Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams
Kaifeng Zhang School of Artificial Intelligence Nanjing University, China [email protected]
Kai Ming Ting School of Artificial Intelligence Nanjing University, China [email protected]
arXiv:2607.27126v1 [cs.LG] 29 Jul 2026
Abstract Persistence Diagram (PD) is known to capture point cloud topology effectively, but its computation has high time complexity. Expected Persistence Diagram (EPD) has been developed to reduce the time cost by studying the topology of multiple subsets of a point cloud and it serves as a distribution of topological features. Existing EPD vectorizations often rely on predefined point transformations, such as Gaussian or landscape functions. We study an alternative discretization based on Voronoi histograms, which trades smooth functional approximation for adaptive partition-based counting. We propose to use Voronoi Diagram-based histogram as the vectorization of EPD, without imposing an explicit smooth point transformation model. Under stated separation and normalization conditions, we establish stability bounds and characterize when the histogram representation preserves Wassersteinscale variation. We demonstrate the effectiveness of our proposed representation on real-world datasets which have significant topological features for classification and dimensionality reduction tasks.
1
Introduction
Topological Data Analysis (TDA) [62] aims to capture topological features of point cloud X through Persistent Homology (PH) [27]. PH analyzes the topological features of X through a nested sequence of simplicial complexes [54] called filtration. The birth and death of a cycle, e.g., a ring (1-dimensional cycle) or a void (2-dimensional cycle), of the sequence are encoded in Persistence Diagram (PD) D = {ri = (bi , di ) ∈ Ω|1 ≤ i ≤ N (D)}, where each r ∈ D is referred as a topological feature, Ω = {(t1 , t2 ) ∈ R2 |t2 > t1 } is an open half-plane and N (D) is the number of topological features. PD has been used to address problems in various fields, such as biology [44, 45, 64, 65] and chemistry [42, 57]. Each PD can be utilized as a topological summary of a point cloud and used in various machine learning tasks. The challenge is that PD cannot be directly fed as input into a machine learning algorithm. Vectorization [1, 8, 12, 20, 26, 48] and kernel methods [13, 39, 41, 50] for PD are developed to meet this challenge. A key problem in using PD is its high time complexity. As pointed by 69, the time complexity to obtain PD in the worst case is O(m3 ), where m is the number of simplices in the filtration. When we consider the 1-dimensional PD of point cloud X, i.e., rings, we need to consider up to 2-simplices and the time complexity can be O(|X|9 ). Several works have attempted to reduce the time cost by subsampling [11, 16, 17]. We study a vectorization method for a subsample-induced PD called Expected Persistence Diagram (EPD). Since a PD D = {ri = (bi , di ) ∈ Ω|1 ≤ i ≤ N (D)} can PN (D) be equivalently represented as a measure µ = i=1 δri , where ri is a topological and δr Pfeature n is the Dirac point mass at r ∈ Ω. Empirical EPD is defined as the average µ̄ = n1 j=1 µi , where each µi is the PD of a sampled point cloud Xi ⊂ X of the same size, referred as sampled PD for short. Equivalently, an EPD is a distribution of topological features supported on the open half Preprint.
plane Ω = {(t1 , t2 ) ∈ R2 |t2 > t1 } [16], with each topological feature r = (b, d) ∈ Ω encoding the birth and death of a cycle in the filtration. Current methods [17, 63] buildP a representation of EPD by averaging the vectors of sampled PDs in n their core computations, i.e. n1 i=1 Ψ(µi ). For the choice of Ψ, 17 uses the Persistence Landscape (PL); 63 extends the choice from PL to Persistence Image (PI) [1], Persistence Silhouettes (PS) [18] and Persistence Weighted Gaussian Kernels (PWGK) [40]. In order to vectorize each sampled PD D, a predefined continuous point transformation function f [12], whose value reaches its peak at r ∈ D, is used to transform point r in PD into a function. The discrete form of op({f (r)}r∈D ) is then used as the vectorization of D, where op is a permutation invariant operation. When op is summation, averaging the vectors of sampled PDs is equivalent to averaging the vectors f (r) of each topological feature r in EPD. Existing EPD vectorizations make different design choices. PI, PS, and PL provide smooth and stable functional summaries, but require selecting a point transformation and discretization grid. These choices can blur local mass differences or emphasize certain regions of the (birth,death) plane. We propose to treat the empirical EPD as a distribution of topological features, and utilize Vrep, a simple histogram based on Voronoi Diagrams [49], to represent an EPD effectively. Vrep makes a different choice: it uses a data-dependent Voronoi partition and records empirical mass in each cell. The main contributions of this work are: (1) Propose to represent EPD as a distribution of topological features by using histogram based on an unsupervised Voronoi Diagram partition mechanism. (2) Provide stability analysis and a conditional Wasserstein (an inherent metric applicable to the space of EPDs [25])-related bound that clarifies when Voronoi histograms can separate well-resolved EPDs. (3) Conduct an empirical evaluation of our proposed method on supervised point cloud classification and unsupervised dimensionality reduction tasks.
2
Background
We provide the pertinent information about PD and EPD (refer to 16, 19 for details). Persistence Diagram. Let g : X → R+ denote a function on space X , where R+ stands for positive real numbers, X is a finite and open subset of Euclidean space. g is computed based on a finite point cloud X sampled from manifold M ⊂ X . At scale ϵ ≥ 0, the sublevel set Xϵg = {x ∈ X | g(x) ≤ ϵ} encodes the topological information in X . For γ ≤ ϵ, we can have the nested sublevel sets Xγg ⊆ Xϵg . By increasing scale ϵ from 0, we obtain a filtration, a nested sequence of topological spaces {Xϵg }0≤ϵ<∞ .
Figure 1: Rips filtration on a 2D point cloud and the corresponding 1-dimensional PD. A cycle is considered ‘born’ at b ∈ R when it first emerges in Xbg , and it ‘dies’ at d ∈ R when it ceases to exist in Xpg for any p > d. 0-dimensional cycles are connected components; 1-dimensional cycles are rings or loops; 2-dimensional cycles are voids, etc. Topological feature r = (b, d) is presented in the form of PD D = {ri = (bi , di ) ∈ Ω|1 ≤ i ≤ N (D)}, where N (D) is the number of topological features. In Figure 1, we choose Rips filtration [32] g(·) = 2 minx∈X ℓ(·, x), where ℓ is Euclidean distance, and there is only one ring in the point cloud which is born at ϵ = 0.8 and dies at ϵ = 1.6, i.e., (b, d) = (0.8, 1.6). Expected Persistence Diagram. A PD D = {ri = (bi , di ) ∈ Ω|1 ≤ i ≤ N (D)} can be equivalently PN (D) represented as a counting measure µ on Ω given by A ∈ B → µ(A) = i=1 δri (A), where B is the class of all Borel subsets of Ω and δr denotes the Dirac point mass at r ∈ Ω. When each sampled PD is a random draw from a distribution P , its EPD, denoted as E[µ], is defined as A ∈ B → E[µ](A) = E[µ(A)] [16]. 2
Given a finite set {µ1 , µ2 , ..., µn }, consisting of sampled PDs from P , where n is referred as the number of sampled the empirical EPD is PPDs, n defined as µ̄ = n1 i=1 µi . The support of µ̄ is Sµ̄ = ∪ni=1 Di , where Di = {rj = (bj , dj ) ∈ Ω|1 ≤ j ≤ N (Di )} is the support of the sampled PD µi . EPD can be viewed as a distribution [16] of topological features supported on the open half plane Ω. Quantization [25] has been Figure 2: (a) A torus-shaped point cloud with developed to reduce the support size of EPD. Ex- 20000 points, and (b) its 1-dimensional PD, EPD amples of the PD, EPD and EPD Quantization and EPD Quantization outcomes. The transparent outcomes of a point cloud are shown in Figure blue area contains noise topological features. 2. An inherent metric applicable to the space of EPDs [25] with the same mass, is Wasserstein distance [60]. In a space of EPDs with different masses, the Optimal Partial Transport metric (OTp ) [30] is used to allow any mass transportation from or to the diagonal ∂Ω.
3
Related Work
Current methods [17, 63] vectorize EPD by averaging the vectorization1 of each sampled PD. The vectorization of a sampled PD is in the form of op({f (r)}r∈D ), where op is permutation invariant operation and f is a predefined continuous point transformation function. Current methods can be classified into linear and nonlinear representations according to the choice of op. A linear representation [63] Ψ of PD D = {ri = (bi , di ) ∈ Ω|1 ≤ i ≤ N (D)} is a summary of D in R PN (D) the form Ψ(D) = i=1 f (ri ) = Ω f (u)dµ(u), for a given measurable functionRf on Ω, where µ is the measure form of D. The corresponding vectorization of EPD is E[Ψ(D)] = Ω f (u)dE[µ](u). Pn The vectorization of empirical EPD is then the average value n1 i=1 Ψ(Di ) of the vectorized forms of its sampled PDs. According to 63, Persistence Image (PI) [1], Persistence Silhouettes (PS) [18] and Persistence Weighted Gaussian Kernel’s feature map (PWGK) [40] are linear representations. An existing nonlinear technique is Persistence Landscape (PL) [17]. The nonlinearity roots from the kmax operation in Ψ instead of summation, i.e., Ψ(D) = kmaxr∈D f (r), where f (r) is a function defined on closed interval [0, T ], for t ∈ [0, T ], Ψ(D)(t) = kmax({f (r1 )(t), ..., f (rN )(t)}) and kmax is the operation that takes the k-th largest value in the set. This kmax operation can be viewed as an approximation of weighted summation. The representation of EPD is then the average PL of its sampled PDs. These transformations introduce smoothness, bandwidth, and grid-resolution choices. Such choices are useful for stability and regularization, but may blur local mass differences or emphasize regions that are not optimal for a given dataset.
4
Representation of EPD
We propose to directly treat EPD as a distribution of topological features and develop a representation of EPD based on Voronoi Diagrams built upon a set of random codebooks. A codebook C is a finite set whose element is a point in Ω or a subset of Ω. Our method is motivated by Lemma 2 in 252 , which states that for a measure µ, given any codebook C = {c1 , ..., ck }, the coefficient µ(V (ci )) of a Voronoi cell V (ci ) centered at ci can well represent measure µ. Therefore, we will use this coefficient as the representation. Pk Lemma 2. [25] Let C = {c1 , ..., ck }. Let µ̂(C) = i=1 µ(V (ci ))δci , where V (ci ) is the Pk Voronoi Cell that centered at ci . Let ν = j=1 αj δcj for some coefficients α1 , ..., αk ≥ 0. Then OTp (µ̂(C), µ) ≤ OTp (ν, µ), where OTp is Optimal Partial Transport metric. 1 A detailed description on the vectorization methods of PD is provided in Appendix A. 2 25 tries to reduce the size of measure µ by finding the codebook C that minimize OT (µ̂(C), µ) and then uses the p
optimal C as the new support. The relation between our work and 25 is discussed in Appendix B.
3
4.1
Voronoi Diagram-based Representation
Pn For a codebook C ⊂ Ω, we treat the EPD µ̄ = n1 i=1 µi as a distribution of topological features and define the codebook-based representation of µ̄ in the form of a histogram 3 : Φ(µ̄, C) = [µ̄(V (c1 )), µ̄(V (c2 )), ..., µ̄(V (ck ))], where k is the size of codebook C and V (ci ) represents the Voronoi Cell centered at ci , i.e., V (ci ) = {x ∈ Ω|∀j ̸= i, ∥x − ci ∥2 ≤ ∥x − cj ∥2 }. Although Φ(µ̄, C) is not designed for each sampled PD, it still can be equivalently expressed in an averaged Pn form, that is, Φ(µ̄, C) = 1 i=1 Ψ(Di ), where Ψ(Di ) = n [µi (V (c1 )), ..., µi (V (ck ))] and µi is the corresponding measure form of Di . In order to align with the setting in 25, we define a variant of Φ which has a codebook of size k, with one Voronoi Cell centered at the diagonal ∂Ω, i.e. boundary of the open half plane Ω = {(t1 , t2 ) ∈ R2 |t2 > t1 }. Figure 3: Example of EPD (a) and Voronoi Diagram built Voronoi Cell centered at the diagonal from codebook (b). The red crossed points are codebook is defined as V (ck ) = {x ∈ Ω|∀j ̸= C. The solid black lines are the boundary of the Voronoi k, ∥x − cj ∥2 ≥ miny∈ck ∥x − y∥2 }, Diagram built from C. The blue dashed curve is the boundwhere ck = ∂Ω and each of the other ary between the Voronoi Cell due to the diagonal and other ci for i < k is a point located inside Voronoi Cells. Ω. An illustration of Voronoi Diagram with a Voronoi Cell center at diagonal is given in Figure 3. A normalization of Φ(µ̄, C) to unit l1 -norm can be achieved by a preprocessing step which normalizes the EPD, i.e., µ̄ ← µ̄/µ̄(Ω). We can treat this normalized measure as a distribution since its integral over Ω is 1. In addition, this normalization ensures that any two EPDs have the same mass. Hence it allows us to measure their dissimilarity by Wasserstein distance directly, without using the Optimal Partial Transport metric. So in order to simplify the analysis, we will consider the representation of the normalized EPD in the rest of this paper. A more detailed discussion on the advantage of normalization is given in Appendix D.4. Given a dataset of EPDs {µ̄1 , µ̄2 , ..., µ̄m }, we sample t codebooks Cij ⊂ Sµ̄i , j = 1...t. Each codebook Cij is a set of size k, with each element being sampled from EPD (distribution of topological features) µ̄i . This produces a random codebook set Si = {Cij }1≤j≤t for each EPD µ̄i . The final codebook set S = ∪m i=1 Si is used to represent the given dataset of EPDs. From the codebook-based representation Φ, we define the Voronoi-based Representation (Vrep) of EPD as following. Definition 4.1 (Voronoi-based Representation). Given a codebook set S = ∪m i=1 Si built from an EPD dataset {µ̄1 , µ̄2 , ..., µ̄m }. The Voronoi-based Representation (Vrep) of an EPD µ̄i is defined as L L Φ̂(µ̄i ) = C∈S Φ(µ̄i , C), where stands for a concatenation operation. An illustration is given in Figure 4. As an alternative to Vrep, if each codebook C ∈ S has diagonal ck = ∂Ω while the other k − 1 points are in the open half plane Ω, we denote it as Vrepd . Vrep is data-dependent because for an EPD µ̄i , Φ̂(µ̄i ) is influenced by both the codebook set Si from EPD µ̄i and the codebook set ∪j̸=i Sj from other EPDs in the dataset. We will elaborate this dependence in the Appendix D.3. There are three possible choices of codebook for sampling a codebook C = {c1 , ..., ck } from an EPD µ̄ with support Sµ̄ = ∪ni=1 Di , where Di = {rj = (bj , dj ) ∈ Ω|1 ≤ j ≤ N (D)}: (1) Default: each ci is sampled from Sµ̄ with each r ∈ Sµ̄ having the same weight; (2) Persistence-Weighted: each ci is sampled from Sµ̄ with each r = (b, d) ∈ Sµ̄ having weight d − b; (3) Uniform: each 3 Similar technique has been used as part of a PD representation named Persistence Bag of Word [68], where the assignment
of topological features to each bin is based on Gaussian Mixture Model [51].
4
Figure 4: Illustration of Vrep Φ̂(µ̄i ) of EPD µ̄i . Each Si consists of t codebooks, with each codebook C sampled from EPD µ̄i .
ci is sampled from a uniform distribution supported on a fixed rectangle above the diagonal ∂Ω. The former two choices are relevant to EPD and the last one is not. For Vrepd , these three choices determine the sampling method for the remaining k − 1 points in codebook C, while ck is fixed as the diagonal ∂Ω. Unless otherwise stated, the default choice of codebook is utilized in the proposed EPD representation. We examine the effects of the other two codebook choices in Experiments Section. Like all finite-dimensional vectorizations of PD/EPD, Vrep is lossy. Its information loss is controlled by the codebook size, Voronoi cell geometry, and normalization. Compared with PI/PS, Vrep does not smooth each point by a fixed analytic kernel, but instead aggregates mass inside data-dependent cells. This makes it potentially more faithful to empirical EPD mass allocation. ATOL [52] is another unsupervised PD vectorization method based on codebook; Appendix B clarifies how Vrep differs as an EPD-specific Voronoi histogram representation. 4.2
Stability
We show that Vrep is stable under the perturbation of EPD. We start with the definition of EPD perturbation. For an EPD µ̄, a perturbation refers to a minor movement of location of the Dirac point mass of its sampled PDs. We first provide the stability result of the codebook-based representation Φ and then extend it to Vrep Φ̂. PN (D ) Let µi = j=1 i δrj be a measure of sampled PD Di = {rj = (bj , dj ) ∈ Ω|1 ≤ j ≤ N (Di )}. Pn Definition 4.2 (EPD Perturbation). For an EPD µ̄ = limn→∞ n1 i=1 µi , a ∆-perturbation of EPD is defined as the perturbation of rj in Di (the support of µi ), and the perturbation of each rj is bounded within a ∆-radius open ball, i.e., ∥r̂j − rj ∥2 < ∆, where r̂j is the perturbed version of rj . And we assume that the random perturbation ∆ ∼ N (0, Σ). The perturbed EPD is denoted as µ̄′ . This definition implies that W1 (µ̄, µ̄′ ) ≤ ∆. Given that Φ(µ̄′ , C ′ ) − Φ(µ̄, C) = Φ(µ̄′ , C ′ ) − Φ(µ̄, C ′ ) + Φ(µ̄, C ′ ) − Φ(µ̄, C), there are two kinds of error in the stability analysis we need to consider,i.e., Measure Error & Codebook Error, as shown in the following inequality: ∥Φ(µ̄′ , C ′ ) − Φ(µ̄, C)∥1 ≤ ∥Φ(µ̄′ , C ′ ) − Φ(µ̄, C ′ )∥1 + ∥Φ(µ̄, C ′ ) − Φ(µ̄, C)∥1 . | {z } | {z } Measure Error
Codebook Error
The Measure Error term is caused by the perturbation of measure µ̄ (with fixed codebook C ′ ). The Codebook Error term is caused by the perturbation of codebook C (with fixed measure µ̄). We consider the Measure and Codebook Error in Lemma 4.3 and 4.4 respectively. Lemma 4.3 (Stability Pn w.r.t. perturbation of measure µ̄ with fixed codebook C). For an EPD µ̄ = limn→∞ n1 i=1 µi and an optimal matching η: support(µ̄) → support(µ̄′ ) in the definition of 1-Wasserstein distance between µ̄ and µ̄′ ,given any codebook C of size k, it holds that ∥Φ(µ̄, C) − Φ(µ̄′ , C)∥1 ≤ M · W1 (µ̄, µ̄′ ), where M is a constant determined by measure µ̄. From Definition 4.2, we have the result on the Measure Error term: ∥Φ(µ̄, C) − Φ(µ̄′ , C)∥1 ≤ M ∆. We consider the Codebook Error term in Lemma 4.4, where the upperbound is related to the dimension k of representation Φ. Perturbation causes a change in a single dimension up to C0 ϵ. A representation with higher dimension is less stable. 5
Lemma 4.4 (Stability w.r.t. codebook C with fixed measure µ̄). For a given codebook C of size k, its Pn perturbation version C ′ , and EPD µ̄ = limn→∞ n1 i=1 µi , it holds that ∥Φ(µ̄, C) − Φ(µ̄, C ′ )∥1 ≤ kC0 ∆, where C0 is a constant determined by µ̄ and C. With Lemma 4.3 and 4.4, we reach the following Theorem 4.5 about the stability of Vrep Φ̂, which shows that when the perturbation ∆ is rather small, the total error is Lipschitz continuous w.r.t. ∆. Theorem 4.5. For an EPD dataset {µ̄1 , ..., µ̄m } and ∆-perturbation of every EPD µ̄i , it holds that ∥Φ̂(µ̄i ) − Φ̂(µ̄′i )∥1 ≤ L∆, where L is a constant determined by m, t, k, EPD dataset {µ̄1 , ..., µ̄m } and codebook set S: L = mt(M max + kC0max ), C0max = maxC∈S,j∈[m] C0 (C, µ̄j ) and M max = maxj∈[m] M (µ̄j ). We demonstrate the stability of Vrep by showing the change in representation under different levels of perturbation. The change in representation is defined as δµ̄ = ∥Φ̂(µ̄′ ) − Φ̂(µ̄)∥1 /∥Φ̂(µ̄)∥1 . We use a 3d dynamical system dataset [26, 43], which describes a discrete food chain model. This dataset contains 9 classes, each class contains 50 point clouds with each point cloud having 2000 points. Each class corresponds to a parameter of the dynamical system. With the dataset of 450 point clouds, for each point cloud Xi , we obtain the EPD µ̄i by computing 50 sampled PDs from a subset of Xi with size 200. We report the average P450 1 change of representation δ = 450 i=1 δµ̄i under different perturbation levels, where the level α stands for the bandwidth of the Gaussian noise4 applied on EPD. The Figure 5: Average change δ in Vrep and result shown in Figure 5 is consistent with our analysis Vrepd under different EPD perturbation that a larger dimension k indicates a less stable vector levels α. in Lemma 4.4 and Theorem 4.5. In addition, Vrep is shown to be less stable than Vrepd . This is because there are less points near the boundary between Voronoi Cell centered at diagonal ∂Ω and other Voronoi Cells, i.e., the blue dashed curve shown in Figure 3 (b), which separates the noise topological features from the significant ones. The analysis of the effect of t on δ is provided in Appendix D.2. 4.3
Dissimilarity of Two EPDs partially controlled by Wasserstein distance under codebook approximation conditions
Here we consider two EPDs (µ̄, ν̄) and demonstrate that the most basic component of Vrep (and Vrepd ): codebook-based representation Φ can admit a Wasserstein-related lower bound under approximation conditions. Theorem 4.6. For any given codebook C with size k and two EPDs (µ̄, ν̄) with the same mass, it holds that ∥Φ(µ̄, C) − Φ(ν̄, C)∥1 ≥ [W1 (µ̄, ν̄) − W1 (µ̄, µ̂(c)) − W1 (ν̄, ν̂(c))]/dmax (C), where dmax (C) = maxci ,cj ∈C ∥ci − cj ∥2 is the diameter of C, W1 is Wasserstein distance 5 , µ̂(C) = Pk Pk i=1 µ̄(V (ci ))δci , and ν̂(C) = i=1 ν̄(V (ci ))δci . The bound is informative only when the Wasserstein separation is large relative to the codebook approximation errors. Therefore, this result should be interpreted as a conditional consistency statement for well-resolved codebooks, not as a general guarantee that Vrep distances monotonically preserve Wasserstein distances6 . Appendix F.7 provides a controlled synthetic study that directly compares representation distances with Wasserstein distances. The results show that Vrep/Vrepd are effective when variation is dominated by coarse mass displacement, while smooth representations 4 Here the noises are from N (0, α2 I). ∆ is the bound to the norm of noise in Definition 4.2. For ϵ ∼ N (0, α2 I), p E(∥ϵ∥) = α π/2. So α is intuitively linear to ∆ in an expected manner. The change in representation (δ) in Figure 5 grows slower when α increases. This is consistent with Theorem 4.5, i.e., the change can be upper bounded by a linear function of ∆. 5 For two measures µ, ν having the same total mass on metric space (M, ρ), Wasserstein distance W p,ρ defined as the 1 R infimum of ( M 2 ρ(x, y)p ) p over all transport plans π between µ and ν, i.e. measures on M × M which have for first (resp. second) marginal µ (resp. ν ). When ρ is the Euclidean distance, we write Wp instead of Wp,ρ . 6When W (µ̄, ν̄) is small, the upper bound in Inequality 2 would be more useful, indicating that a smaller W (µ̄, ν̄) leads 1 1 to a smaller ∥Φ(µ̄, C) − Φ(ν̄, C)∥1 .
6
can be preferable for fine within-cell shifts, supporting the trade-off interpretation of our method. A demo analysis on the Wasserstein separation is given in Appendix D.1. These results do not imply that Vrep dominates PI, PS, or other vectorizations. They show that, for a fixed codebook satisfying separation conditions, the Voronoi histogram changes Lipschitz-continuously under perturbations of the normalized EPD and can preserve certain distributional differences at the cell level. 4.4
Time Complexity
For an EPD dataset of size m, {µ̄1 , ..., µ̄m }, the size of codebook set S is mtk, where k is the codebook size and t is the number of codebooks sampled from each EPD. For a particular EPD µ̄i , the Euclidean distance between each point in Sµ̄i and each point in codebook set S needs to be computed for the construction of Voronoi Diagram. So for an EPD µ̄, the time complexity of building Vrep is O(|Sµ̄ |mtk), where Sµ̄ is the support of µ̄. Speed up by subsampling on Sµ̄ . Since we consider µ̄ as a distribution, we can reduce its support size to speed up the process of building a vector representation. Quantization [25] or simple subsampling can be utilized to reduce the support size. It is more efficient to use subsampling because Quantization involves an expensive optimization technique. We use subsampling to reduce Sµ̄ to a fixed size and the resultant time complexity becomes O(mtk) for building Vrep for one EPD. The time complexity of building Vrep’s for the entire dataset is O(m2 tk). We can further reduce time complexity to O(mtk) by using a random subset of fixed size of codebook set S.
5
Experiments
We compare Vrep with existing representation methods of EPD: Average Persistence Image (PI), Persistence Silhouettes (PS) and Persistence Landscape (PL) in point cloud classification task. We calculate the EPD for each point cloud, convert it into a vector, and then feed it into a Random Forest classifier [6]. We follow the setting in [52] and use Random Forests as a ready-to-use tool. Comparable performances can be obtained from using a neural network classifier or other classifiers, depending on the problem. This is a light choice that requires no particular infrastructure or tuning efforts that would produce overly design-dependent results. In addition, we use PointNet [15], an end-to-end network for point cloud input, to measure the difficulty of the classification task. Note that PointNet is not a fair contender for the compared methods because the feature extraction in PointNet is a supervised method but others are unsupervised. Code is available in the Supplementary Material. We also compare Vrep with two more conventional vectorization methods, Betti and Euler Curve in Appendix F.3. For kernel methods, we compare Vrep with Persistence Weighted Gaussian Kernel and Sliced Wasserstein Kernel in Appendix F.2, where the classifier is Support Vector Machine [56]. We do not compare with supervised vectorization methods like PersLay [12] since we aim to show that an unsupervised method (Vrep) with easy classifier like Random Forest can have a competitive performance in comparison with end-to-end supervised PointNet, with lower runtime in our reported settings. The unsupervised nature also enables Vrep to be useful in unsupervised task like dimensionality reduction, as shown in Appendix F.4. Datasets. For datasets with significant topological features, we choose Protein [14, 31], CAD [38] and Time-Delaying embedding [55] (point cloud) derived from time series data [23]. For the computation of EPD, each sampled PD is obtained through Rips filtration [32] for the Protein and time series data. CAD uses a more efficient alpha filtration [28] given its large number of points in each point cloud. The detailed hyperparameter setting and description of datasets are provided in Appendix E. Experiment Settings. We report the mean classification accuracy of 10 random splits where 80% of the dataset is for training and the rest is for testing. 3-fold cross validation on the training set is used to select the best hyperparameters for each approach. For Vrep and Vrepd , the test set’s representation is constructed based on the codebook set S from the training set. The results are shown in Table 1, where CAD0.01 and CAD0.05 stands for the CAD dataset with Gaussian noise of bandwidth being 0.01 and 0.05. The detailed settings and summary of datasets are provided in Appendix E. 7
Table 1: Accuracy of Vrep, Vrepd , Persistence Image (PI), Persistence Silhouettes (PS), Persistence Landscape (PL). PointNet [15] is an end-to-end network designed for point cloud classification. It serves as a means to assess the difficulty of classifying point clouds. The default codebook choice is used in Vrep and Vrepd . A direct computation of PD on the CAD dataset is not feasible, as most point clouds contain too many points. Since the Protein dataset has distance matrix only and PointNet takes point cloud only as input, we use Multidimensional Scaling [5] to transform the distance matrix into a 3d point cloud. Boldface is for the highest accuracy on each dataset (excluding PointNet).
Protein CAD CAD0.01 CAD0.05 Beef BirdChicken DPTW Earthquakes ECG200
Vrep Vrepd EPD 0.985 0.985 0.931 0.900 0.912 0.900 0.923 0.904 0.500 0.483 0.963 0.963 0.776 0.767 0.825 0.824 0.850 0.823
PI PD 0.955 \ \ \ 0.367 0.763 0.640 0.808 0.685
PS EPD 0.790 0.911 0.911 0.900 0.358 0.838 0.742 0.801 0.808
PD 0.880 \ \ \ 0.283 0.875 0.690 0.794 0.728
PL EPD 0.850 0.897 0.881 0.885 0.333 0.875 0.704 0.781 0.795
PD 0.970 \ \ \ 0.317 0.875 0.650 0.805 0.725
EPD 0.950 0.901 0.900 0.900 0.317 0.875 0.650 0.805 0.725
PointNet Point Cloud 0.740 0.950 0.946 0.950 0.550 1.000 0.784 0.805 0.847
Vrep and Vrepd outperform PI, PS and PL over all the datasets. Vrep often produces higher accuracy than Vrepd . But from the result of CAD, CAD0.01 and CAD0.05 , we demonstrate that Vrepd is more stable than Vrep. This is consistent with our stability result shown in Figure 5. The hyperparameter sensitivity results of Vrep and Vrepd are shown in Appendix F.1. We fix the value of t and report the performance under different ks. Vrep (Vrepd ) is rather stable w.r.t. the choice of k when the codebook choice is default or persistence-weighted. On these topology-sensitive datasets, Vrep and Vrepd achieve higher mean accuracy than PI, PS, and PL under the Random Forest pipeline. This result can be interpreted as evidence that adaptive mass aggregation can be effective in these settings. The comparison with PWGK/SWK in Appendix F.2 further shows that smooth kernel methods can be preferable on several datasets, at substantially higher runtime. Compared with PointNet, although Vrep produces comparable or lower accuracy, Vrep can be used in unsupervised task like dimensionality reduction, shown in Section F.4, while PointNet can not be applied to unsupervised tasks due to its requirement of label for the end-to-end training. The time cost comparison with PointNet is provided in the Appendix E.5. Note that Vrep is not uniformly best. In Appendix F.2, PWGK performs better on several CAD and DPTW settings, suggesting that smooth kernel methods can be preferable when the relevant topological variation is better captured by continuous similarity rather than coarse mass aggregation.
Table 2: Accuracy of Vrep and Vrepd under different codebook choices: default, persistence-weighted and uniform. The first two choices are EPD-relevant while the last one is EPD-irrelevant. Boldface is for the highest accuracy on each dataset.
Protein CAD CAD0.01 CAD0.05 Beef BirdChicken DPTW Earthquakes ECG200
default 0.985 0.931 0.912 0.923 0.500 0.963 0.776 0.825 0.850
Vrep persistence 0.920 0.969 0.946 0.942 0.583 0.938 0.756 0.814 0.828
uniform 0.780 0.935 0.904 0.908 0.517 0.925 0.757 0.823 0.830
8
default 0.985 0.900 0.900 0.904 0.483 0.963 0.767 0.824 0.823
Vrepd persistence 0.930 0.961 0.935 0.931 0.558 0.950 0.756 0.812 0.81
uniform 0.665 0.877 0.900 0.896 0.467 0.950 0.735 0.818 0.835
5.1
Study on the codebook choice
We conduct a study on the codebook choice for Vrep and Vrepd . We consider the three codebook choices of the default, persistence-weighted and uniform, stated in the Voronoi Diagram-based Representation Section. As shown in Table 2, the persistence-weighted codebook outperforms the default one on CAD, CAD0.01 , CAD0.05 and Beef, for both Vrep and Vrepd . The persistenceweighted codebook performs better in the noisy case because the codebook is mainly sampled on significant topological features, which reduces the effect of noisy features near the diagonal. The highest accuracy is obtained through either the default or persistence-weighted codebook which consistently outperforms the EPD-irrelevant uniform codebook. Further experiments comparing using Quantization results [25] and ATOL-style centers [52] as codebooks are given in Appendix F.8. 5.2
Scaleup Test
There are two components in an EPD dataset that affect the computation effort for vectorization: the number of EPDs in the dataset and the number n of sampled PDs in each EPD. For the EPD dataset size, as shown in Figure 6 (a), Vrep (Vrepd ) is quadratic to the dataset size because as shown in Definition 4.1, a larger dataset leads to a larger codebook set S. PI, PL and PS all have linear complexity. For the actual time cost, Vrep (Vrepd ) is faster than PL and PS, but slower than PI.
Figure 6: (a) Scaleup test on the dataset size, where the dataset size is 126 at data size ratio =1. The number n of sampled PD in each EPD is set to be 10. The actual time cost is reported here. Corresponding time ratio is reported in Appendix E.6. (b) Scaleup test on CAD w.r.t. the number n of sampled PDs in EPD. The overlapping area between Vrep and Vrepd is enlarged. We perform a scale-up test w.r.t. the number n of sampled PDs in each EPD in Figure 6 (b). The time cost of PI, PS and PL grows as n increases because these methods need to be applied on each sampled PD. Vrep and Vrepd remain almost unchanged over different values of n because the subsampling technique ensures a fixed size of the EPD support Sµ̄ .
6
Conclusion
We proposed Voronoi histograms as an adaptive vectorization of Expected Persistence Diagrams. Instead of applying a fixed smooth point transformation to each sampled persistence diagram, Vrep aggregates normalized empirical EPD mass over data-dependent Voronoi cells. This gives a different approximation bias from PI, PS, and PL: Vrep can efficiently summarize coarse mass allocation with respect to the number of sampled PDs, but it remains a lossy finite-dimensional representation. Our analysis establishes stability under the stated perturbation, normalization, and codebook conditions, and gives a conditional Wasserstein-related bound for well-resolved codebooks. These results should not be interpreted as a general monotonic preservation guarantee or as dominance over existing vectorizations. Empirically, Vrep is competitive on topology-sensitive classification datasets and offers favorable scaling with the number of sampled PDs, while kernel methods such as PWGK and SWK can perform better in some settings at higher computational cost. Limitation is discussed in Appendix D.5. 9
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A
Vectorization methods of PD
Here we introduce unsupervised and supervised vectorization methods for PD. • For unsupervised methods, the simplest category is the basic statistical quantities of PD, including persistence entropy [3, 21, 53], average and standard deviation of birth, death and lifespan (death-birth) [2], etc. The second category aims to construct polynomial maps (template functions) from PD and evaluate the selected maps on PD to get a vector [24, 29, 37, 47]. The third category is representation based on function that uses PD as its parameter, such as Persistence Curve [22], Persistence Landscape [7, 9], Persistence Image [1], Persistence Silhouettes [18] and Persistence B-spline grids [26]. • A representative supervised method is PersLay [12], a neural network layer for PD based on Deep Sets [66], which can be integrated into filtration learning framework for point cloud [46]. Other methods like topological signature [34, 35] are based on learnable exponential functions as structure elements. Similar methods are utilized as part of graph filtration learning framework [33, 36, 67]. Recent work [61] focuses on vectorization via Transformer [59].
B
Relation with Quantization & ATOL
We show the concept difference between Vrep and Quantization/ATOL. The experiments comparing using Quantization results and ATOL-style centers as codebooks are given in Appendix F.8. Quantization. Compared with EPD quantization [25], which aims to reduce the support size of EPD and use the resultant reduced EPD for vectorization in ML tasks, our work focus on directly vectorization of EPD without the middle quantization step. ATOL. ATOL [52] is closely related to our work in that both methods view persistence-based objects as finite measures in a Euclidean space and construct unsupervised vector representations that can be used by standard machine learning models. ATOL learns a collection of representative regions from a training set of persistence diagrams and encodes each measure through responses to localized contrast functions around learned centers. In this sense, ATOL is a general measure-vectorization framework for topological descriptors, with an encoding budget controlling the number of learned regions. Vrep shares the high-level goal of converting measure-valued topological summaries into vectors, but it differs from ATOL in both the object being represented and the form of the representation. ATOL is primarily designed to vectorize individual persistence diagrams or general finite measures. In contrast, Vrep is designed for empirical Expected Persistence Diagrams (EPDs), where each input is already an averaged measure obtained from many sampled persistence diagrams. Thus, Vrep targets the distribution of topological features induced by subsampling, rather than a single persistence diagram. The representation mechanism is also different. ATOL uses learned centers together with contrast functions, producing feature values that depend on the chosen functional responses around these centers. Vrep instead uses a hard Voronoi partition: each coordinate is the normalized mass of the empirical EPD inside a Voronoi cell. Therefore, Vrep can be interpreted as a direct histogram of the EPD over data-dependent cells. This makes the approximation bias explicit: Vrep preserves celllevel mass allocation but discards within-cell geometry, while ATOL can retain smoother localized responses depending on its contrast functions. The codebook construction also plays a different role. In ATOL, centers are learned to summarize discriminative regions of a collection of persistence diagrams under a fixed encoding budget. In Vrep, codebooks are sampled from EPD supports and used to define Voronoi cells for measuring empirical EPD mass. The final Vrep representation concatenates histograms over a set of such codebooks. The 13
diagonal-aware variant Vrepd further includes a Voronoi cell associated with the diagonal, which is useful for separating near-diagonal topological noise from more persistent features. From this perspective, Vrep can be viewed as an EPD-specific, hard-assignment counterpart to general measure vectorization methods such as ATOL. The two methods make different trade-offs. ATOL provides a flexible learned vectorization for persistence measures through contrast functions and an encoding budget, while Vrep focuses on the empirical distributional structure of EPDs and emphasizes direct mass aggregation, stability under EPD perturbations, and computational scaling with respect to the number of sampled persistence diagrams. Our goal is therefore not to replace ATOL, but to provide a complementary representation tailored to EPDs.
C
Proofs
C.1
Proof of Lemma 4.3
Pn Lemma 4.3. For an EPD µ̄ = limn→∞ n1 i=1 µi and an optimal matching η: support(µ̄) → support(µ̄′ ) in the definition of 1-Wasserstein distance between µ̄ and µ̄′ ,given any codebook C of size k, it holds that ∥Φ(µ̄, C) − Φ(µ̄′ , C)∥1 ≤ M · W1 (µ̄, µ̄′ ), where M is a constant. Proof. Denote the pdf of measure µ̄ as p, whose existence is proved in 16 and the pdf of perturbation ϵ ∼ N (0, Σ) as q. And p is k times differentiable if the sample space to construct EPD is k times differentiable (Theorem 3.5 in 16). Denote the corresponding random variable of p as α. Given that α and ϵ is dependent of each other, then the pdf of the perturbed random variable α + ϵ is the convolution of p and q: Z r(y) = (p ∗ q)(y) = p(y − z)q(z)dz. In practice, we use set the max filtration value to be finite, this means that p is a C k function in a closed bounded area Ω′ that is a subset of Ω. So p is Lipschitz continuous. Denote the Lipschitz constant as M , for x ∈ Ω′ , we have that Z |p(x) − r(x)| = |p(x) − q(z)p(x − z)dz| Z = | q(z)[p(x − z) − p(x)]dz| Z ≤ q(z)|p(x − z) − p(x)|dz Z ≤ M q(z)∥z∥dz Based on our definition of the perturbation, it holds that W1 (µ̄, µ̄′ ) =≤
R
q(z)∥z∥dz. So we have
|p(x) − r(x)| ≤ M · W1 (µ̄, µ̄′ ). For any Voronoi Cell V (cj ), we have that |µ̄(V (cj )) − µ̄′ (V (cj ))| = |
Z p(x) − r(x)dx| V (cj )
Z ≤
|p(x) − r(x)|dx V (cj )
≤ M · W1 (µ̄, µ̄′ ) · V ol(V (cj )). 14
As a result, we have ∥Φ(µ̄, C) − Φ(µ̄′ , C)∥1 =
k X
|µ̄(V (cj )) − µ̄′ (V (cj ))|
j=1
≤
k X
M · W1 (µ̄, µ̄′ ) · V ol(V (cj ))
j=1
= M · W1 (µ̄, µ̄′ ).
C.2
Proof of Lemma 4.4.
′ Lemma 4.4. PnFor a given codebook C of size k, its perturbation version C , and EPD µ̄ = 1 limn→∞ n i=1 µi , it holds that
∥Φ(µ̄, C) − Φ(µ̄, C ′ )∥1 ≤ kC0 ∆, where C0 is a constant determined by µ̄. Proof. Similar to Theorem 5.1 [4], we assume that : • β = minj̸=j ∥ci − cj ∥ > 0. • There exists ρ such that B(x, ρ) ∩ C \ {ci } ̸= ∅ for x ∈ Ω′ , where Ω′ is the bounded and closed space we use in the proof of Lemma 4.3. The first assumption means that there are no overlapping elements in the codebook C. For the second assumption, we can choose ρ = D, i.e. the diameter of Ω′ , and B(x, ρ) would cover Ω′ . So it holds that ∀i ∈ [k], B(x, ρ) ∩ C \ {ci } ̸= ∅. According to Theorem 5.1 and Remark 5.3 in 4, let β = mini̸=j ∥c Si − cj ∥ > 0, we have that ∀∆ = O(ϵ) = γϵ, where 0 < ϵ < β/6, ϵ ≤ 8 · d( i∈[k] ci , ∂Ω′ ) and γ = β 1 min{ 16 δ̂( 12ρ+5β ),
d(
′ i∈[k] ci ,∂Ω ) 7 } , it holds that dH (V (ci ), V (c′i )) ≤ ϵ. 8(ρ+β/6)
S
In order to relate Sthe upper bound to ∆, we rewrite the result above as following: for 0 < ∆ < γβ/6 and ∆ ≤ 8 · d( i∈[k] ci , ∂Ω′ ) · γ, it holds that dH (V (ci ), V (cj )) ≤
∆ . γ
V (ci ), V (cj ) ∈ Ω′ is a bounder area in the plane, we denote the total Perimeter of these areas as Li . Then according to dH (A, B) ≤ ∆ γ , it holds that Vol(V (ci )△V (cj )) ≤ Lmax ·
∆ , γ
where Lmax = maxi∈[k] Li . Given the fact that p (pdf of µ̄) is a continuous function in the bounded and closed space Ω′ , we know that p is bounded and let Mp = ∥p∥∞ = supx∈Ω′ p(x). Then it holds that ∆ |µ̄(V (ci ) − µ̄)(V (c′i ))| ≤ Mp · Vol(V (ci )△V (cj )) ≤ Mp · Lmax · . γ Then X ∥Φ(µ̄, C) − Φ(µ̄, C ′ )∥1 = |µ̄(V (ci ) − µ̄)(V (c′i ))| ≤ kC0 ∆, i∈[k]
where C0 = Mp Lmax /γ. 7 Here δ̂ is the modulus of convexity, i.e. a function δ̂ : [0, 2] → [0, 1] defined by δ̂(ϵ) = inf{1 − |(x + y)/2| : |x − y| ≥ ϵ, |x| = |y| = 1}. There are more convent functions for specific cases. The detials are given in page 6 of 4 (https://arxiv.org/pdf/1103.4125).
15
C.3
Proof of Theorem 4.5.
Theorem 4.5. For an EPD dataset {µ̄1 , ..., µ̄m } and ∆-perturbation of every EPD µ̄i , it holds that ∥Φ̂(µ̄i ) − Φ̂(µ̄′i )∥1 ≤ L∆, where L = mt(M max + kC0max ) is a constant determined by m, t, k, EPD dataset {µ̄1 , ..., µ̄m } and codebook set S. Proof. Based on the fact that Φ(µ̄′ , C ′ ) − Φ(µ̄, C) = Φ(µ̄′ , C ′ ) − Φ(µ̄, C ′ ) + Φ(µ̄, C ′ ) − Φ(µ̄, C), it holds that ∥Φ(µ̄′ , C ′ ) − Φ(µ̄, C)∥1 ≤ ∥Φ(µ̄′ , C ′ ) − Φ(µ̄, C ′ )∥1 + ∥Φ(µ̄, C ′ ) − Φ(µ̄, C)∥1 S According to Lemma 4.4 & 4.3, for 0 < ∆ < γβ/6 and ∆ ≤ 8 · d( i∈[k] ci , ∂Ω′ ) · γ, it holds that ∥Φ(µ̄′ , C ′ ) − Φ(µ̄, C)∥1 ≤ (M + kC0 )∆. Take M max = maxC∈S,j∈[m] M (C, µ̄j ) = maxj∈[m] M (µ̄j ), C0max = max SC∈S,j∈[m] C0 (C, µ̄j ), for ∆ that satisfies 0 < ∆ < minC∈S γ(C)β(C)/6 and ∆ ≤ minC∈S 8 · d( i∈[k] ci , ∂Ω′ ) · γ(C), it holds that ∥Φ̂(µ̄i ) − Φ̂(µ̄′i )∥1 = ∥
M
Φ(µ̄i , C) −
C∈S
=
X
M
Φ(µ̄′i , C ′ )∥1
C∈S
∥Φ(µ̄i , C) − Φ(µ̄′i , C ′ )∥1
C∈S
≤ L∆, where L = mt(M max + kC0max ).
C.4
Proof of Theorem 4.6.
Theorem 4.6. For any given codebook C and two EPDs (µ̄, ν̄) with the same total mass, it holds that ∥Φ(µ̄, C) − Φ(ν̄, C)∥1 ≥
W1 (µ̄, ν̄) − W1 (µ̄, µ̂(C)) − W1 (ν̄, ν̂(C)) , dmax (C)
where dmax (C) = maxci ,cj ∈C ∥ci − cj ∥2 is the diameter of C and W1 refers to Wasserstein distance. Proof. By the triangle inequality of Wasserstein distance, ∀ codebook C, it holds that W1 (µ̄, ν̄) ≤ W1 (µ̄, µ̂(C)) + W1 (µ̂(C), ν̄) ≤ W1 (µ̄, µ̂(C)) + W1 (µ̂(C), ν̂(C)) + W1 (ν̄, ν̂(C)). By rearranging the terms, we have W1 (µ̂(C), ν̂(C)) ≥ W1 (µ̄, ν̄) − W1 (µ̄, µ̂(c)) − W1 (ν̄, ν̂(c)). RR By the definition of optimal partial transport metric W1 (µ̂(c), ν̂(c)) = inf π Ω×Ω ∥x − y∥2 dπ corresponds to the minimal cost of all the transportation plans. So it is the lower bound of the cost of a particular transportation plan. We now give a designed transportation plan that link the cost and the distance between Vreps ∥Φ(µ̄, C) − Φ(ν̄, C)∥1 . A particular transportation plan from µ̂(C) to ν̂(C): since µ̂(C) and ν̂(C) have the same support C, we keep the mass of each support point ci as m(ci ) = min(µ̂(C)(ci ), ν̂(C)(ci )) fixed in the transportation. We move the extra mass in {ci : m(ci ) < µ̂(C)(ci )} to the mass-vacant area {ci : m(ci ) > µ̂(C)(ci )}. Since the sum of mass that is needed to be moved is less than ∥Φ(µ̄, C)) − Φ(ν̄, C)∥1 and the cost of each transportation is less than or equal to dmax (C), the total cost of this transportation plan is less than or equal to ∥Φ(µ̄, C) − Φ(ν̄, C)∥1 · dmax (C). 16
Based on that, the cost of this particular transportation plan is larger than W1 (µ̂(C), ν̂(C)). So it holds that ∥Φ(µ̄,C) − Φ(ν̄, C)∥1 · dmax (C) ≥ W1 (µ̂(C), ν̂(C)) ≥W1 (µ̄, ν̄) − W1 (µ̄, µ̂(C)) − W1 (ν̄, ν̂(C)).
C.5
Proof of Theorem D.1
Theorem D.1. For any given codebook C and two EPDs (µ̄, ν̄) with the same total mass, it holds that ∥Φ(µ̄, C) − Φ(ν̄, C)∥1 ≤
k dmin (C)
[W1 (µ̄, ν̄) + W1 (µ̄, µ̂(C)) + W1 (ν̄, ν̂(C))],
where dmin (C) = minci ,ci ∈C ∥ci − cj ∥1 . Proof. By the triangle inequality of Wasserstein distance, ∀ codebook c, it holds that W1 (µ̂(C), ν̂(C)) ≤ W1 (µ̂(C), µ̄) + W1 (µ̄, ν̂(C)) ≤ W1 (µ̂(c), µ̄) + W1 (µ̄, ν̄) + W1 (ν̂(C), ν̄). For W1 (µ̂(C), ν̂(C)), the transportation cost at least contains maxi |µ̂(C)(ci ) − ν̂(C)(ci )| · dmin (C). Since ∥Φ(ν̄, C) − Φ(ν̄, C)∥∞ = maxi |µ̂(C)(ci ) − ν̂(C)(ci )|, it holds that W1 (µ̂(C), ν̂(C)) ≥∥Φ(ν̄, C) − Φ(ν̄, C)∥∞ · dmin (C) ∥Φ(ν̄, C) − Φ(ν̄, C)∥1 ≥ · dmin (C). k Combined with the first inequality derived from triangle inequality of Wasserstein distance, we obtain the final result: ∥Φ(µ̄, C) − Φ(ν̄, C)∥1 ≤
k [W1 (µ̄, ν̄) + W1 (µ̄, µ̂(C)) + W1 (ν̄, ν̂(C))]. dmin (C)
D
Additional Notes
D.1
Effect of the assumption due to the use of point transformation function f .
Here we given two special EPDs (distribution of topological features) µ and ν. In this constructed example, the Voronoi histogram with a suitable codebook better preserves the relevant mass displacement than the selected PI/PS parameterizations. This illustrates one possible trade-off of partition-based aggregation, not a general superiority result. For an EPD µ, a uniform distribution supported on a rectangle with the length of one side of the rectangle being 1 and the total mass being 1, we move µ along the diagonal ∂Ω by 1 − α to get ν, as shown in Figure 7 (a). We denote the support of µ (ν) as Sµ (Sν ) and the intersection area as A = Sµ ∩ Sν . Assume that the translation from µ to ν is very small (α is close to 1) so that the Wasserstein distance between µ and ν is 1 − α, i.e W (µ, ν) = 1 − α. For our proposed codebook-based representation Φ, once the size k of codebook C is very large (∂Ω ∈ / C), given the histogram nature of Φ with each Voronoi Cell as bin, it holds that ∥Φ(µ, C) − Φ(ν, C)∥1 /∥Φ(µ, C)∥1 = 2(1 − α) = 2W (µ, ν) for C sampled from µ or ν. So for Vrep, it holds that ∥Φ̂(µ) − Φ̂(ν)∥1 /∥Φ̂(µ)∥1 = 2(1 − α) = 2W (µ, ν). For linear representation Ψ like PI and PS, it holds that 17
Figure 7: (a) Distribution ν is obtained via the translation of µ. The arrows indicate the direction of translation. The blue and red arrows are for marking the length of rectangle. (b) Illustration of the bijection map η. (c) Illustration of the point transformation function of PS. (d) Illustration of the point transformation function of PI.
= =
∥Ψ(µ) − Ψ(ν)∥1 ∥Ψ(µ)∥1 P P ∥ x∈Sµ \A f (x) − y∈Sν \A f (y)∥1 ∥
∥Ψ(µ)∥1 x∈Sµ \A [f (x) − f (η(x))]∥1
P
(1)
∥Ψ(µ)∥1 ∥f (x) − f (η(x))∥1 x∈Sµ \A
P ≤
P ≤
∥Ψ(µ)∥1 ∥f (x)∥1 + ∥f (η(x))∥1 x∈Sµ \A ∥Ψ(µ)∥1
,
where η is a bijection between Sµ \ A and Sν \ A, for x ∈ Sµ \ A, η(x) is the mirror point that is symmetrical w.r.t. the middle black dashed line in Figure 7 (b), and f is the point transformation function. Assume that for any x, y ∈ Sµ ∪ Sν , it holds that ∥f (x)∥1 = ∥f (y)∥1 , where f (x) and f (y) are the transformed functions. Given the open half plane Ω = {(t1 , t2 ) ∈ R2 |t2 > t1 } and x = (xb , xd ) ∈ Ω, for PI, f (x)(·) = exp(−∥ · −x∥22 /2σ 2 ). For PS, f (x)(·) = max{0, xd − | · −xb |}, x = (xb , xd ). For both PI and PS, there might exist an area B such that for z ∈ B, ψ(x)(z) and ψ(η(x))(z) are both larger than 0, as shown in Figure 7 (c) and (d). Hence, the last line of inequality (1) can not take the equal sign. In consequence, it holds that P ∥Ψ(µ) − Ψ(ν)∥1 x∈Sµ \A ∥f (x)∥1 + ∥f (η(x))∥1 < = 2(1 − α) = 2W (µ, ν). ∥Ψ(µ)∥1 ∥Ψ(µ)∥1 Given the specific distributions µ and ν, Vrep can truthfully reflect Wasserstein distance while PS and PI can not because they assume the distribution is the summation of Gaussian distributions or piecewise linear functions. D.2
Hyperparameter t has no effect on change in Vrep (Vrepd ) representation.
According to Theorem 4.5, we have that for an EPD dataset {µ̄1 , ..., µ̄m }, and∆-perturbation of every EPD µ̄i , it holds that ∥Φ̂(µ̄i ) − Φ̂(µ̄′i )∥1 ≤ L∆, where L = mt(M max + kC0max ) is a constant determined by the hyperparameters of Vrep and µ̄′i is the perturbed version of µ̄i . 18
Combined with the fact that for any EPD µ̄, ∥Φ̂(µ̄)∥1 = mt, we have that δµ̄i =
∥Φ̂(µ̄i ) − Φ̂(µ̄′i )∥1 ∥Φ̂(µ̄i )∥1
≤
L∆ = (M max + kC0max )∆. mt
So hyperparameter t has no effect on change in Vrep (Vrepd ) representation. D.3
Data-dependence of Vrep
We describe the data-dependent nature of Vrep (and Vrepd ). By data-dependence, we mean that the representation of an EPD µ̄ depends more than itself: the codebook C is sampled from another EPD ν̄ and then used to build representations of ν̄ and µ̄. Current vectorization methods just average the vectors of sampled PDs in each EPD, and they do not possess this data-dependence property. Given an EPD dataset with just two EPDs µ̄ and ν̄, our objective here is to clarify precisely what factors ∥Φ(µ̄, C) − Φ(ν̄, C))∥1 depends on when C is sampled from one particular EPD ν̄. Theorem D.1. For any given codebook C with size k and two EPDs (µ̄, ν̄), it holds that ∥Φ(µ̄, C) − Φ(ν̄, C)∥1 ≤ BU (µ̄, ν̄, C), where dmin (C) = minci ,cj ∈C,i̸=j ∥ci − cj ∥2 and the upper bound BU (µ̄, ν̄, C) k dmin (C) [W1 (µ̄, ν̄) + W1 (µ̄, µ̂(C)) + W1 (ν̄, ν̂(C))].
=
Figure 8: Two measures µ̄, ν̄ on Ω and codebook C (red points) sampled from ν̄. We provide an upper bound BU for the l1 distance between Φ(µ̄, C) and Φ(ν̄, C). A larger BU would indicate a possible larger l1 distance. Next we focus on the case where C is sampled from ν̄ and its effect on BU . If the codebook is sampled from ν̄, the key of the analysis is to disentangle µ̄ from C in the W1 (µ̄, µ̂(C)) term of BU . Assume the supports of µ̄ and ν̄ are both compact, denoted as Sµ̄ , Sν̄ and let A = Sµ̄ ∩ Sν̄ . If C is sampled from ν̄, then for W1 (µ̄, µ̂(C)), at least mass of µ̄(Sµ̄ \ A) = µ̄(Ω) − µ̄(A) need to be transported to C ⊂ A, as shown in Figure 8. So we can obtain a simple lower bound of BU as BU (µ̄, ν̄, C) ≥
k [W1 (µ̄, ν̄) + W1 (ν̄, ν̂(C)) + ℓ(C, ∂(Sν̄ ))(µ̄(Ω) − µ̄(A))], dmin (C)
where ℓ(X, Y ) = minx∈X,y∈Y ∥x − y∥2 . This lower bound indicates that when codebook C is sampled from ν̄, two factors 8 (i) less mass of µ̄ in the intersection area A and (ii) a larger ℓ(C, ∂(Sν̄ )) indicate a higher lower bound of BU . Consequently, the distance between the codebook-based representation of µ̄ and ν̄ w.r.t. codebook C are more likely to be large. Combined the results from Theorem 0.5 and D.1, if codebook C ⊂ Sν , we have 1 [W1 (µ̄, ν̄) − W1 (µ̄, µ̂(C)) − W1 (ν̄, ν̂(C))] ≤ ∥Φ(µ̄, C) − Φ(ν̄, C)∥1 dmax (C) k ≤ [W1 (µ̄, ν̄) + W1 (µ̄, µ̂(C)) + W1 (ν̄, ν̂(C))]. dmin (C)
(2)
We denote the lower bound of ∥Φ(µ̄, C) − Φ(ν̄, C)∥1 as BL and the upper bound as BU . With the result W1 (µ̄, µ̂(C)) ≥ ℓ(C, ∂(Sν̄ ))(µ̄(Ω) − µ̄(A)), we can have a upper bound for BL and lower 8 Note that these two factors also affect the lower bound in Theorem 0.5.
19
bound for BU : BL ≤ BU ≥
1 dmax (C)
[W1 (µ̄, ν̄) − ℓ(C, ∂(Sν̄ ))(µ̄(Ω) − µ̄(A)) − W1 (ν̄, ν̂(C))],
k [W1 (µ̄, ν̄) + ℓ(C, ∂(Sν̄ ))(µ̄(Ω) − µ̄(A)) + W1 (ν̄, ν̂(C))]. dmin (C)
The effect of ℓ(C, ∂(Sν̄ ))(µ̄(Ω) − µ̄(A)) has coefficient dmink (C) for BU and dmax1 (C) for BL . So the major effect of ℓ(C, ∂(Sν̄ ))(µ̄(Ω) − µ̄(A)) lies in BU . D.4
Advantage of normalization
As pointed out by [63], normalization is more appropriate when the number of points in PD is not of direct interest but their spatial distribution is. This is typically the case when the PDs contain many points or are obtained from large random filtrations, so that the number of points in PD will mostly account for noisy topological fluctuations due to sampling. D.5
Limitation
Vrep is a finite-dimensional histogram representation of Expected Persistence Diagrams (EPDs), and therefore it is inherently lossy. Its main source of information loss comes from aggregating all mass inside the same Voronoi cell into a single coefficient. As a result, Vrep can preserve coarse mass allocation across cells, but it discards within-cell geometry. When the task-relevant variation is dominated by fine local shifts or changes in the shape of the distribution inside a cell, smooth vectorizations such as PI, PS, or kernel methods may be more sensitive. The quality of Vrep also depends on the choice of codebooks. Codebook size, sampling strategy, and Voronoi cell geometry determine the resolution of the representation. A small or poorly placed codebook may merge distinct topological regions, while a very large codebook increases representation dimension and can reduce stability under perturbations. Our experiments compare default, persistence-weighted, and uniform codebook sampling, but optimal or task-adaptive codebook construction remains open. Vrep also relies on normalization of EPD mass in our theoretical analysis and main experimental setting. This normalization makes Wasserstein distance directly applicable by giving EPDs equal total mass, but it removes information about the total number or total mass of topological features. In tasks where feature count, total persistence mass, or density of topological events is predictive, normalization may discard useful signal. In such cases, unnormalized variants or additional massbased summary features may be preferable. Finally, our empirical evaluation focuses on datasets where topological information is expected to be useful. The results show that Vrep is competitive in these settings and can be computationally attractive when many sampled persistence diagrams are used to form an EPD. However, the experiments do not establish uniform superiority over PI, PS, PL, PWGK, SWK, or supervised topological representations. Broader evaluation on additional domains, larger datasets, and downstream tasks is needed to better characterize when adaptive Voronoi aggregation is preferable to smooth functional or kernel-based representations.
E
Experiment Setting & Summary of Datasets
E.1
Detailed settings
For the Persistence Image (PI), the resolution is set to 10 × 10, resulting in a 100-dimensional vector. The bandwidth is in the range of [0.0001, 0.001, 0.01, 0.1]. For Persistence Landscape (PL), the kmax hyperparameter k is in the range of [2, 4, 6, 8]. The resolution of Persistence Landscape and Persistence Silhouette (PS) is 100 for alignment with Persistence Image. The weight function of PI and PS is the square of persistence. For Persistence Weighted Gaussian Kernel, the bandwidth is also in [0.0001, 0.001, 0.01, 0.1]. For Vrep (Vrepd ), we set t = 10 and k is in the range of [2, 4, 6, 8, 10, 12, 14, 16, 18, 20]. We set the subsample size of each EPD support to be 50, i.e. 20
|Sµ̄ | = 50, The dimension of topological features is also viewed as a hyperparameter. We only consider 0-dimensional or 1-dimensional PD, i.e. connected components or rings. For the computation of EPD, in CAD, each sampled PD is computed from 2% randomly sampled points from point cloud. For Protein, each sampled PD is computed from 50 randomly sampled points. For time series data, each sampled PD is computed from 50% randomly sampled points from point cloud. The number of sampled PD in each EPD is 50. For Random Forest Classifier, we set the size of the forest to be 100 and use Gini impurity to measure the quality of a split. In the study on the codebook choice for Vrep and Vrepd , for a dataset of EPDs, for the uniform choice, we choose to sample codebook from a uniform distribution supported on a rectangle area that covers all the EPDs in the dataset. For scaleup test on the number of sampled PDs in each EPD, we use the CAD dataset with no noise. The time cost of each vectorization is computed on the entire dataset. We report the mean time cost of over different hyperparameter choices. For PointNet, we set batch size to be 8. We use Adam optimizer and set the learning rate to be 0.001. We train the model with 50 epochs in all the datasets except for the Protein dataset, which uses 400 epochs to ensure convergence. In addition, PointNet requires the input to be in the same size but different point clouds in Protein and CAD datasets have different number of points. So for protein, we fix the input size to be 30. For CAD, the input size is 2500. All methods used in the following experiments were run on a machine with i7-12700K CPU (3.60 GHz), 128 GB RAM and NVIDIA GeForce RTX 3060 GPU. PD is computed purely on CPU. PointNet runs on GPU. All the vectorization methods are conducted on CPU. E.2
Summary of datasets
The summary of datasets is provided in Table 3. An example point cloud in CAD dataset is provided in Figure 9. For each protein, a cross-correlation matrix C is provided. Then we use distance matrix D, where Di,j = 1 − |Ci,j |, as input to the EPD computation. CAD is a large-scale dataset of 3D objects of mechanical components. We choose a preselected version [11] of this dataset which contains two classes ‘Bearing’ and ‘Motor’. For time series data (Beef, BirdChicken, DistalPhalanxTW, Earthquakes, ECG200), we follow the setting in [26] and transform a time series into point cloud via Time-Delaying embedding [55]. Table 3: Summary of the point cloud datasets. In each each dataset: the size (number of point clouds), the minimum and maximum numbers of points in one point cloud and the number of classes. dataset Protein CAD Beef BirdChicken DPTW Earthquakes ECG200
size 99 126 60 40 539 461 200
# min 32 30469 468 510 78 510 94
# max 691 206858 468 510 78 510 94
# classes 5 2 5 2 6 2 2
Figure 9: A point cloud in CAD dataset.
21
F
Additional Experiments
F.1
Hyperparameter sensitivity of Vrep (Vrepd ) with different codebook choices
Figure 10: Hyperparameter sensitivity of Vrep (a) and Vrepd (b) with the default codebook choice w.r.t. k. By default, the value of t is 10.
Figure 11: Hyperparameter sensitivity of Vrep (a) and Vrepd (b) with the persistence-weighted codebook choice w.r.t. k. By default, the value of t is 10.
Figure 12: Hyperparameter sensitivity of Vrep (a) and Vrepd (b) with the uniform codebook choice w.r.t. k. By default, the value of t is 10. The hyperparameter sensitivity results of Vrep (Vrepd ) with persistence-weighted and uniform codebook choice are shown in Figure 11 and 12, respectively. The uniform case is less stable w.r.t. k because the codebook area of the uniform case, i.e. the minimal box above the diagonal that bounds all the EPDs, is larger than the codebook area of the persistence-weighted case, which is restricted to the support of EPD. And when a new point is add to the codebook, a larger codebook area contains more variability for the location of this new point. This variability leads to the instability. 22
F.2
Comparison with Persistence Weighted Gaussian Kernel and Sliced Wasserstein Kernel
Persistence Weighted Gaussian Kernel (PWGK) is a kernel for P PD. When employed Pn Pn on EPD, it mean sures the similarity of two EPDs µ̄1 = n1 i=1 µ1i and µ̄2 = n1 i=1 µ2i as n1 i=1 PWGK(µ1i , µ2i ). When compared with PWGK, D Vrep (VrepdE) is used in a linear kernel fashion, i.e. the similarity of two EPDs is defined to be Φ̂(µ̄1 ), Φ̂(µ̄2 ) . The average accuracy and time cost over 10 random split are shown in Tables 4 and 5. The time cost of computing PWGK for EPD is too large for us to conduct scaleup test, so here we only report its average time cost for the datasets. The same goes for Sliced Wasserstein Kernel (SWK). Table 4: Average accuracy and time used to computing kernel value of Vrep, Vrepd , PWGK and SWK. Here we only consider 0-dimensional PD and EPD.
Protein CAD CAD0.01 CAD0.05 Beef BirdChicken DPTW Earthquakes ECG200
EPD 0.975 0.846 0.835 0.838 0.483 0.938 0.646 0.809 0.758
Vrep time (s) 4.24 6.76 4.16 4.16 0.981 0.44 77.98 56.07 10.24
Vrepd EPD time (s) 0.975 2.42 0.823 7.64 0.827 4.86 0.842 4.87 0.5 1.14 0.938 0.51 0.642 90.56 0.786 65.09 0.753 11.87
PD 0.880 \ \ \ 0.250 0.938 0.693 0.803 0.680
PWGK EPD time (s) 0.895 151.35 0.912 1339.71 0.900 1650.24 0.877 2120.78 0.408 82.23 0.938 43.02 0.737 2056.88 0.787 4322.23 0.740 261.57
PD 0.955 \ \ \ 0.267 0.888 0.744 0.803 0.807
SWK EPD 0.865 0.785 0.785 0.662 0.150 0.900 0.750 0.803 0.798
time (s) 25.97 3606.91 3434.44 5200.52 165.99 84.22 1733.85 7716.87 302.59
Table 5: Average accuracy and time used to computing kernel value of Vrep, Vrepd PWGK and SWK. Here we only consider 1-dimensional PD and EPD.
Protein CAD CAD0.01 CAD0.05 Beef BirdChicken DPTW Earthquakes ECG200
EPD 0.915 0.877 0.846 0.858 0.550 0.963 0.756 0.780 0.790
Vrep time (s) 2.18 4.34 4.33 4.24 0.93 0.41 73.75 58.45 10.08
Vrepd EPD time (s) 0.895 2.43 0.865 5.03 0.838 5.06 0.846 4.98 0.517 1.10 0.975 0.49 0.751 85.91 0.783 67.87 0.813 11.95
PD 0.910 \ \ \ 0.367 0.850 0.589 0.745 0.675
PWGK EPD time (s) 0.330 86.12 0.919 162.33 0.919 163.97 0.896 179.90 0.300 17.29 0.850 8.88 0.713 847.07 0.808 2780.98 0.717 130.12
PD 0.520 \ \ \ 0.275 0.900 0.771 0.803 0.813
SWK EPD time (s) 0.165 45.37 0.888 347.89 0.900 386.69 0.888 447.42 0.067 11.70 0.900 8.14 0.723 265.75 0.803 6822.34 0.700 36.11
For 0-dim topological features, PWGK outperforms Vrep and Vrepd in CAD, CAD0.01 , CAD0.05 and DPTW. And SWK outperforms Vrep and Vrepd only on DPTW and ECG200. For 1-dim topological features, PWGK and SWK outperforms Vrep and Vrepd in CAD, CAD0.01 , CAD0.05 and Earthquakes. In summary, PWGK and SWK perform similarly to Vrep and Vrepd . But the time cost of PWGK and SWK is much higher than those of Vrep and Vrepd . In addition, we notice that the accuracy of Vrep and Vrepd used in a linear kernel fashion is lower than the results when they used as vector and input into a Random Forest classifier. So it is recommended that Vrep (Vrepd ) is used in vector form instead of kernel form. F.3
Comparison with more traditional baselines: Betti Curve and Euler Curve
The results of two rather standard method Betti and Euler Curve are provided in Table 6. For Betti Curve, we choose the homology dimension to be 1. Although the time cost of Betti and Euler Curve is low, the accuracies of both Betti and Euler Curve are lower than Vrep over all the datasets.
23
Table 6: Classification accuracy of Betti and Euler Curves. The classifier is Random Forest.
Protein CAD CAD0.01 CAD0.05 Beef BirdChicken DPTW Earthquakes ECG200
F.4
Betti Curve PD EPD 0.95 0.89 \ 0.90 \ 0.90 \ 0.88 0.37 0.38 0.96 0.90 0.74 0.73 0.78 0.79 0.78 0.75
Euler Curve PD EPD 0.21 0.93 \ 0.91 \ 0.90 \ 0.90 0.37 0.50 0.91 0.90 0.74 0.74 0.78 0.77 0.83 0.81
Unsupervised dimensionality reduction task with Vrep on Dynamic System data
Table 7: Dimensionality reduction results of different representation methods and corresponding Calinski-Harabasz (CH) index.
Vrep, CH = 1751
Vrepd , CH = 1702
PS, CH = 1497
PI, CH = 569
PL, CH = 1655
Here we use Vrep, Verpd , PI, PS, and PL in unsupervised dimensionality reduction task. For an EPD dataset, we obtain its high-dimensional representation with different methods and then input into t-SNE [58] for a final 2d representation for visualization. For evaluation on the 2d repesentation, we use Calinski-Harabasz (CH) Index [10], defined as P ni ∥ci − c∥2 /(NC − 1) CH = P Pi , i x∈Ci ∥x − ci ∥2 /(n − NC ) where c is the center of the dataset and ci is the center of cluster Ci . ni is the number of points in cluster Ci and NC is the number of clusters. A higher CH value indicates a better cluster distribution. We use a 3d dynamical system dataset [43, 26], the same as that in the Stability Section, which describes discrete food chain model. This dataset contains 9 classes, each class contains 50 point clouds with each point cloud having 2000 points. Each class corresponds to a parameter of the dynamical system. For EPD computation, we sample 10% points for sampled PD, and each EPD consists of 50 sampled PDs. Each sampled PD is computed via Rips Complex. The perplexity of t-SNE is searched in the range of [5, 10, 15, 20]. The CH result and visualization are shown in Table 7. Vrep outperforms the rest methods, followed by Vrepd , which is consistent with our classification result in Table 1 of the main paper. Compared 24
with PS and PL, Vrep and Vrepd tend to produce clusters with small intracluster distance. Part of class 2 is mixed with class 8 in PS and PL, while in Vrep and Vrepd , class 2 is not split. F.5
Runtime comparison with PointNet
We compare the time cost of Vrep with PointNet in Table 8. For small datasets, the sum of training and vectorization time is significantly smaller than the end-to-end training time of PointNet. For larger datasets like DPTW and Earthquakes, PointNet incurs a comparable time cost to Vrep because the time cost of Vrep increases quadratically with respect to the size of the dataset and the time for Vrep vectorization takes up the majority of the total time cost. Table 8: Time cost (s) of Vrep, Vrepd and PointNet. For Vrep and Vrepd , the vectorization (training) column is the time cost of constructing feature (training the Random Forest classifier). PointNet obtains the hidden vector and classifier in an end-to-end manner. The end-to-end column is the total time cost of the end-to-end training process. Vrep and Vrepd are executed on CPU, while PointNet is on GPU.
Protein CAD CAD0.01 CAD0.05 Beef BirdChicken DPTW Earthquakes ECG200 F.6
Vrep vectorization 3.21 5.55 4.25 4.20 0.96 0.43 75.87 57.26 10.16
train 0.13 0.18 0.15 0.14 0.18 0.11 1.69 1.32 0.30
Vrepd vectorization 2.43 6.34 4.96 4.93 1.12 0.50 88.24 66.48 11.91
train 0.17 0.16 0.15 0.14 0.10 0.07 1.75 1.19 0.30
PointNet end-to-end 23.78 42.77 44.35 43.42 22.74 20.47 69.83 69.35 32.78
Scaleup test on the dataset size
We report the time ratio in the scaleup test on dataset size in Figure 13, corresponding to the actual time cost in Figure 6 in the main paper. Vrep (Vrepd ) has quadratic time complexity while PI, PL and PS all have linear complexity. This is consistent with our analysis in the Time Complexity Section of the main paper.
Figure 13: Scaleup test on the dataset size, where the dataset size is 126 at data size ratio =1. The number n of sampled PD in each EPD is set to be 10. The time ratio is reported here. F.7
Synthetic Correlation with Wasserstein Distance
To isolate the approximation behavior of different vectorizations, we construct synthetic (80) EPDs with controlled distributional changes. Each EPD is an empirical measure with 120 points in the birth-death plane. We consider three settings: (i) coarse mass shift, where one mixture component 25
moves across separated regions of the plane; (ii) within-cell shift, where a compact component undergoes a small local translation; and (iii) local shape change, where the covariance orientation changes while the mean remains fixed. For each setting, we generate 80 EPDs, compute pairwise W1 distances, and report Spearman’s ρ and Kendall’s τ between W1 and representation-space distances. Synthetic EPD construction. We construct three synthetic EPD families to isolate different approximation regimes. Each EPD is an empirical measure with 120 points in the birth-death plane, and all sampled points are projected to satisfy d > b. The parameter θ ∈ [0, 1] controls the distributional change within each family. • Coarse mass shift. Each EPD is sampled from a two-component Gaussian mixture. One component remains fixed, while the other moves as θ increases. This setting creates a coarse displacement of mass across separated regions of the birth-death plane. • Within-cell shift. Each EPD is sampled from a single compact Gaussian component whose mean undergoes a small local translation. This setting tests whether a vectorization can detect fine local movement that may remain inside the same Voronoi cell. • Local shape change. Each EPD is sampled from a Gaussian distribution with fixed mean but rotating anisotropic covariance. This setting changes the local shape of the distribution while keeping the coarse mass location similar.
Figure 14: Synthetic EPD families used in the Wasserstein correlation experiment. Columns show representative EPDs at θ = 0, θ ≈ 0.5, and θ = 1. The coarse mass shift setting mainly changes mass allocation across separated regions; the within-cell shift setting creates small local translations; and the local shape change setting modifies within-region geometry while keeping the mean fixed. The results in Table 9 support the intended trade-off. Vrep and Vrepd obtain the highest correlations in the coarse mass shift setting, where Wasserstein variation is largely explained by mass displacement 26
Table 9: Synthetic correlation with Wasserstein distance. Each entry reports Spearman’s ρ / Kendall’s τ between pairwise representation distances and W1 distances. Synthetic setting
Vrep
Vrepd
PI
PS
PL
Coarse mass shift Within-cell shift Local shape change
0.992 / 0.920 0.989 / 0.909 0.982 / 0.887
0.989 / 0.910 0.989 / 0.909 0.983 / 0.890
0.987 / 0.899 0.991 / 0.918 0.980 / 0.875
0.907 / 0.731 0.978 / 0.875 0.875 / 0.687
0.896 / 0.719 0.661 / 0.471 0.904 / 0.727
across regions that can be captured by Voronoi cells. In the within-cell shift setting, PI is slightly better, indicating that a smooth representation can be more sensitive to local translations when the Voronoi partition is already sufficiently coarse. These synthetic results therefore support the interpretation of Vrep as a complementary representation: it can preserve Wasserstein-scale variation under suitable cell resolution, but it is not a uniformly distance-preserving embedding. F.8
Codebook construction in Vrep/Vrepd : sampling, EPD quantization, and ATOL-style centers
The default Vrep construction samples codebooks from EPD supports. To test how sensitive the empirical behavior is to this choice, we repeat the synthetic Wasserstein-correlation experiment with two alternatives. The first alternative uses the official implementation of the EPD quantization procedure from the GUDHI tutorial utilities (https://github.com/GUDHI/TDA-tutorial/tree/master/ tutorials/utils) for [25], with an additional diagonal Voronoi cell during center optimization. For this variant, we use the same total number of codebooks and the same number of bins per codebook as the sampled-support Vrep construction, and initialize each run from a random subset of the pooled EPD support. This initialization avoids the first-diagram bias of the tutorial’s default high-persistence initializer in multimodal synthetic diagrams. The second alternative uses ATOL-style global centers: we learn a global set of k-means centers from the pooled training supports and use these centers as assignment anchors. In all cases, we keep Vrep’s hard Voronoi histogram readout unchanged, so the experiment isolates the effect of codebook construction rather than changing the downstream representation. Table 10: Effect of codebook construction on synthetic Wasserstein correlation. Each entry reports Spearman’s ρ / Kendall’s τ between pairwise Vrep distances and W1 distances. Synthetic setting Codebook construction Vrep Vrepd Coarse mass shift Coarse mass shift Coarse mass shift
Sampled supports EPD quantization centers ATOL-style centers
0.992 / 0.920 0.993 / 0.922 0.969 / 0.844
0.989 / 0.910 0.992 / 0.918 0.962 / 0.829
Within-cell shift Within-cell shift Within-cell shift
Sampled supports EPD quantization centers ATOL-style centers
0.989 / 0.909 0.985 / 0.895 0.941 / 0.796
0.989 / 0.909 0.986 / 0.898 0.945 / 0.806
Local shape change Local shape change Local shape change
Sampled supports EPD quantization centers ATOL-style centers
0.982 / 0.887 0.970 / 0.850 0.948 / 0.800
0.983 / 0.890 0.970 / 0.850 0.943 / 0.793
The results in Table 10 show that codebook construction matters. Sampled-support codebooks (the default choice of Vrep/Vrepd ) give the strongest and most stable correlations in this experiment. ATOL-style centers remain competitive but are consistently weaker, which is expected because a single global set of centers has lower resolution than the concatenation of multiple sampled codebooks. EPD quantization is competitive on all three settings after matching the sampled-support resolution and using a pooled-support initializer. This comparison also shows that EPD quantization is sensitive to initialization: using only the first diagram’s high-persistence points can miss modes in multimodal diagrams and produce poor Voronoi histograms. With a mode-covering initialization, quantized codebooks provide a principled alternative to sampled supports, while ATOL-style centers give a lower-resolution but still competitive global-center baseline. EPD quantization and ATOL-style have 27
certain objective functions and hence require more time cost for optimization, while the sampled supports do not require such techniques and is faster, as shown in Table 11. Table 11: Average time cost (s). Sample Quantization Coarse mass shift, Vrep Coarse mass shift, Vrepd Local shape change, Vrep Local shape change, Vrepd Within cell shift, Vrep Within cell shift, Vrepd
G
0.0012 ± 0.0001 0.0011 ± 0.0000 0.0015 ± 0.0001 0.0013 ± 0.0002 0.0015 ± 0.0001 0.0014 ± 0.0001
0.5031 ± 0.2604 0.3617 ± 0.0048 0.3900 ± 0.0083 0.3750 ± 0.0131 0.3988 ± 0.0138 0.3704 ± 0.0090
ATOL 0.0912 ± 0.0641 0.0592 ± 0.0036 0.0784 ± 0.0056 0.0715 ± 0.0048 0.0835 ± 0.0049 0.0830 ± 0.0089
On the use of Large Language Models
Large Language Models (LLMs) were utilized in the polishing phase of this paper’s preparation. Specifically, LLMs were employed to optimize linguistic clarity, enhance stylistic coherence, and correct minor grammatical or syntactical inconsistencies. All core intellectual content, including conceptual frameworks, empirical observations, argumentative structure, and citation alignment, was developed, curated, and validated exclusively by the human authors.
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