arXiv subjects

Nikola Milićević

Publications and source records attributed to Nikola Milićević.

7 recordsLinked to original sources

arXiv2026

A McCord-type theorem for pseudotopological spaces and directed graphs

We construct weak homotopy equivalences between the geometric realizations of directed Vietoris-Rips complexes and their underlying directed graphs, seen as pseudotopological spaces. Pseudotopological spaces are a generalization of (Čech) closure spaces which in turn generalize topological spaces, but they also include graphs and directed graphs as full subcategories, making them a natural bridge that connects classical algebraic topology with the more applied side of topology. This weak homotopy equivalence implies that singular homology groups of finite directed graphs can be efficiently calculated from finite combinatorial structures, despite their associated chain groups being infinite dimensional. Along the way, we establish analogues of classical results such as the existence of a long exact sequence for homotopy groups of pairs of pseudotopological spaces and that a weak homotopy equivalence induces isomorphisms for homology groups. This work is similar to the work of McCord for finite topological spaces but in the context of pseudotopological spaces. Our results also give a novel approach for studying (higher) homotopy groups of discrete mathematical structures such as (directed) graphs or digital images.

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arXiv2026

LandscapeSHAP: Which Persistent Homology Class Gets the Credit?

Shapley values, a solution concept from cooperative game theory, have recently become a standard tool for feature credit allocation in machine learning. They provide an axiomatically justified method to fairly distribute a model's prediction among the data features. Shapley values have not yet been applied to explain machine learning models trained on features from topological data analysis. We develop what we believe is the first such approach, focusing on the persistence landscape featurization of persistence diagrams. Because each landscape coordinate is a rank statistic, crediting a model's prediction back to individual persistent homology classes (persistence diagram points) is nontrivial. We introduce LandscapeSHAP, a method for fair credit allocation to persistence diagram points based on a model's prediction. For linear models on persistence landscapes, LandscapeSHAP has a closed form expression that gives the exact Shapley value of every persistence diagram point. In particular, there is no coalition sampling required. We further prove that the four Shapley "fairness" axioms uniquely characterize this credit allocation for any model, not only linear ones. For a general nonlinear model, this unique value can only be calculated exactly from its defining coalition averaging formula, which requires considering all $2^N$ many coalitions, where $N$ is the number of points in the persistence diagram. This is computationally intractable for persistence diagrams of realistic size. We complement the exact linear model result with an efficient Monte Carlo sampling of persistence diagram coalitions. We give convergence rates in terms of number of samples needed to approximate to a desired degree of accuracy. We also prove stability results for the LandscapeSHAP credit allocation, for any model.

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arXiv2026

A continuum of Künneth theorems for persistence modules

We develop new aspects of the homological algebra theory for persistence modules, in both the one-parameter and multi-parameter settings. For a poset $P$ and an order preserving map $φ:P\times P\to P$, we introduce a novel tensor product of persistence modules indexed by $P$, $\otimes_φ$. We prove that each $\otimes_φ$ has a right adjoint, $\mathbf{Hom}^φ$, the internal hom of persistence modules that also depends on $φ$. We prove that every $\otimes_φ$ yields a Künneth short exact sequence of chain complexes of persistence modules. Dually, the $\mathbf{Hom}^φ$ also has an associated Künneth short exact sequence in cohomology. As special cases both of these short exact sequences yield Universal Coefficient Theorems. We show how to apply these to chain complexes of persistence modules arising from filtered CW complexes. For the special case of $P=\mathbb{R}_+$, the $p$-quasinorms for each $p\in (0,\infty]$ yield a distinct $\otimes_{\ell^p_c}$ and its adjoint $\mathbf{Hom}^{\ell^p_c}$. We compute their derived functors, $\mathbf{Tor}^{\ell^p_c}$ and $\mathbf{Ext}_{\ell^p_c}$ explicitly for interval modules. We show that the Universal Coefficient Theorem developed can be used to compute persistent Borel-Moore homology of a filtration of non-compact spaces. Finally, we show that for every $p\in [1,\infty]$ the associated Künneth short exact sequence can be used to significantly speed up and approximate persistent homology computations in a product metric space $(X\times Y,d^p)$ with the distance $d^p((x,y),(x',y'))=||d_X(x,x'),d_Y(y,y')||_p$.

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arXiv2025

Singular homology of roots of unity

We extend some basic results from the singular homology theory of topological spaces to the setting of Čech's closure spaces. We prove analogues of the excision and Mayer-Vietoris theorems and the Hurewicz theorem in dimension one. We use these results to calculate examples of singular homology groups of spaces that are not topological but are often encountered in applied topology, such as simple undirected graphs. We focus on the singular homology of roots of unity with closure structures arising from considering nearest neighbors. These examples can then serve as building blocks along with our Mayer-Vietoris and excision theorems for computing the singular homology of more complex closure spaces.

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arXiv2024

The combinatorial code and the graph rules of Dale networks

We describe the combinatorics of equilibria and steady states of neurons in threshold-linear networks that satisfy Dale's law. The combinatorial code of a Dale network is characterized in terms of two conditions: (i) a condition on the network connectivity graph, and (ii) a spectral condition on the synaptic matrix. We find that in the weak coupling regime the combinatorial code depends only on the connectivity graph, and not on the particulars of the synaptic strengths. Moreover, we prove that the combinatorial code of a weakly coupled network is a sublattice, and we provide a learning rule for encoding a sublattice in a weakly coupled excitatory network. In the strong coupling regime we prove that the combinatorial code of a generic Dale network is intersection-complete and is therefore a convex code, as is common in some sensory systems in the brain.

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arXiv2024

Homotopy, homology, and persistent homology using closure spaces

We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories, and three simplicial singular homology theories. Applied to filtrations of closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance from metric spaces to filtrations of closure spaces and use it to prove that any persistence module obtained from a homotopy-invariant functor on closure spaces is stable.

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arXiv2021

Eilenberg-Steenrod homology and cohomology theories for Čech's closure spaces

We generalize some of the fundamental results of algebraic topology from topological spaces to Čech's closure spaces, also known as pretopological spaces. Using simplicial sets and cubical sets with connections, we define three distinct singular (relative) simplicial and six distinct singular (relative) cubical (co)homology groups of closure spaces. Using acyclic models we show that the three simplicial groups have isomorphic cubical analogues among the six cubical groups. Thus, we obtain a total of six distinct singular (co)homology groups of closure spaces. Each of these is shown to have a compatible homotopy theory that depends on the choice of a product operation and an interval object. We give axioms for an Eilenberg-Steenrod (co)homology theory with respect to a product operation and an interval object. We verify these axioms for three of our six (co)homology groups. For the other three, we verify all of the axioms except excision, which remains open for future work. We also show the existence of long exact sequences of (co)homology groups coming from Künneth theorems and short exact (co)chain complex sequences of pairs of closure spaces.

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