Subspace clustering is the problem of clustering data points into a union of low-dimensional linear affine subspaces. It is the mathematical abstraction of many important problems in computer vision, image processing and machine learning. A line of recent work (4, 19, 24, 20) provided strong theoretical guarantee for sparse subspace clustering (4), the state-of-the-art algorithm for subspace clustering, on both noiseless and noisy data sets. It was shown that under mild conditions, with high probability no two points from different subspaces are clustered together. Such guarantee, however, is not sufficient for the clustering to be correct, due to the notorious "graph connectivity problem" (15). In this paper, we investigate the graph connectivity problem for noisy sparse subspace clustering and show that a simple post-processing procedure is capable of delivering consistent clustering under certain "general position" or "restricted eigenvalue" assumptions. We also show that our condition is almost tight with adversarial noise perturbation by constructing a counter-example. These results provide the first exact clustering guarantee of noisy SSC for subspaces of dimension greater then 3.
As we can see from the second column of Table , SEP guarantees have been quite exhaustively studied and now we understand very well the conditions under which it holds. Specifically, most of the results are now near optimal under the semi-random model: SEP holds in cases even when different subspaces substantially overlap, have canonical angles near @math , the dimension of the subspaces being linear in the ambient dimension, or the number of subspaces to be clustered is exponentially large @cite_5 @cite_15 @cite_21 . In addition, the above results also hold robustly under a small amount of arbitrary perturbation or a large amount of stochastic noise @cite_15 . In particular, it was shown in @cite_15 that the amount of tolerable stochastic noise could even be substantially larger than the signal in both deterministic and semi-random models.
@cite_5 @cite_15 @cite_21
2169232245 2040192961 2962699510 2172028873
We study the problem of online subspace learning in the context of sequential observations involving structured perturbations. In online subspace learning, the observations are an unknown mixture of two components presented to the model sequentially - the main effect which pertains to the subspace and a residual error term. If no additional requirement is imposed on the residual, it often corresponds to noise terms in the signal which were unaccounted for by the main effect. To remedy this, one may impose "structural" contiguity, which has the intended effect of leveraging the secondary terms as a covariate that helps the estimation of the subspace itself, instead of merely serving as a noise residual. We show that the corresponding online estimation procedure can be written as an approximate optimization process on a Grassmannian. We propose an efficient numerical solution, GOSUS, Grassmannian Online Subspace Updates with Structured-sparsity, for this problem. GOSUS is expressive enough in modeling both homogeneous perturbations of the subspace and structural contiguities of outliers, and after certain manipulations, solvable via an alternating direction method of multipliers (ADMM). We evaluate the empirical performance of this algorithm on two problems of interest: online background subtraction and online multiple face tracking, and demonstrate that it achieves competitive performance with the state-of-the-art in near real time. Abstract Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles yield stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4 M − 4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M = 2 , 3 . Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer–Rao lower bounds to identify stability with stronger versions of the injectivity characterizations. In this paper, we consider the cluster estimation problem under the stochastic block model. We show that the semidefinite programming (SDP) formulation for this problem achieves an error rate that decays exponentially in the signal-to-noise ratio. The error bound implies weak recovery in the sparse graph regime with bounded expected degrees as well as exact recovery in the dense regime. An immediate corollary of our results yields error bounds under the censored block model. Moreover, these error bounds are robust, continuing to hold under heterogeneous edge probabilities and a form of the so-called monotone attack. Significantly, this error rate is achieved by the SDP solution itself without any further pre- or post-processing and improves upon existing polynomially decaying error bounds proved using the Grothendieck’s inequality. Our analysis builds on two key ingredients: 1) showing that the graph has a well-behaved spectrum, even in the sparse regime, after discounting an exponentially small number of edges and 2) an order-statistics argument that governs the final error rate. Both arguments highlight the implicit regularization effect of the SDP formulation. We analyze a sublinear RA@?SFA (randomized algorithm for Sparse Fourier analysis) that finds a near-optimal B-term Sparse representation R for a given discrete signal S of length N, in time and space poly(B,log(N)), following the approach given in [A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling, STOC, 2002]. Its time cost poly(log(N)) should be compared with the superlinear @W(NlogN) time requirement of the Fast Fourier Transform (FFT). A straightforward implementation of the RA@?SFA, as presented in the theoretical paper [A.C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, M. Strauss, Near-Optimal Sparse Fourier Representations via Sampling, STOC, 2002], turns out to be very slow in practice. Our main result is a greatly improved and practical RA@?SFA. We introduce several new ideas and techniques that speed up the algorithm. Both rigorous and heuristic arguments for parameter choices are presented. Our RA@?SFA constructs, with probability at least 1-@d, a near-optimal B-term representation R in time poly(B)log(N)log(1 @d) @e^2log(M) such that @?S-R@?"2^2=<(1+@e)@?S-R"o"p"t@?"2^2. Furthermore, this RA@?SFA implementation already beats the FFTW for not unreasonably large N. We extend the algorithm to higher dimensional cases both theoretically and numerically. The crossover point lies at N 70,000 in one dimension, and at N 900 for data on a NxN grid in two dimensions for small B signals where there is noise.