Completing Low-Rank Matrices with Corrupted Samples from Few Coefficients in General Basis

Subspace recovery from corrupted and missing data is crucial for various applications in signal processing and information theory. To complete missing values and detect column corruptions, existing robust Matrix Completion (MC) methods mostly concentrate on recovering a low-rank matrix from few corrupted coefficients w.r.t. standard basis, which, however, does not apply to more general basis, e.g., Fourier basis. In this paper, we prove that the range space of an $m\times n$ matrix with rank $r$ can be exactly recovered from few coefficients w.r.t. general basis, though $r$ and the number of corrupted samples are both as high as $O(\min\{m,n\}/\log^3 (m+n))$. Our model covers previous ones as special cases, and robust MC can recover the intrinsic matrix with a higher rank. Moreover, we suggest a universal choice of the regularization parameter, which is $\lambda=1/\sqrt{\log n}$. By our $\ell_{2,1}$ filtering algorithm, which has theoretical guarantees, we can further reduce the computational cost of our model. As an application, we also find that the solutions to extended robust Low-Rank Representation and to our extended robust MC are mutually expressible, so both our theory and algorithm can be applied to the subspace clustering problem with missing values under certain conditions. Experiments verify our theories.Comment: To appear in IEEE Transactions on Information Theor

Noise-Stable Rigid Graphs for Euclidean Embedding

We proposed a new criterion \textit{noise-stability}, which revised the classical rigidity theory, for evaluation of MDS algorithms which can truthfully represent the fidelity of global structure reconstruction; then we proved the noise-stability of the cMDS algorithm in generic conditions, which provides a rigorous theoretical guarantee for the precision and theoretical bounds for Euclidean embedding and its application in fields including wireless sensor network localization and satellite positioning. Furthermore, we looked into previous work about minimum-cost globally rigid spanning subgraph, and proposed an algorithm to construct a minimum-cost noise-stable spanning graph in the Euclidean space, which enabled reliable localization on sparse graphs of noisy distance constraints with linear numbers of edges and sublinear costs in total edge lengths. Additionally, this algorithm also suggests a scheme to reconstruct point clouds from pairwise distances at a minimum of $O(n)$ time complexity, down from $O(n^3)$ for cMDS