Neural Spline Flows (1906.04032v2)

Abstract: A normalizing flow models a complex probability density as an invertible transformation of a simple base density. Flows based on either coupling or autoregressive transforms both offer exact density evaluation and sampling, but rely on the parameterization of an easily invertible elementwise transformation, whose choice determines the flexibility of these models. Building upon recent work, we propose a fully-differentiable module based on monotonic rational-quadratic splines, which enhances the flexibility of both coupling and autoregressive transforms while retaining analytic invertibility. We demonstrate that neural spline flows improve density estimation, variational inference, and generative modeling of images.

• The paper introduces monotonic rational-quadratic spline transformations to enhance normalizing flows with improved flexibility and analytic invertibility.
• The methodology replaces conventional affine transformations with fully differentiable splines that ensure continuous inversion over segmented intervals.
• Empirical results demonstrate state-of-the-art performance on density estimation benchmarks, validating the approach in both coupling and autoregressive layers.

Neural Spline Flows

The paper "Neural Spline Flows" presents an advanced approach to normalizing flows by introducing a fully differentiable module based on monotonic rational-quadratic splines. This enhancement increases the flexibility of normalizing flows in both coupling and autoregressive transformations, while still maintaining their key property of analytic invertibility.

Introduction

Normalizing flows serve as a robust framework in unsupervised machine learning, allowing for complex density estimation via invertible transformations. They are distinct from other generative models such as VAEs and GANs, as they provide exact density evaluation and sample generation in a single pass. The choice of invertible elementwise transformations is crucial for the flexibility and efficacy of these models.

Methodology

The innovation of the paper is the integration of monotonic rational-quadratic splines into the flow architecture. This transformation acts as a drop-in replacement for traditional affine or additive transformations in coupling and autoregressive layers, significantly enhancing model flexibility. The transformation is defined on an input interval with boundary derivatives matched to ensure continuity. It entails the following steps:

1. Dividing the input space into bins.
2. Constructing a spline defined by rational-quadratic segments.
3. Ensuring invertibility through analytic manipulations inherent to the rational-quadratic formulation.

Results

The numerical results from various datasets demonstrate the superior performance of neural spline flows. For instance:

• On tabular datasets such as Power, Gas, and Hepmass, the method achieves state-of-the-art performance in density estimation.
• Within the field of variational autoencoders, the proposed flows show meaningful improvements in evidence lower bounds and log-likelihood estimates.

Importantly, the use of neural spline flows enables models based on coupling layers to rival, and in some cases surpass, the performance of autoregressive flows, with the added benefit of retaining exact sampling capabilities.

Implications and Future Work

The implications of this work are significant for both practical and theoretical domains. The introduction of monotonic splines offers a method to construct highly flexible and invertible transformations with tractable Jacobians. This advance paves the way for more expressive modeling of complex, high-dimensional data.

Future research may explore extending this framework into domains that require fully differentiable models with efficient training dynamics. It also opens doors for investigating other forms of invertible transformations, potentially leading to new architectures that balance complexity with computational efficiency.

In summary, Neural Spline Flows present a pivotal step forward in the development of expressive, tractable models for unsupervised learning tasks. The modular nature of splines, combined with their exact invertibility, marks a notable stride in advancing normalizing flows towards greater flexibility and applicability.