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Unsupervised Learning: Difference between revisions
→Kernel PCA
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* Generate latent variables <math>z^{(1)},...,z^{(m)} \in \mathbb{R}^r</math> iid where dimension r is less than n. | * Generate latent variables <math>z^{(1)},...,z^{(m)} \in \mathbb{R}^r</math> iid where dimension r is less than n. | ||
** We assume <math>Z^{(i)} \sim N(\mathbf{0},\mathbf{I})</math> | ** We assume <math>Z^{(i)} \sim N(\mathbf{0},\mathbf{I})</math> | ||
* Generate <math>x^{(i)}</math> where <math>X^{(i)} \vert Z^{(i)} \ | * Generate <math>x^{(i)}</math> where <math>X^{(i)} \vert Z^{(i)} \sim N(g_{\theta}(z), \sigma^2 \mathbf{I})</math> | ||
** For some function <math>g_{\theta_1}</math> parameterized by <math>\theta_1</math> | ** For some function <math>g_{\theta_1}</math> parameterized by <math>\theta_1</math> | ||
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;Notes | ;Notes | ||
* Also known as Wasserstein metric | * Also known as Wasserstein metric | ||
==Dimension Reduction== | |||
Goal: Reduce the dimension of a dataset.<br> | |||
If each example <math>x \in \mathbb{R}^n</math>, we want to reduce each example to be in <math>\mathbb{R}^k</math> where <math>k < n</math> | |||
===PCA=== | |||
Principal Component Analysis<br> | |||
Preprocessing: Subtract the sample mean from each example so that the new sample mean is 0.<br> | |||
Goal: Find a vector <math>v_1</math> such that the projection <math>v_1 \cdot x</math> has maximum variance.<br> | |||
Result: These principal components are the eigenvectors of <math>X^TX</math>.<br> | |||
Idea: Maximize the variance of the projections.<br> | |||
<math>\max \frac{1}{m}\sum (v_1 \cdot x^{(i)})^2</math><br> | |||
Note that <math>\sum (v_1 \cdot x^{(i)})^2 = \sum v_1^T x^{(i)} (x^{(i)})^T v_1 = v_1^T (\sum x^{(i)} (x^{(i)})^T) v_1</math>.<br> | |||
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Thus our lagrangian is <math>\max_{\alpha} v_1^T (\sum x^{(i)} (x^{(i)})^T) v_1 + \alpha(\Vert v_1 \Vert^2 - 1)</math><br> | |||
Taking the gradient of this we get:<br> | |||
<math>\nabla_{v_1} (v_1^T (\sum x^{(i)} (x^{(i)})^T) v_1) + \alpha(\Vert v_1 \Vert^2 - 1)</math><br> | |||
<math>= 2(\sum x^{(i)} (x^{(i)})^T) v_1 + 2\alpha v_1</math> | |||
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===Kernel PCA=== | |||
{{main | Wikipedia: Kernel principal component analysis}} | |||
===Autoencoder=== | |||
You have a encoder and a decoder which are both neural networks. | |||