Trainable Joint Time-Vertex Fractional Fourier Transform

TL;DR

This paper introduces a trainable joint time-vertex fractional Fourier transform framework that adaptively learns optimal transform parameters for improved denoising of time-varying graph signals, reducing computational costs.

Contribution

It proposes a hyper-differential form of the JFRFT with differentiable transform orders, enabling neural network-based adaptive learning of transform parameters.

Findings

Enhanced denoising performance on time-varying graph signals.

Reduced computational burden compared to grid search methods.

Demonstrated effectiveness through experimental validation.

Abstract

To address limitations of the graph fractional Fourier transform (GFRFT) Wiener filtering and the traditional joint time-vertex fractional Fourier transform (JFRFT) Wiener filtering, this study proposes a filtering method based on the hyper-differential form of the JFRFT. The gradient backpropagation mechanism is employed to enable the adaptive selection of transform order pair and filter coefficients. First, leveraging the hyper-differential form of the GFRFT and the fractional Fourier transform, the hyper-differential form of the JFRFT is constructed and its properties are analyzed. Second, time-varying graph signals are divided into dynamic graph sequences of equal span along the temporal dimension. A spatiotemporal joint representation is then established through vectorized reorganization, followed by the joint time-vertex Wiener filtering. Furthermore, by rigorously proving the differentiability of the transform orders, both the transform orders and filter coefficients are embedded as learnable parameters within a neural network architecture. Through gradient backpropagation, their synchronized iterative optimization is achieved, constructing a parameters-adaptive learning filtering framework. This method leverages a model-driven approach to learn the optimal transform order pair and filter coefficients. Experimental results indicate that the proposed framework improves the time-varying graph signals denoising performance, while reducing the computational burden of the traditional grid search strategy.