Fisher Flow Matching for Generative Modeling over Discrete Data

• URL: http://arxiv.org/abs/2405.14664v4
• Date: Wed, 30 Oct 2024 11:01:10 GMT
• Title: Fisher Flow Matching for Generative Modeling over Discrete Data
• Authors: Oscar Davis, Samuel Kessler, Mircea Petrache, İsmail İlkan Ceylan, Michael Bronstein, Avishek Joey Bose,
• Abstract summary: We introduce Fisher-Flow, a novel flow-matching model for discrete data. Fisher-Flow takes a manifestly geometric perspective by considering categorical distributions over discrete data. We prove that the gradient flow induced by Fisher-Flow is optimal in reducing the forward KL divergence.
• Score: 12.69975914345141
• License:
• Abstract: Generative modeling over discrete data has recently seen numerous success stories, with applications spanning language modeling, biological sequence design, and graph-structured molecular data. The predominant generative modeling paradigm for discrete data is still autoregressive, with more recent alternatives based on diffusion or flow-matching falling short of their impressive performance in continuous data settings, such as image or video generation. In this work, we introduce Fisher-Flow, a novel flow-matching model for discrete data. Fisher-Flow takes a manifestly geometric perspective by considering categorical distributions over discrete data as points residing on a statistical manifold equipped with its natural Riemannian metric: the $\textit{Fisher-Rao metric}$. As a result, we demonstrate discrete data itself can be continuously reparameterised to points on the positive orthant of the $d$-hypersphere $\mathbb{S}^d_+$, which allows us to define flows that map any source distribution to target in a principled manner by transporting mass along (closed-form) geodesics of $\mathbb{S}^d_+$. Furthermore, the learned flows in Fisher-Flow can be further bootstrapped by leveraging Riemannian optimal transport leading to improved training dynamics. We prove that the gradient flow induced by Fisher-Flow is optimal in reducing the forward KL divergence. We evaluate Fisher-Flow on an array of synthetic and diverse real-world benchmarks, including designing DNA Promoter, and DNA Enhancer sequences. Empirically, we find that Fisher-Flow improves over prior diffusion and flow-matching models on these benchmarks.

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