\begin{algorithmic}
\State \textbf{Input:} Independent initial samples $x_1^{(0)},...,x_N^{(0)}$ from $\mu_0$, momentum parameters $m_l\in[0,1)$ for $l=1,...,L$.
\State Initialize $(v_1,...,v_N)=0$.
\For{$l=1,...,L$}
\State - Set $(\tilde x_1^{(0)},...,\tilde x_N^{(0)})=(x_1^{(l-1)},...,x_N^{(l-1)})$.
\State - Simulate $T_l$ steps of the (momentum) MMD flow:
\For{$t=1,...,T_l$}
\State - Update $v$ by
\begin{align*}
(v_1,...,v_N)\leftarrow \nabla F_d(\tilde x_1^{(t-1)},...,\tilde x_N^{(t-1)}|y_1,...,y_M)+m_l (v_1,...,v_N)
\end{align*}
\State - Update the flow samples:
\begin{align*}
(\tilde x_1^{(t)},...,\tilde x_N^{(t)})=(\tilde x_1^{(t-1)},...,\tilde x_N^{(t-1)})-\tau N\ (v_1,...,v_N)
\end{align*}
\EndFor
\State - Train $\Phi_l$ such that $\tilde x^{(T_l)}\approx \tilde x_i^{(0)}-\Phi_l(\tilde x_i^{(0)}) $ by minimizing the loss
\begin{align*}
\mathcal L(\theta_l)=\frac1N\sum_{i=1}^N \|\Phi_l(\tilde x_i^{(0)})-(\tilde x_i^{(0)}-\tilde x_i^{(T_l)})\|^2.
\end{align*}
\State - Set $(x_1^{(l)},...,x_N^{(l)})=(x_1^{(l-1)},...,x_N^{(l-1)})-(\Phi_l(x_1^{(l-1)}),...,\Phi_l(x_N^{(l-1)}))$.
\EndFor
\end{algorithmic}

\begin{algorithmic}
\State \textbf{Input:} Independent initial samples $x_1^{(0)},...,x_N^{(0)}$ from $\mu_0$, momentum parameters $m_l\in[0,1)$ for $l=1,...,L$.
\State Initialize $(v_1,...,v_N)=0$.
\For{$l=1,...,L$}
\State - Set $(\tilde x_1^{(0)},...,\tilde x_N^{(0)})=(x_1^{(l-1)},...,x_N^{(l-1)})$.
\State - Simulate $T_l$ steps of the (momentum) MMD flow:
\For{$t=1,...,T_l$}
\State - Update $v$ by
\begin{align*}
(v_1,...,v_N)\leftarrow \nabla F_d(\tilde x_1^{(t-1)},...,\tilde x_N^{(t-1)}|y_1,...,y_M)+m_l (v_1,...,v_N)
\end{align*}
\State - Update the flow samples:
\begin{align*}
(\tilde x_1^{(t)},...,\tilde x_N^{(t)})=(\tilde x_1^{(t-1)},...,\tilde x_N^{(t-1)})-\tau N\ (v_1,...,v_N)
\end{align*}
\EndFor
\State - Train $\Phi_l$ such that $\tilde x^{(T_l)}\approx \tilde x_i^{(0)}-\Phi_l(\tilde x_i^{(0)}) $ by minimizing the loss
\begin{align*}
\mathcal L(\theta_l)=\frac1N\sum_{i=1}^N \|\Phi_l(\tilde x_i^{(0)})-(\tilde x_i^{(0)}-\tilde x_i^{(T_l)})\|^2.
\end{align*}
\State - Set $(x_1^{(l)},...,x_N^{(l)})=(x_1^{(l-1)},...,x_N^{(l-1)})-(\Phi_l(x_1^{(l-1)}),...,\Phi_l(x_N^{(l-1)}))$.
\EndFor
\end{algorithmic}
{
    "title": "generative sliced mmd flows with riesz kernels",
    "id": "2305.11463",
    "abstract": "maximum mean discrepancy (mmd) flows suffer from high computational costs in large scale computations. in this paper, we show that mmd flows with riesz kernels $k(x,y) = - \\|x-y\\|^r$, $r \\in (0,2)$ have exceptional properties which allow their efficient computation. we prove that the mmd of riesz kernels, which is also known as energy distance, coincides with the mmd of their sliced version. as a consequence, the computation of gradients of mmds can be performed in the one-dimensional setting. here, for $r=1$, a simple sorting algorithm can be applied to reduce the complexity from $o(mn+n^2)$ to $o((m+n)\\log(m+n))$ for two measures with $m$ and $n$ support points. as another interesting follow-up result, the mmd of compactly supported measures can be estimated from above and below by the wasserstein-1 distance. for the implementations we approximate the gradient of the sliced mmd by using only a finite number $p$ of slices. we show that the resulting error has complexity $o(\\sqrt{d/p})$, where $d$ is the data dimension. these results enable us to train generative models by approximating mmd gradient flows by neural networks even for image applications. we demonstrate the efficiency of our model by image generation on mnist, fashionmnist and cifar10.",
    "categories": "cs.lg math.pr stat.ml",
    "doi": "",
    "created": "2023-05-19",
    "updated": "2024-02-20",
    "authors": [
        "johannes hertrich",
        "christian wald",
        "fabian altekr\u00fcger",
        "paul hagemann"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2305.11463"
}
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