This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from a small number of entries is impossible; but the knowledge that the unknown matrix has low rank radically changes this premise, making the search for solutions meaningful. This paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever (the information theoretic limit). More importantly, the paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). This convex program simply finds, among all matrices consistent with the observed entries, that with minimum nuclear norm. As an example, we show that on the order of nr log(n) samples are needed to recover a random n × n matrix of rank r by any method, and to be sure, nuclear norm minimization succeeds as soon as the number of entries is of the form nrpolylog(n).

On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries, and comes up in many areas of science and engineering including collaborative filtering, machine learning, control, remote sensing, and computer vision to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown low-rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclear-norm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n × n matrix of low rank r from just about nr log 2 n noisy samples with an error which is proportional to the noise level. We present numerical results which complement our quantitative analysis and show that, in practice, nuclear norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout.

Sensor localization from only connectivity information is a highly challenging problem. To this end, our result for the first time establishes an analytic bound on the performance of the popular MDS-MAP algorithm based on multidimensional scaling. For a network consisting of n sensors positioned randomly on a unit square and a given radio range r = o(1), we show that resulting error is bounded, decreasing at a rate that is inversely proportional to r, when only connectivity information is given. The same bound holds for the range-based model, when we have an approximate measurements for the distances, and the same algorithm can be applied without any modification. 1

We present a novel approach to localization of sensors in a network given a subset of noisy inter-sensor distances. The algorithm is based on “stitching” together local structures by solving an optimization problem requiring the structures to fit together in an “As-Rigid-As-Possible ” manner, hence the name ARAP. The local structures consist of reference “patches” and reference triangles, both obtained from inter-sensor distances. We elaborate on the relationship between the ARAP algorithm and other state-of-the-art algorithms, and provide experimental results demonstrating that ARAP is significantly less sensitive to sparse connectivity and measurement noise. We also show how ARAP may be distributed.

Abstract—We present a new approach to simultaneously denoise and parameterize unorganized point cloud data. This is achieved by minimizing an appropriate energy function defined on the point cloud and its parameterization. An iterative algorithm to minimize the energy is described. The key ingredient of our approach is an “as-rigid-as-possible ” meshless parameterization to map a point cloud with disk topology to the plane without building the connectivity of the point cloud. Then 2D triangulation method can be applied to the planar parameterization to provide triangle connectivity for the 2D points, which can be transferred back to the 3D point cloud to form a triangle mesh surface. We also show how to generalize the approach to meshes with closed topology of any genus. Experimental results have shown that our approach can effectively denoise the point cloud and our meshless parameterization can preserve local distances in the point cloud, resulting in a more regular 3D triangle mesh, compared to other methods. Keywords—point clouds; denoising; meshless parameterization; triangulation; surface reconstruction 1.

Kernel learning is a powerful framework for nonlinear data modeling. Using the kernel trick, a number of problems have been formulated as semidefinite programs (SDPs). These include Maximum Variance Unfolding (MVU) (Weinberger et al., 2004) in nonlinear dimensionality reduction, and Pairwise Constraint Propagation (PCP) (Li et al., 2008) in constrained clustering. Although in theory SDPs can be efficiently solved, the high computational complexity incurred in numerically processing the huge linear matrix inequality constraints has rendered the SDP approach unscalable. In this paper, we show that a large class of kernel learning problems can be reformulated as semidefinite-quadratic-linear programs (SQLPs), which only contain a simple positive semidefinite constraint, a second-order cone constraint and a number of linear constraints. These constraints are much easier to process numerically, and the gain in speedup over previous approaches is at least of the order m 2.5, where m is the matrix dimension. Experimental results are also presented to show the superb computational efficiency of our approach. 1

Modern mobile robots navigate uncertain environments using complex compositions of camera, laser, and sonar sensor data. Manual calibration of these sensors is a tedious process that involves determining sensor behavior, geometry and location through model specification and system identification. Instead, we seek to automate the construction of sensor model geometry by mining uninterpreted sensor streams for regularities. Manifold learning methods are powerful techniques for deriving sensor structure from streams of sensor data. In recent years, the proliferation of manifold learning algorithms has led to a variety of choices for autonomously generating models of sensor geometry. We present a series of comparisons between different manifold learning methods for discovering sensor geometry for the specific case of a mobile robot with a variety of sensors. We also explore the effect of control laws and sensor boundary size on the efficacy of manifold learning approaches. We find that ”motor babbling ” control laws generate better geometric sensor maps than mid-line or wall following control laws and identify a novel method for distinguishing boundary sensor elements. We also present a new learning method, sensorimotor embedding, that takes advantage of the controllable nature of robots to build sensor maps.

A mobile robot typically has access to a high dimensional stream of multi-modal sensor information. Modern mobile robots navigate uncertain environments using complex mixtures of camera, laser, and sonar sensor data. Manual calibration of these sensors is a tedious process that involves determining sensor behavior and geometry through model specification. To automate the construction of sensor models requires mining the uninterpreted sensor stream for regularities in both known and unknown environments. One method of doing so involves using the interpoint distances between individual sense element sensor streams as a proxy for the interpoint distances between sense elements themselves. We can then apply dimensionality reduction to generate a low dimensional representation of sensor geometry. We present a series of initial comparisons of manifold learning methods for discovering sensor geometry on a mobile robot. 1 Bootstrap Learning We can view the problem of a robot situated in the world as a dynamical system xt = G(xt−1, µt−1) (1) st = H(xt) (2) νt = C(st) (3)

Abstract—We consider the problem of positioning a cloud of points in the Euclidean space R d, from noisy measurements of a subset of pairwise distances. This task has applications in various areas, such as sensor network localizations, NMR spectroscopy of proteins, and molecular conformation. Also, it is closely related to dimensionality reduction problems and manifold learning, where the goal is to learn the underlying global geometry of a data set using measured local (or partial) metric information. Here we propose a reconstruction algorithm based on a semidefinite programming approach. For a random geometric graph model and uniformly bounded noise, we provide a precise characterization of the algorithm’s performance: In the noiseless case, we find a radius r0 beyond which the algorithm reconstructs the exact positions (up to rigid transformations). In the presence of noise, we obtain upper and lower bounds on the reconstruction error that match up to a factor that depends only on the dimension d, and the average degree of the nodes in the graph. I.

This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from a small number of entries is impossible; but the knowledge that the unknown matrix has low rank radically changes this premise, making the search for solutions meaningful. This paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever (the information theoretic limit). More importantly, the paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). This convex program simply finds, among all matrices consistent with the observed entries, that with minimum nuclear norm. As an example, we show that on the order of nr log(n) samples are needed to recover a random n × n matrix of rank r by any method, and to be sure, nuclear norm minimization succeeds as soon as the number of entries is of the form nrpolylog(n).