Research map / Learning theory and algorithms / Optimal transport
A Universal Approximation Theorem of Deep Neural Networks for Expressing Probability Distributions
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Represent the generator as the gradient of a ReLU DNN potential | Source-to-target distribution approximation · General target distributions on R^d | W1 approximation and network size: n≤ C/ε^2 (d=1); C log^2 (ε)/ε^2 (d=2); C^d/ε^d (d≥3) | Mixed |
| Represent the generator as the gradient of a ReLU DNN potential | Source-to-target distribution approximation · General target distributions on R^d | MMD approximation and network size: n ≤ C/ε^2 | Helped |
| Represent the generator as the gradient of a ReLU DNN potential | Source-to-target distribution approximation · General target distributions on R^d | KSD approximation and network size: n ≤ Cd/ε^2 | Helped |
| Use empirical measures as intermediate targets and semi-discrete optimal transport to map the source to them | ReLU gradient-potential DNN · Discrete target distribution with n support points | exact transport representation: depth L=⌈log2 n⌉ and width N = 2^L = 2^⌈log2 n⌉ | Helped |
| Parameterize the transport potential rather than the transport map directly | Generative neural networks · Targets with disjoint modes or supports | ability to represent discontinuous transport maps | Helped |
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