Sparse Subspace Clustering (SSC) is a state-of-the-art method for clustering high-dimensional data points lying in a union of low-dimensional subspaces. However, while $\ell_1$ optimization-based SSC algorithms suffer from high computational complexity, other variants of SSC, such as Orthogonal Matching Pursuit-based SSC (OMP-SSC), lose clustering accuracy in pursuit of improving time efficiency. In this letter, we propose a novel Active OMP-SSC, which improves clustering accuracy of OMP-SSC by adaptively updating data points and randomly dropping data points in the OMP process, while still enjoying the low computational complexity of greedy pursuit algorithms. We provide heuristic analysis of our approach, and explain how these two active steps achieve a better tradeoff between connectivity and separation. Numerical results on both synthetic data and real-world data validate our analyses and show the advantages of the proposed active algorithm.
Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；Univ. Paris-Sud（巴黎南大学）；Université Paris-Saclay（巴黎萨克雷大学）
Greedy algorithms for feature selection are widely used for recovering sparse high-dimensional vectors in linear models. In classical procedures, the main emphasis was put on the sample complexity, with little or no consideration of the computation resources required. We present a novel online algorithm: Online Orthogonal Matching Pursuit (OOMP) for online support recovery in the random design setting of sparse linear regression. Our procedure selects features sequentially, alternating between allocation of samples only as needed to candidate features, and optimization over the selected set of…
We consider the general dimensionality reduction problem of locating in a high-dimensional data cloud, a $k$-dimensional non-Gaussian subspace of interesting features. We use a projection pursuit approach -- we search for mutually orthogonal unit directions which maximise the 2-Wasserstein distance of the empirical distribution of data-projections along these directions from a standard Gaussian. Under a generative model, where there is a underlying (unknown) low-dimensional non-Gaussian subspace, we prove rigorous statistical guarantees on the accuracy of approximating this unknown subspace by…
Axiomatizing Neural Networks via Pursuit of Subspaces
Tampere University, Faculty of ITC, Finland（芬兰塔尔库大学信息与通信技术学院）；Department of Electrical Engineering, Qatar University, Qatar（卡塔尔大学电气工程系）；Donders Institute, Radboud University, The Netherlands（荷兰拉德堡德大学多纳尔斯研究所）
University of California, Berkeley（加州大学伯克利分校）；Cornell University（康奈尔大学）；Simons Institute for Theory of Computing（西蒙斯计算理论研究所）；University of Pennsylvania（宾夕法尼亚大学）
We consider the problem of learning discriminative representations for data in a high-dimensional space with distribution supported on or around multiple low-dimensional linear subspaces. That is, we wish to compute a linear injective map of the data such that the features lie on multiple orthogonal subspaces. Instead of treating this learning problem using multiple PCAs, we cast it as a sequential game using the closed-loop transcription (CTRL) framework recently proposed for learning discriminative and generative representations for general low-dimensional submanifolds. We prove that the equ…
This paper studies the subspace clustering problem in which data points collected from high-dimensional ambient space lie in a union of linear subspaces. Subspace clustering becomes challenging when the dimension of intersection between subspaces is large and most of the self-representation based methods are sensitive to the intersection between the span of clusters. In sharp contrast to the self-representation based methods, a recently proposed clustering method termed Innovation Pursuit, computed a set of optimal directions (directions of innovation) to build the adjacency matrix. This paper…
Few-shot Multi-Task Learning of Linear Invariant Features with Meta Subspace Pursuit
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University（上海交通大学自然科学研究院）；SJTU-Yale Joint Center for Biostatistics and Data Science, Shanghai Jiao Tong University（上海交通大学-耶鲁大学生物统计与数据科学联合中心）
Texas A&M University（德克萨斯A&M大学）；Cainiao Network（菜鸟网络）；University of California, Davis（加州大学戴维斯分校）；The Chinese University of Hong Kong（香港中文大学）
We consider the problem of maximizing the $\ell_1$ norm of a linear map over the sphere, which arises in various machine learning applications such as orthogonal dictionary learning (ODL) and robust subspace recovery (RSR). The problem is numerically challenging due to its nonsmooth objective and nonconvex constraint, and its algorithmic aspects have not been well explored. In this paper, we show how the manifold structure of the sphere can be exploited to design fast algorithms for tackling this problem. Specifically, our contribution is threefold. First, we present a manifold proximal point …
Indian Institute of Technology, Kharagpur（印度理工学院克勒格布尔分校）
Recovery of an unknown sparse signal from a few of its projections is the key objective of compressed sensing. Often one comes across signals that are not ordinarily sparse but are sparse blockwise. Existing block sparse recovery algorithms like BOMP make the assumption of uniform block size and known block boundaries, which are, however, not very practical in many applications. This paper addresses this problem and proposes a two step procedure, where the first stage is a coarse block location identification stage while the second stage carries out finer localization of a non-zero cluster wit…
Language representations are known to carry stereotypical biases and, as a result, lead to biased predictions in downstream tasks. While existing methods are effective at mitigating biases by linear projection, such methods are too aggressive: they not only remove bias, but also erase valuable information from word embeddings. We develop new measures for evaluating specific information retention that demonstrate the tradeoff between bias removal and information retention. To address this challenge, we propose OSCaR (Orthogonal Subspace Correction and Rectification), a bias-mitigating method th…
