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Statistical Inference for Persistence Diagrams via Landmark Embeddings: Minimax Theory and Finite Approximation

Pramita Bagchi, Sushovan Majhi, Atish Mitra, Žiga Virk

arXiv:2609.07691v1math.STcs.CGstat.TH

TL;DR

Population averaging can erase geometric separation between persistence diagrams, complicating inference from their embeddings. The paper develops diagram-level Hilbert-space inference and structured conditions linking template separation to population signals. Those conditions yield uniform power and matching minimax bounds, while the ABIDE analysis detects no difference under its mean-embedding estimand.

  • Problem

    A lower-distortion bound between individual diagrams need not survive population averaging, so it does not by itself establish detectable population differences.

    Source: §1 · pp. 1–2

  • Method

    The paper develops fixed-map Hilbert-space inference and, for additive landmark embeddings, structured population bounds using certified template separation.

    Source: §3 · p. 9, §3.2 · p. 11, §11 · p. 53

  • Results

    Restricted template conditions yield explicit population-signal margins and uniform power guarantees, with matching minimax sample complexity when a within-class KL comparison holds; ABIDE procedures detect no mean-embedding difference.

    Source: §1 · p. 3, §11 · p. 53

  • Takeaways & Limitations

    For additive embeddings, population means can be interpreted through mean counting measures, while geometric power conclusions require restrictions that keep template separation detectable after averaging.

    Source: §1 · p. 3, §11 · p. 53

    The ABIDE analysis is marginal and does not adjust for subject-level covariates, so it does not isolate diagnosis effects from potential confounding.

    Source: §10.5 · p. 52

Abstract

from arXiv · show

Hilbert-space embeddings enable inference for populations of persistence diagrams, but separation between individual diagrams need not survive population averaging. We develop a framework for inference on population mean embeddings, with particular attention to the additive landmark representations PLACE and PALACE. Treating each diagram as one independent observation, we apply Hilbert-space limit theory to obtain covariance estimators, two-sample tests, and confidence balls under suitable moment conditions, without requiring a lower-distortion bound. For additive embeddings, we identify the population mean as an embedding of the mean counting measure and show that geometric separation of these measures alone cannot guarantee uniform testing power. We then introduce a model with latent template diagrams, missing features, and location perturbations. Under common or feature-specific prevalence conditions, a diagram-level lower-distortion certificate yields explicit lower bounds on population mean separation. These margins provide finite-sample uniform power guarantees, and an additional information-divergence comparison gives matching sample-complexity bounds over restricted scale ranges. Confidence sets yield lower bounds on transport separation of population mean measures and exclusion guarantees for specified structured alternatives. We also quantify how orthogonal truncation changes the certified signal and the approximation allowance needed for confidence sets targeting the full embedding, relating sample size, retained coordinates, and template separation. Simulations examine calibration, power, and coverage, and an analysis of resting-state connectivity from the Autism Brain Imaging Data Exchange illustrates the procedures.

1 Introduction

The paper develops population inference for persistence-diagram embeddings while addressing the gap between diagram-level geometric separation and detectable population signals. It derives uniform power guarantees under structured population conditions, with corresponding confidence and approximation analyses.

Population interpretation

  • For additive embeddings, the population mean equals the embedding of the mean counting measure, and a positive signal lower-bounds transport separation of those measures.Geometric separation alone is insufficient for uniform power: rare expected mass or cancellation can make the signal vanish while transport separation stays fixed.Source: §1 · p. 2

Hilbert-space inference

  • Treating each diagram as one independent Hilbert-valued observation supports limit theory, covariance estimation, two-sample tests, and confidence balls under moment conditions.The fixed-map probability results do not require additivity or a lower-distortion bound.Source: §1 · pp. 2–3

Population power guarantees

  • Under uniform prevalence, response-conditioning, and perturbation restrictions, certified template pairs yield explicit population-signal margins and sufficient sample sizes for finite-sample uniform power.The margins apply across structured alternative classes indexed by minimum template separation.Source: §1 · p. 3

Minimax bounds

  • A matching minimax sample complexity of order g_r(τ)^-2 requires a within-class Kullback–Leibler comparison; the lower-distortion certificate alone does not suffice.The divergence comparison must be bounded by a constant multiple of g_r(τ)^2.Source: §1 · p. 3

Confidence sets and extensions

  • Confidence sets provide lower bounds on transport separation and can exclude specified structured alternatives; nearest-centroid certificates and fixed-representation regression extensions are also developed.The geometric interpretation of confidence sets is distinct from their sampling-uncertainty coverage.Source: §1 · p. 3

Finite approximation

  • Orthogonal truncation changes the certified signal, while inference targeting the full embedding must also allow for omitted-coordinate approximation error.A fixed approximation allowance does not vanish with sample size; a zero lower bound can be inconclusive when the uniform approximation bound is conservative.Source: §1 · pp. 3–4

2 Persistence Diagrams as Statistical Objects

Persistence diagrams encode feature births and deaths but have matching-based geometry, variable cardinality, and no natural linear structure. The section develops counting-measure representations and explains how these properties shape population inference.

2.1 From filtered data to persistence diagrams

  • A persistence diagram records feature birth–death pairs from a filtration; near-diagonal points can be deleted at low bottleneck cost, but persistence alone does not identify noise.The paper restricts analysis to bounded, finite off-diagonal features and notes that cardinality bounds affect later guarantees.Source: §2.1 · pp. 5–6

2.2 Bottleneck geometry

  • Bottleneck distance minimizes the largest matched displacement or diagonal-deletion cost, making it sensitive to the worst mismatch rather than the number of differing features.The bounded-cardinality diagram space is compact, complete, and separable.Source: §2.2 · pp. 6–7

2.3 Random diagrams and counting measures

  • The mean counting measure records expected feature intensity and cardinality, but need not correspond to a diagram or determine the diagram law.The bottleneck distance can also be represented as transport distance between padded counting measures.Source: §2.3 · pp. 7–8

2 Persistence Diagrams as Statistical Objects

  • Diagrams may have dependent points and variable cardinality, so inference treats each diagram—not each point—as the sampling unit.An upper stability bound alone does not ensure that diagram-level separation survives population averaging.Source: §2.4 · p. 8

2.4 Statistical implications of diagram geometry

  • Mean-based inference is available for fixed Hilbert-space maps under moment conditions, while geometric power guarantees require extra structure linking diagram separation to population signal.For additive maps, the mean embedding has a counting-measure interpretation.Source: §2.4 · p. 8

3 Landmark Embeddings as an Interface

The section separates general inference for fixed Hilbert-space embeddings from the additive structure needed to interpret population means as embeddings of mean measures. It shows why diagram-level geometric separation may vanish under averaging, then gives structured conditions that transfer certified diagram separation into population signal.

3.1 Fixed Hilbert embeddings and the additive interface

  • The framework distinguishes fixed-map assumptions, sufficient for limit theory and two-sample procedures, from the additive interface used for population mean-measure analysis.Moment requirements strengthen for central limit, covariance, and quantitative Gaussian approximation results.Source: §3 · p. 9, §3.1 · p. 9

3.2 Landmark constructions

  • PLACE and raw PALACE use pooled point responses and can be additive, whereas the original Mitra–Virk map is a whole-diagram landmark map without the additive mean-measure identity.The distinctive ingredient is an explicit bottleneck lower-distortion certificate for certified landmark constructions.Source: §3.2 · p. 11

3.3 Population mean embeddings and the obstruction

  • For additive PLACE and PALACE maps, the population mean embedding is the embedding of a padded mean counting measure, whose expected feature counts are well defined without feature correspondence.A positive additive mean-embedding signal also lower-bounds transport separation of the mean measures.Source: §3.3 · pp. 11–12
  • A diagram-level lower-distortion certificate alone cannot guarantee uniform population power: separated mass can be arbitrarily rare, and weighted responses can cancel.The additive estimand records transformed mean intensity, not every aspect of the diagram laws.Source: §3.3 · p. 12

3.4 Structured population detectability

  • The template–thinning–perturbation model allows arbitrary joint dependence among feature appearances and perturbations; sub-Gaussianity is used only for an explicit first-moment bound.Other conditions controlling the first moment can replace sub-Gaussianity.Source: §3.4 · pp. 12–13
  • Under common or feature-specific prevalence regimes, the theorem gives population mean-separation lower bounds when structural signal exceeds the perturbation penalty.These are pair-specific guarantees; uniform power requires class-wide bounds on relevant prevalence, retention, response, and perturbation quantities.Source: §3.4 · pp. 15–16
  • The theorem’s sufficient bounds apply only on certified pair classes and admissible scales; failure of a condition or a zero bound is inconclusive about whether the population signal is zero.Uniform or minimax guarantees additionally require class-wide restrictions.Source: §3.4 · p. 16

4 Probabilistic Theory for Embedded Diagrams

This section develops Hilbert-space inference for population mean embeddings, treating each diagram as one observation. It establishes mean and covariance theory, Gaussian approximations, and conditions for finite-coordinate inference to approximate the full embedding.

4.1 Population mean and covariance

  • The target is a Hilbert-space mean embedding, not a mean diagram; distinct diagram laws can share it, so mean-based tests detect mean-separated alternatives.For additive maps, the population mean embedding equals Tν[μP].Source: §4.1 · p. 17, §4.5 · p. 20

4.2 Laws of large numbers and central limit theory

  • The empirical mean has m^-1/2 Gaussian error, and squared-norm statistics have a weighted chi-square null limit; two-sample uncertainty increases with imbalance.The sample size counts independent diagrams, not the persistence points within each diagram.Source: §4 · p. 16, §4.2 · pp. 17–18

4.3 Covariance estimation

  • The empirical covariance supports directional-variance and leading-spectrum estimation, with its rate controlled by independent diagrams and the fourth moment of embedded variation.The results also establish convergence in Hilbert–Schmidt and operator norm.Source: §4.3 · p. 18

4.4 Quantitative Gaussian approximation in finite coordinates

  • Finite-coordinate Berry–Esseen bounds cover centered balls, but approximation can deteriorate with rank and covariance conditioning and excludes covariance-estimation and projection-selection errors.A bounded embedding alone does not control the bound's conditioning term.Source: §4.4 · p. 19

4.5 Finite-coordinate approximation

  • For full-embedding inference, truncation error must be negligible at the m^-1/2 sampling scale; otherwise inference targets the projection unless approximation bias is accounted for.The approximation condition separates truncation error from sampling error.Source: §4.5 · pp. 19–20

5 Two-Sample Inference for Mean Embeddings

This section develops two-sample tests of equality of fixed population mean embeddings, treating each diagram as an independent replicate rather than testing equality of diagram laws. It gives geometric power and matching sample-complexity guarantees under stated structural, identifiability, and model-richness conditions.

5.1 Canonical statistic and null calibration

  • Under equal mean embeddings, the squared-distance statistic has a weighted chi-square null limit and yields asymptotic level-α testing with spectral covariance plug-in calibration.Permutation calibration is finite-sample valid under the stronger null P = Q, but is not generally exact under equal means; any data-dependent fitting must be repeated within permutations unless learned independently.Source: §5.1 · p. 21

5.2 Power in mean-embedding separation

  • With bounded between-diagram variance, testing a fixed mean-embedding alternative requires effective sample size of order Δ^-2, where Δ is the mean-embedding separation.The inverse-square rate reflects signal strength, while the bound also depends on variability and desired Type I and Type II error control.Source: §5.2 · p. 22

5.3 Structured geometric alternatives

  • Common- and feature-specific prevalence conditions yield certified lower bounds on mean-embedding separation and uniform power when the resulting class margin is positive.The restrictions control prevalence, response retention, and perturbation; they exclude rare-mass and cancellation sequences and are not claimed for unrestricted diagram populations.Source: §5.3 · pp. 23–24
  • Geometric separation alone cannot ensure uniform power: without a lower bound on population mass carrying that separation, the alternative class has no finite uniform sample complexity.The result is established for the specified rare-mass construction.Source: §5.3 · p. 25

5.4 Minimax bounds through an information modulus

  • The minimax lower-bound framework defines an information modulus comparing structured alternatives with the mean-null class and requires a regular hard subfamily over the scales considered.The stated common- and feature-specific minimax richness assumptions impose additional restrictions; the linear-floor conditions can be verified for PLACE and PALACE on specified scale ranges.Source: §5.4 · pp. 25–26
  • The minimax richness conditions require regular local alternatives, while the response operator must remain identifiable and well-conditioned on the count and prevalence contrast space.For PLACE, paths must meet its certification condition; for PALACE, they must lie in the covered support and remain admissible over the scale set. The restricted singular-value condition can fail through near-identical landmark responses, prevalence cancellation, or rank loss along the path.Source: §5.4 · p. 27

5.5 Relation to STRAND and scope of the test

  • STRAND pools feature lifetimes and uses an i.i.d.-feature calibration, whereas this framework treats each diagram as one independent observation and estimates covariance across diagrams.STRAND’s pooled approach can be problematic when features within diagrams are dependent, and varying diagram cardinalities cause diagrams with more features to receive greater weight; its diagram-level permutation calibration instead requires exchangeability of complete diagrams.Source: §5.5 · p. 28
  • The embedding retains joint birth–death locations, unlike STRAND’s lifetime-based target, so it can detect location differences despite similar lifetime distributions; neither test alone establishes equality of full diagram distributions.STRAND is invariant to translations along the diagonal and provides Kaplan–Meier localization and an interpretable hazard ratio when proportional hazards holds, or a median-persistence difference otherwise.Source: §5.5 · p. 28

6 Confidence Sets and Geometric Certificates

Confidence sets quantify uncertainty about population mean embeddings and contrasts; deterministic embedding properties then support geometric or decision certificates. The section distinguishes sampling coverage from what those certificates can establish.

6 Confidence Sets and Geometric Certificates

  • The section’s certificates combine statistical uncertainty with deterministic embedding properties, so failure to obtain a positive certificate is inconclusive rather than evidence for the opposite conclusion.The confidence probability comes from the statistical set; its geometric interpretation comes from the embedding interface.Source: §6 · p. 29

6.1 Confidence balls for a population mean embedding

  • For a fixed embedding, moment conditions yield asymptotic confidence balls for population means, with coverage stated pointwise rather than uniformly over diagram distributions.A trace bound also gives a finite-sample ball of radius v/√(mα), though it can be wide or conservative.Source: §6.1 · p. 30

6.2 Pullback and geometric exclusion for mean measures

  • An additive embedding’s confidence ball gives a lower confidence certificate for a candidate mean measure’s transport distance, but a weak certificate does not imply geometric closeness.The bound can be small when the cardinality bound or point-level Lipschitz constant is conservative.Source: §6.2 · p. 31

6.3 Two-sample confidence sets and certified effect sizes

  • Two-sample confidence intervals quantify mean-embedding separation; their lower endpoint also bounds transport separation for additive embeddings, while an upper endpoint below a structured margin can exclude that class.The structured exclusion is pointwise asymptotic and requires a positive, scientifically defensible margin.Source: §6.3 · p. 32

6.4 A secondary certificate for nearest-centroid decisions

  • Nearest-centroid certificates can ensure a query’s assignment agrees with the population-centroid rule and remains unchanged under a bounded bottleneck perturbation.This is a query-specific robustness guarantee, not a classification-risk theorem; it does not eliminate overlap among class-conditional distributions.Source: §6.4 · p. 33

7 Finite Approximation and Inferential Resolution

Finite-coordinate inference can target either the projected mean or the full embedding, but full-target claims must account for omitted coordinates. The section connects truncation error and retained dimension to certified signal, sample size, and confidence-set radii.

7.1 Truncation error and two inferential targets

  • A fixed truncation defines a valid projected estimand; claims about the full embedding require an allowance for omitted coordinates, whose orthogonal truncation reduces certified signal.A zero truncation certificate is inconclusive, and alternatives supported entirely in omitted coordinates may escape projected-mean tests.Source: §7 · p. 34, §7.1 · pp. 34–35

7.2 Testing under finite approximation

  • Uniform power under finite approximation depends on the certified margin after truncation loss and sampling variability, with effective sample size counting diagrams and retained dimension absent from the conservative bound.The certifying truncation level can limit sample-size inflation relative to the full-margin benchmark to (1 − γ^2)^−1.Source: §7.2 · pp. 35–37

7.3 Confidence sets with approximation bias

  • Adding the deterministic approximation error to a projected confidence radius yields coverage for the full mean; for a two-population contrast, the allowance is 2ε_K.A fixed approximation allowance does not shrink with more observations, unlike the statistical radius.Source: §7.3 · p. 37

7.4 Practical selection of the truncation level

  • A theorem-aligned workflow prespecifies the embedding and truncation, obtains a valid deterministic tail bound, and reports statistical uncertainty separately from approximation allowance.Same-sample selection by p-value or studentized separation generally invalidates fixed-K calibration; sample splitting or selection-aware resampling are alternatives.Source: §7.4 · p. 38

8 Extensions

The fixed embedding can serve as a covariate map for regression and classification, but those extensions do not by themselves establish predictability, causal interpretation, or vanishing error. Their guarantees depend on the representation and the relevant statistical and geometric conditions.

8 Extensions

  • With a fixed finite representation, standard regression and classification theory governs estimation; embedding stability alone does not imply predictable outcomes or vanishing classification error.The extensions describe associational prediction and do not provide causal interpretation of topological features.Source: §8 · p. 38

8.1 Regression with a diagram-valued predictor

  • For fixed retained dimension, least-squares inference targets the best linear predictor in those coordinates and uses a heteroskedasticity-robust sandwich covariance.Identifiability requires a positive-definite design condition; growing dimension or regularization requires additional assumptions.Source: §8.1 · pp. 39–40

8.2 Prediction stability and geometric interpretation

  • The additive Lipschitz interface bounds score changes under diagram perturbations and lets a large pairwise score difference certify that diagrams cannot be too close.A response contrast additionally requires alignment between the regression direction and the relevant embedding difference.Source: §8.2 · p. 40

8.3 Classification and scope

  • A fixed linear threshold rule has a query-specific perturbation radius, but geometric separation alone does not guarantee a downstream score contrast or eliminate classification error.Translating certified separation into score contrast requires direction alignment; overlapping class-conditional distributions can retain Bayes error.Source: §8.3 · p. 41

9 Simulation Studies

The simulations assess approximation accuracy, test calibration and power, confidence-ball coverage, and the responses of mean-embedding and survival-based tests to different alternatives across two additive constructions. Results depend on signal-to-noise, covariance estimation, and whether features are analyzed jointly by diagram or pooled as lifetimes; truncation-level effects and alternatives changing cardinality without changing lifetime or birth marginals remain untested.

9.1 Design

  • The design used PLACE and PALACE configurations fixed from an independent 2,000-diagram pilot and controlled mixtures of fixed diagram atoms with exactly known mean-embedding separation.Separate point-cloud scenarios generated random diagrams for the central-limit study; an obvious one-loop versus two-loop alternative yielded power 0.446 at m=m′=20 and 0.828 at m=m′=50.Source: §9.1 · pp. 42–43

9.2 Study 1: Gaussian approximation for the two-sample statistic

  • Fourteen of 18 Gaussian-approximation cells fell inside the Monte Carlo band, and the four exceptions missed by at most 0.008, with no sample-size trend.Effective covariance rank was 1.01–1.06 in every cell, consistent with variation dominated by total persistence mass; the study could not estimate a Berry–Esseen rate.Source: Table 3 · p. 44, §9.2 · p. 44

9.3 Study 2: level

  • Under the strict null, every permutation-test size was within three Monte Carlo standard errors of 0.05, including unbalanced designs, while the tuned near-null stress test’s largest rejection frequency was 0.068.The near-null retained appreciable covariance contrast while limiting mean residual to one tenth of sampling noise; these results do not establish validity under the exact weak null.Source: §9.3 · pp. 45–46

9.4 Study 2: power and the detection boundary

  • Power rose from approximately 0.17 to above 0.90 as d increased from 0.1 to 0.3, with PLACE and PALACE curves matching within Monte Carlo error despite different raw certificates.The comparison shows that power tracks scale-free signal-to-noise rather than the certificates’ raw magnitudes; the fixed-sample-size experiment does not estimate the Δ_ν^-2 sample-size rate.Source: Table 5 · p. 46, §9.4 · pp. 46–47

9.5 Coverage of the confidence balls

  • PLACE confidence-ball coverage stayed nominal, while PALACE over-covered at m=25 (0.976) and approached 0.948 by m=400; both arms were in the accurate plug-in-radius regime at m=201 and m′=235.Oracle coverage was nominal at every sample size, including m=25; PALACE’s small-sample over-coverage reflects covariance estimation and a dispersed estimated spectrum.Source: Table 6 · p. 47, §9.5 · p. 47

9.6 What the test can and cannot detect

  • Under a birth shift, Strand rejected at 0.052 versus PALACE at 1.000 and PLACE at 0.886; under a persistence shift, Strand reached 0.910 while PALACE was 0.174.The methods target different alternatives: lifetime-based Strand has no signal when lifetimes are unchanged, while adaptive PALACE landmarks favor location shifts; PLACE was competitive against both.Source: Table 7 · p. 48, §9.6 · p. 48
  • With increased between-subject persistence-scale dispersion, the log-rank test rejected a true null in more than half of replications, whereas diagram-label permutation maintained its level.Pooled features are clustered within diagrams, violating the log-rank independence assumption; permutation moves each subject’s features together.Source: Table 8 · p. 48, §9.6 · pp. 48–49

9.7 Scope of the simulation evidence

  • The simulations cover the two additive constructions; Mitra and Virk’s (2024) whole-diagram map is covered by Section 4’s general theory but is not simulated here.Source: §9.7 · p. 49
  • The simulations examine birth shifts, lifetime shifts, mixture-weight contrasts, and feature-count changes, but not cardinality changes that preserve lifetime marginals and birth locations.Source: §9.7 · p. 49
  • Point-cloud and level experiments have r(Σ) near unity, so the near-identical statistics in Table 4 do not show that statistic choice is generally immaterial.Source: §9.7 · p. 49
  • No simulation varies truncation level K, leaving Theorem 7.1’s three-way relationship unexamined; testing it requires one configuration truncated to its first K coordinate blocks with ε_K evaluated on that map.Source: §9.7 · p. 49

10 Data Analysis: Resting-State Connectivity in ABIDE

The ABIDE analysis illustrates the inference procedures using resting-state connectivity diagrams, with subjects as the sampling units. No embedding-based test detects a group difference, and the confidence sets do not establish that the population means differ.

10.1 Data and construction of the diagrams

  • The 436 subjects comprise 201 autistic participants and 235 controls across 20 sites; each subject contributes a variable-cardinality connectivity diagram.The diagrams have mean cardinality 165.2 and range [71, 294]; the absolute-correlation dissimilarity treats anticorrelation as coupling.Source: §10.1 · pp. 49–50

10.2 Two-sample tests

  • No embedding-based test rejects, and the site-conditioned analysis reaches the same conclusion; the confidence sets do not establish a difference between group means.PLACE and PALACE would require roughly eight times the present cohort to certify a gap as large as the point estimate; for MV, the bias-corrected estimate is zero.Source: §10.2 · pp. 50–51, §10.3 · p. 52
  • PLACE and PALACE spectral and permutation p-values are close, while their norm statistics are strongly correlated (rank correlation 0.936).For PLACE the p-value pair is 0.157 and 0.153; for PALACE it is 0.160 and 0.162. Table 10 reports permutation p-values across the cohort.Source: §10.2 · pp. 50–51, Table 10 · p. 50

10.3 Confidence sets and the resolution of the non-rejection

  • All three confidence-set separation ratios fall below 1, so the group-mean balls are not disjoint.The estimates imply roughly eight times the present cohort would be needed to certify the PLACE or PALACE gap; MV has no finite sample-size target because its corrected separation is zero.Source: §10.3 · pp. 51–52, Table 11 · p. 51

10.4 Comparison with the survival-based test

  • The site-stratified STRAND asymptotic p-value is 0.005, but its diagram-label permutation p-value is 0.561; the proportional-hazards diagnostic fails.STRAND pools persistence values, whereas the permutation procedure resamples diagram labels; its median lifetime-difference interval covers zero.Source: §10.4 · p. 52, Table 12 · p. 52

10.5 Scope and limitations

  • The analysis is an illustration, not a study of autism, and it does not adjust for age, sex, head motion, or other subject-level covariates.Its group comparisons are marginal and do not isolate diagnosis effects from these potential sources of confounding.Source: §10.5 · pp. 52–53

11 Discussion

The discussion separates statistical inference for fixed Hilbert-space embeddings from the additional assumptions needed to interpret mean differences geometrically. The appendix supplies supporting constructions and verifies certificates for particular embedding maps.

11 Discussion

  • Fixed embeddings support mean and covariance inference, two-sample tests, and confidence sets, but geometric conclusions require additional structure because population averaging can cancel diagram-level separation.The ABIDE procedures do not detect a group difference at the resolution available in that sample; the non-rejections do not establish equality.Source: §11 · p. 53

A Technical Details for Landmark Embeddings

  • The appendix separates general fixed-Hilbert inference from additive PLACE/PALACE results and a distinct whole-law transfer theorem for the Mitra–Virk map.The framework conditions on the embedding configuration; an independently fitted reference configuration permits conditional application of the results.Source: A Technical Details for Landmark… · p. 56, §A.3 · p. 57

A.1 Proof of Lemma 2.1

  • The padded representation maps a diagram to a multiset of off-diagonal coordinates, with diagonal coordinates added to represent diagrams of fewer than n points.The padded transport distance represents bottleneck distance for these diagrams.Source: §A.1 · p. 56, §A.2 · p. 56

A.2 Padded transport geometry

  • Padded transport gives a transport-geometry representation of bottleneck distance, including the empty diagram as the diagonal class.The one-point metric treats matching two points directly or sending both to the diagonal, whichever yields lower cost.Source: §A.2 · p. 56

A.3 General fixed-Hilbert inference scope and additive-interface proofs

  • For fixed-map inference, first moments ensure Bochner means and second moments support Hilbert-space limit theory; higher-moment conditions apply only when a result requires them.Same-sample selection of the embedding configuration is a separate selection problem.Source: §A.3 · p. 57

A.4 Construction-specific verification and certified scale sets

  • PLACE and PALACE certificates apply on construction-specific scale domains and pair classes, rather than sharing one universal numerical floor.For PLACE, the step certificate requires separation at least 3R1 and coherence at active scales; its affine certificate has a distinct stated range.Source: §A.4 · pp. 58–59

A.5 Rare-mass obstruction

  • A diagram-pair certificate alone cannot guarantee population-mean separation: the residual mass may vanish, or embedded responses may cancel after averaging.In the rare-mass example, padded mean measures remain transport-separated while linearity makes the embedding-mean difference proportional to the small mass.Source: §A.5 · p. 60

A.6 Template–thinning–perturbation proofs and secondary corollaries

The appendix derives population-separation bounds for template diagrams under thinning and perturbation, then distinguishes additive mean-measure results from whole-law transfer for the Mitra–Virk map. It also records the moment conditions used for general Hilbert-space inference.

A.6 Template–thinning–perturbation proofs and secondary corollaries

  • The direct heterogeneous-response bound does not require a nonzero response contrast or an Assumption 3.5 certificate.It applies under the additive-map, template-thinning, and perturbation assumptions.Source: §A.6 · p. 61
  • Heterogeneous-prevalence bounds use a restricted response margin, with a direct-response variant that can be numerically stronger.The appendix also gives uniform perturbation-scale bounds by substituting the cardinality and prevalence limits.Source: §A.6 · p. 62
  • With common prevalence, a certified template separation yields a population mean gap of at least p times the certificate margin.The proof factors both the certified template response and perturbation term by the common prevalence.Source: §A.6 · p. 62
  • Bounded movement of appearing features alone does not guarantee whole-diagram closeness under thinning, because template features can be absent.For sub-Gaussian perturbations, a high-probability bound requires choosing a tail level.Source: §A.6 · p. 63

A.7 Whole-law population transfer for the Mitra–Virk map

  • For the Mitra–Virk map, diagram-level separation alone does not ensure population-mean separation: residual mass can vanish and responses can cancel.The theorem gives a structured sufficient bound for a fixed pair; uniform power requires class-wide lower bounds on residual mass and response margin.Source: §A.7 · pp. 64–65
  • Tail-controlled perturbations replace bounded residual radii with a first-moment bound, while bounded-support perturbations retain direct radius bounds.The appendix transfers these residual-law bounds into the stated population-separation result.Source: §A.7 · pp. 64–65
  • The whole-law Mitra–Virk result is distinct from the additive template-mass decomposition and does not assert a characteristic embedding of diagram laws.Its population estimand is not generally an embedding of a padded mean counting measure.Source: §A.7 · pp. 63, 65

B Technical Proofs for Section 4

  • Bochner integrability follows from finite first moments; finite second moments make the covariance operator trace class in the fixed Hilbert space.The appendix assumes a fixed embedding configuration for these mean and covariance facts.Source: B Technical Proofs for Section 4 · p. 65, §B.1 · p. 65

B.2 Proof of Theorem 4.1

This appendix develops the Hilbert-space limit and approximation results underlying the inference procedures and proves testing results for structured alternatives, specifying conditions and constructing hard alternatives under common and feature-specific prevalence. It also derives an information-divergence bound for the feature-specific-prevalence hard subfamily and begins the proof of Lemma 5.1 using the information modulus definition.

B.2 Proof of Theorem 4.1; B.3 Norm-squared consequence of the central limit theorem; B.4 Proof of Corollary 4.1

  • Under finite second moments, the Hilbert-space central limit theorem gives Gaussian limits; continuous mapping yields the norm-squared limit, and independent samples yield a two-sample Gaussian limit.The Gaussian covariance is trace class, ensuring a Radon law on the Hilbert space.Source: §B.2 · p. 65, §B.3 · p. 66, §B.4 · p. 66

B.5 Proof of Proposition 4.1; B.8 Assumption dependencies

  • Covariance estimators are almost surely consistent in trace norm, with operator- and Hilbert–Schmidt-norm consistency following; the mean-square rate requires a finite fourth moment.The mean strong law needs a finite first moment, while the Hilbert-space CLT and trace-norm covariance consistency need a finite second moment.Source: §B.5 · p. 66, §B.8 · p. 68

B.6 Proof of Theorem 4.2

  • For fixed finite-rank projections, standardization on the covariance range yields a finite-coordinate Berry–Esseen bound; one-dimensional projections give a scalar bound.The bound requires a finite standardized third moment on the covariance range.Source: §B.6 · p. 67

B.7 Proof of Proposition 4.2

  • The truncation proposition bounds the discrepancy between full and truncated empirical embeddings, their centered versions, and the resulting signal separation.The argument uses a deterministic per-diagram approximation bound and the reverse triangle inequality.Source: §B.7 · p. 67

C.1 Null limit and consistency of the canonical statistic; C.2 Finite-sample power bound

  • The canonical statistic has a Gaussian null limit and is consistent under fixed alternatives; a finite-sample power bound controls Type II error when the signal and effective sample size meet the stated condition.The bound follows by applying Markov’s inequality to the squared norm of the estimation error.Source: §C.1 · p. 68, §C.2 · p. 69

C.3 Structured signal margins

  • Structured signal margins yield uniform power bounds under common- or feature-specific prevalence, whereas geometric mean-measure separation alone cannot ensure uniform power over the unrestricted class.The impossibility construction keeps geometric separation fixed while choosing a sufficiently small mixture proportion to make power fall below the target.Source: §C.3 · p. 69

C.4.1 Auxiliary information bounds

  • Diagram formation cannot increase Kullback–Leibler divergence, providing the information-contraction step used in the minimax lower-bound argument.The lemma applies the data-processing inequality to the map from labeled latent features to an unlabeled persistence diagram.Source: §C.4.1 · p. 70

C.4.2 Common-prevalence hard subfamily; C.4.3 Feature-specific-prevalence hard subfamily

  • Under the stated assumptions, hard alternative pairs with finite constants independent of separation are constructed for both common- and feature-specific-prevalence classes.The feature-specific-prevalence construction additionally assumes the covariance envelope satisfies v ≥ Bν.Source: §C.4.2 · p. 70, §C.4.3 · p. 74

C.4.3 Feature-specific-prevalence hard subfamily

  • The divergence argument combines bounds for Bernoulli prevalence and perturbed feature locations, then uses product-law additivity and data processing for observed diagrams.The margin condition is applied in the final step before the bounds are combined.Source: §C.4.3 · pp. 76–77
  • The feature-specific-prevalence hard-subfamily proof concludes by combining its final bounds.It also establishes that the two null laws are identical.Source: §C.4.3 · pp. 75, 77

C.4.4 Proof of Lemma 5.1

  • The proof of Lemma 5.1 begins by applying the definition of the information modulus.Source: §C.4.4 · p. 77

C.4.5 Proof of the minimax theorem

The appendix proves the minimax theorem by establishing upper- and lower-bound arguments, then justifies Gaussian plug-in calibration and mean-measure pullback results. These results support inference on Hilbert-space embeddings and connect embedding-space confidence sets to transport-based conclusions.

C.4.5 Proof of the minimax theorem

  • The minimax proof uses equal group sizes for its upper-bound argument and compares null and alternative joint laws, with Pinsker’s inequality and tensorization used for the lower bound.The proof also treats the cases where K_r(τ) is zero or infinite by limiting conventions.Source: §C.4.5 · p. 78

D.1 Gaussian plug-in calibration

  • Trace-norm convergence of positive covariance operators implies weak convergence of their centered Gaussian laws and convergence in distribution of the squared norms.The proof couples the Gaussian elements through a common isonormal cylindrical Gaussian element.Source: §D.1 · pp. 78–79
  • The Gaussian plug-in calibration argument combines Hilbert-space central limit theory with almost-sure trace-norm consistency of the covariance estimator.When the limiting covariance is nonzero, continuity of the weighted chi-square law supports convergence of the conditional quantile.Source: §D.1 · p. 79

D.2 Mean-measure pullback and exclusion

  • Under Assumption 3.3, the appendix establishes an upper transport bound for arbitrary padded mean measures using the point-level Lipschitz condition.The proof bounds the cost under any coupling by its essential supremum before taking the infimum over couplings.Source: §D.2 · pp. 79–80
  • The mean-measure pullback proof derives its stated conclusion on the event that the true embedded mean belongs to the confidence set, including a closed-ball specialization.The coverage probability of that event completes the argument.Source: §D.2 · p. 80

D.3 Two-sample confidence set and geometric consequences

The two-sample results establish asymptotic confidence-set coverage and derive geometric consequences, including exclusion of structured alternatives. A separate certificate relates empirical nearest-centroid margins to population labels and their stability under diagram perturbations.

D.3 Two-sample confidence set and geometric consequences

  • The two-sample plug-in confidence set has asymptotic coverage when the limiting covariance contrast is nonzero; with zero covariance, it is a singleton with exact coverage one.The scalar interval for the effect norm may cover even when the Hilbert-space confidence ball does not contain the contrast.Source: §D.3 · p. 80
  • For a fixed structured alternative, the uniform separation lower bound yields a pointwise asymptotic exclusion bound, without asserting uniform Gaussian approximation over the alternative class.The scalar effect-size interval follows from ordinary and reverse triangle inequalities.Source: §D.3 · pp. 80–81

D.4 Nearest-centroid certificate

  • When the simultaneous centroid-error bounds hold, the first margin condition makes every empirical competitor gap positive and certifies the population nearest-centroid label.Replacing a diagram by one within ε can decrease the pairwise empirical margin by at most 2nL_ν ε; below the stated certified radius, the population label remains unchanged.Source: §D.4 · p. 81

E Technical Proofs for Section 7

The technical proofs quantify how orthogonal truncation affects signal, testing guarantees, and confidence sets. They establish uniform power under a sample-size condition and show how projected confidence sets can be inflated to cover full targets.

E.1 Signal retained by orthogonal truncation

  • The truncation proof uses orthogonality and a uniform approximation bound to control retained signal under construction-specific certification conditions when diagram distance exceeds R_0(ν).It then specializes the argument to additive embeddings and applies the first part of the proposition.Source: §E.1 · pp. 81–82
  • For additive embeddings, the proof applies the general truncation bound to structured alternatives to obtain the corresponding certified population-separation bound.The proof invokes Theorem 3.1 before applying the proposition’s first part.Source: §E.1 · p. 82

E.2 Testing under finite approximation

  • The finite-approximation test has uniform level control and achieves the stated uniform power guarantee when its sample-size condition holds.The proof combines covariance bounds, the projected separation bound, and a reverse-triangle-inequality argument.Source: §E.2 · pp. 83–84

E.3 Confidence sets under finite approximation

  • For two samples, the inflated ball covers the full contrast with at least the projected ball’s coverage probability.The proof uses the projected coverage event and triangle inequalities.Source: §E.3 · p. 84
  • When K_{m,m′} tends to zero, the centered truncation error converges to zero in L2, while direct centering at the full target requires a stronger condition.Inflating the projected Hilbert-space ball by the truncation allowance preserves at least its coverage probability for the full target; the two-sample argument also yields a confidence interval.Source: §E.3 · pp. 84–85

F.1 Proof of Proposition 8.1

The proof establishes a Gaussian limit for the least-squares estimator and an O_P(m^-1/2) rate. It also verifies estimator consistency and derives a truncation bound using orthogonality.

F.1 Proof of Proposition 8.1

  • Boundedness and the finite second moment ensure integrability for the strong-law and central-limit arguments.Source: §F.1 · pp. 85–86
  • Positive definiteness makes the sample matrix invertible with probability tending to one, yielding an eventually unique least-squares minimizer.Source: §F.1 · p. 86
  • Slutsky’s theorem gives the stated Gaussian limit, with an O_P(m^-1/2) rate.Source: §F.1 · p. 86

F.2 Proof of Proposition 8.2

  • The truncation proof uses orthogonality, takes a supremum over D, and rearranges the bound when ∥Π_Kβ∥ > 0.Source: §F.2 · p. 87

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