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 matrix-vector 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 matrix-vector product for the l_infty case.
Our upper bound results have many
further downstream applications, including the fastest algorithm for computing
a relative error low-rank 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 Johnson-Lindenstrauss lemma.
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This content will become publicly available on October 17, 2024
Matrix Compression via Randomized Low Rank and Low Precision Factorization
Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner. However, modern matrices can involve billions of elements, making their storage and processing quite demanding in terms of computational resources and memory usage. Although prohibitively large, such matrices are often approximately low rank. We propose an algorithm that exploits this structure to obtain a low rank decomposition of any matrix A as A≈LR, where L and R are the low rank factors. The total number of elements in L and R can be significantly less than that in A. Furthermore, the entries of L and R are quantized to low precision formats −− compressing A by giving us a low rank and low precision factorization. Our algorithm first computes an approximate basis of the range space of A by randomly sketching its columns, followed by a quantization of the vectors constituting this basis. It then computes approximate projections of the columns of A onto this quantized basis. We derive upper bounds on the approximation error of our algorithm, and analyze the impact of target rank and quantization bit-budget. The tradeoff between compression ratio and approximation accuracy allows for flexibility in choosing these parameters based on specific application requirements. We empirically demonstrate the efficacy of our algorithm in image compression, nearest neighbor classification of image and text embeddings, and compressing the layers of LlaMa-7b. Our results illustrate that we can achieve compression ratios as aggressive as one bit per matrix coordinate, all while surpassing or maintaining the performance of traditional compression techniques.
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- Award ID(s):
- 2037304
- NSF-PAR ID:
- 10482591
- Publisher / Repository:
- arXiv
- Date Published:
- Journal Name:
- arXivorg
- ISSN:
- 2331-8422
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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