将薛定谔桥拓展至离散数据，解决图像编码等生成问题
Categorical Schrödinger Bridge Matching
- 基于迭代马尔可夫拟合，构建离散空间下的薛定谔桥算法
- 理论证明离散版算法收敛，实测在图像码本上效果优异
- 适合做向量量化表示、文本生成等离散数据建模任务
薛定谔桥（SB）是解决无配对域转换等生成建模任务的强大框架，但现有研究主要针对连续空间 ℝᴰ，对离散空间（如有限集 ℤᴰ）的理论与算法仍不完善。典型例子包括现代自编码器的向量量化（VQ）码本、文本中的词元、分子中原子的类别等。本文基于最近提出的迭代马尔可夫拟合（IMF）方法，为离散空间中的薛定谔桥提供了理论与算法基础。我们严格证明了离散时间迭代马尔可夫拟合（D-IMF）在离散空间中收敛到薛定谔桥。据此提出实用算法——分类薛定谔桥匹配（CSBM）。通过合成数据与图像的VQ表示实验验证了其有效性。代码已开源：https://github.com/gregkseno/csbm。
原文摘要 · Abstract (English)
The Schrödinger Bridge (SB) is a powerful framework for solving generative modeling tasks such as unpaired domain translation. Most SB-related research focuses on continuous data space $\mathbb{R}^{D}$ and leaves open theoretical and algorithmic questions about applying SB methods to discrete data, e.g, on finite spaces $\mathbb{S}^{D}$. Notable examples of such sets $\mathbb{S}$ are codebooks of vector-quantized (VQ) representations of modern autoencoders, tokens in texts, categories of atoms in molecules, etc. In this paper, we provide a theoretical and algorithmic foundation for solving SB in discrete spaces using the recently introduced Iterative Markovian Fitting (IMF) procedure. Specifically, we theoretically justify the convergence of discrete-time IMF (D-IMF) to SB in discrete spaces. This enables us to develop a practical computational algorithm for SB, which we call Categorical Schrödinger Bridge Matching (CSBM). We show the performance of CSBM via a series of experiments with synthetic data and VQ representations of images. The code of CSBM is available at https://github.com/gregkseno/csbm.
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