On Representing (Anti)Symmetric Functions

07/30/2020
by   Marcus Hutter, et al.
13

Permutation-invariant, -equivariant, and -covariant functions and anti-symmetric functions are important in quantum physics, computer vision, and other disciplines. Applications often require most or all of the following properties: (a) a large class of such functions can be approximated, e.g. all continuous function, (b) only the (anti)symmetric functions can be represented, (c) a fast algorithm for computing the approximation, (d) the representation itself is continuous or differentiable, (e) the architecture is suitable for learning the function from data. (Anti)symmetric neural networks have recently been developed and applied with great success. A few theoretical approximation results have been proven, but many questions are still open, especially for particles in more than one dimension and the anti-symmetric case, which this work focusses on. More concretely, we derive natural polynomial approximations in the symmetric case, and approximations based on a single generalized Slater determinant in the anti-symmetric case. Unlike some previous super-exponential and discontinuous approximations, these seem a more promising basis for future tighter bounds. We provide a complete and explicit universality proof of the Equivariant MultiLayer Perceptron, which implies universality of symmetric MLPs and the FermiNet.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/04/2019

Universal approximation of symmetric and anti-symmetric functions

We consider universal approximations of symmetric and anti-symmetric fun...
research
03/22/2023

Anti-symmetric Barron functions and their approximation with sums of determinants

A fundamental problem in quantum physics is to encode functions that are...
research
07/13/2019

ND-Wavelets Derived from Anti-symmetric Systems of Isolated Particles using the Determinant of Slater

Wavelets are known to be closely related to atomic orbital. A new approa...
research
10/10/2015

Optimal Piecewise Linear Function Approximation for GPU-based Applications

Many computer vision and human-computer interaction applications develop...
research
06/19/2015

Spectral Analysis of Symmetric and Anti-Symmetric Pairwise Kernels

We consider the problem of learning regression functions from pairwise d...
research
06/02/2022

Exponential Separations in Symmetric Neural Networks

In this work we demonstrate a novel separation between symmetric neural ...
research
08/05/2023

Kalai's 3^d-conjecture for unconditional and locally anti-blocking polytopes

Kalai's 3^d-conjecture states that every centrally symmetric d-polytope ...

Please sign up or login with your details

Forgot password? Click here to reset