arXiv subjects

João Paixão

Publications and source records attributed to João Paixão.

6 recordsLinked to original sources

arXiv2026

Least-Squares and Low-Rank Approximation for Linear Relations Using a Diagrammatic Language

We employ the machinery of linear relations to the study of optimization problems in linear algebra. We first show that the relational version of the pseudo-inverse can be realized through a generalization of the least-squares problem. This allows one to prove that the pseudo-inverse realizes the solution of certain relational optimization problems. Our main result is showing that a certain truncation of this pseudo-inverse defines a solution to a relational version of the classical low-rank approximation problem which recovers both the Eckart-Young Theorem and several optimization problems involving pairs of matrices and vector spaces.

cs.SC↗

arXiv2025

Generalizing the Invertible Matrix Theorem with Linear Relations using Graphical Linear Algebra

Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent or spans the whole space; a linear function has a right or a left inverse; a linear function is surjective or injective; and the kernel of a matrix is trivial or the its image is full. The Invertible Matrix Theorem ties all these ideas and many others together. Many modern linear algebra books use this theorem as a guiding principle to explain many connections in linear algebra. The main idea is to separately characterize whether the linear function is surjective or injective. The proof usually uses a matrix decomposition as the key step. However, the invertible matrix theorem deals with a single linear function, a single set of vectors, a single subspace, and a single matrix. In this work, we generalize part of the invertible matrix theorem to results about a pair of linear functions, a pair of sets of vectors, a pair of subspaces, and a single linear relation. The main idea is to separately characterize the linear relation's fundamental properties -- whether it is surjective, injective, deterministic and total. Our proof uses a decomposition of a linear relation as the key step. Unfortunately, reasoning with linear relations in classical notation requires applying many rules besides shuffling quantifiers and variables around, which can obscure the symmetries in the results. Therefore, this work employs graphical linear algebra, a two-dimensional diagrammatic syntax with the fundamental rules of linear relations built-in.

cs.SC↗

arXiv2020

Discrete Line Fields on Surfaces

Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse--Smale decomposition of a (generic) field plays a fundamental role, relating the geometric structure of phase space to a combinatorial object consisting of critical points and separatrices. Such concepts led Forman to a satisfactory theory of discrete vector fields, in close analogy to the continuous case. In this paper, we introduce discrete line fields. Again, our definition is rich enough to provide the counterparts of the basic results in the theory of continuous line fields: a Euler-Poincaré formula, a Morse--Smale decomposition and a topologically consistent cancellation of critical elements, which allows for topological simplification of the original discrete line field.

cs.CG↗

arXiv2015

Random collapsibility and 3-sphere recognition

A triangulation of a $3$-manifold can be shown to be homeomorphic to the $3$-sphere by describing a discrete Morse function on it with only two critical faces, that is, a sequence of elementary collapses from the triangulation with one tetrahedron removed down to a single vertex. Unfortunately, deciding whether such a sequence exist is believed to be very difficult in general. In this article we present a method, based on uniform spanning trees, to estimate how difficult it is to collapse a given $3$-sphere triangulation after removing a tetrahedron. In addition we show that out of all $3$-sphere triangulations with eight vertices or less, exactly $22$ admit a non-collapsing sequence onto a contractible non-collapsible $2$-complex. As a side product we classify all minimal triangulations of the dunce hat, and all contractible non-collapsible $2$-complexes with at most $18$ triangles. This is complemented by large scale experiments on the collapsing difficulty of $9$- and $10$-vertex spheres. Finally, we propose an easy-to-compute characterisation of $3$-sphere triangulations which experimentally exhibit a low proportion of collapsing sequences, leading to a heuristic to produce $3$-sphere triangulations with difficult combinatorial properties.

math.GT↗

arXiv2013

Parameterized Complexity of Discrete Morse Theory

Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds, through a reduction to the erasability problem. Here, we refine the study of the complexity of problems related to discrete Morse theory in terms of parameterized complexity. On the one hand we prove that the erasability problem is W[P]-complete on the natural parameter. On the other hand we propose an algorithm for computing optimal Morse matchings on triangulations of 3-manifolds which is fixed-parameter tractable in the treewidth of the bipartite graph representing the adjacency of the 1- and 2-simplexes. This algorithm also shows fixed parameter tractability for problems such as erasability and maximum alternating cycle-free matching. We further show that these results are also true when the treewidth of the dual graph of the triangulated 3-manifold is bounded. Finally, we investigate the respective treewidths of simplicial and generalized triangulations of 3-manifolds.

cs.CG↗

arXiv2012

Computational topology and normal surfaces: Theoretical and experimental complexity bounds

In three-dimensional computational topology, the theory of normal surfaces is a tool of great theoretical and practical significance. Although this theory typically leads to exponential time algorithms, very little is known about how these algorithms perform in "typical" scenarios, or how far the best known theoretical bounds are from the real worst-case scenarios. Here we study the combinatorial and algebraic complexity of normal surfaces from both the theoretical and experimental viewpoints. Theoretically, we obtain new exponential lower bounds on the worst-case complexities in a variety of settings that are important for practical computation. Experimentally, we study the worst-case and average-case complexities over a comprehensive body of roughly three billion input triangulations. Many of our lower bounds are the first known exponential lower bounds in these settings, and experimental evidence suggests that many of our theoretical lower bounds on worst-case growth rates may indeed be asymptotically tight.

math.GT↗