7,010 research outputs found
Low-rank Matrix Completion using Alternating Minimization
Alternating minimization represents a widely applicable and empirically
successful approach for finding low-rank matrices that best fit the given data.
For example, for the problem of low-rank matrix completion, this method is
believed to be one of the most accurate and efficient, and formed a major
component of the winning entry in the Netflix Challenge.
In the alternating minimization approach, the low-rank target matrix is
written in a bi-linear form, i.e. ; the algorithm then alternates
between finding the best and the best . Typically, each alternating step
in isolation is convex and tractable. However the overall problem becomes
non-convex and there has been almost no theoretical understanding of when this
approach yields a good result.
In this paper we present first theoretical analysis of the performance of
alternating minimization for matrix completion, and the related problem of
matrix sensing. For both these problems, celebrated recent results have shown
that they become well-posed and tractable once certain (now standard)
conditions are imposed on the problem. We show that alternating minimization
also succeeds under similar conditions. Moreover, compared to existing results,
our paper shows that alternating minimization guarantees faster (in particular,
geometric) convergence to the true matrix, while allowing a simpler analysis
In-network Sparsity-regularized Rank Minimization: Algorithms and Applications
Given a limited number of entries from the superposition of a low-rank matrix
plus the product of a known fat compression matrix times a sparse matrix,
recovery of the low-rank and sparse components is a fundamental task subsuming
compressed sensing, matrix completion, and principal components pursuit. This
paper develops algorithms for distributed sparsity-regularized rank
minimization over networks, when the nuclear- and -norm are used as
surrogates to the rank and nonzero entry counts of the sought matrices,
respectively. While nuclear-norm minimization has well-documented merits when
centralized processing is viable, non-separability of the singular-value sum
challenges its distributed minimization. To overcome this limitation, an
alternative characterization of the nuclear norm is adopted which leads to a
separable, yet non-convex cost minimized via the alternating-direction method
of multipliers. The novel distributed iterations entail reduced-complexity
per-node tasks, and affordable message passing among single-hop neighbors.
Interestingly, upon convergence the distributed (non-convex) estimator provably
attains the global optimum of its centralized counterpart, regardless of
initialization. Several application domains are outlined to highlight the
generality and impact of the proposed framework. These include unveiling
traffic anomalies in backbone networks, predicting networkwide path latencies,
and mapping the RF ambiance using wireless cognitive radios. Simulations with
synthetic and real network data corroborate the convergence of the novel
distributed algorithm, and its centralized performance guarantees.Comment: 30 pages, submitted for publication on the IEEE Trans. Signal Proces
Alternating minimization algorithms for graph regularized tensor completion
We consider a low-rank tensor completion (LRTC) problem which aims to recover
a tensor from incomplete observations. LRTC plays an important role in many
applications such as signal processing, computer vision, machine learning, and
neuroscience. A widely used approach is to combine the tensor completion data
fitting term with a regularizer based on a convex relaxation of the multilinear
ranks of the tensor. For the data fitting function, we model the tensor
variable by using the Canonical Polyadic (CP) decomposition and for the
low-rank promoting regularization function, we consider a graph Laplacian-based
function which exploits correlations between the rows of the matrix unfoldings.
For solving our LRTC model, we propose an efficient alternating minimization
algorithm. Furthermore, based on the Kurdyka-{\L}ojasiewicz property, we show
that the sequence generated by the proposed algorithm globally converges to a
critical point of the objective function. Besides, an alternating direction
method of multipliers algorithm is also developed for the LRTC model. Extensive
numerical experiments on synthetic and real data indicate that the proposed
algorithms are effective and efficient
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