817 research outputs found
Matrix inequalities involving the Khatri-Rao product
summary:We extend three inequalities involving the Hadamard product in three ways. First, the results are extended to any partitioned blocks Hermitian matrices. Second, the Hadamard product is replaced by the Khatri-Rao product. Third, the necessary and sufficient conditions under which equalities occur are presented. Thereby, we generalize two inequalities involving the Khatri–Rao product
Fast Exact Leverage Score Sampling from Khatri-Rao Products with Applications to Tensor Decomposition
We present a data structure to randomly sample rows from the Khatri-Rao
product of several matrices according to the exact distribution of its leverage
scores. Our proposed sampler draws each row in time logarithmic in the height
of the Khatri-Rao product and quadratic in its column count, with persistent
space overhead at most the size of the input matrices. As a result, it
tractably draws samples even when the matrices forming the Khatri-Rao product
have tens of millions of rows each. When used to sketch the linear least
squares problems arising in CANDECOMP / PARAFAC tensor decomposition, our
method achieves lower asymptotic complexity per solve than recent
state-of-the-art methods. Experiments on billion-scale sparse tensors validate
our claims, with our algorithm achieving higher accuracy than competing methods
as the decomposition rank grows.Comment: To appear at the 37th Conference on Neural Information Processing
Systems (Neurips'23). 28 pages, 10 figures, 6 table
DFacTo: Distributed Factorization of Tensors
We present a technique for significantly speeding up Alternating Least
Squares (ALS) and Gradient Descent (GD), two widely used algorithms for tensor
factorization. By exploiting properties of the Khatri-Rao product, we show how
to efficiently address a computationally challenging sub-step of both
algorithms. Our algorithm, DFacTo, only requires two sparse matrix-vector
products and is easy to parallelize. DFacTo is not only scalable but also on
average 4 to 10 times faster than competing algorithms on a variety of
datasets. For instance, DFacTo only takes 480 seconds on 4 machines to perform
one iteration of the ALS algorithm and 1,143 seconds to perform one iteration
of the GD algorithm on a 6.5 million x 2.5 million x 1.5 million dimensional
tensor with 1.2 billion non-zero entries.Comment: Under review for NIPS 201
New Holder - Type Inequalities for the Tracy-Singh and Khatri-Rao Products of Positive Matrices
Recently, the authors establised a number of inequalities involving Khatri-Rao product of two positives matrices. Here, in this paper, the result are establised in three ways. First, we find new Holder-type inequalities for Tracy-Singh and Khatri-Rao products products of positive semi-devinite matrices. Second, the result are extended to provide estimates of sums of the Khatri-Rao and Tracy-Singh products of any finite number of positive semi-definite matrices. Three, the result lead to inequalities involving the Hadamard and Kronecker, as a special case
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