59 research outputs found
A Note on Alternating Minimization Algorithm for the Matrix Completion Problem
We consider the problem of reconstructing a low rank matrix from a subset of
its entries and analyze two variants of the so-called Alternating Minimization
algorithm, which has been proposed in the past. We establish that when the
underlying matrix has rank , has positive bounded entries, and the graph
underlying the revealed entries has bounded degree and diameter
which is at most logarithmic in the size of the matrix, both algorithms succeed
in reconstructing the matrix approximately in polynomial time starting from an
arbitrary initialization. We further provide simulation results which suggest
that the second algorithm which is based on the message passing type updates,
performs significantly better.Comment: 8 pages, 2 figure
Weighted -minimization for generalized non-uniform sparse model
Model-based compressed sensing refers to compressed sensing with extra
structure about the underlying sparse signal known a priori. Recent work has
demonstrated that both for deterministic and probabilistic models imposed on
the signal, this extra information can be successfully exploited to enhance
recovery performance. In particular, weighted -minimization with
suitable choice of weights has been shown to improve performance in the so
called non-uniform sparse model of signals. In this paper, we consider a full
generalization of the non-uniform sparse model with very mild assumptions. We
prove that when the measurements are obtained using a matrix with i.i.d
Gaussian entries, weighted -minimization successfully recovers the
sparse signal from its measurements with overwhelming probability. We also
provide a method to choose these weights for any general signal model from the
non-uniform sparse class of signal models.Comment: 32 Page
Giant Component in Random Multipartite Graphs with given Degree Sequences
We study the problem of the existence of a giant component in a random multipartite graph. We consider a random multipartite graph with p parts generated according to a given degree sequence n[superscript d][subscript i](n),nβ₯1 which denotes the number of vertices in part i of the multipartite graph with degree given by the vector d in an n-node graph. We assume that the empirical distribution of the degree sequence converges to a limiting probability distribution. Under certain mild regularity assumptions, we characterize the conditions under which, with high probability, there exists a component of linear size. The characterization involves checking whether the Perron-Frobenius norm of the matrix of means of a certain associated edge-biased distribution is greater than unity. We also specify the size of the giant component when it exists. We use the exploration process of Molloy and Reed Molloy and Reed (1995) to analyze the size of components in the random graph. The main challenges arise due to the multidimensionality of the random processes involved which prevents us from directly applying the techniques from the standard unipartite case. In this paper we use techniques from the theory of multidimensional Galton-Watson processes along with Lyapunov function technique to overcome the challenges
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