WaveletDiff: Multilevel Wavelet Diffusion For Time Series Generation

Abstract

Time series are ubiquitous in many applications that involve forecasting, classification and causal inference tasks, such as healthcare, finance, audio signal processing and climate sciences. Still, large, high-quality time series datasets remain scarce. Synthetic generation can address this limitation; however, current models confined either to the time or frequency domains struggle to reproduce the inherently multi-scaled structure of real-world time series. We introduce WaveletDiff, a new framework that trains diffusion models directly on wavelet coefficients to exploit the inherent multi-resolution structure of time series data. The model combines dedicated transformers for each decomposition level with cross-level attention mechanisms that enable selective information exchange between temporal and frequency scales through adaptive gating. It is also informed by level-specific energy constraints based on Parseval's theorem which preserve time-frequency properties throughout the diffusion process. Comprehensive tests across six real-world datasets from energy, finance, and neuroscience domains demonstrate that WaveletDiff outperforms the diffusion baselines FourierDiffusion, Diffusion-TS, and SigDiffusions on the majority of metrics, with the smallest margin over FourierDiffusion, while still achieving roughly 3x lower discriminative and Context-FID scores. Against the VAE/transformer-based MSDformer, the results are mostly comparable, with WaveletDiff using fewer parameters and less training time on most datasets. The most revealing finding is the significant performance gap on fMRI data (in favor of MSDformer) and EEG (in favor of WaveletDiff). This finding is explained via a careful testing/examination of the properties of wavelet coefficients for generative, as opposed to analyses/decomposition tasks. Our code is available at https://github.com/GarlicWang/WaveletDiff.