arXiv subjects

Luciano Melodia

Publications and source records attributed to Luciano Melodia.

10 recordsLinked to original sources

arXiv2026

Groupoid Homology and Classifying-Space Homology Are Not Isomorphic

For an ample groupoid $\mathcal{G}$, Matui-type groupoid homology $H_\bullet(\mathcal{G};\mathbb{Z})$ is built from the nerve $\mathcal{G}_\bullet$ via the Moore complex of compactly supported locally constant chains $C_c(\mathcal{G}_n,\mathbb{Z})$, with differential the alternating sum of pushforwards along the face maps. For a discrete group the theory agrees with the singular homology of the classifying space. For a totally disconnected locally compact Hausdorff space, viewed as a groupoid of units, it computes compactly supported cohomology instead, which is not a homotopy invariant. We make the resulting discrepancy explicit: for the unit groupoid on the Cantor set $X$ we compute $H_0(\mathcal{G};\mathbb{Z})\cong C(X,\mathbb{Z})$, a countable group, whereas $H^{\text{sing}}_0(B\mathcal{G};\mathbb{Z})\cong\bigoplus_{x\in X}\mathbb{Z}$ has cardinality $2^{\aleph_0}$. Cardinality alone separates the two groups, and it separates them in degree $0$ already. In every positive degree they agree for this groupoid.

math.AT↗

arXiv2026

Persistent Magnitude Homology for Quantitative Equational Theories

A quantitative equational theory $U$ reasons about terms that agree up to a numerical error. It presents a free algebra $T_UA$ over a metric space $A$ of generators, the terms of the syntax at the least distance the axioms derive, and that metric is its semantic content. We give a functorial invariant of it, the persistent magnitude homology of $T_UA$: a barcode where the module is tame, finite linear algebra where $T_UA$ is finite, Lipschitz in each degree. Magnitude homology is graded by length and knows nothing of persistence, its persistent refinement nothing of where its bars begin and end, yet the two are one construction: filtering the length nerve by sublevel sets of the length yields the persistence module, and the associated graded of that filtration is the magnitude complex. A long exact sequence exchanges them, and each side gains what it lacked. Magnitude homology locates the critical values of the barcode, so a graded computation lists the lengths at which an endpoint can occur, and the barcode acquires a stability estimate of $(n+1)δ$ in degree $n$ under a perturbation of size $δ$, and a computed perturbation shows that the factor cannot be dropped. An inclusion of theories induces a morphism of the presenting monads and, where the induced map is bijective and shortens no distance by more than $δ$, a comparison of barcodes under the same bound, so a barcode movement measures the metric-semantic strength of the added axioms. Four examples are computed, one in every degree.

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arXiv2026

Learning Transfers: Kan Extensions for Neural Invariants

A representation transfers if it stays usable once the task has changed. Standard evaluations report target accuracy or a distance between data distributions, but neither says which structure of the representation is meant to survive. Here we make that structure explicit and computable. A task is a small category, a change of task is a functor, and a representation is a functor into a category of invariants. The structure the target has to exhibit is the left Kan extension of the source functor along the change of task. Our transfer discrepancy is the supremum over target objects of the distance between that extension and the observed target invariant, so we score transfer against the invariant the change of task forces rather than against the source. We prove cokernel presentations of the extension over comma categories, in chain complexes and in persistence modules. On one-parameter modules of finite type we show the discrepancy is the bottleneck distance, computed without approximation. We also test on sampled manifolds and on learned latent clouds whether the score recovers the intended change of task and rejects every structural control that we pose.

cs.LG↗

arXiv2026

Discrete Coefficients and Open Invariant Covers in the Homology of Ample Groupoids

The homology of an ample groupoid is computed from the complex of compactly supported continuous functions on the nerve. Two hypotheses routinely imposed on this complex behave in opposite ways. We show that the comparison map from the integral chain complex tensored with the coefficient group to the chain complex with coefficients is always injective, and that it is surjective if and only if every compactly supported continuous function into the coefficient group is locally constant. The universal coefficient theorem for ample groupoids is therefore a discrete coefficient statement, and it already fails for the real numbers on the Cantor set. We then show that a cover of the unit space by two clopen invariant subsets forces the groupoid to split as a disjoint union of three clopen invariant reductions, so that the associated Mayer-Vietoris sequence has vanishing connecting maps and computes nothing. Dropping closedness repairs this, since two open invariant subsets covering the unit space give a Mayer-Vietoris sequence for every topological abelian coefficient group. For an integer action on a two point compactification of the integers we compute that sequence and find that its connecting map is an isomorphism of infinite cyclic groups, which no cover of a unit space by clopen invariant subsets can achieve.

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arXiv2026

Deep Learning Estimation of Absorbed Dose for Nuclear Medicine Diagnostics

In radionuclide therapy with $^{177}\mathrm{Lu}$, the absorbed-dose distribution can be approximated by convolving the time-integrated activity distribution with a dose voxel kernel for a single tissue type. This approximation is fast but inaccurate: it treats the body as homogeneous and therefore ignores the tissue heterogeneity that governs where energy is deposited. The heterogeneity can be recovered by combining computed tomography and single-photon emission computed tomography with a Monte Carlo transport simulation, at a high computational cost. We investigate whether the map from a local density kernel to the corresponding dose voxel kernel can instead be learned from data by a convolutional neural network, so that density-adapted kernels become available without a full transport calculation for each patient. On held-out patient data, the proposed U-residual architecture reaches a continuous intersection-over-union score of $0.86$ after $308$ epochs, with a mean squared error of $1.24\times 10^{-4}$ on the normalised targets. This generalisation to unseen data indicates that the network approximates, rather than merely memorises, the simulation-based map from density kernels to dose voxel kernels. The network does not replace the underlying transport physics; it approximates the density-to-dose association that a full Monte Carlo transport simulation would otherwise have to supply anew for every patient.

stat.ML↗

arXiv2026

Universal Coefficients and Mayer-Vietoris Sequence for Groupoid Homology

We study homology of ample groupoids via the compactly supported Moore complex of the nerve. Let $A$ be a topological abelian group. For $n\ge 0$ set $C_n(\mathcal G;A) := C_c(\mathcal G_n,A)$ and define $\partial_n^A=\sum_{i=0}^n(-1)^i(d_i)_*$. This defines $H_n(\mathcal G;A)$. The theory is functorial for continuous étale homomorphisms. It is compatible with standard reductions, including restriction to saturated clopen subsets. In the ample setting it is invariant under Kakutani equivalence. We reprove Matui type long exact sequences and identify the comparison maps at chain level. For discrete $A$ we prove a natural universal coefficient short exact sequence $$0\to H_n(\mathcal G)\otimes_{\mathbb Z}A\xrightarrow{\ ι_n^{\mathcal G}\ }H_n(\mathcal G;A)\xrightarrow{\ κ_n^{\mathcal G}\ }\operatorname{Tor}_1^{\mathbb Z}\bigl(H_{n-1}(\mathcal G),A\bigr)\to 0.$$ The key input is the chain level isomorphism $C_c(\mathcal G_n,\mathbb Z)\otimes_{\mathbb Z}A\cong C_c(\mathcal G_n,A)$, which reduces the groupoid statement to the classical algebraic UCT for the free complex $C_c(\mathcal G_\bullet,\mathbb Z)$. We also isolate the obstruction for non-discrete coefficients. For a locally compact totally disconnected Hausdorff space $X$ with a basis of compact open sets, the image of $Φ_X:C_c(X,\mathbb Z)\otimes_{\mathbb Z}A\to C_c(X,A)$ is exactly the compactly supported functions with finite image. Thus $Φ_X$ is surjective if and only if every $f\in C_c(X,A)$ has finite image, and for suitable $X$ one can produce compactly supported continuous maps $X\to A$ with infinite image. Finally, for a clopen saturated cover $\mathcal G_0=U_1\cup U_2$ we construct a short exact sequence of Moore complexes and derive a Mayer-Vietoris long exact sequence for $H_\bullet(\mathcal G;A)$ for explicit computations.

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arXiv2026

Persistent Homology as Stopping-Criterion for Voronoi Interpolation

In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact calculating the persistent homology on it after each iteration to capture the changing topology of the data. The boundary points are identified as critical. The Bottleneck and Wasserstein distance serve as a measure of quality between the original point set and the interpolation. If the norm of two distances exceeds a heuristically determined threshold, the algorithm terminates. We give the theoretical basis for this approach and justify its validity with numerical experiments.

cs.CG↗

arXiv2026

Estimate of the Neural Network Dimension using Algebraic Topology and Lie Theory

In this paper we present an approach to determine the smallest possible number of neurons in a layer of a neural network in such a way that the topology of the input space can be learned sufficiently well. We introduce a general procedure based on persistent homology to investigate topological invariants of the manifold on which we suspect the data set. We specify the required dimensions precisely, assuming that there is a smooth manifold on or near which the data are located. Furthermore, we require that this space is connected and has a commutative group structure in the mathematical sense. These assumptions allow us to derive a decomposition of the underlying space whose topology is well known. We use the representatives of the $k$-dimensional homology groups from the persistence landscape to determine an integer dimension for this decomposition. This number is the dimension of the embedding that is capable of capturing the topology of the data manifold. We derive the theory and validate it experimentally on toy data sets.

stat.ML↗

arXiv2026

Homological Time Series Analysis of Sensor Signals from Power Plants

In this paper, we use topological data analysis techniques to construct a suitable neural network classifier for the task of learning sensor signals of entire power plants according to their reference designation system. We use representations of persistence diagrams to derive necessary preprocessing steps and visualize the large amounts of data. We derive deep architectures with one-dimensional convolutional layers combined with stacked long short-term memories as residual networks suitable for processing the persistence features. We combine three separate sub-networks, obtaining as input the time series itself and a representation of the persistent homology for the zeroth and first dimension. We give a mathematical derivation for most of the used hyper-parameters. For validation, numerical experiments were performed with sensor data from four power plants of the same construction type.

cs.LG↗

arXiv2024

Algebraic and Topological Persistence

This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory. Moreover, we present the fundamental concepts of algebraic topology, including exact and short exact sequences and relative homology groups derived from quotienting with subspaces of a topological space. These tools are used to prove the Excision Theorem in algebraic topology. Subsequently, the theorem is applied to demonstrate the equivalence of simplicial and singular homology for triangulable topological spaces, i.e. those topological spaces which admit a simplicial structure. This enables a more general theory of homology to be adopted in the study of filtrations of point clouds. The chapter on homological persistence makes use of these tools throughout. We develop the theory of persistent homology, the homology of filtrations of topological spaces, and the corresponding dual concept of persistent cohomology. This work aims to provide mathematicians with a robust foundation for productive engagement with the aforementioned theories. The majority of the proofs have been rewritten to clarify the relationships between the techniques discussed. The novel aspect of this contribution is the canonical presentation of persistence theory and the associated ideas through a rigorous mathematical treatment for triangulable topological spaces and closing some gaps in the existing literature.

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