Proposition 2. (From main text) The Bayes error of flow models is monotonically increasing in . That is, for 0 < 0, we have that EBayes(ˆp) EBayes(ˆp 0). B.1 Hardness of Classes In addition to measuring the difficulty of classification tasks relative to one another, it also may be of interest to evaluate the relative difficulty of individual classes within a particular task. A natural way to do this is by looking at the error of one-vs-all classification tasks. The optimal Bayes classifier in this task is CBayes(x)= 0 if logpj(x) logp j(x), 1 otherwise .

Evaluating the inherent difficulty of a given data-driven classification problem is important for establishing absolute benchmarks and evaluating progress in the field. To this end, a natural quantity to consider is the Bayes error, which measures the optimal classification error theoretically achievable for a given data distribution. While generally an intractable quantity, we show that we can compute the exact Bayes error of generative models learned using normalizing flows. Our technique relies on a fundamental result, which states that the Bayes error is invariant under invertible transformation. Therefore, we can compute the exact Bayes error of the learned flow models by computing it for Gaussian base distributions, which can be done efficiently using Holmes-Diaconis-Ross integration. Moreover, we show that by varying the temperature of the learned flow models, we can generate synthetic datasets that closely resemble standard benchmark datasets, but with almost any desired Bayes error. We use our approach to conduct a thorough investigation of state-of-the-art classification models, and find that in some -- but not all -- cases, these models are capable of obtaining accuracy very near optimal. Finally, we use our method to evaluate the intrinsic "hardness" of standard benchmark datasets.

This paper presents a simple, self-supervised method for magnifying subtle motions in video: given an input video and a magnification factor, we manipulate the video such that its new optical flow is scaled by the desired amount. To train our model, we propose a loss function that estimates the optical flow of the generated video and penalizes how far if deviates from the given magnification factor. Thus, training involves differentiating through a pretrained optical flow network. Since our model is self-supervised, we can further improve its performance through test-time adaptation, by finetuning it on the input video. It can also be easily extended to magnify the motions of only user-selected objects. Our approach avoids the need for synthetic magnification datasets that have been used to train prior learning-based approaches.

Despite Flow Matching and diffusion models having emerged as powerful generative paradigms for continuous variables such as images and videos, their application to high-dimensional discrete data, such as language, is still limited. In this work, we present Discrete Flow Matching, a novel discrete flow paradigm designed specifically for generating discrete data. Discrete Flow Matching offers several key contributions: (i) it works with a general family of probability paths interpolating between source and target distributions; (ii) it allows for a generic formula for sampling from these probability paths using learned posteriors such as the probability denoiser ($x$-prediction) and noise-prediction ($\epsilon$-prediction); (iii) practically, focusing on specific probability paths defined with different schedulers improves generative perplexity compared to previous discrete diffusion and flow models; and (iv) by scaling Discrete Flow Matching models up to 1.7B parameters, we reach 6.7% Pass@1 and 13.4% Pass@10 on HumanEval and 6.7% Pass@1 and 20.6% Pass@10 on 1-shot MBPP coding benchmarks. Our approach is capable of generating high-quality discrete data in a non-autoregressive fashion, significantly closing the gap between autoregressive models and discrete flow models.

Second, learning the flow model requires sampling from the feasible action space, which is also challenging.

Molecular generation (Stokes et al., 2020) has