\begin{algorithm}[t]
    \caption{MEP-ONMF}
    \label{alg}
\begin{algorithmic}[1]
\Require data matrix $X \in \mathbb{R}^{d \times n}_+$, number of features $k_{\text{max}}$, $\Gamma$, $\beta_{\text{max}}$, $c_j$ [for the $\ell_0$-forced version] 
\State \textbf{Initialization} $\beta_{init}$%\leftarrow\frac{1}{2\lambda_{\text{max}}C_{x}}$
, $k\leftarrow1$, $w_1\leftarrow\sum_{x}xp(x)$, $\lambda_1\leftarrow1$, if $W$-orthogonal $X \leftarrow X^{\intercal}$
\State \textbf{Normalization} $x_j \leftarrow \frac{x_i}{||x_i||_2} \;$ for $i=1...n$
\Loop \, {until $\beta=\beta_{\text{max}}$}
\Repeat
    \State  update $p_{j|i}$, $w_j$ and $\alpha_j$ by (\ref{Gibbs}) and (\ref{alpha}) [or (\ref{optsol}) for optimally-weighted]
  \Until convergence
    \State  $\beta \leftarrow \Gamma \beta$
    \If{$k<k_{\text{max}}$}
    \For {all $w_j$}
    \State check the phase transition condition
    \If{satisfied for $w_t$}
    \State add another feature $w_{k+1}=w_t+\delta$
    \State $\lambda_{w_{k+1}}\leftarrow0.5\lambda_{w_t}$, $\lambda_{w_t} \leftarrow 0.5\lambda_{w_t}$ 
    \EndIf
    \EndFor
    \EndIf
    \EndLoop
\State  do post-processing (\ref{postpro}) on $H$
\State  $W_{:j}\leftarrow w_j$ and $H_{ij}\leftarrow p_{j|i}$ $\forall i,j$
\State if {$W$-orthogonal}: $W\leftarrow H^{\intercal}$, $H\leftarrow W^{\intercal}$\\ 
\Return $W$, $H$
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
[t]
\caption{MEP-ONMF}
\begin{algorithmic}
[1]
\Require data matrix $X \in \mathbb{R}^{d \times n}_+$, number of features $k_{\text{max}}$, $\Gamma$, $\beta_{\text{max}}$, $c_j$ [for the $\ell_0$-forced version]
\State \textbf{Initialization} $\beta_{init}$%\leftarrow\frac{1}{2\lambda_{\text{max}}C_{x}}$
, $k\leftarrow1$, $w_1\leftarrow\sum_{x}xp(x)$, $\lambda_1\leftarrow1$, if $W$-orthogonal $X \leftarrow X^{\intercal}$
\State \textbf{Normalization} $x_j \leftarrow \frac{x_i}{||x_i||_2} \;$ for $i=1...n$
\Loop \, {until $\beta=\beta_{\text{max}}$}
\Repeat
\State update $p_{j|i}$, $w_j$ and $\alpha_j$ by (\ref{Gibbs}) and (\ref{alpha}) [or (\ref{optsol}) for optimally-weighted]
\Until convergence
\State $\beta \leftarrow \Gamma \beta$
\If{$k<k_{\text{max}}$}
\For {all $w_j$}
\State check the phase transition condition
\If{satisfied for $w_t$}
\State add another feature $w_{k+1}=w_t+\delta$
\State $\lambda_{w_{k+1}}\leftarrow0.5\lambda_{w_t}$, $\lambda_{w_t} \leftarrow 0.5\lambda_{w_t}$
\EndIf
\EndFor
\EndIf
\EndLoop
\State do post-processing (\ref{postpro}) on $H$
\State $W_{:j}\leftarrow w_j$ and $H_{ij}\leftarrow p_{j|i}$ $\forall i,j$
\State if {$W$-orthogonal}: $W\leftarrow H^{\intercal}$, $H\leftarrow W^{\intercal}$\\
\Return $W$, $H$
\end{algorithmic}
\end{algorithm}
{
    "title": "a novel maximum-entropy-driven technique for low-rank orthogonal   nonnegative matrix factorization with $\\ell_0$-norm sparsity constraint",
    "id": "2210.02672",
    "abstract": "in data-driven control and machine learning, a common requirement involves breaking down large matrices into smaller, low-rank factors that possess specific levels of sparsity. this paper introduces an innovative solution to the orthogonal nonnegative matrix factorization (onmf) problem. the objective is to approximate input data by using two low-rank nonnegative matrices, adhering to both orthogonality and $\\ell_0$-norm sparsity constraints. the proposed maximum-entropy-principle based framework ensures orthogonality and sparsity of features or the mixing matrix, while maintaining nonnegativity in both. additionally, the methodology offers a quantitative determination of the ``true'' number of underlying features, a crucial hyperparameter for onmf. experimental evaluation on synthetic and a standard datasets highlights the method's superiority in terms of sparsity, orthogonality, and computational speed compared to existing approaches. notably, the proposed method achieves comparable or improved reconstruction errors in line with the literature.",
    "categories": "cs.ds cs.it cs.lg math.it math.pr",
    "doi": "",
    "created": "2022-10-06",
    "updated": "2024-01-18",
    "authors": [
        "salar basiri",
        "srinivasa salapaka"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2210.02672"
}
{
    "ToxicityFilter": {
        "text_to_toxicity_attributes": {
            "root.tex": {
                "toxicity_score": 0.020607091,
                "severe_toxicity_score": 0.0013256073,
                "identity_attack_score": 0.0026453468,
                "insult_score": 0.00983089,
                "profanity_score": 0.014190876,
                "threat_score": 0.006699257,
                "sexually_explicit_score": 0.007902941,
                "flirtation_score": 0.30276403
            },
            "main.bib": {
                "toxicity_score": 0.0068170885,
                "severe_toxicity_score": 0.0015544891,
                "identity_attack_score": 0.001105311,
                "insult_score": 0.00654387,
                "profanity_score": 0.014566627,
                "threat_score": 0.0058966195,
                "sexually_explicit_score": 0.014508383,
                "flirtation_score": 0.1994727
            },
            "ieeeconf.cls": {
                "toxicity_score": 0.0058432184,
                "severe_toxicity_score": 0.00079631805,
                "identity_attack_score": 0.0014799143,
                "insult_score": 0.0062588686,
                "profanity_score": 0.012722028,
                "threat_score": 0.0059613483,
                "sexually_explicit_score": 0.008846575,
                "flirtation_score": 0.43877357
            },
            "main.tex": {
                "toxicity_score": 0.010429831,
                "severe_toxicity_score": 0.0011205673,
                "identity_attack_score": 0.0034038029,
                "insult_score": 0.007303875,
                "profanity_score": 0.014942379,
                "threat_score": 0.007340072,
                "sexually_explicit_score": 0.012857023,
                "flirtation_score": 0.25356454
            },
            "IEEEtran.cls": {
                "toxicity_score": 0.0049635945,
                "severe_toxicity_score": 0.0007343292,
                "identity_attack_score": 0.0012024303,
                "insult_score": 0.006011867,
                "profanity_score": 0.012312118,
                "threat_score": 0.005861019,
                "sexually_explicit_score": 0.008374758,
                "flirtation_score": 0.4981806
            },
            "main.bbl": {
                "toxicity_score": 0.00816794,
                "severe_toxicity_score": 0.0008869171,
                "identity_attack_score": 0.0030338243,
                "insult_score": 0.006999873,
                "profanity_score": 0.013336895,
                "threat_score": 0.0070164283,
                "sexually_explicit_score": 0.01020305,
                "flirtation_score": 0.5114233
            }
        },
        "reason": null
    }
}
{
    "num_done": {
        "figure": 0,
        "algorithm": 2
    }
}
{
    "NonTrivialRenderingFilter": {
        "white_pixels_ratio": 93.82458474439606,
        "hash": "003f3f1f878f3f0f",
        "most_frequent_color_ratio": 93.82458474439606
    }
}