Graded Projection Recursion (GPR): Corrections, Obstructions, and Conservative Approximate Matrix Multiplication

TL;DR

This paper revises the Graded Projection Recursion framework to establish a conservative approximate matrix multiplication method that maintains certain guarantees and local tools, addressing previous global obstructions.

Contribution

It introduces a conservative AMM framework that replaces an earlier exact multiplication theorem, providing unbiased estimates with preserved guarantees for protected subspaces.

Findings

The revised framework is unbiased and preserves protected queries.

It localizes stochastic error to residual subspaces.

It addresses global obstructions with a rank/capacity argument.

Abstract

Earlier versions proposed Graded Projection Recursion (GPR) as a deterministic packed-recursion framework for model-honest near-quadratic dense matrix multiplication. This revised version withdraws the exact dense matrix multiplication theorem and the downstream consequences that depended on it with a conservative AMM framework. The local ingredients remain useful as local tools: the three-band packing identity, scaled middle-band extraction under certified gaps, centering and reconstruction identities, and row/column normalization bounds. The gap in the earlier argument is global: the proof relied on a bounded active-state realization that would remove first-mismatch terms through the recursion. For arbitrary dense inputs this would require an exact equality filter over the inner index. We formulate this obstruction as a target-slice/equality-filter problem and give a rank/capacity argument against the natural separable active-state realization. The positive replacement is a conservative approximate matrix multiplication framework. For chosen protected left and right query subspaces, the low/marginal part of AB is computed exactly and an unbiased AMM primitive is applied only to the high/high residual. The resulting estimator is unbiased, preserves protected queries exactly in every realization, localizes stochastic error to the residual subspace, and inherits residual output-norm or query-risk guarantees from the underlying estimator.