通过条件最优传输学习充分低维结构
Learning sufficient low-dimensional structures through conditional optimal transport
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中文总结 AI 辅助
研究通过条件最优传输学习充分低维结构，引入SDR - COT方法，证明线性降维一致性，用不同方式处理欧几里得和希尔伯特值响应，数值研究显示该方法在特定情况下有竞争力。
中文摘要 AI 辅助
充分降维旨在寻找一个保留响应条件律的低维协变量表示。我们引入了SDR - COT，它通过来自独立参考响应的条件最优传输来表示该定律。在可分希尔伯特空间上，充分性迫使最优三角映射的响应分量通过降维进行分解。对于二次成本，诱导插值在每个截断时间间隔上具有博雷尔流状态速度，终端映射无需全局单射性，且该速度具有相同分解。这些结果激发了一个条件流匹配准则。对于线性降维，我们使用适当调整的松弛经验耦合证明了一致性。欧几里得响应通过逐片Caffarelli界处理；希尔伯特值响应通过高斯索伯列夫正则性、插值压缩和高斯连续性方程的唯一性处理。对欧几里得和函数数据的数值研究显示了具有竞争力的性能，特别是当充分信息不仅仅包含在条件均值中时。
英文摘要
Sufficient dimension reduction seeks a low-dimensional covariate representation that preserves the conditional law of a response. We introduce SDR-COT, which represents that law by conditional optimal transport from an independent reference response. On separable Hilbert spaces, sufficiency forces the response component of the optimal triangular map to factor through the reduction. For quadratic cost, the induced interpolation has a Borel current-state velocity on every truncated time interval, without global injectivity of the terminal map, and this velocity has the same factorisation. These results motivate a conditional-flow-matching criterion. For linear reductions, we prove consistency using a suitably tuned relaxed empirical coupling. Euclidean responses are treated through slicewise Caffarelli bounds; Hilbert-valued responses are treated through Gaussian Sobolev regularity, interpolation compression and uniqueness of a Gaussian continuity equation. Numerical studies with Euclidean and functional data show competitive performance, especially when sufficient information is not solely contained in the conditional mean.
