\section{Related work (literature review)}


\subsection{Gaussian Mixture Model}


The standard GMM can be stated as follows.
$$
p(\boldsymbol{x})=\sum_{k=1}^{K} \pi_{k} \mathcal{N}\left(\boldsymbol{x} | \boldsymbol{\mu}_{k}, \mathbf{\Sigma}_{k}\right)
$$

Given a GMM, the goal is to maximzie the likelihood function with respect to the parameters.

The Expectation-Maximization (EM) algorithm \cite{} is used in maximum likelihood estimation where the problem involves two sets of random variables of which one, X, is observable and the other, Z, is hidden. The goal of the algorithm is to find the parameter vector $\Phi$. The algorithm basically involves two steps: Expectation step and Maximization step which can be summarized as follows.


\par
E-step : $\mathcal{Q}\left(\Phi | \Phi^{l}\right)=E\left[\mathcal{L}_{c}(\Phi | \mathcal{X}, \mathcal{Z}) | \mathcal{X}, \Phi^{l}\right]$
\par
M-step : $\Phi^{l+1}=\arg \max _{\Phi} \mathcal{Q}\left(\Phi | \Phi^{l}\right)$

\par
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\begin{algorithm}
\caption{EM algorithm}\label{euclid}
\label{al}
\hspace*{\algorithmicindent} 
\textbf{Input: data $\mathcal{X}$, number of Gaussian $\mathcal{K}$}  
\\
\hspace*{\algorithmicindent} 
\textbf{Output: the parameters of GMM ($\pi$, $\mu$, $\sigma^{2}$)}

\begin{algorithmic}[1]


\State Initialize the means $\mu_{k}$, variances $\sigma^{2}_{k}$ and mixing coefficients $\pi_{k}$, and evaluate the initial value of the log likelihood.

\State $\boldsymbol{E-step}$
$$
\gamma\left(z_{n k}\right)=\frac{\pi_{k} \mathcal{N}\left(\mathbf{x}_{n} | \boldsymbol{\mu}_{k}, \mathbf{\Sigma}_{k}\right)}{\sum_{j=1}^{K} \pi_{j} \mathcal{N}\left(\mathbf{x}_{n} | \boldsymbol{\mu}_{j}, \mathbf{\Sigma}_{j}\right)}
$$

\State $\boldsymbol{M-step}$

$$
\boldsymbol{\mu}_{k}^{\mathrm{new}} =\frac{1}{N_{k}} \sum_{n=1}^{N} \gamma\left(z_{n k}\right) \mathbf{x}_{n} 
$$

$$
\mathbf{\Sigma}_{k}^{\mathrm{new}} =\frac{1}{N_{k}} \sum_{n=1}^{N} \gamma\left(z_{n k}\right)\left(\mathbf{x}_{n}-\boldsymbol{\mu}_{k}^{\mathrm{new}}\right)\left(\mathbf{x}_{n}-\boldsymbol{\mu}_{k}^{\mathrm{new}}\right)^{\mathrm{T}}
$$

$$
\pi_{k}^{\mathrm{new}} =\frac{N_{k}}{N} 
$$


\State Evaluate the log likelihood

$$
\ln p(\mathbf{X} | \boldsymbol{\mu}, \mathbf{\Sigma}, \boldsymbol{\pi})=\sum_{n=1}^{N} \ln \left\{\sum_{k=1}^{K} \pi_{k} \mathcal{N}\left(\mathbf{x}_{n} | \boldsymbol{\mu}_{k}, \boldsymbol{\Sigma}_{k}\right)\right\}
$$


\end{algorithmic}
\end{algorithm}
{"config": "arxiv", "file": "2006.16845/Related-Work.tex"}