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A theory of meta-factorization

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arxiv 2111.14385 v4 pith:A34KPJZG submitted 2021-11-29 math.NA cs.NA

classification math.NAcs.NA
keywords meta-factorizationfactorizationsmatrixgeneralizedlinearprovidestheoryalgebra
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We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nystr\"{o}m method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. UnWeaving the knots of GraphRAG -- turns out VectorRAG is almost enough

    cs.IR 2026-02 unverdicted novelty 6.0 of 10

    UnWeaver disentangles documents into entities via LLM to retrieve original chunks, yielding a simpler alternative to GraphRAG that still reduces noise and preserves source fidelity.

  2. On the Fundamental Impossibility of Hallucination Control in Large Language Models

    stat.ML 2025-06 reject novelty 5.0 of 10

    The paper claims a mathematical impossibility: every capable LLM must violate at least one of four idealized response properties, so hallucination is structurally inevitable.

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