Topological Distance: Metrics & Applications

• Topological Distance (TD) is a mathematical framework that quantifies dissimilarity between topological objects, maps, and datasets using persistent homology and related tools.
• It incorporates robust methods such as distance-to-a-measure, kernel distances, and bottleneck comparisons to mitigate noise and outlier effects.
• TD finds applications in diverse fields—including geometry, machine learning, quantum systems, and game theory—facilitating rigorous analysis and comparison.

Topological Distance (TD) encompasses a wide range of mathematical constructions and methodologies that quantify “distance” or dissimilarity between topological objects, maps, data sets, or structures. These measures, emerging from topology, persistent homology, applied statistics, computational geometry, and machine learning, formalize the comparison of shapes, data clouds, manifolds, graphs, scalar fields, and even strategic game configurations in rigorous mathematical terms. TD is foundational in both theoretical analysis and data-driven applications where the notion of “closeness” refers not just to geometric proximity, but also to the preservation or disruption of topological, combinatorial, or homotopy-theoretic features. The following sections survey key classes and recent developments in topological distance from the perspective of both pure and applied mathematics.

1. Classical and Robust Function-Based Topological Distances

The foundational model for topological distance is the point-to-set distance function associated to a compact set $S \subset \mathbb{R}^d$, defined as

$\Delta_S(x) = \inf_{y \in S} \|x - y\|.$

Persistent homology tracks the evolution of topological features (components, loops, voids) in the sublevel sets $L_t = \{x : \Delta_S(x) \leq t\}$, yielding a persistence diagram as a multiset of (birth, death) pairs reflecting feature lifetimes. In empirical settings, this is replaced by the empirical distance function $\hat\Delta(x) = \min_{i=1,...,n} \|x - X_i\|$, which has a breakdown point of zero, making it extremely sensitive to noise and outliers.

To address this, robust alternatives have been introduced. The distance-to-a-measure (DTM) function $\delta_{P, m}(x)$ averages over distances to the $k = \lceil mn \rceil$ nearest neighbors for some $0 < m < 1$: $$\delta_{P, m}^2(x) = \frac{1}{m} \int_0^m F_x^{-1}(u)\, du$$ where $F_x(t) = P(\{y : \|y-x\|^2 \leq t\})$. The kernel distance uses a characteristic positive definite kernel $K$ to define

$$D_K(P, \delta_x) = \left(\iint K(z, y)\, dP(z)\, dP(y) + K(x, x) - 2\int K(x, y)\, dP(y) \right)^{1/2},$$

thereby smoothing discontinuities and granting statistical robustness to topological inference.

The central metric for comparing persistent diagrams is the bottleneck distance

$$W_{\infty}(D_1, D_2) = \min_{g: D_1 \rightarrow D_2} \sup_{z \in D_1} \|z - g(z)\|_{\infty},$$

with the stability theorem $W_\infty(D_f, D_g) \leq \|f - g\|_\infty$, establishing that robust function choices lead to stable topological summaries (Chazal et al., 2014).

2. Topological Distance between Maps, Spaces, or Scalar Fields

In homotopy theory, distances between continuous (or simplicial) maps are formalized via cover-based or combinatorial measures. The higher homotopic distance $D(f_1, \ldots, f_n)$ is defined as the minimal $k$ such that the domain admits an open covering by $k + 1$ sets on each of which all maps are homotopic (or c-contiguous in the discrete/simplicial setting): $$D(f_1, \dots, f_n) = \min \{\, k \mid X = \bigcup_{j=0}^k U_j,\, f_i|_{U_j} \simeq f_l|_{U_j}\; \forall i, l \, \}.$$ This unifies with invariants such as Lusternik–Schnirelmann category (cat), sectional category (secat), and topological complexity (TC$_n$) (Borat et al., 2019, İs et al., 2022).

Discrete analogues, such as simplicial distance (Borat, 2020) and higher contiguity distance (Yazici et al., 8 Mar 2024), operate on barycentric subdivisions and combinatorial covers, utilizing chains of contiguous simplicial maps to compare or relate the discrete maps. For simplicial maps $f, g: K \rightarrow L$, the stabilized simplicial distance SimpD$(f,g)$ is the minimal $k$ such that $S_d^b(K)$ is covered by subcomplexes $J_i$ with $c$-contiguous approximations of $f$ and $g$ on $J_i$. These constructions are critical for digital topology, motion planning, and computational topology.

3. Topological Distance for Diagrams, Graphs, and High-dimensional Data

In Topological Data Analysis, distances between persistence diagrams (PDs) are fundamental for statistical analysis of datasets. Classical metrics include the $q$-Wasserstein and bottleneck distances. Recent developments enable faster and theoretically tunable pseudodistances.

Extended Topological Pseudodistances (ETD) (Nuñez et al., 22 Feb 2024) are defined via 1D projections of PDs: $$\operatorname{ETD}_A(P_1, P_2) = \left( \sum_j \left[ \sum_{\theta \in A} W_p\big(\pi_\theta( P_1^j \cup \widetilde{P}_2^j),\, \pi_\theta( P_2^j \cup \widetilde{P}_1^j )\big)^p \right]^{1/p} \right)^{1/p},$$ where $\pi_\theta$ projects to lines at angle $\theta$ and $\widetilde{P}_i^j$ denotes diagram balancing terms. As $|A|$ grows, ETD converges to the Sliced Wasserstein Distance, while for small $|A|$ it mimics computationally efficient Persistence Statistics.

Graph distances informed by algebraic topology replace classical metrics with invariants such as the marked length spectrum, as in the Truncated Non-Backtracking Spectral Distance (TNBSD) (Torres et al., 2018), which compares the complex spectra of non-backtracking matrices encoding counts of non-backtracking cycles (i.e., free homotopy classes).

For scalar functions on graphs or Euclidean spaces, Scalar Function Topology Divergence (SFTD) (Trofimov et al., 11 Jul 2024) quantifies the difference in multi-scale topological structure not only by persistence intervals, but also by localizing features, using a cross-barcode and summing powers of death-birth differences at matched locations.

4. Integrating Geometry and Topology: Simplicial Hausdorff Distance

A recent approach is the simplicial Hausdorff distance (Nnadi et al., 6 Feb 2025), which fuses geometric and topological comparison of data in the form of point clouds represented as simplicial complexes. For two measured complexes $(X, f)$ and $(Y, g)$ with vertex mappings into $\mathbb{R}^d$, the directed distance is

$$\vec{d}((X, f), (Y, g)) = \max_{k} \max_{\sigma \in X: \dim \sigma = k} \min_{\tau \in Y: \dim \tau = k} \left[\, \max_{v \in \sigma} \min_{w \in \tau} d( f(v), g(w) )\, \right],$$

and the symmetric version

$$\delta((X, f), (Y, g)) = \max\{\, \vec{d}((X, f), (Y, g)),\, \vec{d}((Y, g), (X, f)) \}.$$

This construction ensures both facewise topological matching and geometric proximity. Injectivity of the measurement functions $f$ and $g$ is important for positive-definiteness; otherwise, the associated comparison defines a pseudometric rather than a metric.

5. Manifold Distance, Topological Phase Transitions, and Quantum Systems

In quantum information and condensed matter, topological distance can be defined in terms of overlaps (fidelity, trace distance) between ground state wave functions of two Hamiltonians or two parameter regions. The manifold distance formalizes this as

$$d_1 = 1 - |\langle \psi_k | \phi_{k'} \rangle|^2,\quad d_2 = \sqrt{ 1 - |\langle \psi_k | \phi_{k'} \rangle|^2 },$$

integrated over momentum or parameter space: $$\mathcal{D}_1 = \int d\mathbf{k}\ d_1(\mathbf{k}, \mathbf{k}'),\quad \mathcal{D}_2 = \int d\mathbf{k}\ d_2(\mathbf{k}, \mathbf{k}').$$ At topological phase transitions, the manifold distance is continuous, but higher derivatives exhibit universal divergences—critical signatures of phase boundaries. This distance is applicable to a broad class of topological systems: Hermitian, non-Hermitian, and models with topological order, with extensions expected for real-space and mixed-state scenarios (Fang et al., 6 May 2024).

6. Topological Distance in Machine Learning and Strategic Games

In modern machine learning, topological distances serve as objective functions or evaluation metrics sensitive to structural/combinatorial rather than purely statistical differences. Notable examples include:

• Topology Distance (TD) for GANs: Comparing topological summaries (persistence diagrams, longevity vectors) of feature spaces of real and generated data, yielding metrics sensitive to the manifold structure of data clouds. For input feature sets $F_r, F_g$, define the longevity vectors $l(F_r), l(F_g)$ from their persistence diagrams; the distance is $\|l(F_r) - l(F_g)\|_2$ (Horak et al., 2020).
• Topological Distance Games (TDGs): Utility functions for agents placed on a topology graph depend on both inherent agent-to-agent utilities and their pairwise shortest path distances, $u_i(a) = \sum_{j \neq i} f(d_G(a(i), a(j))) u_i(j)$. Stability and equilibrium concepts are defined in terms of this distance, leading to new complexity and existence results for equilibria in settings combining hedonic games, social distance games, and Schelling games (Bullinger et al., 2022).

7. Applications, Performance, and Design Choices

Topological distances are integral to tasks such as clustering, anomaly detection, model selection, motion planning, shape and texture classification, geometric inference under noise, image segmentation, and the detection of phase transitions in quantum materials. Several key features and observations are common:

• Robustness: Techniques based on DTM, kernel distances, and nonbacktracking cycles mitigate the effects of noise and outliers.
• Computational tuning: Methods such as ETDs or SFTD allow regulation of granularity and complexity, interpolating between fast summary statistics and more discriminative optimal transport-based metrics.
• Statistical inference: Central limit theorems, concentration bounds, and bootstrap procedures permit the assignment of confidence to topological summaries, critical for rigorous inference (Chazal et al., 2014).
• Localization: New measures such as SFTD achieve sensitivity to the spatial occurrence of features, surpassing global summary comparisons.
• Algorithmic complexity: Many methods underline the trade-off between theoretical soundness and computational feasibility, with practical algorithms scaling polynomially in the size and dimension of the data, subject to domain constraints.

Topological Distance Type	Principal Formula/Approach	Domain/Main Application
DTM/Kernels & Persistence Diagrams	DTM $\delta_{P,m}(x)$; bottleneck $W_\infty$	Geometry, shape inference, TDA
Simplicial/Simplicial Hausdorff	SimpD, $\delta((X, f), (Y, g))$	Digital topology, point clouds
Graph Non-backtracking Spectra	TNBSD via non-backtracking matrix	Network comparison, clustering
Extended Topological Pseudodistance	ETD via projections and 1D Wasserstein	Fast diagram comparison, machine learning
Manifold Distance (Quantum)	$d_1$, $d_2$ (overlap of ground states)	Phase transitions, quantum systems
Scalar Function Topology Divergence	F–Cross–Barcode, SFTD sum	Computer vision, shape segmentation
Homotopic/Contiguity Distance	$D(f_1, ..., f_n)$, SD$(\phi_1, ..., \phi_n)$	Homotopy theory, robotics
Game-Theory TDG Utility	$u_i(a) = \sum_j f(d_G(a(i), a(j))) u_i(j)$	Coalition/placement games

A plausible implication is that as data and computational resources scale, the design of TDs will increasingly blend geometry, combinatorics, and statistical learning theory, offering tunable and interpretable metrics that remain robust across disparate applications. In all cases, the essential mathematical idea is to construct a metric or pseudometric on topological summaries—functions, diagrams, spaces, or combinatorial structures—so as to compare, classify, or control them in analytic or algorithmic pipelines.