Integer-Quantized Hierarchical Matrices with Adaptive Bit Allocation
Abstract
We present an adaptive bitwidth allocation method for quantizing low-rank factors in hierarchical matrices in order to decrease storage while meeting a given constraint on their reconstruction error. For the bitwidth allocation we formulate a discrete storage minimization problem and show that an optimal assignment can give larger singular values at least as many bits as smaller ones. We further develop a greedy algorithm that meets the specified max-norm quantization error budget. We investigate a hierarchical block partition in which all of the non-diagonal blocks are represented by quantized low-rank factors and only dense blocks remain dense; in this way a global reconstruction bound related to ηk and Rk is obtained, with the former and latter being bounds on the truncation and reconstruction arithmetic errors, respectively. Extensive simulations, carried out with kernel and BEM matrices under block-relative budgets, indicate that the proposed method achieves a reduction in storage of up to 44.3× when compared to storing the complete matrix densely in FP64, and a reduction of up to 5.70× when compared to the FP64 low-rank factors within the same partition, together with measured reconstruction errors that remained below the maximum norm bounds in all the hierarchical tests.
Related
- Joint Quantized Precoding and Bit Allocation for Fronthaul-Constrained Cell-Free Massive MIMO
- Integer symplectic matrices in expanding boxes
- D(N)-quadruples in upper-triangular 2×2 integer matrices
- Factorisability of Low Dimensional Non-Negative Integer Matrices
- Cardinalities of Jordan Diophantine sets of upper triangular integer matrices
BibTeX
@misc{carson2026integerquantized,
title = {{Integer-Quantized Hierarchical Matrices with Adaptive Bit Allocation}},
author = {Erin Carson and Xinye Chen and Ritesh Khan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40134},
eprint = {2610.40134},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40134},
}
RIS
TY - UNPB
TI - Integer-Quantized Hierarchical Matrices with Adaptive Bit Allocation
AU - Erin Carson
AU - Xinye Chen
AU - Ritesh Khan
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40134
UR - https://arxiv.org/abs/2610.40134
AB - We present an adaptive bitwidth allocation method for quantizing low-rank factors in hierarchical matrices in order to decrease storage while meeting a given constraint on their reconstruction error. For the bitwidth allocation we formulate a discrete storage minimization problem and show that an optimal assignment can give larger singular values at least as many bits as smaller ones. We further develop a greedy algorithm that meets the specified max-norm quantization error budget. We investigate a hierarchical block partition in which all of the non-diagonal blocks are represented by quantized low-rank factors and only dense blocks remain dense; in this way a global reconstruction bound related to η_k and R_k is obtained, with the former and latter being bounds on the truncation and reconstruction arithmetic errors, respectively. Extensive simulations, carried out with kernel and BEM matrices under block-relative budgets, indicate that the proposed method achieves a reduction in storage of up to 44.3× when compared to storing the complete matrix densely in FP64, and a reduction of up to 5.70× when compared to the FP64 low-rank factors within the same partition, together with measured reconstruction errors that remained below the maximum norm bounds in all the hierarchical tests.
ER -