PaperlayerView exact revision on arXiv

Source-linked AI summary

Where Do Two Populations of Persistence Diagrams Differ? Calibrated Local Inference at a Fixed Budget

Pramita Bagchi, Edward Bae, Atish Mitra, Alexander D. Silberman, Žiga Virk, Sushovan Majhi

arXiv:2610.08292v1stat.MEcs.LGmath.STstat.TH

TL;DR

Local two-sample tests for persistence diagrams often detect global differences without locating them, while local inference must choose neighborhood radii that trade resolution against precision. This paper uses additive landmark responses and a Gaussian multiplier bootstrap to form simultaneous local-contrast intervals. In fixed-budget simulations, a prespecified single-radius grid using all observations localized changes most often, while pilot fitting helped when the grid poorly resolved the occupied region.

  • Problem

    Local inference faces a radius tradeoff: small neighborhoods separate nearby changes but often have zero response, whereas large ones can merge nearby losses and gains.

    Source: §1 · p. 1

  • Method

    Additive landmark responses estimate local mean contrasts, and a Gaussian multiplier bootstrap calibrates simultaneous intervals across centers and radii.

    Source: §1 · p. 1, §2 · p. 2

  • Results

    In the main fixed-budget simulations, a prespecified single-radius grid using all observations localized changes most often, while pilot fitting helped when the grid poorly resolved the occupied region.

    Source: §6 · p. 7

  • Takeaways & Limitations

    On the simultaneous coverage event, every reported neighborhood lies within twice its radius of the support of the mean-measure difference.

    Source: §1 · p. 2

    The method targets local mean contrasts and can miss differences that cancel under its responses; unselected regions have not been shown to lack changes.

    Source: §6 · p. 7

Abstract

from arXiv · show

Many two-sample tests for populations of persistence diagrams assess global differences without identifying the regions of the birth-death plane that contribute to them. We study simultaneous inference for local mean contrasts when the number of available diagrams is fixed. They are differences in expected weighted feature mass within $\ell_\infty$ neighborhoods at several centers and radii. We estimate these contrasts using additive landmark responses. A Gaussian multiplier bootstrap calibrates simultaneous confidence intervals while allowing unequal group covariances. The neighborhoods whose intervals exclude zero form a map with approximate family-wise error control, and selecting a subset of original intervals for display preserves their joint coverage guarantee. On the simultaneous coverage event, every reported neighborhood lies within twice its radius of the support of the mean-measure difference. A geometric result gives sufficient radius conditions for a displaced feature to produce a nonzero contrast. A comparison of sufficient detection thresholds quantifies the tradeoff between reducing the number of tested coordinates and reserving observations for an independent pilot. In simulations with 40 to 120 diagrams per class, the bands achieved 94%-98% simultaneous coverage under both the strict null and equal means with unequal covariances. In the latter setting, a permutation maximum and the pooled-t implementation of the two-stage persistence-image test of Moon and Lazar rejected in up to 32% and 26% of runs, respectively. In the fixed-budget simulations, spending a third of the observations on a pilot to choose landmarks or radii located changes less often than a prespecified grid at a single radius. On the MUTAG benchmark, the localized region concentrates on rings of fused-ring systems, an exploratory reading.

1 Introduction

Local inference must balance spatial resolution against the precision available when few diagrams are observed. The paper estimates local mean contrasts with additive landmark responses and jointly calibrates intervals across centers and radii, with applications to molecular diagrams.

1 Introduction

  • A small neighborhood can separate nearby changes but often has zero response, while a large one can merge nearby losses and gains; radius choice changes both estimand and precision.This choice is especially consequential when few diagrams are available.Source: §1 · p. 1
  • The method estimates weighted feature-mass contrasts in ℓ∞ neighborhoods using additive landmark responses and calibrates all coordinates jointly with a Gaussian multiplier bootstrap.Figure 1 illustrates the resulting map on MUTAG.Source: §1 · p. 1
  • Figure 1 reports 58 localized coordinates at 26 centers, all with greater mass in the mutagenic class, and 97% of localized neighborhoods matched across random splits.The caption describes a pilot third of 175 deduplicated molecules and five radii per center.Source: Figure 1 · p. 2
  • Simultaneous intervals cover local contrasts across centers and radii, and displaying a data-dependent subset preserves their joint coverage guarantee.On the coverage event, a spatial bound relates reported neighborhoods to the support of the mean-measure difference; a displacement model gives sufficient radius conditions for a nonzero contrast.Source: §1 · p. 2
  • At fixed budget, pilot selection trades fewer tested coordinates against fewer inference observations; the main simulations localized changes most often with a prespecified single-radius grid using all diagrams.The paper also gives a power result: lifetime-multiset statistics have power equal to their level against diagonal shifts.Source: §1 · p. 2

2 Local contrasts

The local contrasts are expected differences in weighted diagram-point mass within neighborhoods around chosen centers. The landmark responses define these contrasts without matching features across diagrams, but opposing changes can cancel.

2 Local contrasts

  • For each diagram law, the mean counting measure gives the expected number of diagram points in any measurable region, without requiring feature matching across diagrams.The diagrams considered have at most n points in the specified birth-death frame.Source: §2 · p. 2
  • A landmark specifies a center, radius, and positive weight; its response is a weighted local count and is bounded by n times the weight and radius.The positive-response region is the frame-clipped open ℓ∞ square around the center, with the radius cap keeping it off the diagonal.Source: §2 · p. 2
  • The contrast is a difference in expected weighted local mass, and moving points within a neighborhood can change it without changing the expected count.Opposing local changes can cancel, yielding a zero contrast.Source: §2 · p. 2
  • Inference uses independent samples from the two diagram populations, and the fixed family must be prespecified or selected using an independent labeled pilot.Coverage statements with a pilot are conditional on that pilot; under the coordinate mean null, group covariances may differ.Source: §2 · pp. 2–3

3 Inference and neighborhood choice

The method constructs a simultaneous band for a fixed family of local contrasts using Gaussian multiplier calibration. It allows joint examination of several radii and pilot-based family selection while retaining the original intervals for valid display.

3 Inference and neighborhood choice

  • The Gaussian multiplier bootstrap calibrates normalized contrasts jointly across centers and radii, using the same multipliers across coordinates and a maximum-statistic critical value.The localization set contains coordinates whose intervals exclude zero, and the contrast sign gives the reported direction.Source: §3 · p. 3
  • Displaying any subset of the original simultaneous intervals preserves their coverage guarantee, including the smallest tested radius at each center whose interval excludes zero.This identifies the finest tested scale with a detected contrast, but does not rule out contrasts at smaller radii.Source: §3 · p. 3
  • Pilot selection can reduce the number of tested coordinates but leaves fewer observations for inference, so its configuration comparisons hold the total diagram budget fixed.The pilot score targets strong anticipated detection, and selection quantities use only the pilot.Source: §3 · p. 3
  • Cross-validation may replace pilot scoring when selection uses only independent pilot data; sample splitting can retain joint coverage by using bands at nominal 1−α/2 and a union bound.The guarantee is at least 1−α minus the sum of the two calibration errors, under the conditions for each band.Source: §3 · pp. 3–4

4 Theory

The theory establishes simultaneous inference and spatial guarantees for local mean contrasts, then gives conditions for detecting displaced features and compares pilot selection with using all observations. It also identifies limitations from sparse responses, finite samples, and the sufficient—not necessary—nature of the radius conditions.

Simultaneous inference

  • Under bounded responses, nondegenerate coordinate variances, balanced group allocation, and valid normalization, the simultaneous band controls false local rejections and supports data-dependent display subsets with joint coverage.For inference-based normalization, the guarantee is asymptotic; in simulations, pooled normalization held its level at every allocation, while groupwise normalization overrejected when one group was three times larger.Source: §4 · p. 4

Detection limits

  • Sparse responses can weaken calibration, so the logarithmic dependence on the number of coordinates does not make arbitrarily small neighborhoods reliable.A restricted lower bound also applies to identifying one changed coordinate from the inference sample alone.Source: §4 · p. 4

The price of a pilot

  • For the five-radius grid, the full family’s sufficient detection threshold is smaller than the pilot-selected subfamily threshold once the pilot fraction exceeds 0.15, under a common κ0ρ0 assumption.This compares sufficient thresholds, not exact powers; selection can still help if it reduces κ0ρ0.Source: §4 · p. 4

Local detectability

  • Equal mean counting measures yield equal mean responses even when diagram laws differ, so a lower-distortion bound for individual diagrams alone does not ensure a population contrast.Under the localized displacement model, the background contributions agree in expectation and the displaced feature determines the contrast.Source: §4 · p. 5
  • A displaced singleton feature yields a positive local contrast when its location and displacement error fit inside a retained landmark radius while the other location remains outside.The resulting contrast is at least q w_k(r_k − d_B(p_k, a) − ϵ); detection follows when this lower bound exceeds the detection threshold.Source: §4 · p. 5

Spatial extent

  • On the simultaneous coverage event, every reported neighborhood lies within twice its maximum radius of the support of the mean-measure difference.A nonzero contrast places its center within its radius of that support, but reported neighborhoods need not cover all of it or lie wholly inside it.Source: §4 · p. 5

Lifetime blindness

  • Every lifetime-based test has the same rejection probability under a diagonal point shift as under the strict null, while a suitable landmark can have a positive contrast.The sufficient landmark condition is that its open square meet the moved points’ original mean-measure support and almost surely miss their images.Source: §4 · p. 5

5 Experiments

Fixed-budget simulations evaluate calibration and localization across landmark and image procedures, including pilot-based choices. The bands generally control error, while localization depends on radius, normalization, and how observations are allocated.

5.1 Design at a fixed total budget

  • At nominal α=0.05, all procedures used the same generated samples and fixed total diagram budget; F0 and F1 used all diagrams, whereas F2, A1, and A2 reserved about one third for pilot selection.The simulations included strict-null, equal-mean unequal-covariance, and feature-displacement designs.Source: §5.1 · p. 5

5.2 Calibration

  • With default normalization, F0–A2 and I1 had FWER 0.022–0.063 across ten settings, while weak-null permutation maximum and pooled-t ML reached 0.32 and 0.26 in the clustered design.Groupwise SD normalization reached FWER 0.29–0.48 under imbalance, including under P=Q; pooled SD normalization was conservative at 0.002–0.036. In a highly unequal-variance design, SD-based normalization located every change while fixed normalization located none, but in the main displacement design fixed normalization performed better.Source: §5.2 · p. 6

5.3 Localization and neighborhood choice

  • The sufficient radius interval for a displaced feature did not ensure high power: at radii 0.035, 0.05, and 0.071, rejection rates were 2%, 60%, and 86%, respectively.Greater jitter narrowed the interval and reduced detection to 0–33%; cross-fitted selection gained power for nearby same-sign changes but lost power for an isolated change, and the positive comparison concerned testing under exchangeability because SD-normalized bands undercovered. The band and an MMD test on the same landmark embedding had similar global power, while the band also provided local intervals.Source: §5.3 · p. 6
  • At 40 diagrams per class and Δ=0.05, F0 locates lifetime-changing and lifetime-preserving shifts in 77% and 92% of runs, respectively; it leads the compared procedures in the main displacement experiments.Across pilot procedures, corresponding localization ranges were 10–29% and 17–47%; testing five radii jointly reduced localization for small shifts. Sensitivity analyses also found that pilot fraction and image resolution affected localization, with image localization rising from 12% at 10×10 to 99.8% at 40×40 under the stated smoothing settings.Source: §5.3 · p. 6, Figure 2 · p. 7

5.4 An exploratory molecular application

  • On MUTAG, A1 produced intervals excluding zero for 58 of 342 coordinates at 26 centers, all indicating greater expected weighted feature mass in the mutagenic class.The map concentrated near birth values 0.06–0.09 and death values 0.09–0.12, and remained nonempty across 20 random pilot/inference splits; COX2 maps were unstable and PTC maps were empty in 16–20 splits.Source: §5.4 · p. 7
  • With 20 molecules per class, A1 yielded nonempty maps in 94% of subsamples versus 28% for the fixed grid; the selected MUTAG region contained 96% of rings in systems with at least three fused rings and 49% of other rings.This structural reading is exploratory: region selection used these data, ring matching depends on the cycle basis, and the authors do not infer a chemical mechanism.Source: §5.4 · p. 7

6 Discussion

The discussion presents a prespecified single-radius grid as a useful starting point when its scale is known, while noting that selection and coverage have important limits.

6 Discussion

  • In fixed-budget simulations, a prespecified single-radius grid using all observations localized changes most often; pilot fitting can help when the grid poorly resolves the occupied region.The latter case is illustrated by MUTAG subsampling results.Source: §6 · p. 7
  • Cross-validated region selection can help for reliably identified nearby same-sign changes, but showed substantial power loss for an isolated change and below-nominal coverage.The positive comparison therefore concerns power rather than validated calibration.Source: §6 · p. 7
  • Reported neighborhoods need not cover the full support of the mean-measure difference, and unselected regions have not been shown to lack changes.Recovering the full changed support and incorporating covariates remain possible extensions.Source: §6 · pp. 7–8
  • Figure 3 plots rejection probability by radius above and the exact contrast, certificate, and band mean half-width on a dense radius sweep below.The shaded band marks the certified interval s + ϵ < r ≤ t − ϵ.Source: Figure 3 · p. 8

AI Use Statement

The authors describe the use of AI tools in producing and checking the paper's code, figures, tables, and text.

AI Use Statement

  • The authors say Claude and ChatGPT were used to write and run experiment and analysis code.Source: AI Use Statement · p. 8
  • The tools were also used to generate tables and figures and to draft and edit text.Source: AI Use Statement · p. 8
  • The authors directed the work, checked its results and arguments, and take responsibility for the paper.Source: AI Use Statement · p. 8

Part I: Technical results

The proof appendix follows the order of the main results and conditions inference statements on an independent pilot when one is used.

A Proofs of the main results

A.1 Simultaneous coverage: proof of Theorem 1

The appendix develops guarantees for simultaneous multiplier-band coverage, including estimated normalization and selected reporting, and establishes bounds for detecting and spatially interpreting local contrasts. It also clarifies the scope of the pilot-threshold comparison and shows that localized responses can detect moved features that lifetime-only tests cannot distinguish.

A.1 Simultaneous coverage: proof of Theorem 1

  • Theorem 1’s simultaneous-coverage approximation has a constant independent of the number of multiplier draws B and level α.The proof also extends the guarantee to any nonempty coordinate subset fixed conditional on the pilot, using that subset’s own multiplier maximum.Source: §A.1 · p. 12

A.2 Estimated normalization

  • Estimated normalization gives simultaneous coverage error at most ε_N + 1/(B + 1), with coverage converging to 1 − α as B → ∞.The guarantee also applies to fixed coordinate subsets and retains joint coverage for any data-dependent selection of original intervals; pooled normalization does not impose equal group covariances.Source: §A.2 · pp. 13, 15

A.3 Error control and selected reporting: proof of Corollary 2

  • Step-down testing controls false rejections using the fixed-subset guarantee, and selecting original intervals after observing the data preserves their joint coverage.The proof shows the step-down procedure rejects at least the coordinates rejected by the single-step procedure; selected reporting requires retaining the original intervals.Source: §A.3 · p. 15

A.4 Detection: proof of Proposition 1

  • The detection proof bounds multiplier responses using a Gaussian tail and union bound, then controls critical-value exceedance with Markov’s inequality.It treats the zero-variance case separately, where all multiplier responses vanish.Source: §A.4 · p. 16
  • The detection proof applies the conditional coverage bound to coordinates meeting the signal condition on G, then uses a union bound over three complementary events.Taking expectations gives the same unconditional bound conditional on the pilot.Source: §A.4 · p. 16

A.5 The pilot comparison: proof of Corollary 3

  • The pilot comparison orders sufficient detection thresholds for each realized retained subfamily, but does not order actual detection probabilities.For retained coordinates, response functions and population variances remain unchanged, and the split preserves group proportions and ωk; the threshold comparison uses a common product κ0ρ0.Source: §A.5 · p. 16

A.6 Local detectability: proof of Theorem 2

  • A displaced feature can produce a nonzero local contrast when its neighborhood contains the feature under P but excludes it under Q, given equal background mean measures and a shared presence probability.If the feature is within η of center a and displaced to b, a sufficient radius condition is η + ϵ < r_k ≤ dB(a, b) − η − ϵ; exceeding the corresponding detection threshold gives the stated guarantee.Source: §A.6 · p. 17
  • Under the additional feature-location condition, either applicable lower bound yields the detection guarantee when it exceeds the corresponding threshold.The condition η + ϵ < r_k ≤ dB(a, b) − η − ϵ implies the first radius condition.Source: §A.6 · p. 17

A.7 Spatial extent: proof of Proposition 2

  • On simultaneous coverage, each rejected neighborhood lies within twice its radius of the support of the mean-measure difference; any displayed subset retains this inclusion but need not cover the support.A nonzero contrast requires the response neighborhood to meet the support of the signed mean-measure difference.Source: §A.7 · p. 17

A.8 Lifetime invariance: proof of Proposition 3

  • A diagonal translation preserves point lifetimes, so every lifetime-based test has the same distribution under (P, Q) as under (P, P).The translated points have unchanged lifetimes, leaving the joint law of lifetime data unchanged.Source: §A.8 · p. 17
  • Localized responses can nevertheless detect moved points when the response square meets the support of their mean measure under P and their images contribute zero under Q.Under the coupling Y = τ(X), the expected response difference is strictly positive.Source: §A.8 · p. 17

B Additional technical results

The appendix develops technical interpretations and limits for the local contrasts, then specifies calibration and comparator procedures. It also examines what plug-in separation estimates and how pilot selection and resampling are implemented.

B.1 Transport separation of mean counting measures

  • The padded transport representation equates diagram bottleneck distance with the W∞,0 distance between padded counting measures.Padding adds absence states so the measures have equal mass; their restriction to the original space recovers the mean measure.Source: §B.1 · p. 18
  • On simultaneous coverage, a transport lower bound certifies separation of padded mean counting measures; a zero bound does not establish equality of mean measures or diagram laws.The certificate uses an upper Lipschitz bound and requires no lower-distortion assumption.Source: §B.1 · p. 19

B.2 A lower bound for identifying one changed coordinate

  • The lower-bound construction concerns identifying one changed coordinate in a fixed disjoint family, not recovering spatial support with overlapping or learned neighborhoods.The Q-sample carries no information about the changed coordinate, so the bound depends on the P-sample size; pilot information would also need to enter the lower-bound experiment.Source: §B.2 · p. 20

B.3 Bias of the plug-in separation

  • The plug-in empirical separation is biased by covariance terms, so maximizing it across configurations can reward variance as well as population differences.For a fixed family, the band instead uses coordinatewise contrast estimates that are unbiased.Source: §B.3 · p. 20

C Implementation details

  • The diagram is the sampling unit, and the implementation section covers calibration, pilot selection, comparators, simulations, and molecular analyses.Fixed-budget studies reserve pilot observations within the total diagram budget.Source: C Implementation details · p. 21

C.1 Calibration and resampling

  • The Gaussian multiplier band separately centers group responses and shares each multiplier across every coordinate of a diagram; permutation instead tests the exchangeable null P = Q.The coordinate mean null can hold with unequal group laws and does not by itself justify permutation calibration.Source: §C.1 · p. 21

C.2 Pilot selection and image comparators at a fixed budget

  • Pilot procedures reserve one third of each group, fit class-aware centers, screen low-activity coordinates, and use validation responses to normalize candidate radii.The persistence-image comparator of Moon and Lazar (2023) uses pooled t-tests and Benjamini–Hochberg adjustment, targeting FDR.Source: §C.2 · p. 21

C.3 The survival-based comparator

  • The survival comparator pools point lifetimes and applies a log-rank statistic; its pooled asymptotic calibration treats lifetimes as independent and ignores within-diagram dependence.Diagram-level label permutation retains the diagram as sampling unit and is valid under P = Q.Source: §C.3 · pp. 21–22

C.4 Selection and the inference sample

Selecting a new family on the inference sample and recalibrating only that family is not covered by the fixed-family argument. Independent pilot or cross-fitting selection separates family choice from inference, while selecting intervals for display after calibration retains the original guarantee.

C.4 Selection and the inference sample

  • Selecting a new family on inference data and recalibrating only that family is not covered by the fixed-family argument; in Appendix F.4, this raised FWER to as much as five times nominal.Maximizing empirical separation can also favor configurations with larger variance.Source: §C.4 · p. 22
  • An independent pilot or cross-fitting separates family selection from inference, whereas displaying a subset of intervals after calibrating the full family preserves simultaneous coverage.Cross-fitting uses every observation once for inference.Source: §C.4 · p. 22

D Fixed-budget studies

  • The fixed-budget studies count pilot-reserved diagrams in the total allocation and examine calibration, localization, radius choice, and selection procedures.The supplementary CV study uses a separate generator and reports its own allocation and calibration.Source: D Fixed-budget studies · p. 22, §C.4 · p. 22

D.1 Simulation design and evaluation

The fixed-budget studies assess calibration, localization, geometric guarantees, and selection, then apply the method to molecular persistence diagrams. Across those studies, results include high simultaneous coverage, geometric checks, and an exploratory MUTAG structural interpretation.

D.1 Simulation design and evaluation

  • The studies vary feature displacement, weak-null covariance structures, landmark grids, and pilot-based procedures; localization requires a rejected nonzero-contrast coordinate near the changed support.The fixed-budget allocation counts pilot observations, and the main displacement designs include both lifetime-changing and lifetime-preserving shifts.Source: §D.1 · pp. 22–23

D.2 Calibration and normalization

  • 94%–98% of bands achieved simultaneous coverage in studies S1 and S4; with no feature present, every procedure held its nominal level in the reported simulations.Table 2 reports S1 simultaneous coverage.Source: §D.2 · p. 23, Table 2 · p. 23

Estimated normalization.

  • Under substantial coordinate variance differences, the change at the feature template was never located, and hybrid floors did not repair the groupwise estimate under imbalance.Table 3 compares fixed, groupwise, pooled, and hybrid normalizations for null FWER.Source: Estimated normalization · p. 23, Table 3 · p. 24

D.3 Localization and geometric checks

  • In 2,395 nonzero-contrast coordinates, every center lay within its changed support by its radius, and every sampled point of its square lay within twice that radius.Across 1,332 certified center–radius cases, the Theorem 2 lower bound did not exceed the exact contrast; these were numerical checks, not coverage guarantees.Source: §D.3 · p. 25

D.4 Sensitivity analyses

  • At 80 diagrams per class, moving the template off-grid lowered F0 localization to 0.93–0.99; per-coordinate permutation references located the change in 97% on the grid and 64–66% on the image at ∆ = 0.10.The permutation references had null rejection rates of 0.036–0.044 on the grid and 0.026–0.028 on the image.Source: §D.4 · p. 25, Table 8 · p. 27

D.5 Cross-validated region and radius selection

  • For four changed centers, cross-fitted region and radius selection gained 12.8 percentage points over joint testing of all radii, while pilot selection gained 1.0 point.The correct region was selected in 93.7% of outer selection halves for four changed centers, versus 11.7% for an isolated change; several larger-design cells had subnominal coverage.Source: §D.5 · pp. 28–29, Table 9 · p. 29

E.1 Data and preprocessing

  • The primary molecular analysis deduplicates graphs and uses a fixed landmark grid or an adaptive family of 100 fitted centers; primary heat-kernel choices were not specified prospectively.The reanalysis uses one graph from each duplicate group and drops groups with conflicting labels.Source: §E.1 · p. 30, §E.2 · p. 30, E Molecular application · p. 30

E.3 Mapping localized regions to molecular structure

  • In the exploratory MUTAG mapping, minimum-cycle-basis rings were matched to degree-one points, and all 486 basis rings matched a point, versus 452 with NetworkX’s cycle basis.The structural reading depends on the cycle basis as well as the filtration; fused systems share bonds between rings.Source: §E.3 · p. 30, Figure 4 · p. 32, Table 13 · p. 32

E.4 Earlier molecular analysis

The earlier molecular analysis reports localized regions in MUTAG and DHFR, with exploratory summaries linking the MUTAG region to fused-ring systems. Its interpretation is limited by data-dependent region construction, while earlier simulation studies examine calibration and the cost of simultaneity.

E.4 Earlier molecular analysis

  • The earlier heat-kernel analysis localized degree-zero contrasts at 72 of 187 landmarks on MUTAG and 55 of 192 on DHFR, and degree-one contrasts at 21 of 145 and 30 of 178, respectively.Another split recovered 89–97% of localized regions, and top-landmark agreement ranged from 85–100%.Source: §E.4 · p. 33, Figure 5 · p. 34, Table 14 · p. 34
  • In the earlier MUTAG map, 82% of rings in fused systems of three or more rings fell in the localized region, compared with 21% of other rings.Mutagenic compounds placed an average of 2.66 rings and 2.76 branchings in the region, compared with 0.35 and 1.03 in the other class.Source: §E.4 · p. 33
  • The earlier MUTAG summaries are exploratory because the region was selected using these data and the pilot helped construct it; they do not provide an independent test of a chemical explanation.COX2 localization was weak and unstable, while the marginal PTC landmark was not reproduced by another split.Source: §E.4 · p. 33

F Additional simulation studies

  • The earlier simulation designs differ from the main fixed-budget studies in pilot allocation, jitter, normalization, and null-coordinate definitions.Except for the cross-fitting arm, the pilot added 200 diagrams per class to the inference sample.Source: F Additional simulation studies · p. 35, §F.1 · p. 35

F.2 Level, coverage, and the price of simultaneity

  • For pilot-fitted landmarks, reported FWER ranged from 0.008 to 0.048 across sample sizes; normal-reference Bonferroni rates reached 0.33–0.48 at the finest resolution.Bonferroni adjustment controls FWER when applied to valid marginal tests; the reported failures concern marginal normal calibration.Source: §F.2 · p. 35
  • At fine resolution, sparse responses raised bootstrap critical values to 5–6 versus about 3.8 for the unit-normal reference, reducing power at m = 20.Pilot-fitted landmarks concentrated on occupied regions and kept the critical value within 0.7 of the Bonferroni reference in this design.Source: §F.2 · p. 35

F.3 Finite-sample behavior of estimated normalization

Across the finite-sample studies, inference-sample selection and variance estimation could miscalibrate the bands, especially with imbalance or sparse responses. Independent pilot selection, cross-fitting, and the multiplier band showed better localization or error control in the reported comparisons.

F.3 Finite-sample behavior of estimated normalization

  • With unequal group sizes, the sample-normalized band’s FWER was 0.51, versus 0.45 for normal-reference Bonferroni; the permutation maximum’s FWER was 0.064.With an independent pilot scale, FWER was 0.023 and coverage was 0.98, while normal-reference Bonferroni remained liberal at 0.12.Source: §F.3 · pp. 35, 37, Table 15 · p. 36

F.4 Selection on the inference sample

  • Selecting the K largest |Zk| values on the inference sample yielded FWER 0.13–0.24, compared with 0.016–0.036 when selection used an independent pilot.Class-aware farthest-point placement on the inference sample gave FWER 0.026–0.068, but neither result gives a general guarantee for an inference-fitted family.Source: §F.4 · p. 37, Table 16 · p. 38

F.5 Localization

  • With pilot-fitted landmarks and K≥100, the band localized the signal in at least 99% of runs for every Δ≥0.10 when m≥40, but only 70–98% when m=20.At the finest grid and m=20, band power fell to 0.01–0.03; with K=50, the displaced feature could lie beyond selected neighborhoods, while K=100 closed that gap.Source: §F.5 · pp. 37, 39, Table 17 · p. 39, Figure 8 · p. 39

F.6 Comparison with pixelwise image inference

  • At Δ=0.10 and m=20, pilot-fitted landmarks had bootstrap power of 70–94% for K∈{100, 200}, versus 15–23% for 111 or 189 image pixels.At Δ=0.20, localization error was 0.02–0.04 for landmarks and 0.03–0.07 for images; with m=60, both maps approached full power once images had at least 189 pixels.Source: §F.6 · p. 40, Table 18 · p. 40

F.7 Lifetime-changing and lifetime-preserving shifts

  • In the lifetime-changing regime, the band’s power was 0.77 at Δ=0.10 and 0.87 at Δ=0.20, while survival tests remained at or below 0.15.For lifetime-preserving shifts, survival tests stayed at their null rejection rates; the band’s power was 0.63 at Δ=0.05 and 0.86 at Δ=0.30.Source: §F.7 · pp. 40–41, Figure 9 · p. 41, Table 19 · p. 41

F.8 Cross-fitting and unequal-covariance nulls

  • With 80 diagrams per class total, cross-fitting localized in at least 96% of runs, and null FWER ranged from 0.014–0.046 across reported settings.Under unequal covariances, the band’s FWER ranged from 0.005–0.057; the permutation maximum reached 9–18% in clustered designs when the smaller group was more variable.Source: §F.8 · pp. 41–42, Table 20 · p. 42
  • Recomputing sample normalization for each permutation still produced rejection rates of 18–33% in the unbalanced clustered comparison and 12–24% in balanced clustered designs.The authors attribute persistent difficulty to sparse-coordinate variance estimation at these sample sizes.Source: §F.8 · p. 42

F.9 Complete tables for the earlier calibration and localization studies

  • The complete earlier-study tables cover grids of 55, 105, 210, and 406 landmarks, inference sizes m∈{20, 40, 60}, and shifts Δ∈{0, 0.10, 0.15, 0.20, 0.30}.The studies used 500 replications and B=2000 throughout.Source: §F.9 · pp. 43–45
Loading 2610.08292v1…