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.
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Sparse Random Khatri-Rao Product Codes for Distributed Matrix Multiplication
We introduce two generalizations to the paradigm of using Random Khatri-Rao Product (RKRP) codes for distributed matrix multiplication. We first introduce a class of codes called Sparse Random Khatri-Rao Product (SRKRP) codes which have sparse generator matrices. SRKRP codes result in lower encoding, computation and communication costs than RKRP codes when the input matrices are sparse, while they exhibit similar numerical stability to other state of the art schemes. We empirically study the relationship between the probability of the generator matrix (restricted to the set of non-stragglers) of a randomly chosen SRKRP code being rank deficient and various parameters of the coding scheme including the degree of sparsity of the generator matrix and the number of non-stragglers. Secondly, we show that if the master node can perform a very small number of matrix product computations in addition to the computations performed by the workers, the failure probability can be substantially improved.
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- Award ID(s):
- 2008714
- PAR ID:
- 10427618
- Date Published:
- Journal Name:
- IEEE Information Theory Workshop
- Page Range / eLocation ID:
- 416 to 421
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ITW54588.2022.9965842
