We consider the problem of selecting non-zero entries of a matrix $A$ in order to produce a sparse sketch of it, $B$, that minimizes $\|A-B\|_2$. For large $m \times n$ matrices, such that $n \gg m$ (for example, representing $n$ observations over $m$ attributes) we give sampling distributions that exhibit four important properties. First, they have closed forms computable from minimal information regarding $A$. Second, they allow sketching of matrices whose non-zeros are presented to the algorithm in arbitrary order as a stream, with $O(1)$ computation per non-zero. Third, the resulting sketch matrices are not only sparse, but their non-zero entries are highly compressible. Lastly, and most importantly, under mild assumptions, our distributions are provably competitive with the optimal offline distribution. Note that the probabilities in the optimal offline distribution may be complex functions of all the entries in the matrix. Therefore, regardless of computational complexity, the optimal distribution might be impossible to compute in the streaming model.
Explaining Deep Network Classification of Matrices: A Case Study on Monotonicity
Institute of Mathematics and Statistics, Federal University of Rio Grande do Sul（数学与统计学学院，里约格朗德杜斯尔大学）；Division of Mathematics and Physics, UKK, Mälardalen University（数学与物理系，马尔达伦大学）
Accumulation of Sub-Sampling Matrices with Applications to Statistical Computation
Departments of Mathematics and Computer Science, Hong Kong Baptist University（香港 Baptist 大学数学与计算机科学系）；Department of Mathematics, University of Maryland, College Park（马里兰大学 College Park 分校数学系）
我们提出了一种新的技术，用于学习从离散动力学获得的采样器中的质量矩阵，保持一些能量函数。现有的自适应采样器使用黎曼预处理技术，其中质量矩阵是被采样的参数的函数。这导致了能量重构和由此产生的动力学的显着复杂性，通常导致隐式方程组，并需要在蛙跳步骤中求高维矩阵的逆。我们的方法提供了一个更简单的替代方案，通过使用现有的动态在抽样步骤的Monte Carlo EM框架，并学习质量矩阵的M步骤与一种新的在线技术。我们还提出了一种自适应地设置在E步骤中收集的样本数量的方法，使用来自蛙跳动态的采样误差估计。随着一种新的随机采样器的基础上Nosé-庞加莱动力学，我们使用这个框架与标准的哈密顿蒙特卡罗（HMC）以及较新的随机算法，如SGHMC和SGNHT，并显示出强大的性能合成和真正的高维采样的情况下，我们实现采样精度相媲美的黎曼采样器，同时显着更快。
Sketching Transformed Matrices with Applications to Natural Language Processing
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of Washington（华盛顿大学）
