发表机构
University of Liverpool（利物浦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对信号解混中基变化问题，提出基于度量空间和最优传输距离的几何方法，通过最小化类内方差优化，在一维及多种数据集上验证有效性。
AI 中文摘要
信号分离问题在科学中很常见。一个突出的例子发生在使用衍射或光谱学通过测量来识别混合物中各个组分时。在最简单的情况下，测量信号是各组成部分对应的基模式的线性组合。解混问题是从不同混合物的测量中推断出全部或部分基模式以及组分的丰度。该任务的核心挑战之一是基在混合物之间因噪声和测量过程的精确物理机制而变化。这通常通过定制的基于模型和参数化的方法来解决，这些方法由于所做假设的性质而局限于特定应用领域。我们提出了一种新颖的几何方法来解决解混问题，该方法通过度量空间的视角来看待测量过程中的数据生成，从而将焦点从参数化模型转移到基变换与相应几何之间的一般关系上。我们利用最优传输距离来捕获常见的基变化，并使用候选解的类内方差最小化来驱动优化。我们特别关注一维情况，因为其实际重要性以及高效的距离和传输图例程的可用性。我们方法的有效性通过一系列解混任务得到证明，包括随机高斯混合模型、模拟粉末X射线衍射和实验室高光谱成像数据集。
英文摘要
Signal separation problems are common in science. A prominent example of this occurs during the use of diffraction or spectroscopy to identify the individual components of a mixture by measuring it. In the simplest case, the measured signal is a linear combination of basis patterns corresponding to the constituent parts. The unmixing problem is to infer all or some of these basis patterns and abundances of components from measurements of distinct mixtures. One of the core challenges of this task is the variation of the basis from mixture to mixture due to noise and the exact physics of the measurement process. This is usually addressed with tailored model-based and parametric methods that are then limited in use to specific application domains by the nature of the assumptions made. We propose a novel geometric approach to unmixing problems which views the generation of data during measurement through a metric space lens, thereby shifting the focus from parametrised models to a general relationship between basis transformations and the corresponding geometry. We take advantage of the optimal transport distances to capture commonly occurring basis variations, and use minimisation of in-class variance of candidate solutions to drive the optimisation. We pay special attention to the one-dimensional case due to its practical importance and availability of efficient distance and transport map routines. The effectiveness of our approach is demonstrated on a range of unmixing tasks using random Gaussian mixture models, simulated powder X-ray diffraction, and laboratory hyperspectral imaging datasets.
Comments40 pages, 16 figures, 5 tables
