TRAINING INVERTIBLE LINEAR LAYERS THROUGH RANK-ONE PERTURBATIONS

Abstract

Many types of neural network layers rely on matrix properties such as invertibility or orthogonality. Retaining such properties during optimization with gradientbased stochastic optimizers is a challenging task, which is usually addressed by either reparameterization of the affected parameters or by directly optimizing on the manifold. This work presents a novel approach for training invertible linear layers. In lieu of directly optimizing the network parameters, we train rank-one perturbations and add them to the actual weight matrices infrequently. This P 4 Inv update allows keeping track of inverses and determinants without ever explicitly computing them. We show how such invertible blocks improve the mixing and thus the mode separation of the resulting normalizing flows. Furthermore, we outline how the P 4 concept can be utilized to retain properties other than invertibility.

1. INTRODUCTION

The present work introduces a novel algorithmic concept for training invertible linear layers and facilitate tractable inversion and determinant computation, see Figure 1 . In lieu of directly changing the network parameters, the optimizer operates on perturbations to these parameters. The actual network parameters are frozen, while a parameterized perturbation (a rank-one update to the frozen parameters) serves as a proxy for optimization. Inputs are passed through the perturbed network



Figure 1: Training of deep neural networks (DNN). Standard DNN transform inputs x into outputs y through activation functions and linear layers, which are tuned by an optimizer. In contrast, P 4 training operates on perturbations to the parameters. Those are defined to retain certain network properties (here: invertibility as well as tractable inversion and determinant computation). The perturbed parameters are merged in regular intervals.

