Update README.md

Author: jamgochiana

README.md

@@ -8,26 +8,35 @@ It is particularly useful in high-dimensional settings where the number of featu
 By constraining the covariance matrices in this manner, we achieve a balance between flexibility and computational efficiency while avoiding overfitting.  
 
 We consider a dataset represented as $X \in \mathbb{R}^{n \times m}$ with **$n$** data points and **$m$** features, potentially with $m \gg n$. A Gaussian Mixture Model with $K$ components over elements $x^{(i)} \in \mathbb{R}^m$ is defined as:  
-$$
-p(x) = \sum_{k=1}^{K} \pi_k \mathcal{N}(x \mid \mu_k, \Sigma_k)\text{,} 
-$$  
+
+$$p(x) = \sum_{k=1}^{K} \pi_k \mathcal{N}(x \mid \mu_k, \Sigma_k)\text{,}$$
+
 with non-negative mixture weights $\pi_k$ that sum to one. Each Gaussian component has mean $\mu_k \in \mathbb{R}^m$ and covariance matrix of the form:  
+
 $$
-\Sigma_k = \text{diag}(d_k) + L_k L_k^T
+\Sigma_k = \text{diag}(d_k) + L_k L_k^T\text{,}
 $$ 
+
 where:  
 - $d_k \in \mathbb{R}^m$ is the diagonal vector capturing independent feature noise,  
 - $L_k \in \mathbb{R}^{m \times k}$ (with rank $k \ll m$) captures the dominant covariance structure via a low-rank factor.  
 
-The EM algorithm for fitting GMMs alternates between an **E**xpectation and a **M**aximization:  
+The EM algorithm for fitting GMMs alternates between **E**xpectation and **M**aximization:  
+
+1. **E-step:** Compute responsibilities based on current parameters:
+   
+  $$\gamma_{ik} = \frac{\pi_k \mathcal{N}(x^{(i)} \mid \mu_k, \Sigma_k)}{\sum_j \pi_j \mathcal{N}(x^{(i)} \mid \mu_j, \Sigma_j)}\text{.}$$
+  
+2. **M-step:** Update parameters by maximizing expected complete-data log-likelihood.
+
+  $$\pi_k = \frac{n_k}{n}, \quad \text{where } n_k = \sum_{i=1}^{n} \gamma_{ik}\text{,}$$
+
+  $$\mu_k = \frac{1}{n_k} \sum_{i=1}^{n} \gamma_{ik} x^{(i)}\text{,}$$
+
+  $$\Sigma_k = \frac{1}{n_k} \sum_{i=1}^{n} \gamma_{ik} (x^{(i)} - \mu_k)(x^{(i)} - \mu_k)^\top\text{.}$$
 
-- **E-step:** Compute responsibilities based on current parameters:  
-  $$
-  \gamma_{nk} = \frac{\pi_k \mathcal{N}(x_n \mid \mu_k, \Sigma_k)}{\sum_j \pi_j \mathcal{N}(x_n \mid \mu_j, \Sigma_j)}
-  $$
-  Computing responsibilities requires inverting $\Sigma_k$. 
-  Since this can be prohibitively expensive in large dimensions, we leverage the structure of the covariance matrices to calculate the responsibilities using an $\mathcal{O}(k^3)$ matrix inversion rather than a full $\mathcal{O}(m^3)$ inversion.
-- **M-step:** Update parameters by maximizing expected complete-data log-likelihood. 
-Since updating $\Sigma_k$ directly is intractable in high dimensions, an **inner EM loop** is performed using **Factor Analysis**, where $L_k$ and $D_k$ are iteratively estimated to approximate the full covariance structure.  
+Computing responsibilities requires inverting $\Sigma_k$. 
+Since this can be prohibitively expensive in large dimensions, we leverage the structure of the covariance matrices to calculate the responsibilities using an $\mathcal{O}(k^3)$ matrix inversion rather than a full $\mathcal{O}(m^3)$ inversion.
+Additionally, since updating $\Sigma_k$ directly is intractable in high dimensions, we perform an **inner EM loop** during maximization is performed using **Factor Analysis**, where $L_k$ and $D_k$ are iteratively estimated to approximate the full covariance structure. 
 
-This low-rank plus diagonal structure is particularly advantageous in scenarios such as **text modeling, gene expression analysis, and compressed sensing**, where **\( m \gg n \)** leads to singular or poorly conditioned full covariance estimates. Our package leverages efficient matrix decompositions and batched computations to ensure scalability, making it well-suited for large-scale, high-dimensional datasets.  
+This low-rank plus diagonal structure is particularly advantageous in settings such as **time-series analysis (e.g. for finance), text modeling, gene expression analysis, and compressed sensing**, where $m \gg n$ leads to singular or poorly conditioned full covariance estimates. Our package leverages efficient matrix decompositions and batched computations to ensure scalability, making it well-suited for large-scale, high-dimensional datasets.