Beyond the Manifold Hypothesis: Hybrid Spectral Parameterizations for Flow Matching
cs.LG
Submitted: 2026-09-26
Updated: 2026-09-26
License: http://creativecommons.org/licenses/by/4.0/
The gist: 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.
Terminology
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 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.
Sources
- Balancing Frequencies and Pixels in Flow Matching
- Spectrally-Guided Diffusion Noise Schedules
- A Fourier Space Perspective on Diffusion Models
- Where the Score Lives: A Wavelet View of Diffusion
- Equivalent Flows, Unequal Learning: Clean-Latent Prediction in Transformers
- Revisiting Diffusion Model Predictions Through Dimensionality
- Better Source, Better Flow: Learning Condition-Dependent Source Distribution for Flow Matching
- LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop
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