Flow-based generative model

A flow-based generative model is a generative model used in machine learning that explicitly models a probability distribution by leveraging normalizing flow^[1], which is a statistical method using the change-of-variable law of probabilities to transform a simple distribution into a complex one, which is usually the distribution (i.e. likelihood function) of observed data, ${\displaystyle p(x)}$.

The direct modeling of likelihood provides many advantages. For example, the negative log-likelihood can be directly computed and minimized as the loss function. Additionally, novel samples can be generated by sampling from the initial distribution, and applying the flow transformation.

In contrast, many alternative generative modeling methods such as variational autoencoder (VAE) and generative adversarial network do not explicitly represent the likelihood function.

Method

Let ${\displaystyle z_{0}}$  be a (possibly multivariate) random variable with distribution ${\displaystyle p_{0}(z_{0})}$ .

For ${\displaystyle i=1,...,K}$ , let ${\displaystyle z_{i}=f_{i}(z_{i-1})}$  be a sequence of random variables transformed from ${\displaystyle z_{0}}$ . The functions ${\displaystyle f_{1},...,f_{K}}$  should be invertible, i.e. the inverse function ${\displaystyle f_{i}^{-1}}$  exists. The final output ${\displaystyle z_{K}}$  models the target distribution.

The log likelihood of ${\displaystyle z_{K}}$  is (see derivation):

${\displaystyle \log p_{K}(z_{K})=\log p_{0}(z_{0})-\sum _{i=1}^{K}\log \left|\det {\frac {df_{i}(z_{i-1})}{dz_{i-1}}}\right|}$ 

To efficiently compute the log likelihood, the functions ${\displaystyle f_{1},...,f_{K}}$  should be 1. easy to invert, and 2. easy to compute the determinant of its Jacobian. In practice, the functions ${\displaystyle f_{1},...,f_{K}}$  are modeled using deep neural networks, and are trained to minimize the negative log-likelihood given data samples from the target distribution.

Derivation of log likelihood

Consider ${\displaystyle z_{1}}$  and ${\displaystyle z_{0}}$ . Note that ${\displaystyle z_{0}=f_{1}^{-1}(z_{1})}$ .

By the change of variable formula, the distribution of ${\displaystyle z_{1}}$  is:

${\displaystyle p_{1}(z_{1})=p_{0}(z_{0})\left|\det {\frac {df_{1}^{-1}(z_{1})}{dz_{1}}}\right|}$ 

Where ${\displaystyle \det {\frac {df_{1}^{-1}(z_{1})}{dz_{1}}}}$  is the determinant of the Jacobian matrix of ${\displaystyle f_{1}^{-1}}$ .

By the inverse function theorem:

${\displaystyle p_{1}(z_{1})=p_{0}(z_{0})\left|\det \left({\frac {df_{1}(z_{0})}{dz_{0}}}\right)^{-1}\right|}$ 

By the identity ${\displaystyle \det(A^{-1})=\det(A)^{-1}}$  (where ${\displaystyle A}$  is an invertible matrix), we have:

${\displaystyle p_{1}(z_{1})=p_{0}(z_{0})\left|\det {\frac {df_{1}(z_{0})}{dz_{0}}}\right|^{-1}}$ 

The log likelihood is thus:

${\displaystyle \log p_{1}(z_{1})=\log p_{0}(z_{0})-\log \left|\det {\frac {df_{1}(z_{0})}{dz_{0}}}\right|}$ 

In general, the above applies to any ${\displaystyle z_{i}}$  and ${\displaystyle z_{i-1}}$ . Since ${\displaystyle \log p_{i}(z_{i})}$  is equal to ${\displaystyle \log p_{i-1}(z_{i-1})}$  subtracted by a non-recursive term, we can infer by induction that:

${\displaystyle \log p_{K}(z_{K})=\log p_{0}(z_{0})-\sum _{i=1}^{K}\log \left|\det {\frac {df_{i}(z_{i-1})}{dz_{i-1}}}\right|}$ 

Examples

TODO description

• RealNVP
• TODO more items, and citation

Applications

TODO description

• Point-cloud modeling
• Music generation
• TODO more items, and citation

References

1. . arXiv:1505.05770. {{cite arXiv}}: Missing or empty |title= (help) A bot will complete this citation soon. Click here to jump the queue

External links

• Flow-based Deep Generative Models

Category:Machine learning Category:Statistical models Category:Probabilistic models