GPTQ-2D：三次時間雙向自適應舍入

摘要

自適應舍入方法（如 GPTQ，或等價地 Babai 的最近平面演算法）在二次度量下將實數矩陣舍入為整數矩陣。它們按固定順序逐項處理矩陣項，並透過三角回饋矩陣將每次舍入誤差傳播到尚未處理的項。我們研究此任務的雙邊版本，其中固定的非奇異基底矩陣同時作用於殘差的左側和右側；常見的單邊情形是右基底為單位矩陣的特例。將矩陣向量化可把雙邊目標轉換為二次度量，其 Gram 矩陣為 Kronecker 乘積，因此一維演算法可以原樣套用，但需要矩陣維度的四次方時間。我們提出 GPTQ-2D，能以三次方時間產生相同的舍入矩陣。它逐條反對角線地舍入矩陣項；同一條反對角線上的項彼此獨立，可並行舍入。

English

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.