Risk-Entropic Flow Matching

• URL: http://arxiv.org/abs/2512.03078v1
• Date: Fri, 28 Nov 2025 00:37:37 GMT
• Title: Risk-Entropic Flow Matching
• Authors: Vahid R. Ramezani, Benjamin Englard,
• Abstract summary: Tilted (entropic) risk, obtained by applying a log-exponential transform to a base loss, is a well established tool in statistics and machine learning.<n>We apply the standard risk-sensitive (log-exponential) transform to the conditional FM loss and show that the resulting tilted risk loss is a natural upper-bound on a meaningful conditional entropic FM objective.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Tilted (entropic) risk, obtained by applying a log-exponential transform to a base loss, is a well established tool in statistics and machine learning for emphasizing rare or high loss events while retaining a tractable optimization problem. In this work, our aim is to interpret its structure for Flow Matching (FM). FM learns a velocity field that transports samples from a simple source distribution to data by integrating an ODE. In rectified FM, training pairs are obtained by linearly interpolating between a source sample and a data sample, and a neural velocity field is trained to predict the straight line displacement using a mean squared error loss. This squared loss collapses all velocity targets that reach the same space-time point into a single conditional mean, thereby ignoring higher order conditional information (variance, skewness, multi-modality) that encodes fine geometric structure about the data manifold and minority branches. We apply the standard risk-sensitive (log-exponential) transform to the conditional FM loss and show that the resulting tilted risk loss is a natural upper-bound on a meaningful conditional entropic FM objective defined at each space-time point. Furthermore, we show that a small order expansion of the gradient of this conditional entropic objective yields two interpretable first order corrections: covariance preconditioning of the FM residual, and a skew tail term that favors asymmetric or rare branches. On synthetic data designed to probe ambiguity and tails, the resulting risk-sensitive loss improves statistical metrics and recovers geometric structure more faithfully than standard rectified FM.

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