Continuous function

As explained in Deep Learning with R, Third Edition

Continuity is the property that sufficiently small changes in a function's input produce sufficiently small changes in its output.

A continuous function changes smoothly: sufficiently small changes in its input produce sufficiently small changes in its output. In a variational autoencoder (VAE), continuity makes nearby latent points decode into similar images. section 17.1.2

Continuity gives a VAE latent space two practical properties:

• Nearby points represent related outputs rather than isolated examples. section 17.1.2
• Together with low dimensionality, it makes latent directions meaningful and supports manipulation through concept vectors. section 17.1.2

Sampling a continuous latent neighborhood

The encoder maps an input image to z_mean and z_log_var. A random sample z = z_mean + exp(0.5 * z_log_var) * epsilon is decoded back into an image. Because sampling varies around z_mean, nearby latent points can produce images similar to the input. section 17.1.2

Figure 17.4 A VAE maps an image to two vectors, z_mean and z_log_var, which define a probability distribution over the latent space, used to sample a latent point to decode.

Use the diagram to trace how an image becomes a probability distribution over latent points before decoding. section 17.1.2

How Deep Learning with R, Third Edition covers it

How deep it goes

Defines continuity intuitively and connects it to latent sampling, stopping short of formal mathematical conditions.

Examples

Uses random latent sampling around an encoded image and shows the encoder-to-decoder transformation.

What you can do after

Evaluate why nearby latent points should decode to similar images in a variational autoencoder.

How it connects

Connects continuity with variational autoencoders, latent spaces, decoders, reconstruction loss, regularization loss, and Kullback–Leibler divergence.

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Also explained in

• 

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  Geometry for ProgrammersContinuity is the property of a function having no excluded domain points within its intended domain and no abrupt changes in value.

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This explainer was generated by Manning's knowledge graph from the text of Deep Learning with R, Third Edition. The section links point to the parts of the book it draws on.