This paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. This generalizes the standard low-rank matrix completion problem to situations in which the matrix rank can be quite high or even full rank. Since the columns belong to a union of subspaces, this problem may also be viewed as a missing-data version of the subspace clustering problem. Let X be an n x N matrix whose (complete) columns lie in a union of at most k subspaces, each of rank > kn. The main result of the paper shows that under mild assumptions each column of X can be perfectly recovered with high probability from an incomplete version so long as at least CrNlog^2(n) entries of X are observed uniformly at random, with C>1 a constant depending on the usual incoherence conditions, the geometrical arrangement of subspaces, and the distribution of columns over the subspaces. The result is illustrated with numerical experiments and an application to Internet distance matrix completion and topology identification.
Our work builds upon the results of @cite_14 , which quantifies the deviation of an incomplete vector norm with respect to the incoherence of the sampling pattern. While this work also examines subspace detection using incomplete data, it assumes complete knowledge of the subspaces.
@cite_14
2102628473 2147276092 1561053739 2267569616
We consider the problem of deciding whether a highly incomplete signal lies within a given subspace. This problem, Matched Subspace Detection, is a classical, well-studied problem when the signal is completely observed. High-dimensional testing problems in which it may be prohibitive or impossible to obtain a complete observation motivate this work. The signal is represented as a vector in ℝn, but we only observe m ≪ n of its elements.We show that reliable detection is possible, under mild incoherence conditions, as long as m is slightly greater than the dimension of the subspace in question. Traditional sampling theories consider the problem of reconstructing an unknown signal x from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that x lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which x lies in a union of subspaces. In this paper, we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which x lies in a sum of k subspaces, chosen from a larger set of m possibilities. The samples are modeled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a block-sparse vector whose nonzero elements appear in fixed blocks. We then propose a mixed lscr2 lscr1 program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP, we also prove stability of our approach in the presence of noise and modeling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently. In this paper, the Subspace Pursuit (SP) recovery of signals with sensing matrix perturbations is analyzed. Previous studies have only considered the robustness of Basis pursuit and greedy algorithms to recover the signal in the presence of additive noise with measurement and or signal. Since it is impractical to exactly implement the sampling matrix A in a physical sensor, precision errors must be considered. Recently, work has been done to analyze the methods with noise in the sampling matrix, which generates a multiplicative noise term. This new perturbed framework (both additive and multiplicative noise) extends the prior work of Basis pursuit and greedy algorithms on stable signal recovery from incomplete and inaccurate measurements. Our works show that, under reasonable conditions, the stability of the SP solution of the completely perturbed scenario was limited by the total noise in the observation. Subspace recovery from noisy or even corrupted data is critical for various applications in machine learning and data analysis. To detect outliers, Robust PCA (R-PCA) via Outlier Pursuit was proposed and had found many successful applications. However, the current theoretical analysis on Outlier Pursuit only shows that it succeeds when the sparsity of the corruption matrix is of O(n r), where n is the number of the samples and r is the rank of the intrinsic matrix which may be comparable to n. Moreover, the regularization parameter is suggested as 3 (7√γn), where γ is a parameter that is not known a priori. In this paper, with incoherence condition and proposed ambiguity condition we prove that Outlier Pursuit succeeds when the rank of the intrinsic matrix is of O(n logn) and the sparsity of the corruption matrix is of O(n). We further show that the orders of both bounds are tight. Thus R-PCA via Outlier Pursuit is able to recover intrinsic matrix of higher rank and identify much denser corruptions than what the existing results could predict. Moreover, we suggest that the regularization parameter be chosen as 1√log n, which is definite. Our analysis waives the necessity of tuning the regularization parameter and also significantly extends the working range of the Outlier Pursuit. Experiments on synthetic and real data verify our theories.