Hao Wu

Hao Wu, Raj Rao Nadakuditi

We describe a method for unmixing mixtures of freely independent random variables in a manner analogous to the independent component analysis (ICA) based method for unmixing independent random variables from their additive mixtures. Random matrices play the role of free random variables in this context so the method we develop, which we call Free component analysis (FCA), unmixes matrices from additive mixtures of matrices. Thus, while the mixing model is standard, the novelty and difference in unmixing performance comes from the introduction of a new statistical criteria, derived from free probability theory, that quantify freeness analogous to how kurtosis and entropy quantify independence. We describe the theory, the various algorithms, and compare FCA to vanilla ICA which does not account for spatial or temporal structure. We highlight why the statistical criteria make FCA also vanilla despite its matricial underpinnings and show that FCA performs comparably to, and sometimes better than, (vanilla) ICA in every application, such as image and speech unmixing, where ICA has been known to succeed. Our computational experiments suggest that not-so-random matrices, such as images and short time fourier transform matrix of waveforms are (closer to being) freer "in the wild" than we might have theoretically expected.

Zihao Wang, Datong Zhou, Yong Zhang et al.

As a fundamental problem of natural language processing, it is important to measure the distance between different documents. Among the existing methods, the Word Mover's Distance (WMD) has shown remarkable success in document semantic matching for its clear physical insight as a parameter-free model. However, WMD is essentially based on the classical Wasserstein metric, thus it often fails to robustly represent the semantic similarity between texts of different lengths. In this paper, we apply the newly developed Wasserstein-Fisher-Rao (WFR) metric from unbalanced optimal transport theory to measure the distance between different documents. The proposed WFR document distance maintains the great interpretability and simplicity as WMD. We demonstrate that the WFR document distance has significant advantages when comparing the texts of different lengths. In addition, an accelerated Sinkhorn based algorithm with GPU implementation has been developed for the fast computation of WFR distances. The KNN classification results on eight datasets have shown its clear improvement over WMD.