GPTQ-2D: Cubic-Time Two-Sided Adaptive Rounding

2026-07-29

AI Summary

This study addresses the problem of two-sided adaptive rounding, which involves rounding a real matrix to an integer matrix under a quadratic metric induced by left and right nonsingular basis matrices. The authors propose a parallel rounding algorithm based on anti-diagonal ordering, exploiting the independence of elements along each anti-diagonal to enable efficient computation. This approach reduces the time complexity from $O(n^4)$ to $O(n^3)$ for the first time while producing results identical to those of the original algorithm. By integrating a quadratic metric constructed via Kronecker products with the Babai nearest-plane heuristic, the method achieves accuracy equivalent to vectorized one-dimensional GPTQ but with substantially lower computational overhead.

Abstract

Adaptive rounding methods such as GPTQ, or equivalently Babai's nearest plane algorithm, round a real matrix to integers under a quadratic metric. They process the entries in a fixed order, one at a time, propagating each rounding error to the entries not yet processed through a triangular feedback matrix. We study the two-sided version of this task, in which fixed nonsingular basis matrices act on both the left and the right of the residual; the familiar one-sided case is the special case of an identity right basis. Vectorizing the matrix turns the two-sided objective into a quadratic metric whose Gram matrix is a Kronecker product, so the one-dimensional algorithm applies verbatim, but takes quartic time in the matrix dimension. We present GPTQ-2D, which produces the identical rounded matrix in cubic time. It rounds the entries anti-diagonal by anti-diagonal; entries on the same anti-diagonal are independent and are rounded in parallel.

Problem

• adaptive rounding
• two-sided rounding
• GPTQ
• Kronecker product
• matrix quantization

Innovation

• GPTQ-2D
• two-sided adaptive rounding
• cubic-time algorithm
• Kronecker product
• anti-diagonal parallelization