div class="arithmatex">\[\begin{equation}
    \boldsymbol{\hat Y} = \bb{X}(\bb{X}^T\bb{X})^{-1}\bb{X}^T\bb{Y} = \bb{X}\hat B\in \mathbb{R}^{N\times k},\non
 \end{equation}\]</div>
<p>and <span class="arithmatex">\(\hat y = \hat B^Tx\)</span> for a single training sample <span class="arithmatex">\(x\)</span>. We estimate the new parameters of the Gaussian distributions from transformed data, denoted by <span class="arithmatex">\(\pi_k^{\text{new}}\)</span>, <span class="arithmatex">\(\hat\mu_k^{\text{new}}\)