Matrix Completion

In this paper it is shown that a partialsign symmetric P -matrix, whose digraph of specified entries is a symmetric n-cycle with n ≥ 6, can be completed to a sign symmetric P - matrix. The analogous completion property is also established for a partial weakly sign symmetric P -matrix and for a partialweakl y sign symmetric P0-matrix. Patterns of

The non-negative P0 – matrix completion is considered for 5x5 matrices specifying digraphs for p = 5,
q = 3, where p is number of vertices and q is number of arcs by performing zero completion on the
matrices. The study establishes that all digraphs for p = 5, q = 3 specifying 5x5 partial matrices which
are either cycles or acyclic digraphs have non-negative P0 –completion.

Completion problems arise in a variety of applications, such as statistics (e.g. entropy methods for missing data), chemistry (the molecu-lar conformation problem), systems theory, discrete optimization (relaxation methods), data compression, etc., as well as in operator theory and within matrix theory (e.g. determinantal inequalities). In addition to applications, completion problems have provided an excellent mechanism for understanding matrix structure more deeply. In this article, we survey the recent works on matrix completion problems.

Background model initialization is commonly the first step of the background subtraction process. In practice, several challenges appear and perturb this process, such as dynamic background, bootstrapping, illumination changes, noise image, etc. In this context, we investigate the background model initialization as a reconstruction problem from missing data. This problem can be formulated as a matrix or tensor completion task where the image sequence (or video) is revealed as partially observed data. In this paper , the missing entries are induced from the moving regions through a simple joint motion-detection and frame-selection operation. The redundant frames are eliminated, and the moving regions are represented by zeros in our observation model. The second stage involves evaluating twenty-three state-of-the-art algorithms comprising of thirteen matrix completion and ten tensor completion algorithms. These algorithms aim to recover the low-rank component (or background model) from partially observed data. The Scene Background Initialization data set was selected in order to evaluate this proposal with respect to the background model challenges. Our experimental results show the good performance of LRGeomCG method over its direct competitors.

We look at the real positive (semi)definite matrix completion problem from the convex optimization viewpoint. The problem is introduced via relative entropy minimization, transformed into the standard max-det from, and conditions are sought for existence of positive definite and positive semidefinite completions. Using basic tools of convex optimization, a unifying view of the existence and uniqueness problem for positive (semi)definite matrix completions is presented. Some results previously established using functional-analytic techniques are recovered and some new are given. In particular, the maximum determinant completion is generalized to the positive semidefinte matrices with an arbitrary sparsity pattern.

The main challenge faced by wireless sensor networks today is the problem of power consumption at the sensor nodes. Over time, researchers have developed different strategies to address this issue. Such strategies are strongly model dependent and/or application specific. In this work, we take a fresh look at the problem of power consumption in wireless sensor networks from a signal processing perspective. The main idea is simple. Sample only a subset of all the sensor nodes at a given instant and transmit them (this reduces both sampling and communication cost for all the nodes combined). At the central unit (sink) use smart mathematical tools (matrix completion algorithms) to estimate the data for the entire network. We have showed that, if about 1% reconstruction error is allowed, only 20% of the sensors need to sample and transmit at a given instant. This means on an average the life of the network is increased 5-fold. If more error reconstruction error is allowed, even lesser number of sensors need to be active at a given instant leading to more prolonged life of the network.

In this paper, we show some necessary and/or sufficient conditions
so that AB and/or BA are core matrices, whenever A and B are core
complex matrices (a matrix A is a core matrix, that is a matrix of index
one, if Im(A) ∩ Ker(A) = {0}). More specifically, we also analyse some
similarsituationswhereA,B,AB and/orBAarediagonalizablematrices
or EP matrices. Moreover, we provide, as an application, two results
concerning the reverse order law for the Moore–Penrose inverse of
the product of two matrices.

The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solu- tion techniques relax this problem to a weighted, nearest, (positive) semidefinite programming, SDP,completion problem, by using the linear

Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization with affine constraints (ARMP) and show that SVP recovers the minimum rank solution for affine constraints that satisfy the "restricted isometry property" and show robustness of our method to noise. Our results improve upon a recent breakthrough by Recht, Fazel and Parillo (RFP07) and Lee and Bresler (LB09) in three significant ways: 1) our method (SVP) is significantly simpler to analyze and easier to implement, 2) we give recovery guarantees under strictly weaker isometry assumptions 3) we give geometric convergence guarantees for SVP even in presense of noise and, as demonstrated empirically, SVP is significantly faster on real-world and synthetic problems. In addition, we address the practically important problem of low-rank matrix completion (MCP), which can be seen as a special case of ARMP. We empirically demonstrate that our algorithm recovers low-rank incoherent matrices from an almost optimal number of uniformly sampled entries. We make partial progress towards proving exact recovery and provide some intuition for the strong performance of SVP applied to matrix completion by showing a more restricted isometry property. Our algorithm outperforms existing methods, such as those of \cite{RFP07,CR08,CT09,CCS08,KOM09,LB09}, for ARMP and the matrix-completion problem by an order of magnitude and is also significantly more robust to noise.

We derive a spectral method for unsupervised learning ofWeighted Context Free Grammars. We frame WCFG induction as finding a Hankel matrix that has low rank and is linearly constrained to represent a function computed by inside-outside recursions. The proposed algorithm picks the grammar that agrees with a sample and is the simplest with respect to the nuclear norm of the Hankel matrix.

In this paper, the nonnegative Q-matrix completion problem is studied. A real $n × n$ matrix is a $Q-$matrix if for $k ∈ {1,. .. , n}$, the sum of all $k × k$ principal minors is positive. A digraph $D$ is said to have nonnegative $Q-$completion if every partial nonnegative $Q-$matrix specifying $D$ can be completed to a nonnegative $Q-$matrix. For nonnegative $Q-$completion problem, necessary conditions and sufficient conditions for a digraph to have nonnegative Q-completion are obtained. Further, the digraphs of order at most four that have nonnegative $Q-$completion have been studied.

An innovative electrocardiogram compression algorithm is presented in this paper. The proposed method is based on matrix completion, a new paradigm in signal processing that seeks to recover a low-rank matrix based on a small number of observations. The low-rank matrix is obtained via normalization of electrocardiogram records. Using matrix completion, the ECG data matrix is recovered from a few number of entries, thereby yielding high compression ratios comparable to those obtained by existing compression techniques. The proposed scheme offers a low-complexity encoder, good tolerance to quantization noise, and good quality reconstruction.

... When secondary users experience multi-path fading or happen to be shadowed, the reports ... with probability ϵ/ √ p × m. Given Ep×m, the partial observation of M is ... MENG et al.: COLLABORATIVE SPECTRUM SENSING FROM SPARSE OBSERVATIONS IN COGNITIVE RADIO ...

— A method is proposed to interpolate the electromagnetic near field when no information on the radiating source is available. In absence of a priori knowledge, general properties of the electromagnetic field are exploited to estimate the field, namely the minimum complexity of the field and the continuity of the first derivatives. These properties are enforced by minimizing the nuclear norm and using the Thin Plate Spline interpolation results, respectively. The proposed procedure is validated experimentally by interpolating the planar electrical near field radiated by three antennas. The quality of the interpolation and its robustness to noise is investigated. Despite its simplicity, the interpolation method is able to properly estimate the near field from a random coarse sampling of 2 λ. The quality of the near field interpolation is also confirmed by deriving the far field. These promising results pave the way for the development of fast antenna measurement procedures.

Given a matrix M of low-rank, we consider the problem of reconstructing it from noisy observations of a small, random subset of its entries. The problem arises in a variety of applications, from collaborative filtering (the `Netflix problem') to structure-from-motion and positioning. We study a low complexity algorithm introduced by Keshavan et al.(2009), based on a combination of spectral techniques and manifold optimization, that we call here OptSpace. We prove performance guarantees that are order-optimal in a number of circumstances.

Continuing to estimate the Direction-of-arrival (DOA) of the signals impinging on the antenna array, even when a few elements of the underlying Uniform Linear Antenna Array (ULA) fail to work will be of practical interest in RADAR, SONAR and Wireless Radio Communication Systems. This paper proposes a new technique to estimate the DOAs when a few elements are malfunctioning. The technique combines Singular Value Thresholding (SVT) based Matrix Completion (MC) procedure with the Direct Data Domain (D3) based Matrix Pencil (MP) Method. When the element failure is observed, first, the MC is performed to recover the missing data from failed elements, and then the MP method is used to estimate the DOAs. We also, propose a very simple technique to detect the location of elements failed, which is required to perform MC procedure. We provide simulation studies to demonstrate the performance and usefulness of the proposed technique. The results indicate a better performance, of the proposed DOA estimation scheme under different antenna failure scenarios.