A Tensor Decomposition Perspective on Second-order RNNs

TL;DR

This paper explores tensor decomposition in second-order RNNs, specifically CPRNNs, analyzing their capacity and demonstrating that they outperform other models under fixed parameter constraints on the Penn Treebank dataset.

Contribution

It introduces the CPRNN model using CP decomposition for second-order RNNs and analyzes how rank and hidden size influence model capacity and performance.

Findings

CPRNNs outperform RNNs, 2RNNs, and MIRNNs with fixed parameters.

Model capacity depends on rank and hidden size.

Empirical results on Penn Treebank support theoretical analysis.

Abstract

Second-order Recurrent Neural Networks (2RNNs) extend RNNs by leveraging second-order interactions for sequence modelling. These models are provably more expressive than their first-order counterparts and have connections to well-studied models from formal language theory. However, their large parameter tensor makes computations intractable. To circumvent this issue, one approach known as MIRNN consists in limiting the type of interactions used by the model. Another is to leverage tensor decomposition to diminish the parameter count. In this work, we study the model resulting from parameterizing 2RNNs using the CP decomposition, which we call CPRNN. Intuitively, the rank of the decomposition should reduce expressivity. We analyze how rank and hidden size affect model capacity and show the relationships between RNNs, 2RNNs, MIRNNs, and CPRNNs based on these parameters. We support these results empirically with experiments on the Penn Treebank dataset which demonstrate that, with a fixed parameter budget, CPRNNs outperforms RNNs, 2RNNs, and MIRNNs with the right choice of rank and hidden size.