Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transport

Matteo Pegoraro

stat.MLMar 16, 2026

Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transport

Authors: Matteo Pegoraro

Organizations: Institute of Computing, Faculty of Informatics, Università della Svizzera Italiana

Abstract

We improve and extend persistence spheres, introduced in~\cite{pegoraro2025persistence}. Persistence spheres map an integrable measure $μ$ on the upper half-plane, including persistence diagrams (PDs) as counting measures, to a function $S(μ)\in C(\mathbb{S}^2)$ , and the map is stable with respect to 1-Wasserstein partial transport distance $\mathrm{POT}_1$ . Moreover, to the best of our knowledge, persistence spheres are the first explicit representation used in topological machine learning for which continuity of the inverse on the image is established at every compactly supported target. Recent bounded-cardinality bi-Lipschitz embedding results in partial transport spaces, despite being powerful, are not given by the kind of explicit summary map considered here. Our construction is rooted in convex geometry: for positive measures, the defining ReLU integral is the support function of the lift zonoid. Building on~\cite{pegoraro2025persistence}, we refine the definition to better match the $\mathrm{POT}_1$ deletion mechanism, encoding partial transport via a signed diagonal augmentation. In particular, for integrable $μ$ , the uniform norm between $S(0)$ and $S(μ)$ depends only on the persistence of $μ$ , without any need of ad-hoc re-weightings, reflecting optimal transport to the diagonal at persistence cost. This yields a parameter-free representation at the level of measures (up to numerical discretization), while accommodating future extensions where $μ$ is a smoothed measure derived from PDs (e.g., persistence intensity functions~\citep{wu2024estimation}). Across clustering, regression, and classification tasks involving functional data, time series, graphs, meshes, and point clouds, the updated persistence spheres are competitive and often improve upon persistence images, persistence landscapes, persistence splines, and sliced Wasserstein kernel baselines.

Figures & tables

Figure 1: Example of the lift zonoid construction for a discrete measure $\mu=\sum_{i}c_{i}\delta_{p_{i}}$ . Each atom contributes the segment $c_{i}[0,(1,p_{i})]\subset\mathbb{R}^{3}$ , and $Z_{\mu}$ is obtained as their Minkowski sum.

	Shared across (i)–(iii)				(i) Baseline		(ii) $\pers^{2}$		(iii) Step
$\sigma$	POT 1	SW	PL	PSpl	PI	PSph	PI	PSph	PI	PSph
$10$	$0.760\pm 0.228$	$0.527\pm 0.044$	$\mathbf{0.984\pm 0.086}$	$0.964\pm 0.111$	$0.783\pm 0.238$	$0.530\pm 0.076$	$0.958\pm 0.129$	$0.762\pm 0.206$	$0.933\pm 0.163$	$0.672\pm 0.200$
$15$	$0.631\pm 0.152$	$0.522\pm 0.039$	$\mathbf{0.949\pm 0.140}$	$0.771\pm 0.238$	$0.581\pm 0.165$	$0.512\pm 0.045$	$0.867\pm 0.202$	$0.700\pm 0.188$	$0.554\pm 0.122$	$0.565\pm 0.130$
$20$	$0.547\pm 0.068$	$0.526\pm 0.048$	$\mathbf{0.846\pm 0.218}$	$0.641\pm 0.215$	$0.527\pm 0.053$	$0.502\pm 0.010$	$0.812\pm 0.214$	$0.642\pm 0.163$	$0.523\pm 0.050$	$0.513\pm 0.027$

Table 1: FDA unsupervised simulations (i)–(iii). Mean

\pm

standard deviation of Rand index over

100

independent runs. Since only persistence images (PI) and persistence spheres (PSph) change across scenarios (i)–(iii), the values of POT 1 , SW, PL, and PSpl are reported only once for each noise level.

$\sigma$	POT 1	SW	PSph	PL	PI	PSpl
$10$	$\mathbf{0.975\pm 0.107}$	$0.616\pm 0.172$	$0.641\pm 0.186$	$0.515\pm 0.071$	$0.795\pm 0.240$	$0.496\pm 0.001$
$15$	$\mathbf{0.907\pm 0.191}$	$0.625\pm 0.181$	$0.653\pm 0.195$	$0.518\pm 0.081$	$0.753\pm 0.243$	$0.496\pm 0.003$
$20$	$\mathbf{0.888\pm 0.205}$	$0.650\pm 0.191$	$0.645\pm 0.190$	$0.511\pm 0.054$	$0.704\pm 0.240$	$0.496\pm 0.002$

Table 2: FDA unsupervised simulation (iv). Mean

\pm

standard deviation of Rand index over

100

independent runs for scenario (iv), where the two classes share the largest oscillations and the sample size is fixed to

n=200

Panel of Figure 2 — Figure 2: FDA simulations. Left: realizations from the baseline generative model used in scenarios (i)–(iii), where the discriminative signal is more likely to be carried by large oscillations while noise induces abundant low-persistence clutter. Right: realizations from scenario (iv), where the largest oscillations are shared across the two classes, so that an excessive bias toward high-persistence features becomes detrimental.

	POT 1	SW	PSph	PL	PI	PSpl
$H_{0}$	$0.859\pm 0.010$	$0.859\pm 0.010$	$0.858\pm 0.010$	$0.596\pm 0.052$	$\mathbf{0.950\pm 0.058}$	$0.283\pm 0.015$
$H_{1}$	$0.872\pm 0.006^{\dagger}$	$\mathbf{0.873\pm 0.009}$	$0.861\pm 0.013$	$0.630\pm 0.211$	$0.754\pm 0.019$	$0.285\pm 0.018$

Table 3: PP unsupervised. Mean

\pm

standard deviation of Rand index over

100

runs. Bold denotes the best mean in each row. A dagger † marks methods whose 95% confidence interval for the mean (with

n=100

runs) overlaps with that of the best method.

Method	Corr. with POT 1
SW	$0.996\pm 0.004$
PSph	$0.935\pm 0.055$
PL	$0.822\pm 0.071$
PI	$0.886\pm 0.092$
PSpl	$0.527\pm 0.102$

Table 4: PP correlation. Mean

\pm

standard deviation of the per-iteration, per-homological-degree correlation between POT 1 and each competing distance, computed over repeated runs. Since

H_{0}

and

H_{1}

are computed on the same data in each iteration, the two correlations within an iteration are not independent; accordingly, we only report descriptive statistics.

Panel of Figure 3 — Figure 3: PP simulation. Top row: one simulated point pattern for each point process. Bottom row: corresponding persistence diagrams in degrees $H_{0}$ (blue) and $H_{1}$ (red). Ordering: Poisson, Thomas, Matérn, jittered lattice.

Panel of Figure 4 — Figure 4: Examples of data, persistence diagrams, and PSph representations appearing in the supervised experiments of Section 9 .

	PSph	PSph*	PI	PL	PSpl	PersLay	SWK
Regression
Eyeglasses	$0.960\pm 0.004^{\dagger}$	$0.966\pm 0.003^{\dagger}$	$0.969\pm 0.006^{\dagger}$	$0.955\pm 0.018^{\dagger}$	$\mathbf{0.971\pm 0.011^{\dagger}}$	$0.248\pm 0.031$	$\mathbf{0.971\pm 0.003^{\dagger}}$
Tecator	$\mathbf{0.973\pm 0.007^{\dagger}}$	$0.969\pm 0.009^{\dagger}$	$0.940\pm 0.018$	$0.954\pm 0.011$	$\mathbf{0.973\pm 0.009^{\dagger}}$	$0.895\pm 0.029$	$0.953\pm 0.010$
Classification
Growth	$\mathbf{0.900\pm 0.047^{\dagger}}$	$0.850\pm 0.052^{\dagger}$	$0.836\pm 0.056$	$0.768\pm 0.060$	$0.836\pm 0.043$	$0.807\pm 0.043$	$0.768\pm 0.058$
NOx	$0.863\pm 0.044^{\dagger}$	$\mathbf{0.869\pm 0.041^{\dagger}}$	$0.803\pm 0.043$	$0.789\pm 0.062$	$0.860\pm 0.058^{\dagger}$	$0.717\pm 0.078$	$0.840\pm 0.055^{\dagger}$

Table 5: Supervised case studies. We report average

R^{2}

for regression and average accuracy for classification, across

5

runs for “Eyeglasses” and

10

runs for the remaining tasks. Values are reported as mean

\pm

standard deviation. PSph* denotes the weighted persistence-sphere construction introduced in Pegoraro (2026) . All benchmarks are taken from Pegoraro (2026) ; in the present paper, only the PSph and PI columns are recomputed, while the remaining columns are reported from that work. Bold entries indicate the best-performing method in each row. A dagger † marks methods whose 95% confidence interval, computed using the appropriate number of repetitions for the corresponding row, overlaps with that of the best method.

Panel of Figure 5 — Figure 5: One-point diagrams drifting along $(1,1)$ . Left: attenuation of a pure diagonal offset between $D_{k}$ and $D_{k}^{\prime}$ . Right: behavior of the deletion-to-diagonal cost for $D_{k}$ . The updated definition matches the $\POT_{1}$ geometry by construction, whereas in Pegoraro (2026) both effects are mediated by the decay of the reweighting $\omega_{K}$ along diagonal lines.

Panel of Figure 6 — Figure 6: Two elementary mechanisms by which standard topological summaries deform the $\POT_{1}$ geometry. Persistence landscapes amplify variability at high persistence, while Gaussian persistence images induce a kernel-dependent saturation of pairwise distances. In the latter case, for fixed bandwidth $\sigma$ , the discrepancy with $\POT_{1}$ becomes more pronounced at high persistence, since the persistence-image distance saturates at a fixed kernel scale whereas the $\POT_{1}$ flattening occurs only at the much larger deletion scale determined by the sum of the persistences.

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Sierpiński--Knopp Wasserstein Distance for Persistence Diagrams and Applications to 2-Wasserstein Approximation

Sebastien Tchitchek, Julien Tierny

CNRS and Sorbonne Université

Persistent Homology Wasserstein Distance

Open paper AskPDF arXiv

This paper introduces the Sierpiński-Knopp (SK) Wasserstein distance, a fast metric between persistence diagrams. The SK-Wasserstein distance, denoted

d_{\mathrm{SK}}

, maps diagram points and their diagonal projections to the unit interval via the Sierpiński-Knopp space-filling curve on the upper diagonal triangle. The encoded point sets are then efficiently matched via one-dimensional optimal assignment, in

O(N\log N)

steps, yielding an explicit diagonal-aware point assignment between the two input persistence diagrams. We show that the SK-Wasserstein distance controls the classical

2

-Wasserstein distance between diagrams, admits an explicit isometric embedding into a Hilbert space, and induces a positive-definite Gaussian kernel, making the resulting geometry directly compatible with Euclidean and kernel-based learning methods. A tighter surrogate dissimilarity, noted

W_Γ

, is also introduced based on the point assignments along the curve. Experiments on 12 scientific collections comprising 227 diagrams show median per-collection speedup of

d_{\mathrm{SK}}

over state-of-the-art approximations of

W_2

626\times

, while the aggregate speedup over the full benchmark is

2100\times

. Average-linkage partitions obtained from

d_{\mathrm{SK}}

and

W_Γ

each exactly match the corresponding

W_2

partition on 8 of the 12 collections. Hilbert

k

-means and Gaussian spectral clustering, both based on

d_{\mathrm{SK}}

, achieve mean adjusted Rand indices (ARI) of

0.756

and

0.800

, respectively, with respect to the benchmark reference partitions, compared to

0.750

obtained by average linkage on

W_2

. The Gaussian

d_{\mathrm{SK}}

kernel supports other kernel-based analysis tasks, as illustrated by its use for contiguous segmentation of ordered diagram collections in our experiments.

Abstract

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