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Facundo Mémoli

Publications and source records attributed to Facundo Mémoli.

At least 19 recordsLinked to original sources
arXiv2026

Persistence and Topological Complexity

Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower bound, the zero-divisor-cup-length, for persistent topological spaces, and establish their stability. For Vietoris--Rips filtrations of metric spaces, we show that the erosion distances between these persistent invariants are bounded above by twice the Gromov--Hausdorff distance. We develop metric models of classical topological spaces based on cube boundaries with the $\ell_\infty$ metric and their products and wedges, allowing explicit computations of persistent invariants and their distances. These examples show that, in certain cases, persistent topological complexity and zero-divisor-cup-length distinguish spaces that persistent homology, persistent LS-category, and persistent cup-length fail to distinguish.

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arXiv2026

Critical Diameters and Vietoris-Rips Filtrations

We relate critical diameters of finite configurations to the topology of Vietoris-Rips complexes. For a compact metric space and $0<r<s$, we prove that the canonical inclusion from scale $r$ to scale $s$ is a homotopy equivalence whenever $[r,s)$ contains no diameter of a finite labelled configuration at which the diameter function has zero weak slope. On Riemannian manifolds, weak-slope stationarity implies Clarke criticality. At positive diameter, when the distances realizing the diameter are smooth, both are equivalent to first-order stationarity: the absence of a direction decreasing all these distances to first order. For closed connected smooth manifolds, the Clarke critical diameter spectrum has Hausdorff dimension zero, even at the cut locus. In the real-analytic case, it is finite at each fixed number of labels and countable over all label numbers. On the unit round sphere $S^m$, $m\geq1$, positive nonantipodal weak-slope stationary configurations are characterized by nonzero nonnegative equilibrium stresses. The least positive critical diameter is $\arccos(-1/(m+1))$, and the first accumulation point is $\arccos(-1/m)$. We lift every such stress through a spherical stack construction adapted from Lovász, producing stationary diameters that approach the original value from below in the next dimension as the number of layers increases. We also give example in general metric space showing that weak-slope stationary diameters need not correspond to changes in the homotopy type of Vietoris-Rips complexes.

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arXiv2026

Vietoris--Rips Coindex Thresholds for Round Spheres: Spherical Joins and Chromatic Obstructions for Gromov--Hausdorff Distances

For $0\le m\le n$, let $c_{m,n}$ be the infimum of scales at which the Vietoris--Rips filtration of the round sphere $\mathbb{S}^m$ admits a continuous odd map from $\mathbb{S}^n$. A quantitative Borsuk--Ulam theorem gives $c_{m,n}/2\le d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)$, and it was asked whether equality always holds. Using a synchronized product-measure lift of the spherical join, we construct continuous odd maps between Vietoris--Rips metric thickenings with target scale equal to the maximum input scale. Iteration gives $c_{m+d,n+d}\le c_{m,n}$ for every integer $d\ge0$. More generally, the finite join law adds $m_i+1$ and $n_i+1$ separately and bounds the resulting $c$-value by $\max_i c_{m_i,n_i}$. For $0\le r<π$, the pairs $(k,\ell)$ with $c_{k-1,\ell-1}\le r$ are closed under addition, so Fekete's lemma gives a limit for the maximal admissible $\ell/k$ as $k\to\infty$. This structure, exact values, and projective-code estimates give finite and asymptotic bounds. If $1\le m_j c_{m_j,n_j}/2$ for all sufficiently large $j$, producing infinitely many counterexamples.

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arXiv2026

Metric Geometry of Lebesgue, Wasserstein, and Gromov-Wasserstein Spaces: Submetries, Curvature, and Geodesics

A metric space $Z$ gives rise to three natural classes of infinite-dimensional metric spaces associated to $Z$: $p$-Wasserstein spaces of probability measures on $Z$, nonlinear Lebesgue $L^p$-spaces of $Z$-valued maps, and $p$-Gromov-Wasserstein spaces of $Z$-valued kernels. The latter class, referred to as $Z$-Gromov-Wasserstein ($Z$-GW) spaces, extends the classical Gromov-Wasserstein framework from metric measure spaces to more general, possibly attributed, network-like structures, and unifies many GW-type distances that nowadays play a significant role in metric geometry, data science and machine learning. In this article we develop a unified metric-geometric theory of these three classes of spaces, with a particular focus on the $Z$-GW spaces. Our first main result identifies a fundamental submetry structure linking them: the nonlinear Lebesgue space maps via a submetry onto the $Z$-GW space, which in turn maps via a submetry onto the Wasserstein space. This structure provides a mechanism for transferring geometric information among the three spaces. We apply this framework to geodesics and Alexandrov curvature. For $1<p<\infty$, we prove that geodesicity of $Z$ is equivalent to geodesicity of each of the three associated spaces; in the endpoint case $p=1$, all three associated spaces are geodesic, even when $Z$ is not. We also characterize geodesics in the $Z$-GW space as generalized interpolations, extending a known characterization in the classical setting due to Sturm. Finally, we give a complete classification of Alexandrov curvature bounds for these spaces in terms of the curvature of $Z$. Thus, while the main focus of the paper is a new metric-geometric theory of $Z$-GW spaces, the submetry framework also extends classical theorems for Wasserstein and Gromov-Wasserstein spaces and yields new geometric consequences for nonlinear Lebesgue spaces.

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arXiv2026

The Wasserstein transform

We introduce the Wasserstein Transform (WT), a general unsupervised framework for updating distance structures on given data sets with the purpose of enhancing features and denoising. Our framework represents each data point by a probability measure reflecting the neighborhood structure of the point, and then updates the distance by computing the Wasserstein distance between these probability measures. The Wasserstein Transform is a general method which extends the mean shift family of algorithms. We study several instances of WT, and in particular, in one of the instances which we call the Gaussian Transform (GT), we utilize Gaussian measures to model neighborhood structures of individual data points. GT is computationally cheaper than other instances of WT since there exists closed form solution for the $\ell^2$-Wasserstein distance between Gaussian measures. We study the relationship between different instances of WT and prove that each of the instances is stable under perturbations. We devise iterative algorithms for performing the above-mentioned WT and propose several strategies to accelerate GT, such as an observation from linear algebra for reducing the number of matrix square root computations. We examine the performance of the Wasserstein Transform method in many tasks, such as denoising, clustering, image segmentation and word embeddings.

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arXiv2025

The Z-Gromov-Wasserstein Distance

The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze datasets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. With a view toward establishing a general framework for the theory of GW-like distances, this paper considers a vast generalization of the notion of a metric measure space: for an arbitrary metric space $Z$, we define a $Z$-network to be a measure space endowed with a kernel valued in $Z$. We introduce a method for comparing $Z$-networks by defining a generalization of GW distance, which we refer to as $Z$-Gromov-Wasserstein ($Z$-GW) distance. This construction subsumes many previously known metrics and offers a unified approach to understanding their shared properties. This paper demonstrates that the $Z$-GW distance defines a metric on the space of $Z$-networks which retains desirable properties of $Z$, such as separability, completeness, and geodesicity. Many of these properties were unknown for existing variants of GW distance that fall under our framework. Our focus is on foundational theory, but our results also include computable lower bounds and approximations of the distance which will be useful for practical applications.

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arXiv2025

Geometry and Stability of Supervised Learning Problems

We introduce a notion of distance between supervised learning problems, which we call the Risk distance. This distance, inspired by optimal transport, facilitates stability results; one can quantify how seriously issues like sampling bias, noise, limited data, and approximations might change a given problem by bounding how much these modifications can move the problem under the Risk distance. With the distance established, we explore the geometry of the resulting space of supervised learning problems, providing explicit geodesics and proving that the set of classification problems is dense in a larger class of problems. We also provide two variants of the Risk distance: one that incorporates specified weights on a problem's predictors, and one that is more sensitive to the contours of a problem's risk landscape.

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arXiv2025

Ephemeral persistence features and the stability of filtered chain complexes

We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the level of filtered chain complexes includes points with zero persistence which provide additional information to that present at homology level. The resulting invariant, called verbose barcode, which has a stronger discriminating power than the usual barcode, is proved to be stable under certain metrics that are sensitive to these ephemeral points. In some situations, we provide ways to compute such metrics between verbose barcodes. We also exhibit several examples of finite metric spaces with identical (standard) VR barcodes yet with different verbose VR barcodes thus confirming that these ephemeral points strengthen the standard VR barcode.

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arXiv2025

Geometric Bounds for Persistence

In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset $\mathcal{X}$ of a Banach space $\mathbf{Y}$, we analyze the topological features arising in the family $\mathcal{N}_\bullet(\mathcal{X} \subset \mathbf{Y})$ of nested neighborhoods of $\mathcal{X}$ in $\mathbf{Y}$ and provide several geometric bounds on their persistence (lifespans). We begin by examining the lifespans of these homology classes in terms of their filling radii in $\mathbf{Y}$, establishing connections between these lifespans and fundamental invariants in metric geometry, such as the Urysohn width. We then derive bounds on these lifespans by considering the $\ell^\infty$-principal components of $\mathcal{X}$, also known as Kolmogorov widths. Additionally, we introduce and investigate the concept of extinction time of a metric space $\mathcal{X}$: the critical threshold beyond which no homological features persist in any degree. We propose methods for estimating the Čech and Vietoris-Rips extinction times of $\mathcal{X}$ by relating $\mathcal{X}$ to its convex hull and to its tight span, respectively.

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arXiv2025

Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the distortion of (possibly discontinuous) functions between them. The more these functions must distort the metrics, the larger the Gromov--Hausdorff distance must be. Topology has few tools to obstruct the existence of discontinuous functions. However, an arbitrary function $f\colon X\to Y$ induces a continuous map between their Vietoris--Rips simplicial complexes, where the allowable choices of scale parameters depend on how much the function $f$ distorts distances. We can then use equivariant topology to obstruct the existence of certain continuous maps between Vietoris--Rips complexes. With these ideas we bound how discontinuous an odd map between spheres $S^k\to S^n$ with $k>n$ must be, generalizing a result by Dubins and Schwarz (1981), which is the case $k=n+1$. As an application, we recover or improve upon all of the lower bounds from Lim, M{é}moli, and Smith (2022) on the Gromov--Hausdorff distances between spheres of different dimensions. We also provide new upper bounds on the Gromov--Hausdorff distance between spheres of adjacent dimensions.

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arXiv2025

Grassmannian Persistence Diagrams: Special Properties in the 1-Parameter Setting

In this paper, we explore the discriminative power of Grassmannian persistence diagrams of 1-parameter filtrations, examine their relationships with other related constructions, and study their computational aspects. Grassmannian persistence diagrams are defined through Orthogonal Inversion, a notion analogous to Möbius inversion. We focus on the behavior of this inversion for the poset of segments of a linear poset. We demonstrate how Grassmannian persistence diagrams of 1-parameter filtrations are connected to persistent Laplacians via a variant of orthogonal inversion tailored for the reverse-inclusion order on the poset of segments. Additionally, we establish an explicit isomorphism between Grassmannian persistence diagrams and Harmonic Barcodes via a projection. Finally, we show that degree-0 Grassmannian persistence diagrams are equivalent to treegrams, a generalization of dendrograms. Consequently, we conclude that finite ultrametric spaces can be recovered from the degree-0 Grassmannian persistence diagram of their Vietoris-Rips filtrations.

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arXiv2025

Grassmannian Persistence Diagrams

We introduce Orthogonal Möbius Inversion $\mathsf{OI}$, a concept analogous to Möbius inversion on finite posets, which is applicable to order-preservings functions from a finite poset to the Grassmannian $\mathsf{Gr}(V)$ of an inner product space $V$. This notion critically relies on the inner product structure on $V$ enabling it to capture much finer information than standard integer-valued persistence diagrams. Orthogonal Inversion is a special case of the broader concept of Orthomodular Inversion, where the target space is any orthomodular lattice, which we also identify. We apply Orthogonal Inversion in order to construct a "non-negative" persistence diagram for any given multiparameter filtration $\mathsf{F}$ of a finite simplicial complex $K$, indexed over an arbitrary finite poset $P$. This is done by applying it to the birth-death spaces of $\mathsf{F}$. Analogously to $1$-parameter classical persistence diagrams, these multiparameter Grassmannian persistence diagrams offer straightforward interpretability. Specifically, to a segment $(b, d) \in \mathsf{Seg}(P)$, (1) the Grassmannian persistence diagram canonically assigns a vector subspace of $C_ρ^K$ consisting of cycles that are born at $b$ and become boundaries at $d$ and (2) this assignment is exhaustive at the homology level. Finally, we relate our Grassmannian persistence diagrams to the recently introduced notion of Möbius homology, thus enhancing its interpretability through the lens of our framework.

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arXiv2025

$\ell^p$-Stability of Weighted Persistence Diagrams

We introduce the concept of weighted persistence diagrams and develop a functorial pipeline for constructing them from finite metric measure spaces. This builds upon an existing functorial framework for generating classical persistence diagrams from finite pseudo-metric spaces. To quantify differences between weighted persistence diagrams, we define the $p$-edit distance for $p\in [1,\infty]$, and-focusing on the weighted Vietoris-Rips filtration-we establish that these diagrams are stable with respect to the $p$-Gromov-Wasserstein distance as a direct consequence of functoriality. In addition, we present an Optimal Transport-inspired formulation of the $p$-edit distance, enhancing its conceptual clarity. Finally, we explore the discriminative power of weighted persistence diagrams, demonstrating advantages over their unweighted counterparts.

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arXiv2024

Extremal spherical polytopes and Borsuk's conjecture

We generate anti-self-polar polytopes via a numerical implementation of the gradient flow induced by the diameter functional on the space of all finite subsets of the sphere, and prove related results on the critical points of the diameter functional as well as results about the combinatorics of such polytopes. We also discuss potential connections to Borsuk's conjecture.

math.CO↗
arXiv2024

Comparison Results for Gromov-Wasserstein and Gromov-Monge Distances

Inspired by the Kantorovich formulation of optimal transport distance between probability measures on a metric space, Gromov-Wasserstein (GW) distances comprise a family of metrics on the space of isomorphism classes of metric measure spaces. In previous work, the authors introduced a variant of this construction which was inspired by the original Monge formulation of optimal transport; elements of the resulting family are referred to Gromov-Monge (GM) distances. These GM distances, and related ideas, have since become a subject of interest from both theoretical and applications-oriented perspectives. In this note, we establish several theoretical properties of GM distances, focusing on comparisons between GM and GW distances. In particular, we show that GM and GW distances are equal for non-atomic metric measure spaces. We also consider variants of GM distance, such as a Monge version of Sturm's $L_p$-transportion distance, and give precise comparisons to GW distance. Finally, we establish bi-Hölder equivalence between GM distance and an isometry-invariant Monge optimal transport distance between Euclidean metric measure spaces that has been utilized in shape and image analysis applications.

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arXiv2024

The Generalized Rank Invariant: Möbius invertibility, Discriminating Power, and Connection to Other Invariants

In addition to inherent computational challenges, the absence of a canonical method for quantifying `persistence' in multi-parameter persistent homology remains a hurdle in its application. One of the best known quantifications of persistence for multi-parameter persistent homology is the rank invariant, which has recently evolved into the generalized rank invariant (GRI) by naturally extending its domain. This extension enables us to quantify persistence across a broader range of regions in the indexing poset compared to the rank invariant. However, the size of the domain of the GRI is generally formidable, making it desirable to restrict its domain to a more manageable subset for computational purposes. The foremost questions regarding such a restriction of the domain are: (1) How to restrict, if possible, the domain of the GRI without any loss of information? (2) When can we more compactly encode the GRI as a `persistence diagram'? (3) What is the trade-off between computational efficiency and the discriminating power of the GRI as the amount of the restriction on the domain varies? (4) What proxies exist for persistence diagrams in the multi-parameter setting that can be derived from the GRI? To address the first three questions, we generalize and axiomatize the classic fundamental lemma of persistent homology via the notion of Möbius invertibility of the GRI which we propose. This extension also contextualizes known results regarding the (generalized) rank invariant within the classical theory of Möbius inversion. We conduct a comprehensive comparison between Möbius invertibility and other existing concepts related to the structural simplicity of persistence modules. We address the fourth question through the notion of motivic invariants. We demonstrate that many invariants from the literature can be both derived from the GRI and recast as motivic invariants.

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