Beyond the Manifold Hypothesis: Hybrid Spectral Parameterizations for Flow Matching

2026-09-26

AI Summary

This study addresses the significant performance divergence among different prediction targets in flow matching, which arises from signal-to-noise ratio (SNR) discrepancies and information bottlenecks. By analyzing the SNR across individual directions of the data covariance, this work proposes a spectrum-mixing parameterization method that adapts to both time and direction. Departing from conventional manifold assumptions, it reveals that optimal parameterization depends on the directional SNR distribution rather than solely on intrinsic dimensionality. The theoretical optimality of this approach is rigorously proven on Gaussian data. Furthermore, the proposed method substantially accelerates optimization convergence without introducing additional training overhead while demonstrating strong robustness, thereby offering a novel theoretical perspective and practical framework for parameterization design in generative models.

Abstract

Flow matching and diffusion can be trained to predict different quantities, most commonly the data $x_1$, the source noise $x_0$, or the velocity~$v$. Although theoretically equivalent, these can lead to substantially different performances. We identify two main drivers for these differences: the source--data signal-to-noise ratio, and the information bottleneck induced by the neural architecture. We show that, beyond intrinsic data dimension, the factor affecting the optimal parametrization the most is a certain signal-to-noise ratio in each data covariance direction. From this analysis, we introduce new \emph{spectral hybrid} parameterizations that adapt across time and data covariance directions; we show that these are optimal for Gaussian data. We also show that architecture-induced compression changes which parameterization is easier to learn, with $v$-prediction being more sensitive to discarded directions than $x_1$-prediction. Experiments across architectures and source scales show that our spectral parameterizations are robust across regimes, can substantially accelerate optimization, while incurring essentially no additional training cost compared with standard parameterizations.

Problem

• Flow Matching
• Diffusion Models
• Parameterization
• Signal-to-Noise Ratio
• Information Bottleneck

Innovation

• Flow Matching
• Spectral Hybrid Parameterization
• Information Bottleneck
• Signal-to-Noise Ratio
• Diffusion Models