Lie Group Auto-Encoder

TL;DR

This paper introduces a Lie group-based auto-encoder that encodes data into tangent space representations of Gaussian distributions on a Lie group manifold, enhancing generative modeling by leveraging geometric properties.

Contribution

It proposes a novel auto-encoder model using Lie group theory to encode Gaussian distributions, including an exponential mapping layer and intrinsic loss for improved generative performance.

Findings

Effective generation of Gaussian samples from Lie group representations

Enhanced encoding by using Lie algebra vectors instead of raw parameters

Experimental validation shows improved generative quality

Abstract

In this paper, we propose an auto-encoder based generative neural network model whose encoder compresses the inputs into vectors in the tangent space of a special Lie group manifold: upper triangular positive definite affine transform matrices (UTDATs). UTDATs are representations of Gaussian distributions and can straightforwardly generate Gaussian distributed samples. Therefore, the encoder is trained together with a decoder (generator) which takes Gaussian distributed latent vectors as input. Compared with related generative models such as variational auto-encoder, the proposed model incorporates the information on geometric properties of Gaussian distributions. As a special case, we derive an exponential mapping layer for diagonal Gaussian UTDATs which eliminates matrix exponential operator compared with general exponential mapping in Lie group theory. Moreover, we derive an intrinsic loss for UTDAT Lie group which can be calculated as l-2 loss in the tangent space. Furthermore, inspired by the Lie group theory, we propose to use the Lie algebra vectors rather than the raw parameters (e.g. mean) of Gaussian distributions as compressed representations of original inputs. Experimental results verity the effectiveness of the proposed new generative model and the benefits gained from the Lie group structural information of UTDATs.