Matrix sensing is the problem of reconstructing a low-rank matrix from a few linear measurements. In many applications such as collaborative filtering, the famous Netflix prize problem, and seismic data interpolation, there exists some prior information about the column and row spaces of the ground-truth low-rank matrix. In this paper, we exploit this prior information by proposing a weighted optimization problem where its objective function promotes both rank and prior subspace information. Using the recent results in conic integral geometry, we obtain the unique optimal weights that minimize the required number of measurements. As simulation results confirm, the proposed convex program with optimal weights requires substantially fewer measurements than the regular nuclear norm minimization.
In @cite_6 , the authors heuristically propose the following optimization problem to exploit prior subspace information: where @math and @math with dimension @math are the estimates of column and row subspaces of the rank @math ground-truth matrix @math , @math , and @math is an upper-bound for @math . However, they did not answer how to explicitly find @math and @math .
@cite_6
2099953425 2132743051 2089784543 2244863692
We study the basic problem of robust subspace recovery. That is, we assume a data set that some of its points are sampled around a fixed subspace and the rest of them are spread in the whole ambient space, and we aim to recover the fixed underlying subspace. We first estimate "robust inverse sample covariance" by solving a convex minimization procedure; we then recover the subspace by the bottom eigenvectors of this matrix (their number correspond to the number of eigenvalues close to 0). We guarantee exact subspace recovery under some conditions on the underlying data. Furthermore, we propose a fast iterative algorithm, which linearly converges to the matrix minimizing the convex problem. We also quantify the effect of noise and regularization and discuss many other practical and theoretical issues for improving the subspace recovery in various settings. When replacing the sum of terms in the convex energy function (that we minimize) with the sum of squares of terms, we obtain that the new minimizer is a scaled version of the inverse sample covariance (when exists). We thus interpret our minimizer and its subspace (spanned by its bottom eigenvectors) as robust versions of the empirical inverse covariance and the PCA subspace respectively. We compare our method with many other algorithms for robust PCA on synthetic and real data sets and demonstrate state-of-the-art speed and accuracy. We consider the problem of reconstructing a data stream from a small subset of its entries, where the data stream is assumed to lie in a low-dimensional linear subspace, possibly corrupted by noise. It is also important to track the change of underlying subspace for many applications. This problem can be viewed as a sequential low-rank matrix completion problem in which the subspace is learned in an online fashion. The proposed algorithm, called Parallel Estimation and Tracking by REcursive Least Squares (PETRELS), identifies the underlying low-dimensional subspace via a recursive procedure for each row of the subspace matrix in parallel, and then reconstructs the missing entries via least-squares estimation if required. PETRELS outperforms previous approaches by discounting observations in order to capture long-term behavior of the data stream and be able to adapt to it. Numerical examples are provided for direction-of-arrival estimation and matrix completion, comparing PETRELS with state of the art batch algorithms. We consider the problem of approximating a set P of η points in Rd by a j-dimensional subspace under the lp measure, in which we wish to minimize the sum of lp distances from each point of P to this subspace. More generally, the Fq (lp)-subspace approximation problem asks for a j-subspace that minimizes the sum of qth powers of lp-distances to this subspace, up to a multiplicative factor of (1 + e). We develop techniques for subspace approximation, regression, and matrix approximation that can be used to deal with massive data sets in high dimensional spaces. In particular, we develop coresets and sketches, i.e. small space representations that approximate the input point set P with respect to the subspace approximation problem. Our results are: • A dimensionality reduction method that can be applied to Fq (lp)-clustering and shape fitting problems, such as those in [8, 15]. • The first strong coreset for F1 (l2)-subspace approximation in high-dimensional spaces, i.e. of size polynomial in the dimension of the space. This coreset approximates the distances to any j-subspace (not just the optimal one). • A (1 + e)-approximation algorithm for the j-dimensional F1 (l2)-subspace approximation problem with running time ηd(j e)O(1) + (η + d)2poly(j e). • A streaming algorithm that maintains a coreset for the F1 (l2)-subspace approximation problem and uses a space of d [EQUATION] (weighted) points. • Streaming algorithms for the above problems with bounded precision in the turnstile model, i.e, when coordinates appear in an arbitrary order and undergo multiple updates. We show that bounded precision can lead to further improvements. We extend results of [7] for approximate linear regression, distances to subspace approximation, and optimal rank-j approximation, to error measures other than the Frobenius norm. Given a set of objects @math , each with @math numeric attributes, a top- @math preference scores these objects using a linear combination of their attribute values, where the weight on each attribute reflects the interest in this attribute. Given a query preference @math , a top- @math query finds the @math objects in @math with highest scores with respect to @math . Given a query object @math and a set of preferences @math , a reverse top- @math query finds all preferences @math for which @math becomes one of the top @math objects with respect to @math . Previous solutions to these problems are effective only in low dimensions. In this paper, we develop a solution for much higher dimensions (up to high tens), if many preferences exhibit sparsity —i.e., each specifies non-zero weights for only a handful (say @math - @math ) of attributes (though the subsets of such attributes and their weights can vary greatly). Our idea is to select carefully a set of low-dimensional core subspaces to “cover” the sparse preferences in a workload. These subspaces allow us to index them more effectively than the full-dimensional space. Being multi-dimensional, each subspace covers many possible preferences; furthermore, multiple subspaces can jointly cover a preference, thereby expanding the coverage beyond each subspace’s dimensionality. Experimental evaluation validates our solution’s effectiveness and advantages over previous solutions.