Update 2021JUL21.md

2021JUL21.md

@@ -23,11 +23,11 @@ In summary, in addition to finding the the hyper-parameters $`\pi, \mu, \Sigma_{
  3. Initialise the transformation matrix $`\mathbf{U} = \mathbf{MN}^{T}`$, where $`\mathbf{MDN}^{T}`$ is the SVD of $`\sum_{k=1}^{K}\mu_{k}^{H}(\mu_{k}^{L})^{T}`$.
  4. For iteration = $`1, 2, ...`$, do
     - **E-step.** Evaluate the responsibilities using the current parameter values
-       - $`\gamma(y_{nk}) = \frac{\pi_{k}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}, \Sigma_{k}^{H})}{\sum_{j=1}^{K}\pi_{j}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}, \Sigma_{k}^{H})}`$
+       - $`\gamma(y_{nk}) = \frac{\pi_{k}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}^{L}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}^{L}, \Sigma_{k}^{H})}{\sum_{j=1}^{K}\pi_{j}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}^{L}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}^{L}, \Sigma_{k}^{H})}`$
     - **M-step.** Re-estimate the parameters using the current responsibilities by setting the derivatives to zero
-       - $`\mu_{k}^{\text{new}} = \frac{1}{N_{k}}((\Sigma^{H}_{k})^{-1} + (\Sigma^{L}_{k})^{-1} )^{-1}\sum_{n=1}^{N}\gamma(y_{nk})((\Sigma_{k}^{H})^{-1}\mathbf{Ux}^{H}_{n} + (\Sigma_{k}^{L})^{-1}\mathbf{x}_{n}^{L} )`$
-       - $`\Sigma_{k}^{L} = \frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(y_{nk})(\mathbf{x}^{L}_{n} - \mathbf{\mu}_{k})(\mathbf{x}^{L}_{n} - \mathbf{\mu}_{k})^{T}`$
-       - $`\Sigma_{k}^{H} = \frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(y_{nk})(\mathbf{Ux}^{H}_{n} - \mathbf{\mu}_{k})(\mathbf{Ux}^{H}_{n} - \mathbf{\mu}_{k})^{T}`$
+       - $`\mu_{k}^{L} = \frac{1}{N_{k}}((\Sigma^{H}_{k})^{-1} + (\Sigma^{L}_{k})^{-1} )^{-1}\sum_{n=1}^{N}\gamma(y_{nk})((\Sigma_{k}^{H})^{-1}\mathbf{Ux}^{H}_{n} + (\Sigma_{k}^{L})^{-1}\mathbf{x}_{n}^{L} )`$
+       - $`\Sigma_{k}^{L} = \frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(y_{nk})(\mathbf{x}^{L}_{n} - \mathbf{\mu}_{k}^{L})(\mathbf{x}^{L}_{n} - \mathbf{\mu}_{k}^{L})^{T}`$
+       - $`\Sigma_{k}^{H} = \frac{1}{N_{k}}\sum_{n=1}^{N}\gamma(y_{nk})(\mathbf{Ux}^{H}_{n} - \mathbf{\mu}_{k}^{L})(\mathbf{Ux}^{H}_{n} - \mathbf{\mu}_{k}^{L})^{T}`$
        - $`\pi_k = \frac{N_{k}}{N}`$
        - $`\mathbf{U}=`$ 
     - Evaluate the likelihood and check for convergence.