Experimental Analysis of Legendre Decomposition in Machine Learning

TL;DR

This paper investigates Legendre decomposition for non-negative tensors, analyzing its theoretical properties and practical applications, including experiments on tensor projection, clustering, and potential connections to neural networks.

Contribution

It provides a comprehensive theoretical analysis of Legendre decomposition and explores its practical use in tensor representation and clustering, highlighting its limitations and future prospects.

Findings

Parameters on submanifold cannot be directly used as low-rank representations

Legendre decomposition's properties are linked to neural networks and low-rank applications

Experimental verification shows limited effectiveness of submanifold parameters for low-rank tensor approximation

Abstract

In this technical report, we analyze Legendre decomposition for non-negative tensor in theory and application. In theory, the properties of dual parameters and dually flat manifold in Legendre decomposition are reviewed, and the process of tensor projection and parameter updating is analyzed. In application, a series of verification experiments and clustering experiments with parameters on submanifold were carried out, hoping to find an effective lower dimensional representation of the input tensor. The experimental results show that the parameters on submanifold have no ability to be directly used as low-rank representations. Combined with analysis, we connect Legendre decomposition with neural networks and low-rank representation applications, and put forward some promising prospects.