163,971 research outputs found
On the Power of Adaptivity in Matrix Completion and Approximation
We consider the related tasks of matrix completion and matrix approximation
from missing data and propose adaptive sampling procedures for both problems.
We show that adaptive sampling allows one to eliminate standard incoherence
assumptions on the matrix row space that are necessary for passive sampling
procedures. For exact recovery of a low-rank matrix, our algorithm judiciously
selects a few columns to observe in full and, with few additional measurements,
projects the remaining columns onto their span. This algorithm exactly recovers
an rank matrix using observations,
where is a coherence parameter on the column space of the matrix. In
addition to completely eliminating any row space assumptions that have pervaded
the literature, this algorithm enjoys a better sample complexity than any
existing matrix completion algorithm. To certify that this improvement is due
to adaptive sampling, we establish that row space coherence is necessary for
passive sampling algorithms to achieve non-trivial sample complexity bounds.
For constructing a low-rank approximation to a high-rank input matrix, we
propose a simple algorithm that thresholds the singular values of a zero-filled
version of the input matrix. The algorithm computes an approximation that is
nearly as good as the best rank- approximation using
samples, where is a slightly different coherence parameter on the matrix
columns. Again we eliminate assumptions on the row space
Convergence of Alternating Least Squares Optimisation for Rank-One Approximation to High Order Tensors
The approximation of tensors has important applications in various
disciplines, but it remains an extremely challenging task. It is well known
that tensors of higher order can fail to have best low-rank approximations, but
with an important exception that best rank-one approximations always exists.
The most popular approach to low-rank approximation is the alternating least
squares (ALS) method. The convergence of the alternating least squares
algorithm for the rank-one approximation problem is analysed in this paper. In
our analysis we are focusing on the global convergence and the rate of
convergence of the ALS algorithm. It is shown that the ALS method can converge
sublinearly, Q-linearly, and even Q-superlinearly. Our theoretical results are
illustrated on explicit examples.Comment: tensor format, tensor representation, alternating least squares
optimisation, orthogonal projection metho
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