Pith. sign in

REVIEW 2 major objections 5 minor 79 references

The paper claims that a kernel-based permutation test detects differences in persistence intensity functions at the minimax-optimal separation rate, with matching upper and lower bounds of order (n+m)^{-s̄/(2s̄+1)} over anisotropic Sobolev

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 09:12 UTC pith:F3DRBZCD

load-bearing objection First minimax power analysis for persistence intensity functions, but the upper-bound rate rests on an unproved permutation-quantile bound for unbounded-cardinality diagrams. the 2 major comments →

arxiv 2607.20893 v1 pith:F3DRBZCD submitted 2026-07-23 math.ST stat.TH

A Two-Sample Test on Weighted Persistence Intensity Functions in Topological Data Analysis

Yeongung Han , Ilmun Kim , Jisu Kim This is my paper
classification math.ST stat.TH MSC 62G1062G2055N3160D05
keywords persistence intensity functiontwo-sample testpermutation testkernel methodminimax optimalityanisotropic Sobolev ballČech complextopological data analysis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that a kernel-based permutation test can tell two persistence intensity functions apart as soon as their weighted L2 difference is of size roughly (n+m)^{-s̄/(2s̄+1)}, where s̄ is the harmonic mean of the two smoothness parameters of an anisotropic Sobolev ball. It proves an upper bound on the test's separation rate and a matching lower bound showing no level-α test can do better. The technical crux is that persistence diagrams can have unbounded cardinality; the paper introduces a conditional-moment assumption that controls this and yields a sharp variance bound. It also gives a complete description of the 1-dimensional Čech persistence diagram of point clouds on a circle, used to make the lower-bound construction realistic. A bandwidth-aggregated version of the test is implemented and shows high empirical power.

Core claim

The central claim is that the proposed permutation test achieves the minimax optimal separation rate for testing equality of persistence intensity functions over anisotropic Sobolev balls: the upper-bound rate (Theorem 4.4) and the lower-bound rate (Theorem 4.5) match at (n+m)^{-s̄/(2s̄+1)}, with s̄ the harmonic mean of s1 and s2. The test statistic is an unbiased U-statistic estimator of the squared RKHS distance between weighted kernel embeddings of the two unknown intensity functions; its variance is controlled by a new assumption (A3) that handles unbounded diagram cardinality through a conditional moment bound. The lower bound is built by embedding arbitrary probability densities on the

What carries the argument

The load-bearing object is the weighted kernel embedding µ_p = ∫ p(x) w(·) w(x) k_λ(·, x) dx into the RKHS of the kernel k_λ(x,y) = (1/(λ1λ2)) k1((x1−y1)/λ1) k2((x2−y2)/λ2). The squared RKHS distance between µ_p and µ_q admits an unbiased two-sample U-statistic estimator, and the paper's variance bound for that estimator is what makes the upper-bound rate possible. The bandwidth is set to λ_i = (n+m)^{−τ/(2s_i)} with τ = (1 + 1/(4s1) + 1/(4s2))^{-1}, which optimally balances bias and variance. On the lower-bound side, the key mechanism is an explicit characterization of the Čech complex on a circle: a point cloud on S^1(d) has 1-dimensional persistence diagram consisting of a single point (b

Load-bearing premise

The upper-bound proof relies on a quoted permutation-quantile bound that is asserted to hold for persistence diagrams with unbounded cardinality; if that bound fails in this setting, the claimed minimax separation rate for the test is not established.

What would settle it

Simulate the permutation test under a Poisson-process model with unbounded diagram cardinality and compare the empirical (1−α) quantile of the permuted test statistics to the bound C δ^{-1/2} ln(1/α)/((n+m)√(λ1λ2)) of Lemma E.4; if the quantile grows faster than this as the mean cardinality increases, the upper-bound rate collapses.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the bounds are correct, any level-α test on persistence diagrams—not just the proposed one—requires roughly ε^{-(2s̄+1)/s̄} total observations to detect weighted intensity differences of size ε, so the rate is a fundamental limit.
  • The bandwidth aggregation version removes the need to know the smoothness parameters s1, s2, at only an iterated-logarithmic price, making the optimal-rate procedure usable without oracle information.
  • The explicit Čech-on-circle characterization offers a direct geometric construction of point clouds with prescribed single-feature persistence diagrams, which can serve as building blocks for other TDA constructions.
  • The variance bound extends kernel two-sample testing from bounded-cardinality multisets to random measures with unbounded cardinality, subject to the conditional-moment assumption (A3).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the lower-bound construction embeds all densities on Ω into the diagram model, so minimax hardness results for ordinary nonparametric two-sample testing transfer to TDA; no diagram-specific representation can avoid the rate.
  • Editorial inference: the same U-statistic variance decomposition should apply to one-sample goodness-of-fit and independence tests for persistence intensity functions.
  • Editorial inference: the empirical dependence on the weight function suggests a data-driven choice of w, analogous to bandwidth aggregation, as a natural extension.
  • Editorial inference: the closed-form circle construction could provide exact calibration examples for TDA tests, not just theoretical lower bounds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes a kernel-based permutation two-sample test for persistence intensity functions. It embeds weighted intensity functions in a weighted RKHS, estimates the squared RKHS distance between the two embeddings by a U-statistic, and analyzes the uniform separation rate over anisotropic Sobolev balls. Under assumptions (A1)-(A3) on the random persistence diagram model, a variance bound for the test statistic is derived; this is combined with a permutation-quantile bound to obtain an upper separation rate of order (n+m)^{-\bar{s}/(2\bar{s}+1)}. A matching minimax lower bound is constructed by embedding probability densities on a rectangle into singleton persistence diagrams, using an explicit characterization of the Čech persistence diagram on a circle. The paper also presents a bandwidth-aggregated test and numerical comparisons on simulated and real data.

Significance. If the upper-bound proof is completed, this would be the first minimax-optimality result for two-sample testing on persistence diagrams and a meaningful bridge between kernel MMD testing and TDA. The lower-bound embedding is conceptually elegant, and the Čech-on-circle characterization is of independent interest. The paper is detailed, with a full supplement and public code. However, the central upper bound relies on an imported, unproved permutation-quantile bound for unbounded-cardinality diagrams, and the lower-bound embedding has a repairable but real gap. These issues prevent acceptance in the present form.

major comments (2)
  1. [Supplementary Material, §E.2 (Lemma E.4); used in Proposition 4.3 and Theorem 4.4] The permutation-quantile bound is the second key ingredient of the upper bound, but it is asserted for the present unbounded-cardinality setting with the proof attributed to Kim et al. (2022, Thm 6.1) via Schrab et al. (2023, Prop. 4). No argument is given that the hypotheses of those Euclidean, bounded-kernel results carry over to the conditional distribution of the U-statistic (10) when the diagram cardinalities are unbounded. The quantile q̂_{1−α} is conditional on the realized diagrams, so a single unusually large diagram can inflate it; Assumption (A3) controls an expectation weighted by ‖w²p_ℓ‖∞ and does not by itself imply the requisite tail control for the conditional quantile. Since Theorem 4.4 depends on Lemma E.4 through Proposition 4.3, the matching upper bound is not established as written.
  2. [Supplementary Material, §F.5, Part A, and Proposition 5.1] The lower bound requires that for each (b,d)∈Ω there is a point cloud X(b,d)⊂S¹(d) with PD₁(Čech(X(b,d)))={(b,d)}. The text gives a verbal construction and invokes Proposition 5.1, but it does not specify n or the exact positions as a function of (b,d), nor does it verify that the map x↦X(x) can be chosen measurably so that the pushforward of f yields a genuine random point cloud satisfying (A1). This is likely repairable, but as written it is a gap in the embedding step of Theorem 4.5.
minor comments (5)
  1. [Abstract and Section 2.4] The claim that the model is 'broad enough to include all probability densities on the subset of R² where y>x≥0' overstates the assumptions: the model class P imposes (A2), bounded death time, and the weighted-intensity bounds in (A3). Lemma F.5 only provides an embedding for densities satisfying those restrictions. Suggest rewording.
  2. [Section 5, Proposition 5.1] In boundary configurations where the maximal adjacent distance equals 2r (for example, three points on a closed semicircle with an antipodal pair), the formula gives birth = death = r. It should be clarified whether a point on the diagonal is counted in PD₁. This does not affect the lower-bound construction, where b<d, but it affects the statement of Proposition 5.1.
  3. [Supplementary Material, proof of Lemma 4.2 (Lemma F.1)] The proof uses Jensen's inequality with X/|X| as a probability measure. If |X| can be zero, this needs a separate convention or an explicit handling of the empty-diagram case.
  4. [Proposition 4.3 and Lemma E.4] The stated lower bound on the number of permutations B differs between Proposition 4.3 (factor 12) and Lemma E.4 (factor 3), and the logarithmic arguments are not aligned. Please reconcile the constants.
  5. [Section 6] Theorem 4.4 establishes the separation rate for the oracle bandwidth λ*. The bandwidth-aggregated Aggtest is not shown in this paper to inherit that minimax rate; it is justified by a citation to Schrab et al. (2026). This should be stated explicitly so that the theoretical and practical claims are not conflated.

Circularity Check

0 steps flagged

No significant circularity: the minimax upper and lower bounds are derived from explicit variance, embedding, and topological arguments; cited permutation-quantile results are external published theorems, not definitions of the target rate.

full rationale

Walking the derivation chain, I find no load-bearing step in which a claimed prediction or first-principles result equals its input by construction. The test statistic (10) is derived in Section C as an unbiased estimator of the squared RKHS distance, and Lemma 4.2 bounds its variance using Assumption (A3) through the conditional-cardinality decomposition shown in the proof (F.2), which is a new argument rather than a restatement of the target result. Theorem 4.4's upper separation rate follows from Lemma 4.2, the standard sufficient condition Lemma E.3, and the permutation-quantile bound Lemma E.4. Lemma E.4 is genuinely load-bearing and is quoted from Schrab et al. (2023, Prop. 4), with proof attributed to Kim et al. (2022, Thm 6.1); since Ilmun Kim is a co-author, this is a self-citation. It is, however, an external published theorem, not a definition of the separation rate, and it does not assert the paper's minimax conclusion. The real caveat is that the paper does not prove that the Euclidean theorem's conditions transfer to persistence diagrams with unbounded cardinality; the lemma is simply stated with a constant C2(M,N,w,k) in Supplementary Material Section E.2. That is a support gap or correctness risk, not a circularity. The lower bound (Theorem 4.5) embeds an independent Euclidean hard-testing problem via Phi and Proposition 5.1, and it verifies the membership of the embedded measures in P separately; it does not reuse the upper-bound assumptions to define the alternatives. Proposition 5.1 is an explicit topological computation, not a renamed known fit, and the bandwidth-aggregation framework is imported for implementation rather than used to prove the oracle minimax theorem. Thus no step reduces to its own inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

The central minimax claims rest on standard RKHS/U-statistics results, the model assumptions (A2)-(A3), and two substantial external results: the permutation quantile bound (Schrab et al. 2023) and the Ingster-type lower-bound lemma (Albert et al. 2022). No new physical entities are introduced.

free parameters (3)
  • Kernel bandwidth λ=(λ1,λ2) = oracle λ_i^*=(n+m)^{-τ/(2s_i)}; practical grid Λ={(2^{-k},2^{-k})}
    The optimal bandwidth depends on unknown smoothness parameters s1,s2; the paper uses bandwidth aggregation in practice.
  • Weight function w = user-selected, e.g. (y-x)^q with q∈{0,1/4,1/2,3/4,1} or arctan(y-x)
    The test and the lower-bound constant c* depend on the chosen weight; no data-driven selection is proposed.
  • Model-class constants M and N = M bounds death times; lower bound requires N>16||w^2||∞/M^2
    Define the class P(M,N); upper-bound constants depend on M,N; the minimax lower bound holds only for sufficiently large N.
axioms (7)
  • domain assumption Permutation quantile bound (Schrab et al. 2023, Prop 4) extends to persistence-diagram U-statistics under Assumption (A3)
    Used in Proposition 4.3 to control q̂; proof is cited, not re-derived for multisets with unbounded cardinality.
  • domain assumption Model class P: (A1) diagrams from a filtration of a random point cloud; (A2) bounded death time y<M; (A3) conditional control of cardinality and weighted conditional intensities
    These assumptions are necessary for the variance bound Lemma 4.2 and the quantile bound.
  • domain assumption Kernel factors k_i∈L1∩L2, symmetric, unit integral; weight w positive, nondecreasing in y−x, bounded on Ω(M)
    Standing conditions in Section 2.4; used throughout.
  • standard math Two-sample U-statistic variance decomposition (Lee 1990)
    Used in Lemma 4.2's proof.
  • standard math Moore–Aronszajn RKHS theorem and reproducing property
    Used to construct the test statistic as an RKHS distance.
  • domain assumption Ingster-type two-point construction (Albert et al. 2022, Lemma 5)
    Used in Theorem 4.5's lower-bound proof; the adaptation is outlined but the core lemma is external.
  • standard math Nerve theorem and homotopy invariance
    Used in the Čech-on-circle characterization (Section G).

pith-pipeline@v1.3.0-alltime-deepseek · 90 in / 34649 out tokens · 290437 ms · 2026-08-01T09:12:37.560825+00:00 · methodology

0 comments
read the original abstract

The intensity function, defined as the Lebesgue density of the expected measure of a persistence diagram, is a fundamental summary of the probability distribution of persistence diagrams in topological data analysis (TDA). Although several methods have been proposed for estimating intensity functions, statistical hypothesis testing for intensity functions remains largely unexplored. In particular, little is known about the power properties of hypothesis tests based on persistence diagrams. We propose a kernel-based permutation test and analyze its power against alternatives characterized by differences in persistence intensity functions. We introduce assumptions that control the effect of the possibly unbounded cardinality of persistence diagrams and yield a sharp variance bound for the test statistic. We also show that our probability model is broad enough to include all probability densities on the subset of $\mathbb{R}^2$ where $y>x\geq 0$. Using these results, we establish minimax optimality of the proposed test. Along the way, we derive an explicit characterization of the persistence diagram of the \v{C}ech complex on the circle. Since the optimal bandwidth is not directly accessible in practice, we adopt a bandwidth aggregation framework. Simulations and real-data applications demonstrate validity and high empirical power.

Figures

Figures reproduced from arXiv: 2607.20893 by Ilmun Kim, Jisu Kim, Yeongung Han.

Figure 1
Figure 1. Figure 1: (Left) the sphere S 2 , (Middle) a point cloud on S 2 , and (Right) the persistence diagram obtained using the Vietoris-Rips complex. time of σ. Moreover, if σ is nontrivial precisely for r1 ď r ď r2, then r1 is called the birth time of σ. Persistent homology records the birth and death times of all topological features of the space across different scales. 2.2 Persistence diagram A persistence diagram is … view at source ↗
Figure 2
Figure 2. Figure 2: Left: Three persistence diagrams randomly sampled from a random persistence diagram D. Right: The estimated persistence intensity function of D, which is constructed using the three samples. 2.4 Assumptions In this section, we state the assumptions used throughout the paper. We assume that the sample sizes n and m satisfy n — m, that is, there exist constants 0 ă c ď C ă 8 such that cm ď n ď Cm. Conditions… view at source ↗
Figure 3
Figure 3. Figure 3: An example of Φ. Each point in Ω is mapped to the persistence diagram consisting of that point, inducing a distribution P from the probability density f on Ω. We briefly explain how density functions on Ω are embedded into the class P. Define a measurable map Φ : Ω Ñ PD by Φpxq – δx. For a probability density function f on Ω, we push forward f dx to obtain a probability distribution P on PD, namely PpAq – … view at source ↗
Figure 4
Figure 4. Figure 4: (left) Torus with radii R “ 2, r “ 1 (middle) Point cloud from Torus (right) Persistence diagrams for H1 features by Vietoris-Rips complex from the point cloud. ing greater weight to persistence therefore enhances the ability of the test to distinguish between the intensity functions. Consistent with this intuition, larger weight functions lead to higher empirical power. ORBIT5K data simulation We conduct … view at source ↗
Figure 5
Figure 5. Figure 5: Results of the torus simulation: (left) empirical powers of tests over different noise [PITH_FULL_IMAGE:figures/full_fig_p031_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of two critical regions C1 and C2. We now discuss permutation tests. If the probability distribution of Tˆ under the null H0 is known, then selecting the critical region is straightforward. However, deriving the distribution of the test statistic under the null typically requires strong assumptions on the underlying data distributions, such as assuming that F and G are Gaussian. One way to avoid… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison between the true null distribution and its permutation-based approx [PITH_FULL_IMAGE:figures/full_fig_p039_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: An example of a homotopic pair. In topology, a fundamental question is whether two spaces are equivalent. The notion of homotopy between two continuous maps provides a criterion for the equivalence of topolog￾ical spaces. A continuous map f : X Ñ Y is called a homotopy equivalence if there exists a continuous map g : Y Ñ X such that f ˝ g » IdY , and g ˝ f » IdX, where IdX : X Ñ X 7 [PITH_FULL_IMAGE:figur… view at source ↗
Figure 9
Figure 9. Figure 9: Examples of point clouds constructed on the circle [PITH_FULL_IMAGE:figures/full_fig_p078_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Illustrations of the radial projection map [PITH_FULL_IMAGE:figures/full_fig_p081_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Visual illustrations of Claim 4. Proof. Since ρX pxq P BX “ X Ť ´Ť x,yPX,x„y rx, ys ¯ , by Lemma G.2, either ρX pxq P X or ρX pxq P Ť x,yPX,x„y rx, ys holds. If ρX pxq P X , then ρX pxq “ x0. For this case, we obviously have }ρX pxq ´ x0}2 ď }x ´ x0}2 . 53 [PITH_FULL_IMAGE:figures/full_fig_p086_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: First row: base shapes, second row: 50 points from each shape, third row: Per [PITH_FULL_IMAGE:figures/full_fig_p101_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Results of the circle data simulation: (left) empirical powers of tests over differ [PITH_FULL_IMAGE:figures/full_fig_p102_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Examples of the truncated orbits tpxn, ynq : n “ 1, . . . , 1, 000u of the linked twist map and their corresponding persistence diagrams for r “ 2.5, 4.0, and 4.3. 71 [PITH_FULL_IMAGE:figures/full_fig_p104_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: (Upper) Raw time series of two instruments. (Lower left) Point cloud of [PITH_FULL_IMAGE:figures/full_fig_p107_15.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

79 extracted references · 2 canonical work pages

  1. [1]

    2000 , publisher=

    Asymptotic Statistics , author=. 2000 , publisher=

    2000

  2. [2]

    2008 , publisher=

    Introduction to empirical processes and semiparametric inference , author=. 2008 , publisher=

    2008

  3. [3]

    Journal of Computational Geometry , author=

    The density of expected persistence diagrams and its kernel based estimation , volume=. Journal of Computational Geometry , author=. 2020 , pages=

    2020

  4. [4]

    and Borgwardt, K

    Gretton, A. and Borgwardt, K. M. and Rasch, M. J. and Sch. A Kernel Two-Sample Test , journal =. 2012 , volume =

    2012

  5. [5]

    Transactions of the American Mathematical Society , volume=

    Theory of reproducing kernels , author=. Transactions of the American Mathematical Society , volume=. 1950 , publisher=

    1950

  6. [6]

    Integral probability metrics and their generating classes of functions , journal =

    M. Integral probability metrics and their generating classes of functions , journal =. 1997 , publisher =

    1997

  7. [7]

    Proceedings of the 35th International Conference on Machine Learning , volume=

    On the expectation of a persistence diagram by the persistence weighted kernel , author=. Proceedings of the 35th International Conference on Machine Learning , volume=. 2018 , publisher=

    2018

  8. [8]

    Journal of Machine Learning Research , volume=

    Persistence images: A stable vector representation of persistent homology , author =. Journal of Machine Learning Research , volume=

  9. [9]

    Lee, A. J. , series =. 1990 , address =

    1990

  10. [10]

    Applied Statistics , volume=

    Limiting Distribution of Two-Sample Degenerate U-Statistic under Contiguous Alternatives and Applications , author=. Applied Statistics , volume=. 2023 , publisher=

    2023

  11. [11]

    Bernoulli , volume=

    Random matrix approximation of spectra of integral operators , author =. Bernoulli , volume=. 2000 , publisher=

    2000

  12. [12]

    and Kim, I

    Schrab, A. and Kim, I. and Albert, M. and Laurent, B. and Guedj, B. and Gretton, A. , journal =

  13. [13]

    Romano, J. P. and Wolf, M. , title =. Journal of the American Statistical Association , year =. doi:10.1198/016214504000000539 , publisher =

    doi:10.1198/016214504000000539

  14. [14]

    Lazar , title =

    Chul Moon and Nicole A. Lazar , title =. Journal of the Royal Statistical Society: Series C , volume =. 2023 , doi =

    2023

  15. [15]

    The Annals of Statistics , volume=

    Minimax optimality of permutation tests , author=. The Annals of Statistics , volume=. 2022 , publisher=

    2022

  16. [16]

    The Annals of Probability , volume=

    The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality , author=. The Annals of Probability , volume=. 1990 , publisher=

    1990

  17. [17]

    Journal of Machine Learning Research , volume =

    Tong Li and Ming Yuan , title =. Journal of Machine Learning Research , volume =

  18. [18]

    The Annals of Statistics , volume =

    An Asymptotically Optimal Window Selection Rule for Kernel Density Estimates , author =. The Annals of Statistics , volume =. 1984 , publisher =

    1984

  19. [19]

    2004 , note =

    Testing for Equal Distributions in High Dimension , author =. 2004 , note =

    2004

  20. [20]

    Journal of Machine Learning Research , year =

    Kernel Method for Persistence Diagrams via Kernel Embedding and Weight Factor , author =. Journal of Machine Learning Research , year =

  21. [21]

    Proceedings of the 33rd International Conference on Machine Learning , year =

    Kusano, Genki and Hiraoka, Yasuaki and Fukumizu, Kenji , title =. Proceedings of the 33rd International Conference on Machine Learning , year =

  22. [22]

    2016 , publisher =

    An Introduction to the Theory of Reproducing Kernel Hilbert Spaces , author =. 2016 , publisher =

    2016

  23. [23]

    2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) , pages =

    A Stable Multi-Scale Kernel for Topological Machine Learning , author =. 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) , pages =. 2015 , publisher =

    2015

  24. [24]

    Proceedings of the 34th International Conference on Machine Learning , editor =

    Sliced Wasserstein Kernel for Persistence Diagrams , author =. Proceedings of the 34th International Conference on Machine Learning , editor =. 2017 , publisher =

    2017

  25. [25]

    Journal of Machine Learning Research , year =

    A Kernel Two‐Sample Test for Functional Data , author =. Journal of Machine Learning Research , year =

  26. [26]

    Journal of Applied and Computational Topology , volume =

    Mauricio Che and Fernando Galaz-García and Luis Guijarro and Ingrid Amaranta Membrillo Solis , title =. Journal of Applied and Computational Topology , volume =. 2024 , doi =

    2024

  27. [27]

    Proceedings of the American Mathematical Society , volume =

    Che, Mauricio and Galaz-García, Fernando and Guijarro, Luis and Membrillo Solis, Ingrid and Valiunas, Motiejus , title =. Proceedings of the American Mathematical Society , volume =. 2024 , doi =

    2024

  28. [28]

    Advances in Neural Information Processing Systems , year =

    Rahimi, Ali and Recht, Benjamin , title =. Advances in Neural Information Processing Systems , year =

  29. [29]

    Journal of Machine Learning Research , volume=

    Universality, characteristic kernels and RKHS embedding of measures , author=. Journal of Machine Learning Research , volume=. 2011 , publisher=

    2011

  30. [30]

    2007 , publisher =

    A Course in Functional Analysis , author =. 2007 , publisher =. doi:10.1007/978-1-4757-4383-8 , isbn =

    doi:10.1007/978-1-4757-4383-8 2007

  31. [31]

    Bernoulli , volume =

    Yannick Baraud , title =. Bernoulli , volume =. 2002 , ISSN =

    2002

  32. [32]

    Proceedings of the 27th International Conference on Artificial Intelligence and Statistics , year =

    Weichen Wu and Jisu Kim and Alessandro Rinaldo , title =. Proceedings of the 27th International Conference on Artificial Intelligence and Statistics , year =

  33. [33]

    1998 , publisher =

    Asymptotic Statistics , author =. 1998 , publisher =

    1998

  34. [34]

    Journal of Applied and Computational Topology , year =

    Andrew Robinson and Katharine Turner , title =. Journal of Applied and Computational Topology , year =

  35. [35]

    Journal of Machine Learning Research , year =

    Peter Bubenik , title =. Journal of Machine Learning Research , year =

  36. [36]

    Small , volume =

    Hertzsch, Jan-Martin and Sturman, Rob and Wiggins, Stephen , title =. Small , volume =

  37. [37]

    Advances in Neural Information Processing Systems , year =

    Kwangho Kim and Jisu Kim and Manzil Zaheer and Joon Sik Kim and Frederic Chazal and Larry Wasserman , title =. Advances in Neural Information Processing Systems , year =

  38. [38]

    Discrete & Computational Geometry , volume =

    Partha Niyogi and Stephen Smale and Shmuel Weinberger , title =. Discrete & Computational Geometry , volume =. 2008 , doi =

    2008

  39. [39]

    Discrete & Computational Geometry , volume =

    Chazal, Frédéric and David Cohen-Steiner and André Lieutier , title =. Discrete & Computational Geometry , volume =. 2009 , doi =

    2009

  40. [40]

    Computational Geometry: Theory and Applications , volume =

    Attali Dominique and André Lieutier and David Salinas , title =. Computational Geometry: Theory and Applications , volume =. 2013 , doi =

    2013

  41. [41]

    Persistent homology—a survey , volume =

    Edelsbrunner, Herbert and Harer, John , year =. Persistent homology—a survey , volume =. Discrete & Computational Geometry , doi =

  42. [42]

    2010 , publisher =

    Edelsbrunner, Herbert and Harer, John , title =. 2010 , publisher =

    2010

  43. [43]

    Barcodes: The persistent topology of data , volume =

    Ghrist, Robert , year =. Barcodes: The persistent topology of data , volume =. Bulletin (New Series) of the American Mathematical Society , doi =

  44. [44]

    The Annals of Statistics , volume=

    Confidence sets for persistence diagrams , author =. The Annals of Statistics , volume=. 2014 , publisher=

    2014

  45. [45]

    Estimation and Quantization of Expected Persistence Diagrams , year =

    Divol, Vincent and Leclerc, Th. Estimation and Quantization of Expected Persistence Diagrams , year =

  46. [46]

    Frontiers in Artificial Intelligence , volume =

    An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists , author =. Frontiers in Artificial Intelligence , volume =. 2021 , doi =

    2021

  47. [47]

    Topology and Data , journal =

    Gunnar Carlsson , year =. Topology and Data , journal =

  48. [48]

    2021 , isbn =

    Gunnar Carlsson and Mikael Vejdemo-Johansson , title =. 2021 , isbn =. doi:10.1017/9781108975704 , address =

    work page doi:10.1017/9781108975704 2021

  49. [49]

    and Broomhead, D

    Stark, J. and Broomhead, D. S. and Davies, M. E. and Huke, J. , title =. 1997 , issue_date =. doi:10.1016/S0362-546X(96)00149-6 , journal =

    doi:10.1016/s0362-546x(96)00149-6 1997

  50. [50]

    Dynamical Systems and Turbulence, Warwick 1980 , series =

    Takens, Floris , title =. Dynamical Systems and Turbulence, Warwick 1980 , series =. 1981 , publisher =

    1980

  51. [51]

    Foundations of Computational Mathematics , volume =

    Perea, Jose and Harer, John , title =. Foundations of Computational Mathematics , volume =. 2015 , doi =

    2015

  52. [52]

    Escolar and Kaname Matsue and Yasumasa Nishiura , title =

    Yasuaki Hiraoka and Takenobu Nakamura and Akihiko Hirata and Emerson G. Escolar and Kaname Matsue and Yasumasa Nishiura , title =. Proceedings of the National Academy of Sciences , volume =. 2016 , doi =

    2016

  53. [53]

    doi:10.1088/0957-4484/26/30/304001 , year =

    Nakamura, Takenobu and Hiraoka, Yasuaki and Hirata, Akihiko and Escolar, Emerson G and Nishiura, Yasumasa , title =. doi:10.1088/0957-4484/26/30/304001 , year =

    doi:10.1088/0957-4484/26/30/304001

  54. [54]

    Scientific Reports , volume =

    Masao Kimura and Ippei Obayashi and Yasuo Takeichi and Reiko Murao and Yasuaki Hiraoka , title =. Scientific Reports , volume =. 2018 , doi =

    2018

  55. [55]

    Herring, A. L. and Robins, V. and Sheppard, A. P. , title =. Water Resources Research , volume =. 2018 , keywords =

    2018

  56. [56]

    Water Resources Research , volume =

    Jiang, Fei and Tsuji, Takeshi and Shirai, Tomoyuki , title =. Water Resources Research , volume =. 2018 , pages =

    2018

  57. [57]

    Paul Bendich and J. S. Marron and Ezra Miller and Alex Pieloch and Sean Skwerer , title =. The Annals of Applied Statistics , number =. 2016 , doi =

    2016

  58. [58]

    Sholl and J

    Peter Lawson and Andrew B. Sholl and J. Quincy Brown and Brittany Terese Fasy and Carola Wenk , title =. Scientific Reports , year =

  59. [59]

    The Annals of Applied Statistics , number =

    Chul Moon and Qiwei Li and Guanghua Xiao , title =. The Annals of Applied Statistics , number =. 2023 , doi =

    2023

  60. [60]

    A Kernel Method for the Two-Sample-Problem , volume =

    Gretton, Arthur and Borgwardt, Karsten and Rasch, Malte and Sch\". A Kernel Method for the Two-Sample-Problem , volume =. Advances in Neural Information Processing Systems , editor =

  61. [61]

    Proceedings of the 36th International Conference on Neural Information Processing Systems , articleno =

    Schrab, Antonin and Guedj, Benjamin and Gretton, Arthur , title =. Proceedings of the 36th International Conference on Neural Information Processing Systems , articleno =. 2022 , isbn =

    2022

  62. [62]

    Bernoulli , number =

    Xiuyuan Cheng and Yao Xie , title =. Bernoulli , number =. 2024 , doi =

    2024

  63. [63]

    Journal of Machine Learning Research , volume =

    A Kernel Two-Sample Test for Functional Data , author =. Journal of Machine Learning Research , volume =. 2022 , eprint =

    2022

  64. [64]

    and Suslina, Irina A

    Ingster, Yuri I. and Suslina, Irina A. , title =. 2003 , doi =

    2003

  65. [65]

    2023 , issn =

    Topological data analysis of noise: Uniform unimodal distributions , journal =. 2023 , issn =. doi:https://doi.org/10.1016/j.cnsns.2023.107216 , author =

    arXiv 2023

  66. [66]

    2021 , issn =

    On limit theorems for persistent Betti numbers from dependent data , journal =. 2021 , issn =. doi:https://doi.org/10.1016/j.spa.2021.04.013 , author =

    work page doi:10.1016/j.spa.2021.04.013 2021

  67. [67]

    Annals of Statistics , volume =

    Mélisande Albert and Béatrice Laurent and Amandine Marrel and Anouar Meynaoui , title =. Annals of Statistics , volume =. 2022 , doi =

    2022

  68. [68]

    Journal of Machine Learning Research , year =

    Henry Adams and Tegan Emerson and Michael Kirby and Rachel Neville and Chris Peterson and Patrick Shipman and Sofya Chepushtanova and Eric Hanson and Francis Motta and Lori Ziegelmeier , title =. Journal of Machine Learning Research , year =

  69. [69]

    2022 , eprint =

    Comparing Multiple Latent Space Embeddings Using Topological Analysis , author =. 2022 , eprint =

    2022

  70. [70]

    Lehmann and Joseph P

    Erich L. Lehmann and Joseph P. Romano , title =

  71. [71]

    Casella, George and Berger, Roger L , publisher =

  72. [72]

    and Raghupathi, Mrinal , year=

    Paulsen, Vern I. and Raghupathi, Mrinal , year=. An Introduction to the Theory of Reproducing Kernel Hilbert Spaces , publisher=

  73. [73]

    Coxeter, H. S. M. , title =. The Mathematical Gazette , year =

  74. [74]

    I, II, III , author=

    Asymptotically minimax hypothesis testing for nonparametric alternatives. I, II, III , author=. Math. Methods Statist , volume=

  75. [75]

    The Annals of Statistics , volume =

    Cristina Butucea , title =. The Annals of Statistics , volume =. 2007 , month = oct, doi =

    2007

  76. [76]

    Algebraic topology , username =

    Hatcher, Allen , biburl =. Algebraic topology , username =

  77. [77]

    2026 , eprint =

    Aggregation of Statistical Evidence under Exchangeability , author =. 2026 , eprint =

    2026

  78. [78]

    2010 , isbn =

    Edelsbrunner, Herbert and Harer, John , title =. 2010 , isbn =

    2010

  79. [79]

    The Annals of Applied Probability , volume=

    Limit theorems for persistence diagrams , author=. The Annals of Applied Probability , volume=. 2018 , publisher=

    2018