In this paper, we propose an efficient numerical scheme for solving some large‐scale ill‐posed linear inverse problems arising from image restoration. In order to accelerate the computation, two different hidden structures are exploited. First, the coefficient matrix is approximated as the sum of a small number of Kronecker products. This procedure not only introduces one more level of parallelism into the computation but also enables the usage of computationally intensive matrix–matrix multiplications in the subsequent optimization procedure. We then derive the corresponding Tikhonov regularized minimization model and extend the fast iterative shrinkage‐thresholding algorithm (FISTA) to solve the resulting optimization problem. Because the matrices appearing in the Kronecker product approximation are all structured matrices (Toeplitz, Hankel, etc.), we can further exploit their fast matrix–vector multiplication algorithms at each iteration. The proposed algorithm is thus called
Faster Linear Algebra for Distance Matrices
The distance matrix of a dataset X of n points with respect to a distance function
f represents all pairwise distances between points in X induced by f. Due to their
wide applicability, distance matrices and related families of matrices have been
the focus of many recent algorithmic works. We continue this line of research
and take a broad view of algorithm design for distance matrices with the goal of
designing fast algorithms, which are specifically tailored for distance matrices, for
fundamental linear algebraic primitives. Our results include efficient algorithms
for computing matrixvector products for a wide class of distance matrices, such
as the l1 metric for which we get a linear runtime, as well as a quadratic lower
bound for any algorithm which computes a matrixvector product for the l_infty case.
Our upper bound results have many
further downstream applications, including the fastest algorithm for computing
a relative error lowrank approximation for the distance matrix induced by l1
and l2 functions and the fastest algorithm for computing an additive error lowrank
approximation for the l2 metric, in addition to applications for fast matrix
multiplication among others. We also give algorithms for constructing distance
matrices and show that one can construct an approximate l2 distance matrix in
time faster than the bound implied by the JohnsonLindenstrauss lemma.
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 Award ID(s):
 2022448
 NSFPAR ID:
 10430252
 Date Published:
 Journal Name:
 Conference on Neural Information Processing Systems
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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