Non-negative matrix factorization (NMF) is a natural model of admixture and is widely used in science and engineering. A plethora of algorithms have been developed to tackle NMF, but due to the non-convex nature of the problem, there is little guarantee on how well these methods work. Recently a surge of research have focused on a very restricted class of NMFs, called separable NMF, where provably correct algorithms have been developed. In this paper, we propose the notion of subset-separable NMF, which substantially generalizes the property of separability. We show that subset-separability is a natural necessary condition for the factorization to be unique or to have minimum volume. We developed the Face-Intersect algorithm which provably and efficiently solves subset-separable NMF under natural conditions, and we prove that our algorithm is robust to small noise. We explored the performance of Face-Intersect on simulations and discuss settings where it empirically outperformed the state-of-art methods. Our work is a step towards finding provably correct algorithms that solve large classes of NMF problems.
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Tensor Vision Technologies（张量视觉技术公司）
视觉对象是由一个递归层次的知觉整体和部分，其属性，如形状，反射率和颜色，构成了一个层次的内在因果因素的对象外观。然而，物体外观是物体的内在和外在因果因素的组成结果，其中外在因果因素与照明和成像条件有关。因此，本文提出了一个统一的张量模型的整体和部分，并介绍了一个组成层次张量分解，解开层次因果结构的对象图像形成，并纳入多线性块张量分解作为一个特殊情况。所得到的对象表示是整体和部分表示的可解释的组合选择，其使得对象识别对遮挡鲁棒并且减少训练数据要求。我们通过在一个极其简化的合成图像数据集上进行训练，在人脸识别的背景下演示了我们的方法，并在两个数据集上报告了人脸验证结果-Freiburg数据集和由真实世界图像组成的Labeled Face in the Wild（LFW）数据集，从而证实了我们的方法适用于数据匮乏的领域。
Supervised Deep Multimodal Matrix Factorization for Interpretable Brain Network Analysis
Dept. of Mathematics & Operational Research University of Mons（数学与运筹学系蒙斯大学）；Dept. of Computer Science & Engineering Lehigh University（计算机科学与工程系莱斯大学）；Dept. of Industrial & Systems Engineering Lehigh University（工业与系统工程系莱斯大学）
SL$^2$MF: Predicting Synthetic Lethality in Human Cancers via Logistic Matrix Factorization
Joint NTU-UBC Research Centre of Excellence in Active Living for the Elderly (LILY)（南洋理工大学-不列颠哥伦比亚大学活跃老年生活卓越联合研究中心（LILY））；Alibaba-NTU Singapore Joint Research Institute（阿里巴巴-南洋理工大学新加坡联合研究院）；Nanyang Technological University（南洋理工大学）；ShanghaiTech University（上海科技大学）；A*Star（新加坡科技研究局）；Singapore Management University（新加坡管理大学）
Chem-NMF: Multi-layer $α$-divergence Non-Negative Matrix Factorization for Cardiorespiratory Disease Clustering, with Improved Convergence Inspired by Chemical Catalysts and Rigorous Asymptotic Analysis
Electrical and Computer Engineering Department, McMaster University（麦斯特大学电气与计算机工程系）；L.R. Wilson/Bell Canada Chair in Data Communications（数据通信LR. Wilson/Bell Canada chair）
