4,291 research outputs found
1-Bit Matrix Completion under Exact Low-Rank Constraint
We consider the problem of noisy 1-bit matrix completion under an exact rank
constraint on the true underlying matrix . Instead of observing a subset
of the noisy continuous-valued entries of a matrix , we observe a subset
of noisy 1-bit (or binary) measurements generated according to a probabilistic
model. We consider constrained maximum likelihood estimation of , under a
constraint on the entry-wise infinity-norm of and an exact rank
constraint. This is in contrast to previous work which has used convex
relaxations for the rank. We provide an upper bound on the matrix estimation
error under this model. Compared to the existing results, our bound has faster
convergence rate with matrix dimensions when the fraction of revealed 1-bit
observations is fixed, independent of the matrix dimensions. We also propose an
iterative algorithm for solving our nonconvex optimization with a certificate
of global optimality of the limiting point. This algorithm is based on low rank
factorization of . We validate the method on synthetic and real data with
improved performance over existing methods.Comment: 6 pages, 3 figures, to appear in CISS 201
Chebyshev Polynomial Approximation for Distributed Signal Processing
Unions of graph Fourier multipliers are an important class of linear
operators for processing signals defined on graphs. We present a novel method
to efficiently distribute the application of these operators to the
high-dimensional signals collected by sensor networks. The proposed method
features approximations of the graph Fourier multipliers by shifted Chebyshev
polynomials, whose recurrence relations make them readily amenable to
distributed computation. We demonstrate how the proposed method can be used in
a distributed denoising task, and show that the communication requirements of
the method scale gracefully with the size of the network.Comment: 8 pages, 5 figures, to appear in the Proceedings of the IEEE
International Conference on Distributed Computing in Sensor Systems (DCOSS),
June, 2011, Barcelona, Spai
A Channel Coding Perspective of Collaborative Filtering
We consider the problem of collaborative filtering from a channel coding
perspective. We model the underlying rating matrix as a finite alphabet matrix
with block constant structure. The observations are obtained from this
underlying matrix through a discrete memoryless channel with a noisy part
representing noisy user behavior and an erasure part representing missing data.
Moreover, the clusters over which the underlying matrix is constant are {\it
unknown}. We establish a sharp threshold result for this model: if the largest
cluster size is smaller than (where the rating matrix is of size
), then the underlying matrix cannot be recovered with any
estimator, but if the smallest cluster size is larger than , then
we show a polynomial time estimator with diminishing probability of error. In
the case of uniform cluster size, not only the order of the threshold, but also
the constant is identified.Comment: 32 pages, 1 figure, Submitted to IEEE Transactions on Information
Theor
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