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<h1 class="title toc-ignore">An alternate algorithm for FLASH loadings updates</h1>
<h4 class="author"><em>Jason Willwerscheid</em></h4>
<h4 class="date"><em>7/18/2018</em></h4>
</div>
<p><strong>Last updated:</strong> 2018-07-20</p>
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<p></details></p>
<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>If the expression for the KL divergence derived in the <a href="obj_notes.html">previous note</a> is correct, then it seems likely that the FLASH objective could be optimized in a more direct fashion.</p>
</div>
<div id="notation" class="section level2">
<h2>Notation</h2>
<p>I parametrize the posteriors for, respectively, the <span class="math inline">\(i\)</span>th element of the <span class="math inline">\(k\)</span>th loading and the <span class="math inline">\(j\)</span>th element of the <span class="math inline">\(k\)</span>th factor as <span class="math display">\[ q_{l_i} \sim (1 - w_i^{(l)}) \delta_0 + w_i^{(l)} N(\mu_i^{(l)}, \sigma_i^{2(l)}) \]</span> and <span class="math display">\[ q_{f_j} \sim (1 - w_j^{(f)}) \delta_0 + w_j^{(f)} N(\mu_j^{(f)}, \sigma_j^{2(f)}) \]</span> I parametrize the priors as <span class="math display">\[ g_{l_i} \sim \pi_0^{(l)} \delta_0 + (1 - \pi_0^{(l)}) N(0, 1/a_l) \]</span> and <span class="math display">\[ g_{f_j} \sim \pi_0^{(f)} \delta_0 + (1 - \pi_0^{(f)}) N(0, 1/a_f) \]</span></p>
</div>
<div id="objective" class="section level2">
<h2>Objective</h2>
<p>Using the expression for KL divergence derived in the <a href="obj_notes.html">previous note</a>, the objective can be written: <span class="math display">\[\begin{aligned}
\sum_{i, j} \left[ \frac{1}{2} \log \frac{\tau_{ij}}{2 \pi}
- \frac{\tau_{ij}}{2} \left( (R_{ij}^{-k})^2
- 2 R_{ij}^{-k} w_i^{(l)} \mu_i^{(l)} w_j^{(f)} \mu_j^{(f)}
+ w_i^{(l)} (\mu_i^{(l)2} + \sigma_i^{2(l)})
w_j^{(f)} (\mu_j^{(f)2} + \sigma_j^{2(f)}) \right) \right] \\
+\sum_i \left[ (1 - w_i^{(l)}) \log \frac{\pi_0^{(l)}}{1 - w_i^{(l)}}
+ w_i^{(l)} \log \frac{1 - \pi_0^{(l)}}{w_i^{(l)}}
+ \frac{w_i^{(l)}}{2} \left( \log(a_l \sigma_i^{2(l)})
- a_l (\mu_i^{(l)2} + \sigma_i^{2(l)}) + 1 \right)
\right] \\
+ \sum_j \left[ (1 - w_j^{(f)}) \log \frac{\pi_0^{(f)}}{1 - w_j^{(f)}}
+ w_j^{(f)} \log \frac{1 - \pi_0^{(f)}}{w_j^{(f)}}
+ \frac{w_j^{(f)}}{2} \left( \log(a_f \sigma_j^{2(f)})
- a_f (\mu_j^{(f)2} + \sigma_j^{2(f)})+ 1 \right)
\right],
\end{aligned} \]</span></p>
<p>where <span class="math inline">\(R_{ij}^{-k}\)</span> denotes the matrix of residuals obtained by using all factor/loading pairs but the <span class="math inline">\(k\)</span>th.</p>
</div>
<div id="prior-parameter-updates" class="section level2">
<h2>Prior parameter updates</h2>
<p>I derive an algorithm for loadings updates by differentiating with respect to each variable <span class="math inline">\(a_l\)</span>, <span class="math inline">\(\pi_0^{(l)}\)</span>, <span class="math inline">\(\mu_1^{(l)}, \ldots, \mu_n^{(l)}\)</span>, <span class="math inline">\(\sigma_1^{2(l)}, \ldots, \sigma_n^{2(l)}\)</span>, and <span class="math inline">\(w_1^{(l)}, \ldots, w_n^{(l)}\)</span>, and setting each result equal to zero.</p>
<p>The updates for the prior parameters <span class="math inline">\(a_l\)</span> and <span class="math inline">\(\pi_0^{(l)}\)</span> turn out to be very simple. First, differentiating with respect to <span class="math inline">\(a_l\)</span> gives <span class="math display">\[ \sum_i \left[ \frac{w_i^{(l)}}{2} \left( \frac{1}{a_l} - (\mu_i^{(l)2} + \sigma_i^{2(l)}) \right) \right] \]</span> Setting this equal to zero gives <span class="math display">\[ a_l = \frac{\sum_i w_i^{(l)}}{\sum_i w_i^{(l)} (\mu_i^{(l)2} + \sigma_i^{2(l)})} = \frac{\sum_i w_i^{(l)}}{\sum_i E_ql_i^2} \]</span></p>
<p>Next, differentiating with respect to <span class="math inline">\(\pi_0^{(l)}\)</span> gives <span class="math display">\[ \sum_i \left[ \frac{1 - w_i^{(l)}}{\pi_0^{(l)}} - \frac{w_i^{(l)}}{1 - \pi_0^{(l)}} \right] \]</span> Setting this equal to zero gives <span class="math display">\[\begin{aligned}
\pi_0^{(l)} \sum_i w_i^{(l)} &= (1 - \pi_0^{(l)}) \sum_i (1 - w_i^{(l)}) \\
\pi_0^{(l)} &= \frac{1}{n} \sum_i (1 - w_i^{(l)})
\end{aligned}\]</span></p>
</div>
<div id="posterior-parameter-updates" class="section level2">
<h2>Posterior parameter updates</h2>
<p>The updates for the posterior parameters <span class="math inline">\(\mu_i^{(l)}\)</span> and <span class="math inline">\(\sigma_i^{2(l)}\)</span> also turn out to be quite manageable. Differentiating with respect to <span class="math inline">\(\mu_i^{(l)}\)</span> gives <span class="math display">\[ \sum_j \tau_{ij} \left[ R_{ij}^{-k} w_i^{(l)} w_j^{(f)} \mu_j^{(f)}
- w_i^{(l)} \mu_i^{(l)} w_j^{(f)} (\mu_j^{(f)2} + \sigma_j^{2(f)}) \right] - w_i^{(l)} a_l \mu_i^{(l)} \]</span> Setting this equal to zero gives <span class="math display">\[ \mu_i^{(l)}
= \frac{\sum_j \tau_{ij} R_{ij}^{-k} w_j^{(f)} \mu_j^{(f)}}
{a_l + \sum_j \tau_{ij} w_j^{(f)} (\mu_j^{(f)2} + \sigma_j^{2(f)})}
= \frac{\sum_j \tau_{ij} R_{ij}^{-k} Ef_j}
{a_l + \sum_j \tau_{ij} Ef_j^{2}} \]</span></p>
<p>Next, differentiating with respect to <span class="math inline">\(\sigma_i^{2(l)}\)</span> gives <span class="math display">\[ -\frac{1}{2} \sum_j \tau_{ij} w_i^{(l)} w_j^{(f)} (\mu_j^{(f)2} + \sigma_j^{2(f)})
+ \frac{w_i^{(l)}}{2\sigma_i^{2(l)}} - \frac{w_i^{(l)} a_l}{2} \]</span> Setting this equal to zero gives <span class="math display">\[ \sigma_i^{2(l)}
= \frac{1}{a_l + \sum_j \tau_{ij} w_j^{(f)} (\mu_j^{(f)2} + \sigma_j^{2(f)})}
= \frac{1}{a_l + \sum_j \tau_{ij} Ef_j^2} \]</span></p>
<p>It remains to derive the update for <span class="math inline">\(w_i^{(l)}\)</span>. Differentiating gives <span class="math display">\[ \begin{aligned}
\sum_j \tau_{ij} \left[ R_{ij}^{-k} \mu_i^{(l)} w_j^{(f)} \mu_j^{(f)}
- \frac{1}{2}(\mu_i^{(l)2} + \sigma_i^{2(l)})
w_j^{(f)} (\mu_j^{(f)2} + \sigma_j^{2(f)}) \right] \\
- \log \frac{\pi_0^{(l)}}{1 - w_i^{(l)}}
+ \log \frac{1 - \pi_0^{(l)}}{w_i^{(l)}}
+ \frac{1}{2} \left( \log (a_l \sigma_i^{2(l)})
- a_l (\mu_i^{(l)2} + \sigma_i^{2(l)}) + 1 \right)
\end{aligned}\]</span> Setting this equal to zero gives <span class="math display">\[ \begin{aligned}
\log \frac{w_i^{(l)}}{1 - w_i^{(l)}}
&= \log \frac{1 - \pi_0^{(l)}}{\pi_0^{(l)}}
+ \frac{1}{2} \left( \log (a_l \sigma_i^{2(l)}) - a_l (\mu_i^{(l)2} + \sigma_i^{2(l)}) + 1 \right) \\
&+ \sum_j \tau_{ij} \left[R_{ij}^{-k} \mu_i^{(l)} w_j^{(f)} \mu_j^{(f)}
- \frac{1}{2}(\mu_i^{(l)2} + \sigma_i^{2(l)})
w_j^{(f)} (\mu_j^{(f)2} + \sigma_j^{2(f)}) \right],
\end{aligned}\]</span> where the last sum can also be written <span class="math display">\[\sum_j \tau_{ij} \left[R_{ij}^{-k} \mu_i^{(l)} Ef_j
- \frac{1}{2}(\mu_i^{(l)2} + \sigma_i^{2(l)}) Ef_j^2 \right]\]</span></p>
</div>
<div id="algorithm" class="section level2">
<h2>Algorithm</h2>
<p>I suggest that the loadings could be updated by</p>
<ol start="0" style="list-style-type: decimal">
<li>Choosing starting values for <span class="math inline">\(a_l\)</span>, <span class="math inline">\(\pi_0^{(l)}\)</span>, and <span class="math inline">\(w_1^{(l)}, \ldots, w_n^{(l)}\)</span>,</li>
</ol>
<p>and then repeating the following two steps until convergence:</p>
<ol style="list-style-type: decimal">
<li><p>Update <span class="math inline">\(\mu_1^{(l)}, \ldots, \mu_n^{(l)}\)</span> and <span class="math inline">\(\sigma_1^{2(l)}, \ldots, \sigma_n^{2(l)}\)</span>, and then update <span class="math inline">\(w_1^{(l)}, \ldots, w_n^{(l)}\)</span>.</p></li>
<li><p>Update <span class="math inline">\(a_l\)</span> and <span class="math inline">\(\pi_0^{(l)}\)</span>.</p></li>
</ol>
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<p>
This reproducible <a href="http://rmarkdown.rstudio.com">R Markdown</a>
analysis was created with
<a href="https://github.com/jdblischak/workflowr">workflowr</a> 1.0.1
</p>
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