mhjensen
241 lines
12 KiB
HTML
241 lines
12 KiB
HTML
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<meta name="description" content="Week 43: Solving Differential Equations with Deep Learning and Dimensionality Reduction methods">
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<title>Week 43: Solving Differential Equations with Deep Learning and Dimensionality Reduction methods</title>
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Recurrent Neural Networks', 2, None, '___sec0'),
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('Solving ODEs with Deep Learning', 2, None, '___sec1'),
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('Why should we think of reducing the dimensionality',
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2,
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('Basic ideas of the Principal Component Analysis (PCA)',
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('Introducing the Covariance and Correlation functions',
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('Rewriting the Covariance and/or Correlation Matrix',
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('Towards the PCA theorem', 2, None, '___sec11'),
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('The Algorithm before the Theorem', 2, None, '___sec12'),
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('Writing our own PCA code', 2, None, '___sec13'),
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('Compute the sample mean and center the data',
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('Compute the sample covariance', 3, None, '___sec15'),
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('Classical PCA Theorem', 2, None, '___sec17'),
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('Proof of the PCA Theorem', 2, None, '___sec18'),
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('PCA Proof continued', 2, None, '___sec19'),
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('The final step', 2, None, '___sec20'),
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('Geometric Interpretation and link with Singular Value '
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('Principal Component Analysis', 2, None, '___sec22'),
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('Back to the Cancer Data', 2, None, '___sec24'),
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('More on the PCA', 2, None, '___sec25'),
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('Incremental PCA', 2, None, '___sec26'),
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('Randomized PCA', 2, None, '___sec27'),
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('Kernel PCA', 2, None, '___sec28'),
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('Other techniques', 2, None, '___sec30')]}
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<a class="navbar-brand" href="week43-bs.html">Week 43: Solving Differential Equations with Deep Learning and Dimensionality Reduction methods</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week43-bs002.html#___sec0" style="font-size: 80%;"><b>Recurrent Neural Networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs003.html#___sec1" style="font-size: 80%;"><b>Solving ODEs with Deep Learning</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs004.html#___sec2" style="font-size: 80%;"><b>Why should we think of reducing the dimensionality</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs005.html#___sec3" style="font-size: 80%;"><b>Basic ideas of the Principal Component Analysis (PCA)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs006.html#___sec4" style="font-size: 80%;"><b>Introducing the Covariance and Correlation functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs007.html#___sec5" style="font-size: 80%;"><b>Correlation Function and Design/Feature Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs008.html#___sec6" style="font-size: 80%;"><b>Covariance Matrix Examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs009.html#___sec7" style="font-size: 80%;"><b>Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs010.html#___sec8" style="font-size: 80%;"><b>Correlation Matrix with Pandas</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs011.html#___sec9" style="font-size: 80%;"><b>Correlation Matrix with Pandas and the Franke function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs012.html#___sec10" style="font-size: 80%;"><b>Rewriting the Covariance and/or Correlation Matrix</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs013.html#___sec11" style="font-size: 80%;"><b>Towards the PCA theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs014.html#___sec12" style="font-size: 80%;"><b>The Algorithm before the Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec13" style="font-size: 80%;"><b>Writing our own PCA code</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec14" style="font-size: 80%;"> Compute the sample mean and center the data</a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec15" style="font-size: 80%;"> Compute the sample covariance</a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs015.html#___sec16" style="font-size: 80%;"> Diagonalize the sample covariance matrix to obtain the principal components</a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs016.html#___sec17" style="font-size: 80%;"><b>Classical PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs017.html#___sec18" style="font-size: 80%;"><b>Proof of the PCA Theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs018.html#___sec19" style="font-size: 80%;"><b>PCA Proof continued</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs019.html#___sec20" style="font-size: 80%;"><b>The final step</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs020.html#___sec21" style="font-size: 80%;"><b>Geometric Interpretation and link with Singular Value Decomposition</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs021.html#___sec22" style="font-size: 80%;"><b>Principal Component Analysis</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs022.html#___sec23" style="font-size: 80%;"><b>PCA and scikit-learn</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs023.html#___sec24" style="font-size: 80%;"><b>Back to the Cancer Data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs024.html#___sec25" style="font-size: 80%;"><b>More on the PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="#___sec26" style="font-size: 80%;"><b>Incremental PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs026.html#___sec27" style="font-size: 80%;"><b>Randomized PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs027.html#___sec28" style="font-size: 80%;"><b>Kernel PCA</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs028.html#___sec29" style="font-size: 80%;"><b>LLE</b></a></li>
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<!-- navigation toc: --> <li><a href="._week43-bs029.html#___sec30" style="font-size: 80%;"><b>Other techniques</b></a></li>
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</li>
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<!-- !split -->
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<h2 id="___sec26" class="anchor">Incremental PCA </h2>
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<p>
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One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
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memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have
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been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch
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at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new
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instances arrive).
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