arXiv subjects

Jan Jendrysiak

Publications and source records attributed to Jan Jendrysiak.

3 recordsLinked to original sources

arXiv2026

Computing Homomorphisms of Poset Representations with Applications to Multiparameter Persistence

We present algorithms to compute the vector space of homomorphisms Hom(X,Y) between finitely generated representations of the partially ordered set Z^d. Our results generalise to any partially ordered set. Our main theoretical contribution is a uniqueness result for lifts of homomorphisms along free resolutions, which we use to obtain an algorithm running in O(n^4 (thick(Y) + thick(Omega^1 Y))^2 + T_ker(d,n)) time, where thick(Y) denotes the maximal pointwise dimension of Y and T_ker is the time it takes to compute the kernel of a map between projective Z^d-modules. We also apply and analyse a classical approach due to Green, Heath, and Struble (J. Symbolic Comput., 2001), achieving O(n^3 thick(Y)^3 + n^4). Both improve on the naive O(n^6) bound when thick(Y) is small. Applied to the decomposition algorithm AIDA (Dey-J-Kerber, SoCG '25), the classical approach improves the asymptotic runtime the most, strengthening the result of Dey and Xin (J. Appl. Comput. Topology, 2022) for uniquely graded modules. We implement all algorithms in the Persistence Algebra C++ library and benchmark them on the persistent homology of density-alpha bi-filtrations of immune-cell locations. The classical approach has the best worst-case complexity, yet for 2-parameter modules, the lifting algorithm is fastest in practice.

math.AT↗

arXiv2026

Computing the Skyscraper Invariant

We develop the first algorithms for computing the Skyscraper Invariant [FJNT24]. This is a filtration of the classical rank invariant for multiparameter persistence modules defined by the Harder-Narasimhan filtrations along every central charge supported at a single parameter value. Cheng's algorithm [Cheng24] can be used to compute HN filtrations of arbitrary acyclic quiver representations in polynomial time in the total dimension, but in practice, the large dimension of persistence modules makes this direct approach infeasible. We show that by exploiting the additivity of the HN filtration and the special central charges, one can get away with a brute-force approach. For $d$-parameter modules, this produces an FPT $\varepsilon$-approximate algorithm with runtime dominated by $O( 1/\varepsilon^d \cdot T_{\mathsc{dec}})$, where $T_{\mathsc{dec}}$ is the time for decomposition, which we compute with \textsc{aida} [DJK25]. We show that the wall-and-chamber structure of the module can be computed via lower envelopes of degree $d - 1$ polynomials. This allows for an exact computation of the Skyscraper Invariant whose runtime is roughly $O(n^d \cdot T_{\mathsc{dec}})$ for $n$ the size of the presentation of the modules and enables a faster hybrid algorithm to compute an approximation. For 2-parameter modules, we have implemented not only our algorithms but also, for the first time, Cheng's algorithm. We compare all algorithms and, as a proof of concept for data analysis, compute a filtered version of the Multiparameter Landscape for biomedical data.

cs.DS↗

arXiv2025

Decomposing Multiparameter Persistence Modules

Dey and Xin (J.Appl.Comput.Top., 2022, arXiv:1904.03766) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on \emph{all} finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed parameter tractable with respect to the maximal number of relations with the same degree and with further optimisation we obtain an $O(n^3)$ algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library \textsc{aida} which is the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.

math.RT↗