusion Forcing, a new probabilistic sequence model that has the flexibility of next-token prediction models while being able to perform long-horizon guidance like full-sequence diffusion models.
(2)  Taking advantage of Diffusion Forcing’s unique capabilities, we introduce a novel decision-making framework that allows us to use Diffusion Forcing as simultaneously apolicy()  and as aplanner() .
(3)  We formally prove that, under appropriate conditions, optimizing our proposed training objective maximizes a lower bound on the likelihood of the joint distribution ofall sub-sequencesobserved at training time.
(4)  We empirically evaluate CDF across diverse domains such as video generation, model-based planning, visual imitation learning, and time series prediction, and demonstrate CDF’s unique capabilities, such as stabilizing long-rollout autoregressive video generation, composing sub-sequences of those observed at training time with user-determined memory horizon, Monte Carlo Tree Guidance, and more.
introduction
Diffusion Forcing: Next-token PredictionMeets Full-Sequence Diffusion
1Introduction
2407.01392
2Related Work and PreliminariesWe discuss related work and preliminaries for our core application, sequence generative modeling; seeAppendixCfor further literature review.Our method unifies two perspectives on sequence modeling: Bayesian filtering along the time axis, denoted by subscript, and diffusion along an “uncertainty” (or noise level)  axis denoted by superscript. In the following, we denote observations asand latent states as.Bayesian Filtering.Given a Hidden Markov Model (HMM)  defined by latent statesand observations, a Bayes filter is a probabilistic method for estimating latent states recursively over time from incoming observations. A prior modelinfers a belief over the next state given only the current state, and an observation model infers a belief over the next observation given the current latent state. When a new observation is made, a posterior modelprovides an updated estimation of the next latent state. When trained end-to-end with neural networks, latent states are not an estimate of any physical quantity, but a sufficiently expressive latent that summarizes past observations for predicting future observationsin the sequence.Figure 2:Method Overview.Diffusion Forcing trains causal sequence neural networks (such as an RNN or a masked transformer)  to denoise flexible-length sequences where each frame of the sequence can have adifferentnoise level.
In contrast, next-token prediction models, common in language modeling, are trained to predict a single next token from aground-truthsequence (teacher forcing) , and full-sequence diffusion, common in video generation, train non-causal architectures to denoise all frames in a sequence at once with thesamenoise level.
Diffusion Forcing thusinterleavesthe time axis of the sequence and the noise axis of diffusion, unifying strengths of both alternatives and enabling completely new capabilities (see Secs.3.1,3.2) .Diffusion Models.Diffusion modelshave proven to be highly expressive and reliable generative models.
We review their essentials here. Letdenote a data distribution of interest, and let. We consider a forward diffusion process that gradually adds Gaussian noise to a data point over a series of time steps. This process is modeled as a Markov chain, where the data at each stepis noised incrementally:(2.1) whereis the normal distribution andis the variance of the noise added at each step controlled by a schedule. The process continues until the data is converted into pure noise at.
The reverse process is also a Markov chain and attempts to recreate the original data from the noise with a parameterized model:(2.2) where the meanis model with a neural network, and where it is shownthat one can set the covariance to the identity scaled by a fixed constantdepending on. Adopting the standard exposition, we reparametrize the meanin terms of noise prediction. This leadsto the following least squares objective:(2.3) whereand. One can then sample from this model via Langevin dynamics.Guidance of Diffusion Models.Guidanceallows biasing diffusion generation towards desirable predictions at sampling time.
We focus on classifier guidance: given a classifierof some desired(e.g. class or success indicator) , one modifies the Langevin samplinggradientto be. This allows sampling from the joint distribution ofand class labelwithout the need to train a conditional model. Other energies such as a least-squares objective comparing the model output to a desirable ground-truth have been explored in applications such as decision making.Next-Token Prediction Models.Next-token prediction models are sequence models that predict the next framegiven past frames. At training time, one feeds a neural network withand minimizesfor continuous data or a cross-entropy loss for discrete data. At sampling time, one samples the next framefollowing. If one treatsas, one can use the same model to predictand repeat until a full sequence is sampled. Unlike full-sequence diffusion models, next-token models do not accept multi-step guidance, as prior frames must be fully determined to sample future frames.Diffusion Sequence Models.Diffusion has been widely used in sequence modeling.use full-sequence diffusion models to achieve controllable text generation via guidance, such as generating text following specified parts of speech.trains full-sequence diffusion models to synthesize short videos and uses a sliding window to roll out longer conditioned on previously generated frames.uses full-sequence diffusion models as planners in offline reinforcement learning. This is achieved by training on a dataset of interaction trajectories with the environment and using classifier guidance at sampling time to sample trajectories with high rewards towards a chosen goal.modifies auto-regressive models to denoise the next token conditioned on previous tokens. It trains with teacher forcingand samples next-token auto-regressively for time series data. Most similar to our work is AR-Diffusion, which trains full-sequence text diffusion with a causal architecture with linearly dependent noise level along the time axis. We provide a detailed comparision between this approach and ours inAppendixC.
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2Related Work and Preliminaries
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3Diffusion ForcingNoising as partial masking.Recall thatmaskingis the practice of occluding a subset of data, such as patches of an imageor timesteps in a sequence, and training a model to recover unmasked portions.
Without loss of generality, we can view any collection of tokens, sequential or not, as an ordered set indexed by.
Training next-token prediction with teacher forcing can then be interpreted as masking each tokenat timeand making predictions from the past. Restricted to sequences, we refer to all these practices asmasking along the time axis. We can also view full-sequence forward diffusion, i.e., gradually adding noise to the data, as a form ofpartial masking, which we refer to asmasking along the noise axis. Indeed, aftersteps of noising,is (approximately)  pure white noise without information about the original data.We establish a unified view along both axes of masking (see Fig.2) . We denotefor a sequence of tokens, where the subscript indicates the time axis. As above,denotesat noise levelunder the forward diffusion process (2.1) ;is the unnoised token, andis white noise. Thus,denotes a sequence of noisy observations where each token has adifferentnoise level, which can be seen as the degree ofpartial maskingapplied to each token through noising.Diffusion Forcing: Different noise nevels for different tokens.Diffusion Forcing(DF)  is a framework for training and sampling arbitrary sequence lengths of noisy tokens, where critically,the noise levelof each token can vary by time step.
In this paper, we focus on time series data, and thus instantiate Diffusion Forcing with causal architectures (wheredepends only on past noisy tokens) , which we callCausal Diffusion Forcing(CDF) .
For simplicity, we focus on a minimal implementation with a vanilla Recurrent Neural Network (RNN) .Algorithm 1Diffusion Forcing Training1:loop2:Sample tajectory of observations.3:fordo4:Sample independent noise level5:ForwardDiffuse6:Define7:Update.8:Set9:endfor10:MSELoss11:Backprop withand update12:endloopAlgorithm 2DF Sampling with Guidance1:Input:Model, scheduling matrix, initial latent, guidance cost.2:Initialize.3:forrowdo4:fordo5:.6:,.7:8:Update.9:endfor10:AddGuidance11:endfor12:Return.The RNN with weightsmaintains latentscapturing the influence of past tokens, and these evolve via dynamicswith a recurrent layer. When an incoming noisy observationis made, the hidden state is updated in a Markovian fashion111We implementto be deterministic, withrepresenting a distribution over beliefs rather than a sample from it. This allows training by backpropogating through the latent dynamics in Eq.(3.1) .. When, this is the posterior update in Bayes filtering; whereas when(andis pure noise and thus uninformative) , this is equivalent to modeling the “prior distribution”in Bayes filtering. Given latent, an observation modelpredicts;
this unit has the same input-output behavior as a standard conditional diffusion model, using a conditioning variableand a noisy tokenas input to predict the unnoisedand thus, indirectly, the noisevia affine reparametrization.
We can thus directly train (Causal)  Diffusion Forcing with the conventional diffusion training objective. We parameterize the aforementioned unit in terms of noise prediction.
We then find parametersby minimizing the loss(3.1) where we sampleuniformly from,from our training data, andin accordance with the forward diffusion process (see Algorithm1for pseudocode) . Importantly, the loss (3.1)  captures essential elements of Bayesian filtering and conditional diffusion.
In AppendixD.3, we further re-derive common techniques in diffusion model training for Diffusion Forcing. Finally, we prove the validity of this objective stated informally in the following Theorem3.1in AppendixA.Theorem 3.1(Informal) .The Diffusion Forcing training procedure (Algorithm1)  optimizes a reweighting of an Evidence Lower Bound (ELBO)  on the expected log-likelihoods, where the expectation is averaged over noise levelsandnoised according to the forward process. Moreover, under appropriate conditions, optimizing (3.1)  also maximizes a lower bound on the likelihood forall sequences of noise levels, simultaneously.We remark that a special case of ‘all sequences of noise levels’ are those for which eitheror; thus, one can mask outany prior tokenand DF will learn to sample from the correct conditional distribution, modeling the distribution of all possible sub-sequences of the training set.3.1Diffusion Forcing Sampling and Resulting CapabilitiesSampling is depicted inAlgorithm2and is defined by prescribing a noise schedule on a 2Dgrid; columns correspond to time stepand rows indexed bydetermine noise-level.represents the desired noise level of the time-steptoken for row.
To generate a whole sequence of length, initialize the tokensto be white noise, corresponding to noise level. We iterate down the grid row-by-row, denoising left-to-right across columns to the noise levels prescribed by. By the last row, the tokens are clean, i.e. their noise level is.SectionD.5discusses corner cases of this scheme; the hyperparametersare set to their standard values. We now explain new capabilities this sampling paradigm has to offer.Stabilizing Auto-Regressive Generation.For high-dimensional, continuous sequences such as video, auto-regressive architectures are known to diverge, especially when sampling past the training horizon.
In contrast, Diffusion Forcing can stably roll out long sequences even beyond the training sequence length by updating the latents using the previous latent associated with slightly “noisy tokens” for some small noise level. Our experiments (Sec.4.1)  illustrates the resulting marked improvements in long-horizon generation capabilities; App.B.2provides further intuition.Keeping the future uncertain.Beginning from a sequence of white noise tokens, we may denoise the first token fully and the second token partially, yielding, then, and finally denoising all tokens fully to.
Interpreting the noise level as uncertainty, this “zig-zag” sampling scheme intuitively encodes the immediate future as more certain than the far future. Sec.3.2describes how this leads to more effective sequence guidance.Long-horizon Guidance.In Line 10 of Algorithm2, one may add guidance to the partially diffused trajectoryas in Sec.2. Due to the dependency of future tokens on the past, guidance gradients from future tokens can propagate backwards in time. The unique advantage of Diffusion Forcing is that, because we can diffuse future tokens without fully diffusing the past, the gradient guides the sampling ofpasttokens, thereby achieving long-horizon guidance while respecting causality. We elaborate on implementation details in AppendixB.1. As we show inSection4.2, planning in this manner significantly outperforms guided full-sequence diffusion models.3.2Diffusion Forcing for Flexible Sequential Decision MakingThe capabilities offered by Diffusion Forcing motivate our novel framework for sequential decision making (SDM) , with key applications to robotics and autonomous agents.
Consider a Markov Decision Process defined by an environment with dynamics, observationand reward. The goal is to train a policysuch that the expected cumulative reward of a trajectoryis maximized.
We assign tokens. A trajectory is a sequence, possibly of variable length; training is conducted as in Algorithm1.
At each stepof execution, past (noise-free)  tokensare summarized by a latent. Conditioned on this latent, we sample, via Algorithm2, a plan, withcontaining predicted actions, rewards and observations.is a lookahead window, analogous to future predictions in model predictive control. After taking planned action, the environment produces a rewardand next observation, yielding next token. The latent is updated according to the posterior. Our framework enables functionality as bothpolicyandplanner:Flexible planning horizon.Diffusion Forcing (a)  can be deployed ontasks of variable horizon, because each new action is selected sequentially, and
(b)  its lookahead windowcan be shortened to lower latency (using Diffusion Forcing as apolicy) , or lengthened to perform long-horizonplanning(via guidance described below) , without re-training or modifications of the architecture. Note that (a)  is not possible for full-sequence diffusion models like Diffuserwith full-trajectory generation horizons, whereas diffusion policiesneed fixed, small lookahead sizes, precluding (b) .Flexible reward guidance.As detailed in AppendixB.1,
Diffusion Forcing can plan via guidance using any reward (in place of)  specified over future steps: this includes dense per-time step rewards on the entire trajectory trajectory, dense rewards on a future lookahead, and sparse rewards indicating goal completion. Per-time step policies cannot take advantage of this latter, longer horizon guidance.Monte Carlo Tree Guidance (MCTG) , future uncertainty.Causal Diffusion Forcing allows us to influence the generation of a tokenby guidance on the whole distribution of future. Instead of drawing a single trajectory sample to calculate this guidance gradient, we can draw multiple samples and average their guidance gradients. We call this Monte Carlo Tree Guidance, where “tree” comes from the fact that the the denoising of the currentis influenced by gradients through many paths through the future. In the spirit of so-called shooting methods like MPPI,is then guided by the expected reward over the distribution of all future outcomes instead of one particular outcome. The effect of MCTG is enhanced when combined with sampling schedules that keep the noise level of future tokens high when denoising immediate next tokens (e.g. the zig-zag schedule described in Sec.3.1) , accounting for greater uncertainty farther into the future. AppendixB.3further justifies the significance of MCTG, and why Diffusion Forcing uniquely takes advantage of it.Figure 3:Video Generation.Among tested methods, Diffusion Forcing generations are uniquely temporally consistent and do not diverge even when rolling out well past the training horizon. Please see theproject websitefor video results.
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Diffusion Forcing: Next-token PredictionMeets Full-Sequence Diffusion
3Diffusion Forcing
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4ExperimentsWe extensively evaluate Diffusion Forcing’s merits as a generative sequence model across diverse applications in video and time series prediction, planning, and imitation learning. Please find dataset and reproducibility details in the Appendix, as well as video results on theproject website.4.1Video Prediction: Consistent, Stable Sequence Generation and Infinite Rollout.We train a convolutional RNN implementation of Causal Diffusion Forcing for video generative modeling on videos of Minecraft gameplayand DMLab navigation. At sampling time, we perform auto-regressive rollout with stabilization proposed in Sec.3.1.
We consider two baselines, both leveraging the same exact RNN architecture: a next-frame diffusion baseline trained with teacher forcingas well as a causal full-sequence diffusion model.Figure3displays qualitative results of roll-outs generated by Diffusion Forcing and baselines starting from unseen frames for both datasets. While Diffusion Forcing succeeds at stably rolling out even far beyond its training horizon (e.g.frames) , teacher forcing and full-sequence diffusion baselines diverge quickly. Further, within the training horizon, we observe that full-sequence diffusion suffers from frame-to-frame discontinuity where video sequences jump dramatically, while Diffusion Forcing roll-outs show ego-motion through a consistent 3D environment. This highlights the ability of Diffusion Forcing to stabilize rollouts of high-dimensional sequences without compounding errors.\FLEnvironmentMPPICQLIQLDiffuser*Diffuser w/ diffused actionOurs wo/ MCTGOurs\MLMaze2DU-Maze33.25.747.4113.93.16.32.1110.13.9116.72.0\NNMaze2DMedium10.25.034.9121.52.713.52.3136.110.2149.47.5\NNMaze2DLarge5.112.558.6123.06.46.32.1142.85.6159.02.7\MLSingle-task Average16.27.747.0119.58.7129.67141.7\ML[0.11em]
Multi2DU-Maze41.2-24.8128.91.832.81.7107.74.9119.14.0\NNMulti2DMedium15.4-12.1127.23.422.02.7145.66.5152.39.9\NNMulti2DLarge8.0-13.9132.15.86.91.7129.81.5167.12.7\MLMulti-task Average21.5-16.9129.420.57127.7146.17\LLTable 1:Diffusion Forcing for Planning.(top)  During sampling, Diffusion Forcingallows each time step to be denoised on different noise schedules, enabling us to account for causal uncertainty during guided planning. Diffusion Forcingkeeps the far future more uncertain than the near future while Diffuserputs them at the same noise level during sampling. (bottom) Quantitatively, Diffusion Forcingachieves the highest average reward across runs. Diffuser fails dramatically when executing the actually generated actions, requiring a hand-crafted PD controller (indicated by the asterisk)  and ignoring generated actions.4.2Diffusion Planning: MCTG, Causal Uncertainty, Flexible Horizon Control.Decision-making uniquely benefits from Diffusion Forcing’s capabilities. We evaluate our proposed decision-making framework in a standard offline RL benchmark, D4RL. Specifically, we benchmark Diffusion Forcing on a set of 2D maze environments with sparse reward. An agent is tasked with reaching a designated goal position starting from a random starting position. In Appendix16we provide a detailed description of the environment. The benchmark provides a dataset ofrandom walksthrough mazes (thus stochastic) . We train one model per maze.We benchmark the proposed decision-making framework3.2with state-of-the-art offline RL methods and the recently introduced Diffuser, a diffusion planning framework. See Fig.4.1for qualitative and quantitative reuslts: DF outperforms Diffuser and all baselines across allenvironments.Benefit of Monte Carlo Tree Guidance.The typical goal for an RL problem is to find actions that maximize theexpectedfuture rewards, which we achieve through MCTG. Full-sequence diffusion models such as Diffuser do not support sampling to maximize expected reward, as we formally derive inSectionB.3. To understand MCTG’s importance, we ablate it inSection4.1. Removing MCTG guidance degrades our performance, though Diffusion Forcing remains competitive even then.Benefit of Modeling Causality.Unlike pure generative modeling, sequential decision-making takes actions and receives feedback. Due to compounding uncertainty, the immediate next actions are more important than those in the far future.
Though Diffuser and subsequent models are trained to generate sequences of action-reward-state tuples, directly executing the actions will lead to a trajectory that deviates significantly from the generated states. In other words, the generated states and actions are not causally consistent with each other. To address this shortcoming, Diffuser’s implementation ignores the generated actions and instead relies on a hand-crafted PD controller to infer actions from generated states. In Table4.1, we see that Diffuser’s performance drops dramatically when directly executing generated actions. In contrast, Diffusion Forcing’s raw action generations are self-consistent, outperforming even actions selected by combining Diffuser’s state predictions with a handcrafted PD controller.Benefit of Flexible Horizon.Many RL tasks have a fixed horizon, requiring the planning horizon to shrink as an agent makes progress in the task. Diffusion Forcing accomplishes this by design, while full-sequence models like Diffuser perform poorly even with tweaks, as we explain in AppendixB.4.4.3Controllable Sequential Compositional GenerationWe demonstrate that by only modifying the sampling scheme, we can flexibly compose sub-sequences of sequences observed at training time.
We consider a dataset of trajectories on a 2D, square plane, where all trajectories start from one corner and end up in the opposite corner, forming a cross shape. As shown in Fig.1, when no compositional behavior is desired, one can let DF keep full memory, replicating the cross-shaped distribution. When one desires compositionality, one can let the model generate shorter plans without memory using MPC, leading to stitching of the cross’s sub-trajectories, forming a V-shaped trajectory. Due to limited space, we defer the result to AppendixE.2.Figure 4:In our real robot task, a robot arm is asked to swap the slots of two fruits using a third slot. Since the fruits are input in random slots at the beginning, one cannot determine the next steps from a single observation without knowledge of the initial placement of the fruits. As illustrated in (a)  and (b) , the upper observation is the same but the desired outcome illustrated below can vary—the task thus requires remembering the initial configuration. In addition, as shown in (c) , the same model that generates actions also synthesizes realistic video from just a single frame.4.4Robotics: Long horizon imitation learning and robust visuomotor controlFinally, we illustrate that Diffusion Forcing (DF)  opens up new opportunities in visuomotor control of real-world robots. Imitation learningis a popular technique in robotic manipulation where one learns an observation-to-action mapping from expert demonstrations. However, the lack of memory often prevents imitation learning from accomplishing long-horizon tasks. DF not only alleviates this shortcoming but also provides a way to make imitation learning robust.Imitation Learning with Memory.We collect a dataset of videos and actions by teleoperating a Franka robot. In the chosen task, one needs to swap the position of an apple and an orange, using a third slot. See Fig.4for an illustration. The initial positions of the fruits are randomized such that there are two possible goal states. As illustrated in Fig.4, when one fruit is in the third slot, the desired outcome cannot be inferred from the current observation—a policy must remember the initial configuration to determine which fruit to move. In contrast to common behavior cloning methods, DF naturally incorporates memory in its latent state. We found that DF achievessuccess rate while diffusion policy, a state-of-the-art imitation learning algorithm without memory, fails.Robustness to missing or noisy observations.Because it incorporates principles from Bayes filtering,
Diffusion Forcing can perform imitation learning while being robust to noisy or missing observations. We demonstrate this by adding visual distractions and even fully occluding the camera during execution.
DF allows us to easily indicate these observations as “noisy” by using, in which case DF relies heavily on its prior model to predict actions. Consequently, the succes rate is only lowered byto.
In contrast, a next-frame diffusion model baseline attains a success rate of: it must treat perturbed observations as ground truth and suffers out-of-distribution error.Potential for pre-training with video.Finally in parallel to generating actions, Fig.4illustrates that Diffusion Forcing is capable of generating a video of the robot performing the task given only an initial frame, unifying diffusion policy / imitation learning and video generative modeling and paving the way to pre-training on unlabeled video.4.5Time Series Forecasting: Diffusion Forcing is a Good General-purpose Sequence ModelIn AppendixE, we show that DF is competitive with prior diffusionand transformer-basedwork on multivariate time series forecasting, following the experimental setup of.5DiscussionLimitations.Our current causal implementation is based on a small RNN, and applications to higher-resolution video or more complex distributions likely require large transformer models. We do not investigate the scaling behavior of Diffusion Forcing to internet-scale datasets and tasks.Conclusion.In this paper, we introduced Diffusion Forcing, a new training paradigm where a model is trained to denoise sets of tokens with independent, per-token noise levels.
Applied to time series data, we show how a next-token prediction model trained with Diffusion Forcing combines benefits of both next-token models and full-sequence diffusion models.
We introduced new sampling and guidance schemes that lead to dramatic performance gains when applied to tasks in sequential decision making.
Future work may investigate the application of Diffusion Forcing to domains other than time series generative modeling, and scale up Diffusion Forcing to larger datasets.Acknowledgements.This work was supported by the National Science Foundation under Grant No. 2211259, by the Singapore DSTA under DST00OECI20300823 (3D Self-Supervised Learning for Label-Efficient Vision) , by the Intelligence Advanced Research Projects Activity (IARPA)  via Department of Interior/ Interior Business Center (DOI/IBC)  under 140D0423C0075, and by the Amazon Science Hub. Special thanks to Kiwhan Song (MIT)  for 3D-Unet and transformer reimplementation of the paper after initial release.References[1]A. Ajay, Y. Du, A. Gupta, J. Tenenbaum, T. Jaakkola, and P. Agrawal.Is conditional generative modeling all you need for decision-making?arXiv preprint arXiv:2211.15657, 2022.[2]A. Alexandrov, K. Benidis, M. Bohlke-Schneider, V. Flunkert, J. Gasthaus, T. Januschowski, D. C. 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We first establish this in full abstraction (TheoremA.1) , and then specialize to the form of Gaussian diffusion (CorollaryA.2) . We show that the resulting terms decouple in such a fashion that, in the limit of a fully expressive latent and model, makes the reweighting terms immaterial.•We show that the expected likelihood overanydistribution over sequences of noise levels can be lower bounded by a sum over nonnegative terms which, when reweighted, correspond to the terms optimized in the Diffusion Forcing training objective maximizes. Thus, for a fully expressive network that can drive all terms to their minimal value, Diffusion Forcing optimizes a valid surrogate of the likelihood forlikelihood of all sequences of noise levels simultaneously.We begin by stating an ELBO for general Markov forward processes, and generative models, and then specialize to Gaussian diffusion, thereby recovering our loss. We denote our Markov forward processas(A.1) and a parmeterized probability model(A.2) We assume thatsatisfies theMarkov propertythat(A.3) that is, the latent codesis a sufficient statistic forgiven the history. We say thathasdeterministic latentsifis a Dirac delta.Remark 1.In order forto have deterministic latents and correspond to a valid probability distribution, we need to view the latentsnot as individual variables, but as a collections of variablesindexed byand and thehistoryof noise levels.. In this case, simply settingtautologically produces deterministic latents. The reason for indexingwiththen arises because, otherwise,would be ill-defined unlessfor all, and thus,would not correspond to a joint probability measure. The exposition and theorem that follows allowsto be indexed on past noise levels, but suppresses dependence onto avoid notational confusion.A.1Main ResultsWe can now state our main theorem, which provides an evidence lower bound (ELBO)  on the expected log likelihood of partially-noised sequences, under uniform levelsandobtained by noising according toas in (A.1) . Notice that this formulation does not require an explicit for ofor, but we will specialize to Gaussian diffusion in the sequel.Theorem A.1.Fix. Define the expectation over the forward process with random noise levelas(A.4) and the expectation over the latents underconditioned onas(A.5) Then, as long assatisfies the Markov property,whereis a constant depending only on(the unnoised data) . Moreover, if the latents are deterministic (i.e.is a Dirac distribution) , then the inequality holds with inequality if and only if, i.e. the variational approximation is exact.The proof of the amove theorem is given inSectionA.2. Remarkably, it involvesonly twoinequalities! The first holds with equality under deterministic latents and the second holds if and only if variational approximation is exact:. This tightness of the ELBO suggests that the expression inTheoremA.1is a relatively strong surrogate objective for optimizing the likelihoods.A.1.1Specializing to Gaussian diffusionWe now specialTheoremA.1to Gaussian diffusion. For now, we focus on the “-prediction” formulation of diffusion, which is the one used in our implementation. The “-prediction” formalism, used throughout the main body of the text, can be derived similarly (see Section 2 offor a clean exposition) . The following theorem follows directly by apply standard likelihood and KL-divergence computations for the DDPMtoTheoremA.1.Corollary A.2.Let(A.6) and define,. Suppose that we parameterize, where further,Then, as long assatisfies the Markov property, we obtainedwhere above, we define.Proof.The first inequality follows from the standard computations for the “-prediction” formulation of Diffusion (see Section 2.7 ofand references therein) . The second follows by replacing the sum overwith an expectation over.
∎We make a couple remarks:•As noted above,CorollaryA.2can also be stated for-prediction, or the so-called “-prediction” formalism, as all are affinely related.•Define an idealized latentconsisting of all past tokensas well as of their noise levels. This is a sufficient statistic for, and thus we can always view, whereis just compressing. When applying the expectation ofto both sides of the bound inCorollaryA.2, and taking an infimum over possible function approximator, we obtainThis leads to a striking finding: with expressive enough latents and, we can view the maximization of each term inCorollaryA.2separately across time steps. The absence of this coupling means that the weighting terms are immaterial to the optimization, and thus can be ignored.•Given the above remarks, we can optimize the ELBO by taking gradients through the objective specified byCorollaryA.2, and are free to drop any weighting terms (or rescale them)  as desired. Backpropogating throughis straightforward due to deterministic latents. This justifies the correctness of our training objective (3.1)  and protocolAlgorithm1.A.1.2Capturing all subsequencesTheoremA.1stipulates that, up to reweighting, the Diffusion Forcing objective optimizes a valid ELBO on the expected log-likelihoods over uniformly sampled noise levels. The following theorem can be obtained by a straightforward modification of the proof ofTheoremA.1generalizes this to arbitrary (possible temporally correlated)  sequences of noise.Theorem A.3.Letbe an arbitrary distribution over, and define.
Fix. Define the expectation over the forward process with random noise levelas(A.7) and the expectation over the latents underconditioned onas(A.8) Then, as long assatisfies the Markov property,whereis a constant depending only on(the unnoised data) , and where the inequality is anequalityunder the conditions that (a) is a dirac distribution (determinstic latents) , and (b) , i.e. the variational approximation is sharp.In particular, in the Gaussian case ofCorollaryA.2, we haveThe most salient case for us is the restriction ofto fixed sequences of noise(i.e, Dirac distributions on) . In this case,for all but, and thus our training objective need not be a lower bound on. However, the terms in the lower bound are, up to reweighting, ansubsetof those terms optimized in the training objective. Thus, in light of the remarks followingCorollaryA.2, a fully expressive network can optimize all the terms in the loss simultaneously. We conclude that, for a fully expressive neural network, optimizing the training objective (3.1)  is a valid surrogate for maximizing the likelihood of all possible noise sequences.A.2Proof ofTheoremA.1Defineas shorthand for. We begin with the following claimClaim 1(Expanding the latents) .The following lower bound holds:(A.9) Moreover, this lower bound holds with equality ifis a Dirac distribution (i.e., deterministic latents) .Proof.Let’s fix a sequence. It holds that(Markov Property) (Importance Sampling) Thus, by Jensen’s inequality,where the inequality is and equality whenis a Dirac distribution. By applyingto both sides of the above display, and invoking the Markov property of the latents, we conclude that∎We now unpack the terms obtained from the preceding claim.Claim 2(ELBO w.r.t.) .It holds thatwhereis a constant depending only onand, and where the inequality holds with equality if and only if.Proof.We have that((Jensen’s inequality) ) (Markov property of) where the constantdepends only onand. To check the conditions for equality, note that if, thenSinceis strictly concave,if and only if.
∎Claim 3(Computing the expected ELBO) .whereis some other constant depending onand.Proof.The proof invokes similar manipulations to the standard ELBO derivation for diffusion, but with a few careful modifications to handle the fact that we only noise to level. As is standard, we require the identity(A.10) Part 1: Expanding the likelihood ratios. Using the above identity, we obtainwhereusesA.10,invokes a cancellation in the telescoping sum, and the final display follows from the computation(A.11) Observe that, because we don’t parameterize,can be regarded as some constant. Thus,(A.12) Part 2: Taking expecations.We can now simplify to taking expectations. Observe thatand similarly,Finally,is a constantdepending only on.
Thus, from (A.12) ∎Completing the proof of the ELBO.We are now ready to complete the proof. By combining the previous two claims, we havewhereand where again, the above is an equality when. Taking an expectation over, we(A.13) and consequently,Invoking1,We conclude by observing thatis a constant, and thatsince both terms only depend onand. We conclude then thatas needed. Lastly, we recall that the above is anequalityunder the conditions that (a) is a dirac distribution, and (b) , and we reindexto ensure consistency with indexing in standard expositions of the diffusion ELBO.∎Appendix BAdditional Intuitions and ExplainationsB.1Guidance as planningAs stated in Section2, one can use the gradient of the logarithmic of a classiferto guide the sampling process of diffusion model towards samples with a desired attribute. For example,can refer to the indicator of a success event. However, we can consider the logarithmic of a more generalenergy function. This has the interpretation as, where. Some popular cadidate energies include(B.1) corresponding to a cost-to-go; we can obtain unbiased estiames of this gradient by using cumulative reward. We can also use goal distanceas a terminal reward. We provide details about the guidance function deployed in the maze2d planning experiment in AppendixD.5.B.2Noising and stabilizing long-horizon generationsHere explain in detail how we use noising to stabilize long horizon generation. At each time, during the denosing, we maintain a latentfrom the previous time step, withcorresponding to some small amount of noise. We then donext tokendiffusion to diffuse the tokenacross noise levels(corresponding toAlgorithm2with horizon, initial latent, and noise schedule) ; this process also produces latentsassociated with each noise level. From these, we use the latentto repreat the process. This noised latent can be interpreted as an implementation of conditioning onin an autoregressive process.It is widely appreciated that adding noise to data ameliorates long term compounding error in behavior cloning applications, and even induces robustness to non-sequential adversarial attacks. In autoregressive video generation, the noisedis in-distribution for training, because Diffusion Forcing trains from noisy past observation in its training objective.
Hence, this method can be interpreted as a special case of the DART algorithm for behavior cloning, where the imitiator (in our case, video generator)  is given actions (in our case, next video frames)  from noisy observations (in our case, noised previous frames) . Somewhat more precisely, because we are both tokens at training time to train Diffusion Forcing, and using slighlty noised tokens for autoregression at test time, our approach inherits the theoretical guarantess of the HINT algorithm.B.3Why Monte Carlo Tree Guidance relies on Diffusion ForcingMonte Carlo Tree Guidance provides substantial variance reduction in our estimate of a cost-to-go guidance (B.1) . This technique crucially relies on the ability to rollout future tokens from current ones to use these sample rollouts to get Monte Carlo estimates for gradients. This is not feasible with full-sequence diffusion, because this requires denoising all tokens in tandem; thus, for a given fixed noise level, there is no obvious source of randomness to use for the Monte Carlo estimate. It may be possible to achieve variable horizon via the trick proposed in the following subsection to simulate future rollouts, but to our knowledge, this approach is nonstandard.B.4Does the replacement technique lead to flexible horizons in full-sequence diffusion?A naive way to obtain flexible horizon generation in full-sequence diffusion is via the “replacement trick”: consider a full sequence model trained to diffuse, which we partition into. Having diffused tokens, we can attempt to denoise tokens of the form, where wefixto be the previously generated token, and only have score gradients update the remaining. One clear disadvantage of this method is inefficiency - one still needs to diffuse the whole sequence even when there is one step left at. What’s more,points out that this approach of conditioning, named “conditioning by replacement”, is both mathematically unprincipled and can lead to inconsistency in the generated sequence. The best fix proposed byincorporates an additional gradient term with respect toat every diffusion step; this is still an incomplete fix and suffers the computation cost of an extra backward propagation for every sampling step.B.5Further connection to Bayesian filteringThe core idea of Diffusion Forcing can be interpreted as using diffusion to construct an interpolation between prior distribution and posterior distribution of a Bayes filter. Consider the hybrid distribution. When, this hybrid distribution becomes the posterior. On the other hand, when, the hybrid distribution becomesfor. Since the independent Gaussian noise termcontains no information about, this is exactly the prior distribution. By varyingbetweenand, the same neural network can parameterize everything between prior and posterior.Figure 5:Diffusion Forcing is trained on independent level of noises at different timesteps. As a result, we can control the noise levelto achieve different effects on conditioning and prediction.B.6Connection to other sequence training schemesNoise as masking provides a unified view of different sequence training schemes. The following exposition uses a lengthsequence as an example: We always start with fully masked sequencewith the goal of denoising it a “clean sequence” of zero noise.. Assume all diffusions are sampled with-step DDIM.Autoregressive.In teacher forcing, one trains a model to predict the next token conditioned on prior observations. One can train next-token diffusion models with teacher forcing such as: feed neural network with past observations as well as a current observation and ask it to predict clean current observation. A typical training pair can have the input ofand target of.At sampling time, one fully diffuses the next token before adding the diffused observation to history to perform an autoregressive rollout. The diffusion process would thus look likeNotably, Diffusion Forcing can also perform this sampling scheme at sampling time for applications like imitation learning, when one wants to diffuse the next action as fast as possible.Full Sequence Diffusion.Full sequence diffusion models accepts a noisy sequence and denoises level-by-levelNotably, Diffusion Forcing can also perform this sampling scheme at sampling time.Diffusion Forcing with causal uncertaintyAs shown in Figure2, to model causal uncertainty, Diffusion Forcing keeps the far future more uncertain than the near future by having larger noise level, at any time of diffusion. An example pattern looks like this:Notable,is the first one to propose such a linear uncertainty sampling scheme for causal diffusion models, although Diffusion Forcing provides a generalization of such scheme in combination of other abilities.Diffusion Forcing with stablizationPreviously we introduced the autoregressive sampling scheme that Diffusion Forcing can also do. However, such a scheme can accumulate single-step errors because it treats predictedas ground truth observation. Diffusion Forcing addresses this problem by telling the model that generated images should be treated as noisy ground truth, as shown in2.It first fully diffuse the first token,Then, it feed the diffusedinto the model but tell it is of a slightly higher noise level, asto diffuse.Then, it feed the diffusedinto the model but tell it is of a higher noise level, as.Appendix CExtended Related WorkReconstructing masked tokens.Masked Autoencoders for imagesand videosare a popular method for representation learning in pixel space. They have been extended to perform diffusion to generate masked patches conditioned on unmasked ones.Casting Image Generation as Sequence Generation.show that even generative modeling of non-sequential data, such as images, can be fruitfully cast as sequence generative modeling.Non-Diffusion Probabilistic Sequence Models.parameterize token-to-token transitions via a variational auto-encoder. This makes them probabilistic, but does not directly maximize the joint probability of sequences, but rather, enables sampling from the distribution of single-step transitions.AR-Diffusion.Most similar to our work is AR-Diffusionwhich similarly aim to train next-token prediction models for sequence diffusion. Key differences are that AR-Diffusion proposes a noise level that islinearlydependent on the position of each word in the sequence, while our critical contribution is to have each noise level beindependent, as this uniquely enables our proposed sampling schemes, such as stabilizing auto-regressive generation and conditioning on corrupted observations. Further, AR-Diffusion only explores language modeling and does not explore guidance, while we investigate Diffusion Forcing as a broadly applicable sequence generative model with particular applications to sequential decision making. In particular, we introduce Monte-Carlo Tree Guidance as a novel guidance mechanism.Appendix DAdditional Method DetailsD.1ArchitectureVideo DiffusionWe choose both the raw imageand latent stateto be 2D tensors with channel, width, and height. For simplicity, we use the same width and height forand. We then implement the transition modelwith a typical diffusion U-net. We use the output of the U-net as the input to a gated recurrent unit (GRU)  and useas the hidden state feed into a GRU. The output of GRU is treated as. For observation model, we use a-layer resnetfollowed by a conv layer. We combine these two models to create an RNN layer, where the latent of a particular time step is, input isand output is. One can potentially obtain better results by training Diffusion Forcing with a causal transformer architecture. However, since RNN is more efficient for online decision-making, we also stick with it for video prediction and it already gives us satisfying results.We choose the number of channels into befor DMlab andfor Minecraft. In total, our Minecraft model consists ofmillion parameters and our DMlab model consists ofmillion parameters. We can potentially obtain a better Minecraft video prediction model with more parameters, but we defer that to future works to keep the training duration reasonable (day) . In maze planning, the number of total parameters ismillion.Non-Video DiffusionFor non-spatialthat is not video nor images, we use residue MLPsinstead of Unet as the backbone for the dynamics model. Residue MLP is basically the ResNetequivalent for MLP. Similar to video prediction, we feed the output of resMLP into a GRU along withto get. Another ResMLP serves as the observation model.D.2Diffusion parameterizationIn diffusion models, there are three equivalent prediction objectives,,, andparameterization. Different objectives lead to different reweighting of loss at different noise levels, together with SNR reweighting. For example,parameterization andparameterization are essential in generating pixel data that favors high-frequency details.In our experiments, we useparameterization for video prediction and found it essential to both convergence speed and quality.We observe thatparameterization is strongly favorable in planning and imitation learning, likely because they don’t favor an artificial emphasis on high-frequency details. We observe the benefits of v-parameterization in time-series prediction.D.3SNR reweightingSNR reweightingis a widely used technique to accelerate the convergence of image diffusion models. In short, it reweighs the diffusion loss proportional to the signal-to-noise ratio (SNR)  of noisy. In Diffusion Forcing, conditioning variablecan also contain a non-trivial amount of information about, in addition to. For example, in a deterministic markovian system, ifhas its noise level, the posterior statecontains all the information needed to predictregardless of the noise level of.Therefore we re-derive SNR reweighting to reflect this change in Diffusion Forcing. We follow the intuition of original SNR reweighting to loosely define SNR in a sequence with independent levels of noises at different time steps. Denoteas the normalized SNR reweighting factor forfollowing its normal derivation in diffusion models. For example, if one uses min snr strategy, its reweighting factor will always fall betweenwhich we divide byto get. Define signal decay factor, measuring what proportion of signal incontribute to denoising. This is the simple exponential decay model of sequential information. Now, define cumulated SNR recursively as the running mean of:to account for signals contributed by the entire noisy history to the denoising at time step. The other factor that contributes to the denoising isof noisy observation. To combine them, we use a simplified model for independent events. Noticeandalways falls in range, and therefore can be reinterpreted as probabilities of having all the signal one needs to perfect denoise. Since the noise level atis independent of prior noise levels, we can viewandas probabilities of independent events, and thus can composed to define a joint probability, and we use thisas our fused SNR reweighting factor for diffusion training.In our experiments, we choose to follow the min-SNR reweighting strategyto derive the. Our new SNR reweighting proves extremely useful to accelerate the convergence of video prediction, while we didn’t observe a boost on non-image domains so we didn’t use it there.D.4Noise scheduleWe use sigmoid noise schedulefor video prediction, linear noise schedule for maze planning, and cosine schedule for everything else.D.5Implementation Details of Sampling with GuidanceCorner case of sampling noiseIn our sampling algorithm, due to the flexibility of the scheduling matrix, there are corner cases whenis required to stay at its same noise level during a sampling step. The core question of this corner case is whether we should updateat all. One option is just copying over the old value. The other option is to run a backward diffusion followed by a forward diffusion back to its old noise level to resample under the diffusion process. While we conclude this can be an open question, we prefer the later approach, resampling, and use it in Monte Carlo Guidance to generate multiple samples. We note that even if one take the first approach, the guidance gradient can still flow back in the time steps beforeas the dynamics modelcan still propagate the guidance gradient to.Other than Monte Carlo Guidance, this corner case only happens whenorthroughout our experiments. That is, we chose oursuch that once any token gets diffused slightly, it will keep diffusing. In the case of, keepingat the same noise level implies it will stay as white noise, and we don’t even need to sample another white noise. In case, the time step is already completely diffused either approach should give us the same result so we just opt for copying over for simplicity.Guidance for maze planningIn maze planning, our main baseline Diffuerdiscards the reward from the dataset and directly plans with the goal position and velocity. We adopt the same convention for Diffusion Forcing. One can perform guidance on goal position using log-likelihood, but a flexible horizon model should not require users to manually specify ato reach its goal, instead we want it to try to reach the goal for any possible horizon. Therefore we use the reward modelso any time step can be the final step to reach the goal. This objective is challenging due to the non-convex nature of 2D maze, but we found Diffusion Forcing can still reliably find plans without bumping into walls. However, we also observe that the agent tend to leave the goal location due to the nature of the provided dataset - the goal location is just one possible waypoint for the robot to pass through, and there are no trajectories that simply stay at the goal. We also tried this reward for guidance with Diffuser, but it didn’t work even with a good amount of tuning.D.6Performance OptimizationAccelerating the diffusion sampling of Diffusion Forcing is similar to that of normal diffusion models. We adopt DDIMsampling for the diffusion of each token. While we usesteps of diffusion, we sample with onlyDDIM for video prediction andfor non-video domains.While Diffusion Forcing can be implemented with transformers, we use an RNN as the backbone for Diffusion Forcing experiments it’s widely used in decision-making for its flexibility and efficiency in online decision-making systems. To further reduce training time and GPU memory usage, we use frame-stacking to stack multiple observed images as a single. This is due to the fact that adjacent tokens can be very similar - e.g. recording the same motion at higher fps can lead to this. We deem that it’s wasteful if we rollout the dynamics model multiple times to generate almost identical tokens. For video datasets, we manually examine how many time steps it takes to require a minimal level of prediction power instead of copying frames over. There is another reason why we use frame stacking - many diffusion model techniques such as different noise schedules are designed to modelwith correlated elements or redundancy. Low-dimensional systems may need drastically different hyperparameters when they lack the data redundancy these techniques are tested on. Frame stacking is thus also helpful for our non-image experiments so we can start with canonical hyperparameters of diffusion models. We use a frame stack offor DMlab video prediction,for Minecraft andfor maze planning.At sampling time, we also have a design choice to reduce compute usage, as reflected in line 8 of Algorithm2. In line 8, we directly assignto, instead of recalculatingwith posterior model. Since the model is trained to condition onestimated from arbitrary noisy history, we recognize that both are valid approaches. The reason why the choose lineis two fold. First, it cuts the compute by half, avoiding computing posterior every step. Second, this happens to be what we want for stabilization -already contains the information of the cleanunder our simplified observation model, and happens to be estimated with, a noise level higher than that of. This happens to implement the behavior we want for stabilization.D.7Sampling schedule for causal uncertaintyInference is depicted inAlgorithm2and Figure2. In EquationD.1, we illustrate a specific instantiation of thematrix we used for causal planning. For simplicity, we denote the case where a latentis given and aim to generate.(D.1) Diffusion Forcing begins by sampling our sequences as white noise with noise level. It then denoises along each rowofin decreasing order. It does so by proceeding sequentially through frames, updating the latent (Line 5 of Algorithm2) , and then partially applying the backward process to noise leveldictated by the scheduling matrix(Line 6-7 of Algorithm2) . We call alike this pyramid scheduling, as the tokens in the far future are kept at higher noise level than near future.D.8Implementation Details of Timeseries RegressionWe follow the implementation of pytorch-ts, where the validation set is a random subset of the training set with the same number of sequences as the test set. We use early stopping when validation crpssum hasn’t increased for 6 epochs. We leverage the same architecture (1 mlp and 4 grus)  as well as a batch size of 32.D.9Compute ResourcesAll of our experiments usemixed precision training. Time series, maze planning, compositionally, visual imitation experiments can be trained with a singlewithGB of memory. We tune the batch size such that we fully use the memory of GPUs. This translates to a batch size offor maze planning and compositional experiments, andfor visual imitation learning. While we use early stopping on the validation set for time series experiments, we did not carefully search for the minimal number of training steps required, though the model usually converges betweenk tosteps. The above environments thus usually takehours to train although there is without doubt a significant potential for speed up.Video prediction is GPU intensive. We useA100 GPUs for both video prediction datasets. We train forsteps with a batch size of. It usually takehours to converge atsteps of training (occasional validation time also included) .Appendix EAdditional Experiment ResultsE.1Multivariate Probabilistic Time Series ForecastingFigure 6:Prediction intervals of Diffusion Forcing for the first prediction window of the test set in the Electricity time series dataset. Only the first 16 features out of 370 are plotted.To illustrate Diffusion Forcing’s new training objective does not degrade it as a generic sequence model, we evaluate Diffusion Forcing on high-dimensional and long-horizon sequence prediction tasks in time series prediction. We adopt multiple time series datasets with real-world applications from GluonTSand evaluate Diffusion Forcing with strong baselines with standard metrics in this domain. In this section, we mainly focus on the results and analysis. For a detailed description of datasets and the metric, we refer the reader to AppendixF.4.Problem FormulationLetbe a sequence (multivariate time series)  of-dimensional observationsof some underlying dynamical process, sampled in discrete time steps, where. In the problem setting of probabilistic time series forecasting, the sequenceis split into two subsequences at time stepwith: the context window(also called history or evidence)  of length, and the prediction windowof length(also known as the prediction horizon) . Then, the task is to model the conditional joint probability distribution(E.1) over the samples in the prediction window. If we know the distribution in (E.1) , we can sample forecast prediction sequences given some initial context from the evidence sequence. However, most time-dependent data generation processes in nature have complex dynamics and no tractable formulation of. Instead, we construct a statistical model that approximates the generative process in (E.1)  and estimates quantiles via Monte Carlo sampling of simulated trajectories. In this way, confidence levels or uncertainty measures can be calculated, and point forecasts can be produced as the mean or median trajectory.Results.\ctable[
caption = Results for time series forecasting. We report the test set(the lower, the better)  of comparable methods on six time series datasets. We measure the mean and standard deviation of our method from five runs trained with different seeds.,
label = tab:results_ts,
pos = ht,
doinside =,
center,
star
]lccccccc\FLMethod & Exchange  Solar  Electricity  Traffic  Taxi  Wikipedia\MLVES0.0050.000  0.9000.003  0.8800.004  0.3500.002  -  -\NNVAR0.0050.000  0.8300.006  0.0390.001  0.2900.001  -  -\NNVAR-Lasso0.0120.000  0.5100.006  0.0250.000  0.1500.002  -  3.1000.004\NNGARCH0.0230.000  0.8800.002  0.1900.001  0.3700.001  -  -\NNDeepAR-  0.3360.014  0.0230.001  0.0550.003  -  0.1270.042\NNLSTM-Copula0.0070.000  0.3190.011  0.0640.008  0.1030.006  0.3260.007  0.2410.033\NNGP-Copula0.0070.000  0.3370.024  0.0250.002  0.0780.002  0.2080.183  0.0860.004\NNKVAE0.0140.002  0.3400.025  0.0510.019  0.1000.005  -  0.0950.012\NNNKF-  0.3200.020  0.0160.001  0.1000.002  -  0.0710.002\NNTransformer-MAF0.0050.003  0.3010.014  0.0210.000  0.0560.001  0.1790.002  0.0630.003\NNTimeGrad0.0060.001  0.2870.020  0.0210.001  0.0440.006  0.1140.020  0.0490.002\NNScoreGrad sub-VP SDE0.0060.0010.2560.0150.0190.0010.0410.004  0.1010.0040.0430.002\NNOurs0.0030.0010.2890.002  0.0230.0010.0400.0040.0750.0020.0850.007\LLWe evaluate the effectiveness of Diffusion Forcing as a sequence model on the canonical task of multivariate time series forecasting by following the experiment setup ofConcretely, we benchmark Diffusion Forcing on the datasets Solar, Electricity, Traffic, Taxi, and Wikipedia. These datasets have different dimensionality, domains, sampling frequencies, and capture seasonal patterns of different lengths. The features of each dataset are detailed in TableLABEL:tab:ts_data. We access the datasets from GluonTS, and set the context and prediction windows to the same length for each dataset. Additionally, we employ the same covariates as.
We evaluate the performance of the model quantitatively by estimating the Summed Continuous Ranked Probability Scorevia quantiles. As a metric,measures how well a forecast distribution matches the ground truth distribution. We provide detailed descriptions of the metric in AppendixF.4.
We benchmark with other diffusion-based methods in time series forecasting, such as TimeGradand the transformer-based Transformer-MAF. In particular, the main baseline of interest, TimeGrad, is a next-token diffusion sequence model trained with teacher forcing.
We track themetric on the validation set and use early stopping when the metric has not improved for 6 consecutive epochs, while all epochs are fixed to 100 batches across datasets. We then measure theon the test set at the end of training, which we report in TableLABEL:tab:results_ts. We use the exact same architecture and hyperparameters for all time series datasets and experiments.
Diffusion Forcing outperforms all prior methods except forwith which Diffusion Forcing is overall tied, except for the Wikipedia dataset, on which Diffusion Forcing takes fourth place.
Note that time series is not the core application of Diffusion Forcing, and that we merely seek to demonstrate that the Diffusion Forcing objective is applicable to diverse domains with no apparent trade-off in performance over baseline objectives.E.2Additional results in compositional generationSince Diffusion Forcing models the joint distribution of any subset of a sequence, we can leverage this unique property to achieve compositional behavior - i.e., Diffusion Forcing can sample from the distribution ofsubsetsof the trajectory and compose these sub-trajectories into new trajectories.In particular, we show that we can also have flexible control over how compositional Diffusion Forcing is. As shown in7, consider a dataset of trajectories on a 2D, square plane, where all trajectories start from one corner and end up in the opposite corner, forming a cross shape. When no compositional behavior is desired, one can let the models replicate the cross-shaped distribution by allowing full memory of the HMM model.
When one desires compositional such as generating a V-shaped trajectory, which stitches two sub trajectories together, one can let the model generate shorter plans with no-memory context using MPC. (Add figures) .(a) Dataset(b) W/ memory(c) W/o memoryFigure 7:Given a dataset of trajectories (7(a) ) , Diffusion Forcing models the joint distribution of all subsequences of arbitrary length. At sampling time, we can sample from the trajectory distribution by sampling Diffusion Forcing with full horizon (7(b) )  or recover Markovian dynamics by disregarding previous states (7(c) ) .E.3Additional results in video prediction (wo/ cherry picking) Infinite Rollout without sliding windowDiffusion Forcing can rollout longer than maximum training horizonwithout sliding window. That is, we run Diffusion Forcing’s RNN continuously without ever reinitializing. This is a surprising effect we observed from the rollout stabilization property of Diffusion Forcing. In Figure8,10, we use Diffusion Forcing to generate video sequences of lengthand visualize subsampled sequences. Notably, Diffusion Forcing used in these visualizations is trained with a maximum length offrames for Minecraft andframes for DMLab, illustrating it can rollout 2x-5x times longer than it’s trained onwithout sliding window. In addition, we also tried rolling these models out forframes and without seeing the model blowing up on both datasets. There are occasional cases where the Minecraft agent gets stuck and the entire screen is the “dirt” block, but this is more of a dataset issue16and the agent is able to recover after it turns around.Figure 8:Visualization shows Diffusion Forcing trained onframes is able to rolloutframes on Minecraft datasetwithout sliding window. The visualization shows a non-cherry-picked subsampling of theseframes, although Diffusion Forcing can roll out much longer (such as 2000 frames)  on this dataset.Figure 9:Diffusion Forcing trained onframes is able to rolloutframes on Minecraft datasetwithout sliding window. The visualization shows a non-cherry-picked subsampling of theseframes, although Diffusion Forcing can roll out much longer (such as 2000 frames)  on this dataset. The first few frames marked in red are the ground truth images of the dataset used for conditioning.Figure 10:Visualization shows Diffusion Forcing trained onframes is able to rolloutframes on DMLab datasetwithout sliding window. The visualization shows a non-cherry-picked subsampling of theseframes, although Diffusion Forcing can roll out almost infinitely on this dataset. The first few frames marked in red are the ground truth images of the dataset used for conditioning.Figure 11:Visualization shows Diffusion Forcing trained onframes is able to rolloutframes on DMLab datasetwithout sliding window. The visualization shows a non-cherry-picked subsampling of theseframes, although Diffusion Forcing can roll out almost infinitely on this dataset. The first few frames marked in red are the ground truth images of the dataset used for conditioning.ConsistencyWe also present additional results where we only generate within our maximum training length. As shown in figure1312, Diffusion Forcing can generate consistent videos. Results are not cherry-picked.Figure 12:Additional non-cherry-picked video prediction results on DMLab dataset, generated within maximum training length. The first few frames marked in red are the ground truth images of the dataset used for conditioning.Figure 13:Additional non-cherry-picked video prediction results on Minecraft dataset, generated within maximum training length. The first few frames marked in red are the ground truth images of the dataset used for conditioning.E.4Additional results in planningWe provide some additional visualizations of causal planning in15. We also present additional visualization of Diffusion Forcing performing model predictive control in action. As shown in figure14, Diffusion Forcing can generate plans of shorter horizons since it’s flexible horizon.Figure 14:Example MPC planning for maze medium environment. Blue indicated trajectories actually executed already. Red is the plan.Figure 15:Example plans generated for maze medium (above)  and maze large (below)  environments.E.5Real robot experiment setupIn Figure16we visualize our robot experiment setup with corruption on observation. The dataset is collected when the target bag isn’t present, while we test with such a bag in the scene zero-shot for the imitation learning experiment with observation corruption. The typical failure mode is when the robot no longer reacts to the visual clues of the randomized location of objects. We didn’t observe the robot to act wildly due to visual distractors.Figure 16:We randomly throw a target bag on the table as a strong visual distractor. Diffusion Forcing can be prompted to treat observation as corrupted than ground truth.Appendix FAdditional details about datasetsF.1Dataset for video diffusionWe adopt the video prediction dataset Minecraft and DMlab used by TECO.Minecraft NavigationThe Minecraft navigation dataset consists of first-person-view videos of random walks in the Minecraft ‘swamp‘ biome. The agent walks via a technique called ‘sprint jump‘ which allows it to jump across blocks without getting stuck at 1 block obstacles. The agent walks straight most of the time, with small chances of turning left or right. The height and width of the video ispixels and we trim long videos to subsequences offrames. The dataset comes with paired action data but we discard them to bring more stochasticity to the prediction task. Due to limited compute, we only train on aboutof the total subsequences.One problem we noticed about the dataset is when the agent runs into obstacles with a height of 2 blocks or more. In this case, the agent will get stuck and the entire video sequence will consist of grey granite patterns or brown dirty patterns. This leads to a huge amount of frames with these patterns, making video models predict meaningless frames. Yet, we deem this as a problem of this dataset itself.DMLab NavigationDeepmind Lab navigation dataset consists of random walks in a 3D maze environment. For DMLab, the resolution ispixels and we use subsequences offrames. We also disregard the provided actions due to training.We note that the VQ-VAE latent that stable video diffusiondiffuses is also only, indicating Diffusion Forcing has the potential to scale up to higher resolution images with pre-trained image encoder and decoders. Due to the sheer size of the datasets, we only use aboutof the total data sequences for training due to limited computing, as we observe that doing so already allows us to make good generations from initial frames from the test set.F.2Dataset for planningD4RLis a standard offline RL benchmark featuring a wide range of reinforcement learning environments. Each environment is associated with a provided dataset of offline interactions with the environment featuring state, action and reward trajectories.Like Diffuer, we choose the 3 maze environments as they are challenging long-horizon, multi-modal, sparse reward problems uniquely suited for visualization and evaluating planning algorithms. The ID for the 3 used environments are “maze2d-medium-v1”, “maze2d-large-v1”, “maze2d-umaze-v1”. In each environment, one controls the acceleration of a robot to walk it towards a goal. The observation space isdimensional, featuring 2D location and velocity. The action space is 2D acceleration. The agent always receives a random start location and the goal is to reach a fixed goal position for each maze. The agent receives a reward of 1 if it is within a circle of radius 0.5 centered at the goal state, and 0 otherwise.The offline RL dataset for the maze environments consists of random walks in the maze. Specifically, the authors first designate all intersections and turns in the maze as waypoints and code an agent to navigate between waypoints with some randomization. As a result, the random walks are generated in a way that the path is collision-free with the walls. The random walks introduces stochasticity to the dataset, as trajectories in the dataset are never towards a specific goal.There are a few choices adopted from our main baseline Diffuser: we disregard the reward in the dataset and plan with goals only. We also evaluate a multi-gaol variant of each environment (labeled as “multi” in Table4.1) , where the goal is randomized just like the starting position.F.3Dataset for robot learningWe choose a long horizon robotic manipulation task as described in Section4.4: Consider a tabletop with three slots where we can place objects. One places an apple at slot A or slot B randomly, and then places an orange at the other slot between A and B. A robot is challenged to swap the position of two fruits using the third slot C. That is, it can only move a fruit to an empty slot at a time. For example, when the apple is at slot A and the orange is at slot B, it may move the apple to slot C, leaving slot A empty. Then move the orange to slot A and finally move the apple from slot C to slot B. In figure4, we illustrate the non-markovian property of the task: When the apple is at slot B and the orange is at slot C, one cannot tell what the immediate action is without knowing the initial positions of objects.We put stickers on the table indicating a circular region occupied by any slot. Each circular region is designed to be about double the diameter of a fruit. To make sure the task requires visual feedback, we also randomize the location of a fruit inside the slot. We collectedexpert demonstrations of a franka robot performing the task using VR teleoperation and impedance control. Among them, each initial slot configuration makes up half of the dataset. We record videos from two camera views, one from hand camera and one in the front capturing all three slots. Each demonstration also comes withdof actions of the robot hand. During the data collection, since one successful demonstration will swap the position of two objects, its end configuration will naturally serve as the starting configuration of the other randomized location, which we leverage to save time.Each demonstration comprisesframes and actions. We train Diffusion Forcing on the entire sequence. However, since adjacent frames are visually close, we pad and downsample the videos toframes where each frame is bundled withactions.F.4Dataset for time series\ctable[
caption = Characteristics of the GluonTS datasets used to benchmark Diffusion Forcing in the domain of time series forecasting.,
label = tab:ts_data,
pos = ht,
doinside = ,
center,
star
]lccccc\FLDataset & Dimension  Domain  Frequency  Steps  Prediction length\MLExchange  8BUSINESS DAY  6,071  30\NNSolar  137HOUR  7,009  24\NNElectricity  370HOUR  5,833  24\NNTraffic  963  (0,1)   HOUR  4,001  24\NNTaxi  1,21430-MIN  1,488  24\NNWikipedia  2,000DAY  792  30\LLWe use a set of time series datasets accessible via GluonTS, which are adopted from prior works like. These datasets capture real-world data of high-dimensional dynamics like monetary exchange rates or the electricity grid. In TableLABEL:tab:ts_data, we provide a summary of the features of these datasets, such as the dimensionality, the domains, the sampling frequency, the length of the mulitvariate sequence in the training set, and the prediction length. We access the datasets in TableLABEL:tab:ts_datavia GluonTS and wrap the data processing functions implemented in GluonTS in our own dataloaders. Each dataset consists of one long multivariate sequence, which is the training split, and a set of short sequences that make up the test split. We construct a validation set of the same cardinality as the held-out test set as a randomly sampled subset of subsequences from the training set. All splits are normalized by the mean and the standard deviation of the features in the training split.CovariatesOften, statistical models that approximate (E.1)  benefit from manually curated features as additional input to the observations. A sequence of covariatescan be constructed to help the model recognize seasonal patterns and other temporal dependencies. We follow the implementation into construct the covariate sequence as a function of the frequency of each dataset in TableLABEL:tab:ts_data. As such, our covariates are composed of lagged inputs, as well as learned embeddings and handcrafted temporal features that encode information such as the hour of the day or the day of the month, depending on the sampling rate of the particular time series that is being modeled. Therefore, covariates are known for the entire interval, even at inference. We can easily incorporate covariates into the probabilistic framework as(F.1) The benefit obtained from covariates is highly dependent on the characteristics of both the dataset and the model used, as well as the feature engineering practices followed.MetricThe Continuous Ranked Probability Score (CRPS) is a scoring function that measures how good the forecast distribution matches the ground truth distribution:whereis the univariate cumulative distribution function (CDF)  over the predicted value,is a ground truth observation, andis the indicator function that is one ifand zero otherwise. By summing the-dimensional time series along the feature dimension for simulated samples (resulting in)  and ground truth data (as) , we can report theas the average over the prediction window. The lower thevalue, the better does the predicted distribution match the data distribution.First, we manually sum the time series along the feature dimension and estimate the CDFvia 19 quantile levels at each time stepfrom 100 sampled trajectories. We then use the implementation in GluonTsto compute the, which we report asin TableLABEL:tab:results_ts. While we aggregate the data manually, we verify that the numerical error relative to the GluonTS implementation remains orders of magnitude below the precision threshold of the reported metric.
body
Diffusion Forcing: Next-token PredictionMeets Full-Sequence Diffusion
4Experiments
2407.01392
5DiscussionLimitations.Our current causal implementation is based on a small RNN, and applications to higher-resolution video or more complex distributions likely require large transformer models. We do not investigate the scaling behavior of Diffusion Forcing to internet-scale datasets and tasks.Conclusion.In this paper, we introduced Diffusion Forcing, a new training paradigm where a model is trained to denoise sets of tokens with independent, per-token noise levels.
Applied to time series data, we show how a next-token prediction model trained with Diffusion Forcing combines benefits of both next-token models and full-sequence diffusion models.
We introduced new sampling and guidance schemes that lead to dramatic performance gains when applied to tasks in sequential decision making.
Future work may investigate the application of Diffusion Forcing to domains other than time series generative modeling, and scale up Diffusion Forcing to larger datasets.Acknowledgements.This work was supported by the National Science Foundation under Grant No. 2211259, by the Singapore DSTA under DST00OECI20300823 (3D Self-Supervised Learning for Label-Efficient Vision) , by the Intelligence Advanced Research Projects Activity (IARPA)  via Department of Interior/ Interior Business Center (DOI/IBC)  under 140D0423C0075, and by the Amazon Science Hub. Special thanks to Kiwhan Song (MIT)  for 3D-Unet and transformer reimplementation of the paper after initial release.References[1]A. Ajay, Y. Du, A. Gupta, J. Tenenbaum, T. Jaakkola, and P. Agrawal.Is conditional generative modeling all you need for decision-making?arXiv preprint arXiv:2211.15657, 2022.[2]A. Alexandrov, K. Benidis, M. Bohlke-Schneider, V. Flunkert, J. Gasthaus, T. Januschowski, D. C. 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We first establish this in full abstraction (TheoremA.1) , and then specialize to the form of Gaussian diffusion (CorollaryA.2) . We show that the resulting terms decouple in such a fashion that, in the limit of a fully expressive latent and model, makes the reweighting terms immaterial.•We show that the expected likelihood overanydistribution over sequences of noise levels can be lower bounded by a sum over nonnegative terms which, when reweighted, correspond to the terms optimized in the Diffusion Forcing training objective maximizes. Thus, for a fully expressive network that can drive all terms to their minimal value, Diffusion Forcing optimizes a valid surrogate of the likelihood forlikelihood of all sequences of noise levels simultaneously.We begin by stating an ELBO for general Markov forward processes, and generative models, and then specialize to Gaussian diffusion, thereby recovering our loss. We denote our Markov forward processas(A.1) and a parmeterized probability model(A.2) We assume thatsatisfies theMarkov propertythat(A.3) that is, the latent codesis a sufficient statistic forgiven the history. We say thathasdeterministic latentsifis a Dirac delta.Remark 1.In order forto have deterministic latents and correspond to a valid probability distribution, we need to view the latentsnot as individual variables, but as a collections of variablesindexed byand and thehistoryof noise levels.. In this case, simply settingtautologically produces deterministic latents. The reason for indexingwiththen arises because, otherwise,would be ill-defined unlessfor all, and thus,would not correspond to a joint probability measure. The exposition and theorem that follows allowsto be indexed on past noise levels, but suppresses dependence onto avoid notational confusion.A.1Main ResultsWe can now state our main theorem, which provides an evidence lower bound (ELBO)  on the expected log likelihood of partially-noised sequences, under uniform levelsandobtained by noising according toas in (A.1) . Notice that this formulation does not require an explicit for ofor, but we will specialize to Gaussian diffusion in the sequel.Theorem A.1.Fix. Define the expectation over the forward process with random noise levelas(A.4) and the expectation over the latents underconditioned onas(A.5) Then, as long assatisfies the Markov property,whereis a constant depending only on(the unnoised data) . Moreover, if the latents are deterministic (i.e.is a Dirac distribution) , then the inequality holds with inequality if and only if, i.e. the variational approximation is exact.The proof of the amove theorem is given inSectionA.2. Remarkably, it involvesonly twoinequalities! The first holds with equality under deterministic latents and the second holds if and only if variational approximation is exact:. This tightness of the ELBO suggests that the expression inTheoremA.1is a relatively strong surrogate objective for optimizing the likelihoods.A.1.1Specializing to Gaussian diffusionWe now specialTheoremA.1to Gaussian diffusion. For now, we focus on the “-prediction” formulation of diffusion, which is the one used in our implementation. The “-prediction” formalism, used throughout the main body of the text, can be derived similarly (see Section 2 offor a clean exposition) . The following theorem follows directly by apply standard likelihood and KL-divergence computations for the DDPMtoTheoremA.1.Corollary A.2.Let(A.6) and define,. Suppose that we parameterize, where further,Then, as long assatisfies the Markov property, we obtainedwhere above, we define.Proof.The first inequality follows from the standard computations for the “-prediction” formulation of Diffusion (see Section 2.7 ofand references therein) . The second follows by replacing the sum overwith an expectation over.
∎We make a couple remarks:•As noted above,CorollaryA.2can also be stated for-prediction, or the so-called “-prediction” formalism, as all are affinely related.•Define an idealized latentconsisting of all past tokensas well as of their noise levels. This is a sufficient statistic for, and thus we can always view, whereis just compressing. When applying the expectation ofto both sides of the bound inCorollaryA.2, and taking an infimum over possible function approximator, we obtainThis leads to a striking finding: with expressive enough latents and, we can view the maximization of each term inCorollaryA.2separately across time steps. The absence of this coupling means that the weighting terms are immaterial to the optimization, and thus can be ignored.•Given the above remarks, we can optimize the ELBO by taking gradients through the objective specified byCorollaryA.2, and are free to drop any weighting terms (or rescale them)  as desired. Backpropogating throughis straightforward due to deterministic latents. This justifies the correctness of our training objective (3.1)  and protocolAlgorithm1.A.1.2Capturing all subsequencesTheoremA.1stipulates that, up to reweighting, the Diffusion Forcing objective optimizes a valid ELBO on the expected log-likelihoods over uniformly sampled noise levels. The following theorem can be obtained by a straightforward modification of the proof ofTheoremA.1generalizes this to arbitrary (possible temporally correlated)  sequences of noise.Theorem A.3.Letbe an arbitrary distribution over, and define.
Fix. Define the expectation over the forward process with random noise levelas(A.7) and the expectation over the latents underconditioned onas(A.8) Then, as long assatisfies the Markov property,whereis a constant depending only on(the unnoised data) , and where the inequality is anequalityunder the conditions that (a) is a dirac distribution (determinstic latents) , and (b) , i.e. the variational approximation is sharp.In particular, in the Gaussian case ofCorollaryA.2, we haveThe most salient case for us is the restriction ofto fixed sequences of noise(i.e, Dirac distributions on) . In this case,for all but, and thus our training objective need not be a lower bound on. However, the terms in the lower bound are, up to reweighting, ansubsetof those terms optimized in the training objective. Thus, in light of the remarks followingCorollaryA.2, a fully expressive network can optimize all the terms in the loss simultaneously. We conclude that, for a fully expressive neural network, optimizing the training objective (3.1)  is a valid surrogate for maximizing the likelihood of all possible noise sequences.A.2Proof ofTheoremA.1Defineas shorthand for. We begin with the following claimClaim 1(Expanding the latents) .The following lower bound holds:(A.9) Moreover, this lower bound holds with equality ifis a Dirac distribution (i.e., deterministic latents) .Proof.Let’s fix a sequence. It holds that(Markov Property) (Importance Sampling) Thus, by Jensen’s inequality,where the inequality is and equality whenis a Dirac distribution. By applyingto both sides of the above display, and invoking the Markov property of the latents, we conclude that∎We now unpack the terms obtained from the preceding claim.Claim 2(ELBO w.r.t.) .It holds thatwhereis a constant depending only onand, and where the inequality holds with equality if and only if.Proof.We have that((Jensen’s inequality) ) (Markov property of) where the constantdepends only onand. To check the conditions for equality, note that if, thenSinceis strictly concave,if and only if.
∎Claim 3(Computing the expected ELBO) .whereis some other constant depending onand.Proof.The proof invokes similar manipulations to the standard ELBO derivation for diffusion, but with a few careful modifications to handle the fact that we only noise to level. As is standard, we require the identity(A.10) Part 1: Expanding the likelihood ratios. Using the above identity, we obtainwhereusesA.10,invokes a cancellation in the telescoping sum, and the final display follows from the computation(A.11) Observe that, because we don’t parameterize,can be regarded as some constant. Thus,(A.12) Part 2: Taking expecations.We can now simplify to taking expectations. Observe thatand similarly,Finally,is a constantdepending only on.
Thus, from (A.12) ∎Completing the proof of the ELBO.We are now ready to complete the proof. By combining the previous two claims, we havewhereand where again, the above is an equality when. Taking an expectation over, we(A.13) and consequently,Invoking1,We conclude by observing thatis a constant, and thatsince both terms only depend onand. We conclude then thatas needed. Lastly, we recall that the above is anequalityunder the conditions that (a) is a dirac distribution, and (b) , and we reindexto ensure consistency with indexing in standard expositions of the diffusion ELBO.∎Appendix BAdditional Intuitions and ExplainationsB.1Guidance as planningAs stated in Section2, one can use the gradient of the logarithmic of a classiferto guide the sampling process of diffusion model towards samples with a desired attribute. For example,can refer to the indicator of a success event. However, we can consider the logarithmic of a more generalenergy function. This has the interpretation as, where. Some popular cadidate energies include(B.1) corresponding to a cost-to-go; we can obtain unbiased estiames of this gradient by using cumulative reward. We can also use goal distanceas a terminal reward. We provide details about the guidance function deployed in the maze2d planning experiment in AppendixD.5.B.2Noising and stabilizing long-horizon generationsHere explain in detail how we use noising to stabilize long horizon generation. At each time, during the denosing, we maintain a latentfrom the previous time step, withcorresponding to some small amount of noise. We then donext tokendiffusion to diffuse the tokenacross noise levels(corresponding toAlgorithm2with horizon, initial latent, and noise schedule) ; this process also produces latentsassociated with each noise level. From these, we use the latentto repreat the process. This noised latent can be interpreted as an implementation of conditioning onin an autoregressive process.It is widely appreciated that adding noise to data ameliorates long term compounding error in behavior cloning applications, and even induces robustness to non-sequential adversarial attacks. In autoregressive video generation, the noisedis in-distribution for training, because Diffusion Forcing trains from noisy past observation in its training objective.
Hence, this method can be interpreted as a special case of the DART algorithm for behavior cloning, where the imitiator (in our case, video generator)  is given actions (in our case, next video frames)  from noisy observations (in our case, noised previous frames) . Somewhat more precisely, because we are both tokens at training time to train Diffusion Forcing, and using slighlty noised tokens for autoregression at test time, our approach inherits the theoretical guarantess of the HINT algorithm.B.3Why Monte Carlo Tree Guidance relies on Diffusion ForcingMonte Carlo Tree Guidance provides substantial variance reduction in our estimate of a cost-to-go guidance (B.1) . This technique crucially relies on the ability to rollout future tokens from current ones to use these sample rollouts to get Monte Carlo estimates for gradients. This is not feasible with full-sequence diffusion, because this requires denoising all tokens in tandem; thus, for a given fixed noise level, there is no obvious source of randomness to use for the Monte Carlo estimate. It may be possible to achieve variable horizon via the trick proposed in the following subsection to simulate future rollouts, but to our knowledge, this approach is nonstandard.B.4Does the replacement technique lead to flexible horizons in full-sequence diffusion?A naive way to obtain flexible horizon generation in full-sequence diffusion is via the “replacement trick”: consider a full sequence model trained to diffuse, which we partition into. Having diffused tokens, we can attempt to denoise tokens of the form, where wefixto be the previously generated token, and only have score gradients update the remaining. One clear disadvantage of this method is inefficiency - one still needs to diffuse the whole sequence even when there is one step left at. What’s more,points out that this approach of conditioning, named “conditioning by replacement”, is both mathematically unprincipled and can lead to inconsistency in the generated sequence. The best fix proposed byincorporates an additional gradient term with respect toat every diffusion step; this is still an incomplete fix and suffers the computation cost of an extra backward propagation for every sampling step.B.5Further connection to Bayesian filteringThe core idea of Diffusion Forcing can be interpreted as using diffusion to construct an interpolation between prior distribution and posterior distribution of a Bayes filter. Consider the hybrid distribution. When, this hybrid distribution becomes the posterior. On the other hand, when, the hybrid distribution becomesfor. Since the independent Gaussian noise termcontains no information about, this is exactly the prior distribution. By varyingbetweenand, the same neural network can parameterize everything between prior and posterior.Figure 5:Diffusion Forcing is trained on independent level of noises at different timesteps. As a result, we can control the noise levelto achieve different effects on conditioning and prediction.B.6Connection to other sequence training schemesNoise as masking provides a unified view of different sequence training schemes. The following exposition uses a lengthsequence as an example: We always start with fully masked sequencewith the goal of denoising it a “clean sequence” of zero noise.. Assume all diffusions are sampled with-step DDIM.Autoregressive.In teacher forcing, one trains a model to predict the next token conditioned on prior observations. One can train next-token diffusion models with teacher forcing such as: feed neural network with past observations as well as a current observation and ask it to predict clean current observation. A typical training pair can have the input ofand target of.At sampling time, one fully diffuses the next token before adding the diffused observation to history to perform an autoregressive rollout. The diffusion process would thus look likeNotably, Diffusion Forcing can also perform this sampling scheme at sampling time for applications like imitation learning, when one wants to diffuse the next action as fast as possible.Full Sequence Diffusion.Full sequence diffusion models accepts a noisy sequence and denoises level-by-levelNotably, Diffusion Forcing can also perform this sampling scheme at sampling time.Diffusion Forcing with causal uncertaintyAs shown in Figure2, to model causal uncertainty, Diffusion Forcing keeps the far future more uncertain than the near future by having larger noise level, at any time of diffusion. An example pattern looks like this:Notable,is the first one to propose such a linear uncertainty sampling scheme for causal diffusion models, although Diffusion Forcing provides a generalization of such scheme in combination of other abilities.Diffusion Forcing with stablizationPreviously we introduced the autoregressive sampling scheme that Diffusion Forcing can also do. However, such a scheme can accumulate single-step errors because it treats predictedas ground truth observation. Diffusion Forcing addresses this problem by telling the model that generated images should be treated as noisy ground truth, as shown in2.It first fully diffuse the first token,Then, it feed the diffusedinto the model but tell it is of a slightly higher noise level, asto diffuse.Then, it feed the diffusedinto the model but tell it is of a higher noise level, as.Appendix CExtended Related WorkReconstructing masked tokens.Masked Autoencoders for imagesand videosare a popular method for representation learning in pixel space. They have been extended to perform diffusion to generate masked patches conditioned on unmasked ones.Casting Image Generation as Sequence Generation.show that even generative modeling of non-sequential data, such as images, can be fruitfully cast as sequence generative modeling.Non-Diffusion Probabilistic Sequence Models.parameterize token-to-token transitions via a variational auto-encoder. This makes them probabilistic, but does not directly maximize the joint probability of sequences, but rather, enables sampling from the distribution of single-step transitions.AR-Diffusion.Most similar to our work is AR-Diffusionwhich similarly aim to train next-token prediction models for sequence diffusion. Key differences are that AR-Diffusion proposes a noise level that islinearlydependent on the position of each word in the sequence, while our critical contribution is to have each noise level beindependent, as this uniquely enables our proposed sampling schemes, such as stabilizing auto-regressive generation and conditioning on corrupted observations. Further, AR-Diffusion only explores language modeling and does not explore guidance, while we investigate Diffusion Forcing as a broadly applicable sequence generative model with particular applications to sequential decision making. In particular, we introduce Monte-Carlo Tree Guidance as a novel guidance mechanism.Appendix DAdditional Method DetailsD.1ArchitectureVideo DiffusionWe choose both the raw imageand latent stateto be 2D tensors with channel, width, and height. For simplicity, we use the same width and height forand. We then implement the transition modelwith a typical diffusion U-net. We use the output of the U-net as the input to a gated recurrent unit (GRU)  and useas the hidden state feed into a GRU. The output of GRU is treated as. For observation model, we use a-layer resnetfollowed by a conv layer. We combine these two models to create an RNN layer, where the latent of a particular time step is, input isand output is. One can potentially obtain better results by training Diffusion Forcing with a causal transformer architecture. However, since RNN is more efficient for online decision-making, we also stick with it for video prediction and it already gives us satisfying results.We choose the number of channels into befor DMlab andfor Minecraft. In total, our Minecraft model consists ofmillion parameters and our DMlab model consists ofmillion parameters. We can potentially obtain a better Minecraft video prediction model with more parameters, but we defer that to future works to keep the training duration reasonable (day) . In maze planning, the number of total parameters ismillion.Non-Video DiffusionFor non-spatialthat is not video nor images, we use residue MLPsinstead of Unet as the backbone for the dynamics model. Residue MLP is basically the ResNetequivalent for MLP. Similar to video prediction, we feed the output of resMLP into a GRU along withto get. Another ResMLP serves as the observation model.D.2Diffusion parameterizationIn diffusion models, there are three equivalent prediction objectives,,, andparameterization. Different objectives lead to different reweighting of loss at different noise levels, together with SNR reweighting. For example,parameterization andparameterization are essential in generating pixel data that favors high-frequency details.In our experiments, we useparameterization for video prediction and found it essential to both convergence speed and quality.We observe thatparameterization is strongly favorable in planning and imitation learning, likely because they don’t favor an artificial emphasis on high-frequency details. We observe the benefits of v-parameterization in time-series prediction.D.3SNR reweightingSNR reweightingis a widely used technique to accelerate the convergence of image diffusion models. In short, it reweighs the diffusion loss proportional to the signal-to-noise ratio (SNR)  of noisy. In Diffusion Forcing, conditioning variablecan also contain a non-trivial amount of information about, in addition to. For example, in a deterministic markovian system, ifhas its noise level, the posterior statecontains all the information needed to predictregardless of the noise level of.Therefore we re-derive SNR reweighting to reflect this change in Diffusion Forcing. We follow the intuition of original SNR reweighting to loosely define SNR in a sequence with independent levels of noises at different time steps. Denoteas the normalized SNR reweighting factor forfollowing its normal derivation in diffusion models. For example, if one uses min snr strategy, its reweighting factor will always fall betweenwhich we divide byto get. Define signal decay factor, measuring what proportion of signal incontribute to denoising. This is the simple exponential decay model of sequential information. Now, define cumulated SNR recursively as the running mean of:to account for signals contributed by the entire noisy history to the denoising at time step. The other factor that contributes to the denoising isof noisy observation. To combine them, we use a simplified model for independent events. Noticeandalways falls in range, and therefore can be reinterpreted as probabilities of having all the signal one needs to perfect denoise. Since the noise level atis independent of prior noise levels, we can viewandas probabilities of independent events, and thus can composed to define a joint probability, and we use thisas our fused SNR reweighting factor for diffusion training.In our experiments, we choose to follow the min-SNR reweighting strategyto derive the. Our new SNR reweighting proves extremely useful to accelerate the convergence of video prediction, while we didn’t observe a boost on non-image domains so we didn’t use it there.D.4Noise scheduleWe use sigmoid noise schedulefor video prediction, linear noise schedule for maze planning, and cosine schedule for everything else.D.5Implementation Details of Sampling with GuidanceCorner case of sampling noiseIn our sampling algorithm, due to the flexibility of the scheduling matrix, there are corner cases whenis required to stay at its same noise level during a sampling step. The core question of this corner case is whether we should updateat all. One option is just copying over the old value. The other option is to run a backward diffusion followed by a forward diffusion back to its old noise level to resample under the diffusion process. While we conclude this can be an open question, we prefer the later approach, resampling, and use it in Monte Carlo Guidance to generate multiple samples. We note that even if one take the first approach, the guidance gradient can still flow back in the time steps beforeas the dynamics modelcan still propagate the guidance gradient to.Other than Monte Carlo Guidance, this corner case only happens whenorthroughout our experiments. That is, we chose oursuch that once any token gets diffused slightly, it will keep diffusing. In the case of, keepingat the same noise level implies it will stay as white noise, and we don’t even need to sample another white noise. In case, the time step is already completely diffused either approach should give us the same result so we just opt for copying over for simplicity.Guidance for maze planningIn maze planning, our main baseline Diffuerdiscards the reward from the dataset and directly plans with the goal position and velocity. We adopt the same convention for Diffusion Forcing. One can perform guidance on goal position using log-likelihood, but a flexible horizon model should not require users to manually specify ato reach its goal, instead we want it to try to reach the goal for any possible horizon. Therefore we use the reward modelso any time step can be the final step to reach the goal. This objective is challenging due to the non-convex nature of 2D maze, but we found Diffusion Forcing can still reliably find plans without bumping into walls. However, we also observe that the agent tend to leave the goal location due to the nature of the provided dataset - the goal location is just one possible waypoint for the robot to pass through, and there are no trajectories that simply stay at the goal. We also tried this reward for guidance with Diffuser, but it didn’t work even with a good amount of tuning.D.6Performance OptimizationAccelerating the diffusion sampling of Diffusion Forcing is similar to that of normal diffusion models. We adopt DDIMsampling for the diffusion of each token. While we usesteps of diffusion, we sample with onlyDDIM for video prediction andfor non-video domains.While Diffusion Forcing can be implemented with transformers, we use an RNN as the backbone for Diffusion Forcing experiments it’s widely used in decision-making for its flexibility and efficiency in online decision-making systems. To further reduce training time and GPU memory usage, we use frame-stacking to stack multiple observed images as a single. This is due to the fact that adjacent tokens can be very similar - e.g. recording the same motion at higher fps can lead to this. We deem that it’s wasteful if we rollout the dynamics model multiple times to generate almost identical tokens. For video datasets, we manually examine how many time steps it takes to require a minimal level of prediction power instead of copying frames over. There is another reason why we use frame stacking - many diffusion model techniques such as different noise schedules are designed to modelwith correlated elements or redundancy. Low-dimensional systems may need drastically different hyperparameters when they lack the data redundancy these techniques are tested on. Frame stacking is thus also helpful for our non-image experiments so we can start with canonical hyperparameters of diffusion models. We use a frame stack offor DMlab video prediction,for Minecraft andfor maze planning.At sampling time, we also have a design choice to reduce compute usage, as reflected in line 8 of Algorithm2. In line 8, we directly assignto, instead of recalculatingwith posterior model. Since the model is trained to condition onestimated from arbitrary noisy history, we recognize that both are valid approaches. The reason why the choose lineis two fold. First, it cuts the compute by half, avoiding computing posterior every step. Second, this happens to be what we want for stabilization -already contains the information of the cleanunder our simplified observation model, and happens to be estimated with, a noise level higher than that of. This happens to implement the behavior we want for stabilization.D.7Sampling schedule for causal uncertaintyInference is depicted inAlgorithm2and Figure2. In EquationD.1, we illustrate a specific instantiation of thematrix we used for causal planning. For simplicity, we denote the case where a latentis given and aim to generate.(D.1) Diffusion Forcing begins by sampling our sequences as white noise with noise level. It then denoises along each rowofin decreasing order. It does so by proceeding sequentially through frames, updating the latent (Line 5 of Algorithm2) , and then partially applying the backward process to noise leveldictated by the scheduling matrix(Line 6-7 of Algorithm2) . We call alike this pyramid scheduling, as the tokens in the far future are kept at higher noise level than near future.D.8Implementation Details of Timeseries RegressionWe follow the implementation of pytorch-ts, where the validation set is a random subset of the training set with the same number of sequences as the test set. We use early stopping when validation crpssum hasn’t increased for 6 epochs. We leverage the same architecture (1 mlp and 4 grus)  as well as a batch size of 32.D.9Compute ResourcesAll of our experiments usemixed precision training. Time series, maze planning, compositionally, visual imitation experiments can be trained with a singlewithGB of memory. We tune the batch size such that we fully use the memory of GPUs. This translates to a batch size offor maze planning and compositional experiments, andfor visual imitation learning. While we use early stopping on the validation set for time series experiments, we did not carefully search for the minimal number of training steps required, though the model usually converges betweenk tosteps. The above environments thus usually takehours to train although there is without doubt a significant potential for speed up.Video prediction is GPU intensive. We useA100 GPUs for both video prediction datasets. We train forsteps with a batch size of. It usually takehours to converge atsteps of training (occasional validation time also included) .Appendix EAdditional Experiment ResultsE.1Multivariate Probabilistic Time Series ForecastingFigure 6:Prediction intervals of Diffusion Forcing for the first prediction window of the test set in the Electricity time series dataset. Only the first 16 features out of 370 are plotted.To illustrate Diffusion Forcing’s new training objective does not degrade it as a generic sequence model, we evaluate Diffusion Forcing on high-dimensional and long-horizon sequence prediction tasks in time series prediction. We adopt multiple time series datasets with real-world applications from GluonTSand evaluate Diffusion Forcing with strong baselines with standard metrics in this domain. In this section, we mainly focus on the results and analysis. For a detailed description of datasets and the metric, we refer the reader to AppendixF.4.Problem FormulationLetbe a sequence (multivariate time series)  of-dimensional observationsof some underlying dynamical process, sampled in discrete time steps, where. In the problem setting of probabilistic time series forecasting, the sequenceis split into two subsequences at time stepwith: the context window(also called history or evidence)  of length, and the prediction windowof length(also known as the prediction horizon) . Then, the task is to model the conditional joint probability distribution(E.1) over the samples in the prediction window. If we know the distribution in (E.1) , we can sample forecast prediction sequences given some initial context from the evidence sequence. However, most time-dependent data generation processes in nature have complex dynamics and no tractable formulation of. Instead, we construct a statistical model that approximates the generative process in (E.1)  and estimates quantiles via Monte Carlo sampling of simulated trajectories. In this way, confidence levels or uncertainty measures can be calculated, and point forecasts can be produced as the mean or median trajectory.Results.\ctable[
caption = Results for time series forecasting. We report the test set(the lower, the better)  of comparable methods on six time series datasets. We measure the mean and standard deviation of our method from five runs trained with different seeds.,
label = tab:results_ts,
pos = ht,
doinside =,
center,
star
]lccccccc\FLMethod & Exchange  Solar  Electricity  Traffic  Taxi  Wikipedia\MLVES0.0050.000  0.9000.003  0.8800.004  0.3500.002  -  -\NNVAR0.0050.000  0.8300.006  0.0390.001  0.2900.001  -  -\NNVAR-Lasso0.0120.000  0.5100.006  0.0250.000  0.1500.002  -  3.1000.004\NNGARCH0.0230.000  0.8800.002  0.1900.001  0.3700.001  -  -\NNDeepAR-  0.3360.014  0.0230.001  0.0550.003  -  0.1270.042\NNLSTM-Copula0.0070.000  0.3190.011  0.0640.008  0.1030.006  0.3260.007  0.2410.033\NNGP-Copula0.0070.000  0.3370.024  0.0250.002  0.0780.002  0.2080.183  0.0860.004\NNKVAE0.0140.002  0.3400.025  0.0510.019  0.1000.005  -  0.0950.012\NNNKF-  0.3200.020  0.0160.001  0.1000.002  -  0.0710.002\NNTransformer-MAF0.0050.003  0.3010.014  0.0210.000  0.0560.001  0.1790.002  0.0630.003\NNTimeGrad0.0060.001  0.2870.020  0.0210.001  0.0440.006  0.1140.020  0.0490.002\NNScoreGrad sub-VP SDE0.0060.0010.2560.0150.0190.0010.0410.004  0.1010.0040.0430.002\NNOurs0.0030.0010.2890.002  0.0230.0010.0400.0040.0750.0020.0850.007\LLWe evaluate the effectiveness of Diffusion Forcing as a sequence model on the canonical task of multivariate time series forecasting by following the experiment setup ofConcretely, we benchmark Diffusion Forcing on the datasets Solar, Electricity, Traffic, Taxi, and Wikipedia. These datasets have different dimensionality, domains, sampling frequencies, and capture seasonal patterns of different lengths. The features of each dataset are detailed in TableLABEL:tab:ts_data. We access the datasets from GluonTS, and set the context and prediction windows to the same length for each dataset. Additionally, we employ the same covariates as.
We evaluate the performance of the model quantitatively by estimating the Summed Continuous Ranked Probability Scorevia quantiles. As a metric,measures how well a forecast distribution matches the ground truth distribution. We provide detailed descriptions of the metric in AppendixF.4.
We benchmark with other diffusion-based methods in time series forecasting, such as TimeGradand the transformer-based Transformer-MAF. In particular, the main baseline of interest, TimeGrad, is a next-token diffusion sequence model trained with teacher forcing.
We track themetric on the validation set and use early stopping when the metric has not improved for 6 consecutive epochs, while all epochs are fixed to 100 batches across datasets. We then measure theon the test set at the end of training, which we report in TableLABEL:tab:results_ts. We use the exact same architecture and hyperparameters for all time series datasets and experiments.
Diffusion Forcing outperforms all prior methods except forwith which Diffusion Forcing is overall tied, except for the Wikipedia dataset, on which Diffusion Forcing takes fourth place.
Note that time series is not the core application of Diffusion Forcing, and that we merely seek to demonstrate that the Diffusion Forcing objective is applicable to diverse domains with no apparent trade-off in performance over baseline objectives.E.2Additional results in compositional generationSince Diffusion Forcing models the joint distribution of any subset of a sequence, we can leverage this unique property to achieve compositional behavior - i.e., Diffusion Forcing can sample from the distribution ofsubsetsof the trajectory and compose these sub-trajectories into new trajectories.In particular, we show that we can also have flexible control over how compositional Diffusion Forcing is. As shown in7, consider a dataset of trajectories on a 2D, square plane, where all trajectories start from one corner and end up in the opposite corner, forming a cross shape. When no compositional behavior is desired, one can let the models replicate the cross-shaped distribution by allowing full memory of the HMM model.
When one desires compositional such as generating a V-shaped trajectory, which stitches two sub trajectories together, one can let the model generate shorter plans with no-memory context using MPC. (Add figures) .(a) Dataset(b) W/ memory(c) W/o memoryFigure 7:Given a dataset of trajectories (7(a) ) , Diffusion Forcing models the joint distribution of all subsequences of arbitrary length. At sampling time, we can sample from the trajectory distribution by sampling Diffusion Forcing with full horizon (7(b) )  or recover Markovian dynamics by disregarding previous states (7(c) ) .E.3Additional results in video prediction (wo/ cherry picking) Infinite Rollout without sliding windowDiffusion Forcing can rollout longer than maximum training horizonwithout sliding window. That is, we run Diffusion Forcing’s RNN continuously without ever reinitializing. This is a surprising effect we observed from the rollout stabilization property of Diffusion Forcing. In Figure8,10, we use Diffusion Forcing to generate video sequences of lengthand visualize subsampled sequences. Notably, Diffusion Forcing used in these visualizations is trained with a maximum length offrames for Minecraft andframes for DMLab, illustrating it can rollout 2x-5x times longer than it’s trained onwithout sliding window. In addition, we also tried rolling these models out forframes and without seeing the model blowing up on both datasets. There are occasional cases where the Minecraft agent gets stuck and the entire screen is the “dirt” block, but this is more of a dataset issue16and the agent is able to recover after it turns around.Figure 8:Visualization shows Diffusion Forcing trained onframes is able to rolloutframes on Minecraft datasetwithout sliding window. The visualization shows a non-cherry-picked subsampling of theseframes, although Diffusion Forcing can roll out much longer (such as 2000 frames)  on this dataset.Figure 9:Diffusion Forcing trained onframes is able to rolloutframes on Minecraft datasetwithout sliding window. The visualization shows a non-cherry-picked subsampling of theseframes, although Diffusion Forcing can roll out much longer (such as 2000 frames)  on this dataset. The first few frames marked in red are the ground truth images of the dataset used for conditioning.Figure 10:Visualization shows Diffusion Forcing trained onframes is able to rolloutframes on DMLab datasetwithout sliding window. The visualization shows a non-cherry-picked subsampling of theseframes, although Diffusion Forcing can roll out almost infinitely on this dataset. The first few frames marked in red are the ground truth images of the dataset used for conditioning.Figure 11:Visualization shows Diffusion Forcing trained onframes is able to rolloutframes on DMLab datasetwithout sliding window. The visualization shows a non-cherry-picked subsampling of theseframes, although Diffusion Forcing can roll out almost infinitely on this dataset. The first few frames marked in red are the ground truth images of the dataset used for conditioning.ConsistencyWe also present additional results where we only generate within our maximum training length. As shown in figure1312, Diffusion Forcing can generate consistent videos. Results are not cherry-picked.Figure 12:Additional non-cherry-picked video prediction results on DMLab dataset, generated within maximum training length. The first few frames marked in red are the ground truth images of the dataset used for conditioning.Figure 13:Additional non-cherry-picked video prediction results on Minecraft dataset, generated within maximum training length. The first few frames marked in red are the ground truth images of the dataset used for conditioning.E.4Additional results in planningWe provide some additional visualizations of causal planning in15. We also present additional visualization of Diffusion Forcing performing model predictive control in action. As shown in figure14, Diffusion Forcing can generate plans of shorter horizons since it’s flexible horizon.Figure 14:Example MPC planning for maze medium environment. Blue indicated trajectories actually executed already. Red is the plan.Figure 15:Example plans generated for maze medium (above)  and maze large (below)  environments.E.5Real robot experiment setupIn Figure16we visualize our robot experiment setup with corruption on observation. The dataset is collected when the target bag isn’t present, while we test with such a bag in the scene zero-shot for the imitation learning experiment with observation corruption. The typical failure mode is when the robot no longer reacts to the visual clues of the randomized location of objects. We didn’t observe the robot to act wildly due to visual distractors.Figure 16:We randomly throw a target bag on the table as a strong visual distractor. Diffusion Forcing can be prompted to treat observation as corrupted than ground truth.Appendix FAdditional details about datasetsF.1Dataset for video diffusionWe adopt the video prediction dataset Minecraft and DMlab used by TECO.Minecraft NavigationThe Minecraft navigation dataset consists of first-person-view videos of random walks in the Minecraft ‘swamp‘ biome. The agent walks via a technique called ‘sprint jump‘ which allows it to jump across blocks without getting stuck at 1 block obstacles. The agent walks straight most of the time, with small chances of turning left or right. The height and width of the video ispixels and we trim long videos to subsequences offrames. The dataset comes with paired action data but we discard them to bring more stochasticity to the prediction task. Due to limited compute, we only train on aboutof the total subsequences.One problem we noticed about the dataset is when the agent runs into obstacles with a height of 2 blocks or more. In this case, the agent will get stuck and the entire video sequence will consist of grey granite patterns or brown dirty patterns. This leads to a huge amount of frames with these patterns, making video models predict meaningless frames. Yet, we deem this as a problem of this dataset itself.DMLab NavigationDeepmind Lab navigation dataset consists of random walks in a 3D maze environment. For DMLab, the resolution ispixels and we use subsequences offrames. We also disregard the provided actions due to training.We note that the VQ-VAE latent that stable video diffusiondiffuses is also only, indicating Diffusion Forcing has the potential to scale up to higher resolution images with pre-trained image encoder and decoders. Due to the sheer size of the datasets, we only use aboutof the total data sequences for training due to limited computing, as we observe that doing so already allows us to make good generations from initial frames from the test set.F.2Dataset for planningD4RLis a standard offline RL benchmark featuring a wide range of reinforcement learning environments. Each environment is associated with a provided dataset of offline interactions with the environment featuring state, action and reward trajectories.Like Diffuer, we choose the 3 maze environments as they are challenging long-horizon, multi-modal, sparse reward problems uniquely suited for visualization and evaluating planning algorithms. The ID for the 3 used environments are “maze2d-medium-v1”, “maze2d-large-v1”, “maze2d-umaze-v1”. In each environment, one controls the acceleration of a robot to walk it towards a goal. The observation space isdimensional, featuring 2D location and velocity. The action space is 2D acceleration. The agent always receives a random start location and the goal is to reach a fixed goal position for each maze. The agent receives a reward of 1 if it is within a circle of radius 0.5 centered at the goal state, and 0 otherwise.The offline RL dataset for the maze environments consists of random walks in the maze. Specifically, the authors first designate all intersections and turns in the maze as waypoints and code an agent to navigate between waypoints with some randomization. As a result, the random walks are generated in a way that the path is collision-free with the walls. The random walks introduces stochasticity to the dataset, as trajectories in the dataset are never towards a specific goal.There are a few choices adopted from our main baseline Diffuser: we disregard the reward in the dataset and plan with goals only. We also evaluate a multi-gaol variant of each environment (labeled as “multi” in Table4.1) , where the goal is randomized just like the starting position.F.3Dataset for robot learningWe choose a long horizon robotic manipulation task as described in Section4.4: Consider a tabletop with three slots where we can place objects. One places an apple at slot A or slot B randomly, and then places an orange at the other slot between A and B. A robot is challenged to swap the position of two fruits using the third slot C. That is, it can only move a fruit to an empty slot at a time. For example, when the apple is at slot A and the orange is at slot B, it may move the apple to slot C, leaving slot A empty. Then move the orange to slot A and finally move the apple from slot C to slot B. In figure4, we illustrate the non-markovian property of the task: When the apple is at slot B and the orange is at slot C, one cannot tell what the immediate action is without knowing the initial positions of objects.We put stickers on the table indicating a circular region occupied by any slot. Each circular region is designed to be about double the diameter of a fruit. To make sure the task requires visual feedback, we also randomize the location of a fruit inside the slot. We collectedexpert demonstrations of a franka robot performing the task using VR teleoperation and impedance control. Among them, each initial slot configuration makes up half of the dataset. We record videos from two camera views, one from hand camera and one in the front capturing all three slots. Each demonstration also comes withdof actions of the robot hand. During the data collection, since one successful demonstration will swap the position of two objects, its end configuration will naturally serve as the starting configuration of the other randomized location, which we leverage to save time.Each demonstration comprisesframes and actions. We train Diffusion Forcing on the entire sequence. However, since adjacent frames are visually close, we pad and downsample the videos toframes where each frame is bundled withactions.F.4Dataset for time series\ctable[
caption = Characteristics of the GluonTS datasets used to benchmark Diffusion Forcing in the domain of time series forecasting.,
label = tab:ts_data,
pos = ht,
doinside = ,
center,
star
]lccccc\FLDataset & Dimension  Domain  Frequency  Steps  Prediction length\MLExchange  8BUSINESS DAY  6,071  30\NNSolar  137HOUR  7,009  24\NNElectricity  370HOUR  5,833  24\NNTraffic  963  (0,1)   HOUR  4,001  24\NNTaxi  1,21430-MIN  1,488  24\NNWikipedia  2,000DAY  792  30\LLWe use a set of time series datasets accessible via GluonTS, which are adopted from prior works like. These datasets capture real-world data of high-dimensional dynamics like monetary exchange rates or the electricity grid. In TableLABEL:tab:ts_data, we provide a summary of the features of these datasets, such as the dimensionality, the domains, the sampling frequency, the length of the mulitvariate sequence in the training set, and the prediction length. We access the datasets in TableLABEL:tab:ts_datavia GluonTS and wrap the data processing functions implemented in GluonTS in our own dataloaders. Each dataset consists of one long multivariate sequence, which is the training split, and a set of short sequences that make up the test split. We construct a validation set of the same cardinality as the held-out test set as a randomly sampled subset of subsequences from the training set. All splits are normalized by the mean and the standard deviation of the features in the training split.CovariatesOften, statistical models that approximate (E.1)  benefit from manually curated features as additional input to the observations. A sequence of covariatescan be constructed to help the model recognize seasonal patterns and other temporal dependencies. We follow the implementation into construct the covariate sequence as a function of the frequency of each dataset in TableLABEL:tab:ts_data. As such, our covariates are composed of lagged inputs, as well as learned embeddings and handcrafted temporal features that encode information such as the hour of the day or the day of the month, depending on the sampling rate of the particular time series that is being modeled. Therefore, covariates are known for the entire interval, even at inference. We can easily incorporate covariates into the probabilistic framework as(F.1) The benefit obtained from covariates is highly dependent on the characteristics of both the dataset and the model used, as well as the feature engineering practices followed.MetricThe Continuous Ranked Probability Score (CRPS) is a scoring function that measures how good the forecast distribution matches the ground truth distribution:whereis the univariate cumulative distribution function (CDF)  over the predicted value,is a ground truth observation, andis the indicator function that is one ifand zero otherwise. By summing the-dimensional time series along the feature dimension for simulated samples (resulting in)  and ground truth data (as) , we can report theas the average over the prediction window. The lower thevalue, the better does the predicted distribution match the data distribution.First, we manually sum the time series along the feature dimension and estimate the CDFvia 19 quantile levels at each time stepfrom 100 sampled trajectories. We then use the implementation in GluonTsto compute the, which we report asin TableLABEL:tab:results_ts. While we aggregate the data manually, we verify that the numerical error relative to the GluonTS implementation remains orders of magnitude below the precision threshold of the reported metric.
conclusion
Diffusion Forcing: Next-token PredictionMeets Full-Sequence Diffusion
5Discussion