# Risk-Entropic Flow Matching

arXiv: https://arxiv.org/abs/2512.03078 · HTML: https://mlsanity.com/paper/2512.03078/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 12 of its 19 references that are in the map.
- First posted on arXiv 28 Nov 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 12 of 12)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Tilted Empirical Risk Minimization](https://mlsanity.com/paper/2007.01162/) (2020)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [Adaptive Sampling for Stochastic Risk-Averse Learning](https://mlsanity.com/paper/1910.12511/) (2019)
- [Variational Rectified Flow Matching](https://mlsanity.com/paper/2502.09616/) (2025)
- [Learning Bounds for Risk-sensitive Learning](https://mlsanity.com/paper/2006.08138/) (2020)
- [Flow matching achieves almost minimax optimal convergence](https://mlsanity.com/paper/2405.20879/) (2024)
- [A Tunable Loss Function for Robust Classification: Calibration, Landscape, and Generalization](https://mlsanity.com/paper/1906.02314/) (2019)
- [Towards Hierarchical Rectified Flow](https://mlsanity.com/paper/2502.17436/) (2025)
- [Supervised Learning with General Risk Functionals](https://mlsanity.com/paper/2206.13648/) (2022)
- [Generalization and Robustness of the Tilted Empirical Risk](https://mlsanity.com/paper/2409.19431/) (2024)

## Abstract (arXiv)

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.