arXiv subjects

Michał P. Karpowicz

Publications and source records attributed to Michał P. Karpowicz.

5 recordsLinked to original sources

arXiv2026

A theory of meta-factorization

Meta-factorization asks how different factors and transforms describe the same matrix and how a factor can be replaced while preserving that matrix. Projector and reconstruction equations give a common description. For $A=FGH^*$ with outer factors of full column rank $k$, $Y^*F=H^*X=I_k$ certifies generally oblique projectors. Exact reconstruction requires their ranges to contain the column spaces of $A$ and $A^*$. Two design matrices parametrize every certifying pair; the core is unique and nonsingular at $k=\operatorname{rank}(A)$. Detailed constructions recover SVD, column-pivoted QR, UTV, generalized Nystrom, CUR and classical pseudoinverse formulas with their structural hypotheses. The framework makes factor replacement testable from stored factors. For a supplied candidate $\check F$ and transform $\check Y^*$, with $H$ of full column rank, $\check G=(\check Y^*F)G$ preserves $A$ exactly when $[F-\check F(\check Y^*F)]G=0$. Test, repair and exact factored residual require no access to $A$. With the pseudoinverse as transform, the updated product is a closest reconstruction within the candidate's column space in every unitarily invariant norm. We characterize the accepted subspaces under the stated inverse and rank conditions and extend replacement to different stored widths. Nonsingular generalized right-hand sides change certifying coordinates; finite-order choices yield potent operators and periodic reconstruction. Classical Fourier components test preservation of the core at every step, including singular cores. The contribution is a common description with explicit replacement and reconstruction consequences. Numerical acceptance, conditioning and candidate selection require further analysis.

math.NA↗

arXiv2026

On the Fundamental Impossibility of Hallucination Control in Large Language Models

Large language models hallucinate. This paper shows when that is unavoidable and what we can do about it. We model inference as an auction of ideas, in which a model's components, each holding partial knowledge, compete to shape the answer. We then prove Impossibility Theorems showing that whenever a query makes LLM components contest a fact they hold in common, no aggregation of their reports can at once report that knowledge truthfully, avoid manufacturing confidence beyond what it supports, keep the relevant components engaged, and give the best answer. Something must give, and each failure is familiar: a fabricated detail, unearned confidence, ignored knowledge, or a needlessly weak reply. This is no artifact of one design. It reappears when components report probabilities, and inside the transformer itself, where the combined answer is credited more confidence than the internal contributions supplied. The unbalanced semantic budget cannot be settled from within. Factual truth lies outside the model, and in the worst case no internal signal can certify it. What can be certified is support. Given externally authorized evidence, checking that an answer stays within what the evidence entails needs only the answer and the evidence, and we prove when that check is computable. However, a correct answer can lack support, and a supported answer can be false. What counts as evidence, how far beyond it we allow answers to reach, and which failures we can live with are choices no model can make for us.

stat.ML↗

arXiv2026

Generalized Inverses of Matrix Products: From Fundamental Subspaces to Randomized Decompositions

We investigate the Moore-Penrose pseudoinverse and generalized inverse of a matrix product $A=CR$ to establish a unifying framework for generalized and randomized matrix inverses. This analysis is rooted in first principles, focusing on the geometry of the four fundamental subspaces. We examine: (1) the reverse order law, $A^+ = R^+C^+$, which holds when $C$ has independent columns and $R$ has independent rows, (2) the universally correct formula, $A^+ = (C^+CR)^+(CRR^+)^+$, providing a geometric interpretation of the mappings between the involved subspaces, (3) a new generalized randomized formula, $A^+_p = (P^TA)^+P^TAQ(AQ)^+$, which gives $A^+_p = A^+$ if and only if the sketching matrices $P$ and $Q$ preserve the rank of $A$, i.e., $\mathrm{rank}(P^TA) = \mathrm{rank}(AQ) = \mathrm{rank}(A)$. The framework is extended to generalized $\{1,2\}$-inverses and specialized forms, revealing the underlying structure of established randomized linear algebra algorithms, including randomized SVD, the Nyström approximation, and CUR decomposition. We demonstrate applications in sparse sensor placement and effective resistance estimation. For the latter, we provide a rigorous quantitative analysis of an approximation scheme, establishing that it always underestimates the true resistance and deriving a worst-case spectral bound on the error of resistance differences.

math.NA↗

arXiv2024

The Pseudoinverse of $A=CR$ is $A^+=R^+C^+$ (?)

This paper gives three formulas for the pseudoinverse of a matrix product $A = CR$. The first is sometimes correct, the second is always correct, and the third is almost never correct. But that third randomized pseudoinverse $A^+_r$ may be very useful when $A$ is a very large matrix. 1. $A^+ = R^+C^+$ when $A = CR$ and $C$ has independent columns and $R$ has independent rows. 2. $A^+ = (C^+CR)^+(CRR^+)^+$ is always correct. 3. $A^+_r = (P^TCR)^+P^TCRQ(CRQ)^+ = A^+$ only when $\mathrm{rank}(P^TA) = \mathrm{rank}(AQ) = \mathrm{rank}(A)$ with $A = CR$.

math.NA↗

arXiv2023

The secret life of matrix factorizations: how matrix decompositions reveal and keep secrets of linear equations and what we can do about it

This paper explores the relationship between matrix factorizations and linear matrix equations. It shows that every matrix factorization defines two hidden projectors, one for the column space and one for the row space of a matrix, and how to calculate them. The projectors can be applied to solve linear matrix equations, generate low-rank approximations, or design randomized matrix algorithms. But also, as demonstrated, they can be applied in cryptography to encrypt and decrypt messages. The paper discusses some of the security implications of this application and leaves some questions open for further investigation. The basic concepts are illustrated with source code listings. Finally, this work shares some personal reflections on the meaning and importance of understanding in the time of the artificial intelligence revolution.

math.NA↗