Autoencoder: Difference between revisions

Revision as of 06:47, 13 April 2024

Revision as of 06:48, 13 April 2024

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In most situations, the reference distribution is just the [[Empirical measure|empirical distribution]] given by a dataset <math>\{x_1, ..., x_N\} \subset \mathcal X</math>, so that<math display="block">\mu_{ref} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i}</math>

where <math>\delta_{x_i}</math> is the [[Dirac measure]], and the quality function is just L2 loss: <math>d(x, x') = \|x - x'\|_2^2</math>, and <math>\|\cdot\|_2</math> is the Euclidean norm. Then the problem of searching for the optimal autoencoder is just a [[Least squares|least-squares]] optimization:<math display="block">\min_{\theta, \phi} L(\theta, \phi), \text{where } L(\theta, \phi) = \frac{1}{N}\sum_{i=1}^N \|x_i - D_\theta(E_\phi(x_i))\|_2^2</math>

=== Interpretation ===