Graded Projection Recursion (GPR): Corrections, Obstructions, and Conservative Approximate Matrix Multiplication
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909032341569536 |
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| author | Uhlmann, Jeffrey |
| author_facet | Uhlmann, Jeffrey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11988 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graded Projection Recursion (GPR): Corrections, Obstructions, and Conservative Approximate Matrix Multiplication Uhlmann, Jeffrey Computational Complexity Data Structures and Algorithms 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. |
| title | Graded Projection Recursion (GPR): Corrections, Obstructions, and Conservative Approximate Matrix Multiplication |
| topic | Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2511.11988 |