arXiv subjects

Mark Fornace

Publications and source records attributed to Mark Fornace.

5 recordsLinked to original sources

arXiv2026

Direct RNA sequence design under codon constraints using expressive tensor-based secondary structure models

Nucleic acid sequence design via codon optimization is a fundamental task with applications across synthetic biology, mRNA therapeutics, and vaccine design. Given a target protein, it is a major open challenge to navigate the combinatorially large design space of codon sequences mapping to its amino acid sequence. Computational approaches generally seek to optimize simple objectives based on the codon sequence, possibly together with more complicated contributions based on secondary structure analysis. In this work, we demonstrate a direct and efficient algorithm to sample sequences from a suitable Boltzmann distribution defined in terms of the codon sequence and a fully detailed secondary structure free energy model, as well as related algorithms for exact computation of statistical quantities such as free energies, base pairing probabilities, and base and codon marginals. These algorithms draw upon a recently developed tensor-based formulation of secondary structure thermodynamics and demonstrate, for the first time, that global sequence design can be accomplished with respect to a highly accurate free energy model. Moreover, the algorithms can leverage any available CPU and GPU resources in parallel for massive computational speedups.

q-bio.QM↗

arXiv2026

Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop

This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop Linear Systems and Eigenvalue Problems, which was organized at the Simons Institute for the Theory of Computing program on Complexity and Linear Algebra in Fall 2025. The complexity and numerical solution of linear algebra problems is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions. (Updated to reflect the status of the open problems as of August 20, 2026.)

math.NA↗

arXiv2025

An approximation theory for Markov chain compression

We develop a framework for the compression of reversible Markov chains with rigorous error control. Given a subset of selected states, we construct reduced dynamics that can be lifted to an approximation of the full dynamics, and we prove simple spectral and nuclear norm bounds on the recovery error in terms of a suitably interpreted Nystr\"{o}m approximation error. We introduce two compression schemes: a projective compression based on committor functions and a structure-preserving compression defined in terms of an induced Markov chain over the selected states. The Nystr\"{o}m error appearing in our bounds can be controlled using recent results on column subset selection by nuclear maximization. Numerical experiments validate our theory and demonstrate the scalability of our approach.

math.NA↗

arXiv2025

Fast and Accurate Interpolative Decompositions for General, Sparse, and Structured Tensors

In this work, we develop deterministic and random sketching-based algorithms for two types of tensor interpolative decompositions (ID): the core interpolative decomposition (CoreID, also known as the structure-preserving HOSVD) and the satellite interpolative decomposition (SatID, also known as the HOID or CURT). We adopt a new adaptive approach that leads to ID error bounds independent of the size of the tensor. In addition to the adaptive approach, we use tools from random sketching to enable an efficient and provably accurate calculation of these decompositions. We also design algorithms specialized to tensors that are sparse or given as a sum of rank-one tensors, i.e., in the CP format. Besides theoretical analyses, numerical experiments on both synthetic and real-world data demonstrate the power of the proposed algorithms.

math.NA↗

arXiv2024

Column and row subset selection using nuclear scores: algorithms and theory for Nystr\"{o}m approximation, CUR decomposition, and graph Laplacian reduction

Column selection is an essential tool for structure-preserving low-rank approximation, with wide-ranging applications across many fields, such as data science, machine learning, and theoretical chemistry. In this work, we develop unified methodologies for fast, efficient, and theoretically guaranteed column selection. First we derive and implement a sparsity-exploiting deterministic algorithm applicable to tasks including kernel approximation and CUR decomposition. Next, we develop a matrix-free formalism relying on a randomization scheme satisfying guaranteed concentration bounds, applying this construction both to CUR decomposition and to the approximation of matrix functions of graph Laplacians. Importantly, the randomization is only relevant for the computation of the scores that we use for column selection, not the selection itself given these scores. For both deterministic and matrix-free algorithms, we bound the performance favorably relative to the expected performance of determinantal point process (DPP) sampling and, in select scenarios, that of exactly optimal subset selection. The general case requires new analysis of the DPP expectation. Finally, we demonstrate strong real-world performance of our algorithms on a diverse set of example approximation tasks.

math.NA↗