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

stat.MEOct 6, 2026

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

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

Organizations: The George Washington University · Montana Technological University · University of Ljubljana

Abstract

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.

Figures & tables

Figure 1: Local inference on MUTAG (mutagenic against nonmutagenic compounds; heat-kernel signature, $t=10$ , degree one). Class-aware landmarks are fitted on a pilot third of $175$ deduplicated molecules, each with five radii; squares are filled by the standardized contrast $Z_{k}$ of § 3 , and at each center the smallest radius whose interval excludes zero is outlined. All $58$ localized coordinates, at $26$ centers, carry more mass in the mutagenic class. Over $20$ random splits the localized set is never empty and $97\%$ of a split’s localized neighborhoods are matched by another split’s (§ 5.4 ).

design	$T_{P}$ / $T_{Q}$	F0	F1	F2	A1	A2	F1-perm	I1	ML
strict null	40/40	0.032	0.033	0.036	0.055	0.047	0.039	0.027	0.019
strict null	80/80	0.035	0.048	0.046	0.046	0.044	0.053	0.048	0.026
strict null	120/120	0.040	0.043	0.051	0.057	0.049	0.042	0.041	0.020
strict null	40/120	0.033	0.054	0.036	0.038	0.034	0.059	0.042	0.032
clustered	80/80	0.063	0.054	0.059	0.052	0.052	0.051	0.057	0.038
clustered	120/40	0.047	0.041	0.062	0.052	0.057	0.324	0.038	0.262

Table 1: FWER at a fixed total budget,

1000

replications per cell.

T_{P}/T_{Q}

: diagrams per class; in the weak nulls

Q

is the more variable group. Bold: the

95\%

Wilson interval lies above

0.05

Figure 2: Probability of locating the change at a fixed total budget,

1000

replications per point. The pilot methods (F2, A1, A2) infer on two thirds of the diagrams; F0, F1 and the image procedures use all of them.

120

per class and the FWER under alternatives are in Appendix D .

Figure 3: Radius and Theorem 2 ,

80

diagrams per class,

\Delta=0.10

. Top: probability that the coordinate at the grid center nearest the template is rejected by F1, at each of its radii. Bottom: exact contrast and its certificate on a dense radius sweep, with the band’s mean half-width. Shaded: the certified interval

s+\epsilon<r\leq t-\epsilon

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

design	$T_{P}$	$T_{Q}$	F0	F1	F2	A1	A2	F1-perm	I1	ML
shift-pers	40	40	0.97	0.97	0.96	0.94	0.95	–	0.97	–
shift-pers	80	80	0.96	0.95	0.95	0.95	0.96	–	0.95	–
shift-pers	120	120	0.96	0.96	0.95	0.94	0.95	–	0.96	–
shift-pers	40	120	0.97	0.95	0.96	0.96	0.97	–	0.96	–
wn-cluster	80	80	0.94	0.95	0.94	0.95	0.95	–	0.94	–
wn-cluster	120	40	0.95	0.96	0.94	0.95	0.94	–	0.96	–

Appendix

Table 2: S1: simultaneous coverage of the bands.

		F0			F1
design	$T_{P}$ / $T_{Q}$	fixed	group	pooled	fixed	group	pooled	hyb. 1%	hyb. 0.5%
strict null	40/40	0.032	0.034	0.014	0.033	0.040	0.010	0.063	0.047
strict null	80/80	0.035	0.032	0.012	0.048	0.040	0.016	0.061	0.041
strict null	120/120	0.040	0.029	0.024	0.043	0.034	0.023	0.042	0.035
strict null	40/120	0.033	0.375	0.022	0.054	0.392	0.007	0.160	0.341
clustered	80/80	0.063	0.035	0.021	0.054	0.034	0.019	0.067	0.045

Appendix

Table 3: Estimated normalization: FWER at the null,

1000

replications per cell. Fixed:

nw_{k}r_{k}

; group: groupwise SD; pooled: pooled SD; hyb.: groupwise SD floored at

1\%

0.5\%

nw_{k}r_{k}

. Bold: the

95\%

Wilson interval lies above

0.05

design	$T$	$\Delta$	fixed	group	pooled	fixed	group	pooled	hyb. 1%	hyb. 0.5%
			F0			F1
unequal variances	40	0.05	0.00	1.00	1.00	0.00	1.00	1.00	1.00	1.00
unequal variances	40	0.10	0.00	1.00	1.00	0.00	1.00	1.00	1.00	1.00
unequal variances	80	0.05	0.00	1.00	1.00	0.00	1.00	1.00	1.00	1.00
unequal variances	80	0.10	0.00	1.00	1.00	0.00	1.00	1.00	1.00	1.00
unequal variances	120	0.05	0.00	1.00	1.00	0.00	1.00	1.00	1.00	1.00

Appendix

Table 4: Estimated normalization: probability of locating the change,

T

diagrams per class, columns as in Table 3 .

design	$T_{P}$	$T_{Q}$	$\Delta$	F0	F1	F2	A1	A2	F1-perm	I1	ML
shift-pers	40	40	0.05	0.77	0.19	0.29	0.10	0.17	0.19	0.10	0.07
shift-pers	40	40	0.1	0.85	0.60	0.49	0.40	0.47	0.61	0.44	0.36
shift-pers	40	40	0.15	0.97	0.79	0.64	0.52	0.57	0.80	0.62	0.58
shift-pers	80	80	0.05	0.98	0.58	0.72	0.27	0.39	0.59	0.31	0.25
shift-pers	80	80	0.1	1.00	0.97	0.90	0.78	0.80	0.98	0.87	0.85
shift-pers	80	80	0.15	1.00	1.00	0.96	0.87	0.88	1.00	0.96	0.96

Appendix

Table 5: S2: probability of locating the change.

design	$T_{P}$	$T_{Q}$	$\Delta$	F0	F1	F2	A1	A2	F1-perm	I1	ML
shift-pers	40	40	0.05	0.023	0.030	0.019	0.034	0.025	0.034	0.000	0.003
shift-pers	40	40	0.1	0.025	0.026	0.026	0.029	0.022	0.027	0.004	0.008
shift-pers	40	40	0.15	0.015	0.037	0.030	0.035	0.030	0.040	0.001	0.004
shift-pers	80	80	0.05	0.016	0.016	0.018	0.035	0.029	0.017	0.005	0.006
shift-pers	80	80	0.1	0.027	0.033	0.018	0.028	0.025	0.036	0.003	0.010
shift-pers	80	80	0.15	0.023	0.026	0.023	0.030	0.027	0.025	0.001	0.013

Appendix

Table 6: S2: FWER over exact nulls under alternatives.

variant	design	$T_{P}$	$T_{Q}$	$\Delta$	F0	F1	F2	A1	A2	F1-perm	I1	ML
baseline	shift-pers	80	80	0.1	1.00	0.96	0.91	0.80	0.82	0.96	0.87	0.84
q=0.25	shift-pers	80	80	0.1	0.53	0.25	0.18	0.14	0.18	0.26	0.16	0.13
q=1	shift-pers	80	80	0.1	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
eps=0.01	shift-pers	80	80	0.1	1.00	0.99	0.97	0.87	0.89	0.99	0.94	0.95
eps=0.04	shift-pers	80	80	0.1	0.39	0.59	0.31	0.34	0.33	0.59	0.41	0.34
delta=0.05	shift-pers	80	80	0.05	1.00	0.58	0.75	0.28	0.37	0.58	0.31	0.24

Appendix

Table 7: S3 and S4: probability of locating the change (for two features, any component; per-component rates are in the replication files).

variant	design	$T_{P}$	$T_{Q}$	$\Delta$	F0	F1	F2	A1	A2	F1-perm	I1	ML	F0-permHolm	I1-permHolm	I1_10	I1_20	I1_20-permHolm	I1_40	ML_10	ML_20	ML_40
f=1/4	shift-pers	80	80	0.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	–	–	–	–	–	–	–	–	–
f=1/4	shift-pers	80	80	0.1	1.00	0.96	0.90	0.87	0.88	0.96	0.87	0.85	–	–	–	–	–	–	–	–	–
f=1/3	shift-pers	80	80	0.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	–	–	–	–	–	–	–
f=1/3	shift-pers	80	80	0.1	1.00	0.96	0.88	0.75	0.79	0.95	0.87	0.85	0.97	0.64	–	–	–	–	–	–	–
f=1/2	shift-pers	80	80	0.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	–	–	–	–	–	–	–	–	–
f=1/2	shift-pers	80	80	0.1	1.00	0.96	0.78	0.56	0.65	0.96	0.87	0.85	–	–	–	–	–	–	–	–	–

Appendix

Table 8: S5 sensitivity at

80

per class,

500

replications: probability of locating the change (top) and FWER over exact nulls (bottom), by variant and

\Delta

Method	Null FWER	Power, $1$	Power, $4$	Coverage, $4$
All radii jointly	5.5	25.9	69.8	92.7
Pilot: radius	8.2	10.3	41.6	93.9
Pilot: region and radius	5.1	3.2	70.8	94.7
Cross-fitted: radius	6.9	8.1	39.4	91.4
Cross-fitted: region and radius	6.5	4.5	82.6	93.3
Oracle region	5.4	58.2	98.1	93.9

Appendix

Table 9: Additional selection experiment with

25

regions and

120

diagrams per group. Entries are percentages from

1000

replications. Power is the probability that the multiplier band rejects at least one nonzero contrast. Columns labeled

1

and

4

refer to the number of changed centers. Coverage concerns all original intervals reported by a method, including both cross-fitted families.

dataset	filtration	deg.	$n_{0}$ / $n_{1}$	loc.	$p$	loc.	$p$	loc.	$p$	loc.	$p$	loc.	$p$
				F0		F1		F2		A1		A2
MUTAG	HKS $t=10$	1	63/112	1	0.024	3	0.005	1	0.001	58	$<$ 0.001	26	$<$ 0.001
MUTAG	HKS $t=10$	0	63/112	2	$<$ 0.001	7	$<$ 0.001	2	$<$ 0.001	116	$<$ 0.001	58	$<$ 0.001
COX2	HKS $t=10$	1	363/102	0	0.262	1	0.015	0	0.123	0	0.252	0	0.383
DHFR	HKS $t=10$	1	269/449	1	$<$ 0.001	4	$<$ 0.001	1	$<$ 0.001	20	$<$ 0.001	7	$<$ 0.001
COX2	HKS $t=10$	0	363/102	0	0.337	0	0.381	0	0.317	8	0.020	1	0.019

Appendix

Table 10: Primary analyses: localized coordinates and global

p

-values (

n_{0}/n_{1}

diagrams per class used by F0 and F1; the pilot methods infer on two thirds).

dataset	filtration	deg.	method	empty	overlap	matched	loc. (median)
MUTAG	HKS $t=10$	1	F2	2/20	1.00	0.89	1.0
MUTAG	HKS $t=10$	1	A1	0/20	0.62	0.97	50.5
MUTAG	HKS $t=10$	1	A2	0/20	0.65	0.97	23.5
MUTAG	HKS $t=10$	0	F2	0/20	0.66	0.93	2.0
MUTAG	HKS $t=10$	0	A1	0/20	0.59	1.00	57.0
MUTAG	HKS $t=10$	0	A2	0/20	0.20	0.93	14.5

Appendix

Table 11: Stability over

20

random pilot/inference splits: empty maps, median signed mesh overlap between pairs of nonempty maps, mean fraction of a split’s localized neighborhoods matched in sign within their radius by another split, median number localized.

dataset	per class	method	any localized	loc. (median)	overlap with full map
DHFR	40	A1	0.28	0.0	0.00
DHFR	40	A2	0.28	0.0	0.00
DHFR	40	F1	0.30	0.0	0.00
DHFR	80	A1	0.52	1.0	0.07
DHFR	80	A2	0.58	1.0	0.02
DHFR	80	F1	0.48	0.0	0.00

Appendix

Table 12: Learning curves:

50

class-balanced subsamples per size, degree one.

Figure 4: DHFR, heat-kernel signature $t=10$ , degree one, A1 on the primary split: $20$ localized coordinates at $10$ centers.

map	cycle basis	localized	rings in $\geq 3$ -fused systems in region	other rings in region
A1	minimum cycle basis	58	331/346 (0.96)	68/140 (0.49)
A1	cycle basis	58	297/312 (0.95)	68/140 (0.49)
F1	minimum cycle basis	3	7/346 (0.02)	24/140 (0.17)
F1	cycle basis	3	7/312 (0.02)	24/140 (0.17)

Appendix

Table 13: MUTAG: rings in the localized region of the primary maps (exploratory).

dataset	filtration	$H_{d}$	$m{+}m^{\prime}$	$\ell$	$\max_{k}\|Z_{k}\|$	$\hat{c}_{\alpha}$	$p$	$\|\hat{R}\|$	$\|\hat{R}_{\mathrm{sd}}\|$	$\|\hat{R}_{\mathrm{BH}}\|$	recovery	top agree
COX2	hks_t1	$H_{0}$	$243{+}68$	196	3.92	3.23	0.0076	14	14	26	0.42	0.80
COX2	hks_t1	$H_{1}$	$243{+}68$	196	3.11	3.32	0.0820	0	0	0	0.40	1.00
COX2	hks_t10	$H_{0}$	$243{+}68$	182	3.73	4.27	0.1144	0	0	2	0.12	0.00
COX2	hks_t10	$H_{1}$	$243{+}68$	168	7.83	6.78	0.0268	1	3	9	0.34	0.65
COX2	ricci	$H_{0}$	$243{+}68$	164	4.19	4.07	0.0418	1	1	1	0.28	0.60
COX2	ricci	$H_{1}$	$243{+}68$	192	3.96	3.18	0.0048	2	2	2	0.35	0.55

Appendix

Table 14: Earlier molecular analysis under a pilot/inference split. The table reports the global

p

-value, sizes of the three localization sets, and stability over

20

random splits: the fraction of one split’s localized region recovered by another and agreement of the top landmark. These results use the earlier preprocessing and family construction in Appendix E.4 .

Figure 5: DHFR under the heat-kernel filtration at $t=10$ , degree one: $30$ of $178$ landmarks localized, in two clusters of cycles born near $0.06$ – $0.10$ , dying near $0.08$ – $0.13$ and $0.18$ – $0.25$ ; another random pilot/inference split recovers, on average, $94\%$ of the region.

placement	$\ell$	$m$	cover	FWER boot	FWER Bonf	$\hat{c}_{\alpha}$ / $z_{\mathrm{Bonf}}$
FPS	50	20	0.978	0.022	0.070	3.64 / 3.29
FPS	50	60	0.970	0.030	0.054	3.47 / 3.29
FPS	100	20	0.984	0.016	0.086	3.84 / 3.48
FPS	100	60	0.952	0.048	0.082	3.62 / 3.48
FPS	200	20	0.988	0.012	0.086	4.15 / 3.66
FPS	200	60	0.968	0.032	0.056	3.86 / 3.66

Appendix

Table 15: Level and coverage at

\Delta=0

in the earlier additional-pilot studies, for

m\in\{20,60\}

. All sample sizes appear in Appendix F.9 . Every coordinate is null, so FWER of

\widehat{R}

is one minus coverage, and the step-down set makes any rejection exactly when

\widehat{R}

does. The table compares FWER of

\widehat{R}

and normal-reference Bonferroni, and the bootstrap critical value

\widehat{c}_{\alpha}

with

z_{1-\alpha/(2\ell)}

Figure 6: Bootstrap critical values against

\ell

in the earlier additional-pilot design. For pilot-fitted landmarks, the values lie slightly above the normal-reference Bonferroni threshold

z_{1-\alpha/(2\ell)}

, which grows at order

\sqrt{\log\ell}

. Fine grids give larger values because sparse responses yield greater variation in coordinate variances relative to the estimated pilot factors (Proposition 1 ). The normal-reference Bonferroni procedure overrejects in these comparisons (Table 15 ).

Figure 7: Estimated normalization under imbalance. At

\Delta=0

, all coordinates are null. The design uses

20

versus

60

inference diagrams,

\ell=105

, and

300

runs. Left: empirical quantiles of

Z_{k}

against

\mathcal{N}(0,1)

using groupwise sample SDs or independent pilot factors. Right: each run’s maximum and bootstrap critical value, with FWER calculated over these runs.

arm	scale	$\ell$	$m$ / $m^{\prime}$	cover	FWER boot	FWER Bonf	FWER perm	power 0.2	$\hat{c}_{\alpha}$
FPS in-sample	sample	49	20 / 20	–	0.042	0.036	–	0.87	3.16
FPS in-sample	sample	49	60 / 60	–	0.044	0.030	–	0.98	3.23
FPS in-sample	sample	99	20 / 20	–	0.068	0.040	–	0.96	3.33
FPS in-sample	sample	99	60 / 60	–	0.028	0.026	–	1.00	3.41
FPS in-sample	sample	199	20 / 20	–	0.060	0.032	–	1.00	3.49
FPS in-sample	sample	199	60 / 60	–	0.026	0.022	–	1.00	3.57

Appendix

Table 16: Placement, selection, and normalization in the earlier studies. Comparisons include inference-sample versus pilot placement and selection (

m=m^{\prime}\in\{20,60\}

), sample versus pilot normalization in balanced and

20

-versus-

60

designs, and the weak null. Inference-sample arms have no pilot. At

\Delta=0

, every target contrast is zero and FWER is the probability of any rejection. Coverage of a random family is defined, but a family fitted on the inference sample is not covered by the fixed-family theorem. The table reports the available coverage and FWER results for the band, normal-reference Bonferroni, and, where run, the permutation maximum, together with power at

\Delta=0.20

Figure 8: The band on the synthetic task, one run at $\Delta=0.20$ , $m=60$ : class $B$ carries the same features as class $A$ with one feature displaced from the location marked $A$ to the location marked $B$ ; the band localizes the displacement and nothing else.

placement	$\ell$	$\Delta$	$m$	cover	power boot	power Bonf	power BH	FDR BH	FDR boot	loc. err.
FPS	50	0.10	20	0.970	0.46	0.54	0.55	0.025	0.006	0.073
FPS	50	0.10	60	0.976	0.99	0.99	1.00	0.020	0.005	0.043
FPS	50	0.30	20	0.974	0.08	0.15	0.15	0.021	0.013	0.108
FPS	50	0.30	60	0.966	0.49	0.56	0.58	0.027	0.013	0.075
FPS	399	0.10	20	0.994	0.95	0.99	0.99	0.040	0.000	0.007
FPS	399	0.10	60	0.974	1.00	1.00	1.00	0.034	0.002	0.006

Appendix

Table 17: Earlier localization study at

\Delta\in\{0.10,0.30\}

and

m\in\{20,60\}

, for the coarsest and finest placements. Complete tables appear in Appendix F.9 . Power: at least one signal landmark localized. FDR: the reported false-discovery summary using the geometric nulls defined in Appendix F.1 . Localization error: bottleneck distance from the top landmark to the nearer signal. The finest grid retains

384

–

389

of its

406

landmarks after capping and pilot screening, depending on

\Delta

map	$\ell$	$m$	boot	perm	boot	perm	boot	perm	loc. err.
			FWER, $\Delta=0$		power, $\Delta=0.10$		power, $\Delta=0.20$
image pixels	63	20	0.044	0.052	0.09	0.10	0.33	0.34	0.118
image pixels	63	60	0.042	0.042	0.35	0.35	0.90	0.90	0.084
image pixels	111	20	0.066	0.084	0.15	0.19	0.52	0.57	0.065
image pixels	111	60	0.046	0.050	0.67	0.70	0.99	0.99	0.039
image pixels	189	20	0.018	0.052	0.23	0.30	0.66	0.74	0.052

Appendix

Table 18: The mass-univariate baseline: persistence-image pixels against pilot-fitted landmarks, each calibrated by the bootstrap band and by the permutation maximum on the same

500

replications. FWER at

\Delta=0

over all coordinates; power: a signal coordinate localized; loc. err. at

\Delta=0.20

Figure 9: Rejection rate at

\alpha=0.05

against

\Delta

m=30

per class. Left: the signal’s lifetime changes. Right: the signal moves along the diagonal and its lifetime does not change; every lifetime-based procedure is at its level by Proposition 3 .

persistence regime
$\Delta$	0	0.025	0.05	0.075	0.1	0.15	0.2	0.3
band (max-t, bootstrap)	0.02	0.05	0.40	0.66	0.77	0.82	0.87	0.81
embedding MMD (permutation)	0.05	0.09	0.43	0.69	0.76	0.83	0.88	0.87
STRAND (diagram perm.)	0.05	0.05	0.05	0.06	0.08	0.06	0.08	0.13
STRAND (pooled asympt.)	0.05	0.05	0.05	0.06	0.07	0.07	0.09	0.15
landscape MMD	0.07	0.06	0.03	0.04	0.07	0.03	0.07	0.07

Appendix

Table 19: Rejection rates by method and regime.

				pilot scale			sample scale
design	placement	$\ell$	$m$ / $m^{\prime}$	band	Bonf.	perm	band	perm
clustered	FPS	100	20 / 20	0.032	0.024	0.042	0.252	0.201
clustered	FPS	100	20 / 60	0.041	0.028	0.000	0.090	0.024
clustered	FPS	100	60 / 20	0.048	0.035	0.178	0.343	0.184
clustered	FPS	400	20 / 20	0.039	0.014	0.048	0.283	0.219
clustered	FPS	400	20 / 60	0.057	0.010	0.001	0.121	0.024

Appendix

Table 20: Weak nulls with unequal covariance at every landmark. Every

\delta_{k}=0

, so the FWER is the probability of any rejection;

1000

replications per cell. Pilot scale: the band, Bonferroni and the permutation maximum; sample scale: the band and the permutation maximum with each coordinate studentized by its sample variance.

m

m^{\prime}

: the sizes of classes

A

and

B

; class

B

is the more variable. On the grid only the landmarks that see the boxes pass the pilot screen.

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Jun 10, 2026stat.ML

From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features

Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction. We introduce STRAND (Survival Topological Representation ANalysis of Diagrams), which treats (collections of) PDs as survival data: each topological feature with persistence value

p = d - b

is a fully observed time-to-event, and the persistence survival function

S(t) = \mathbb{P}(p > t)

is the central object for comparing diagrams. From this single representation we derive (i) a non-parametric two-sample test with calibrated Type I error and high power from a small number of diagrams; (ii) interpretable effect sizes; and (iii) a 1-Wasserstein-stable feature vector for downstream machine learning. We validate calibration and power on synthetic manifolds with controlled topology, demonstrate competitive vectorisation across 14 graph and 3D point cloud benchmarks, and apply the method to study functional brain connectivity in fMRI/neuroscience data. To our knowledge, STRAND is the first method to provide hypothesis testing and vectorisation for persistence diagrams from a single coherent and interpretable representation.

Juliette Murris, Bernadette Stolz, Karsten Borgwardt

Department of Machine Learning and Systems Biology, Max Planck Institute of Biochemistry, Martinsried, Germany

Persistent Homology Topology Survival Analysis Two-Sample Testing Vectorization Effect Sizes+2

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Aug 2, 2026stat.ML

How fine a change can moments see? A scale law for detecting distribution shift, with a kernel calibration rule

Detecting that a stream of high-dimensional embeddings has changed is usually framed as a choice of statistic. We give a scale law that constrains any moment-based choice and test it against topological alternatives. The law: certifying a feature of spatial scale eps carrying mass fraction f requires polynomial tests of degree N* >= log(1/f)/(2 eps), proved via the Chebyshev extremal problem; a Gauss-quadrature construction gives N* >= 4b-1 for a b-scale topology, so cost is set by feature fineness, not feature count. The law is one-sided: we exhibit an annulus whose mean, covariance and all fourth-order moments equal those of a filled disk, yet H_1 is nonzero. Its practical content is a calibration rule. The upper bound is attained by Gaussian test functions, the RKHS witness of an RBF kernel, so the law predicts which bandwidth an MMD test should use: the feature scale. On real embedding streams we measure sigma*/eps with median 1.12 (IQR 1.01-1.52, n=26) over three settings and three scales, and a data-driven bandwidth reaches AUC >= 0.95. Against an adversary optimised against the defender's statistics (mean, covariance, k-NN, kurtosis), only a bandwidth-matched kernel test still detects. For persistent homology the verdict is mixed and depends on choices usually left implicit. The summary matters more than the filtration: total persistence attains recall 0.75 at FPR 1% where the first persistence landscape attains 0.00. What survives is a cost gap, not a power gap: where persistence works it costs 116x kurtosis, which works at least as well. We conclude not that topological summaries are useless, but that on this task a kernel test whose bandwidth the law sets dominates them.

Adel Kaleche

Independent researcher, France

Persistent Homology Two-Sample Testing Skewness Distribution Shifts Sequential Scaling Scaling Laws+1

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Sep 1, 2026cs.CG

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

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

d_{\mathrm{SK}}

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

O(N\log N)

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

2

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

W_Γ

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

d_{\mathrm{SK}}

over state-of-the-art approximations of

W_2

626\times

, while the aggregate speedup over the full benchmark is

2100\times

. Average-linkage partitions obtained from

d_{\mathrm{SK}}

and

W_Γ

each exactly match the corresponding

W_2

partition on 8 of the 12 collections. Hilbert

k

-means and Gaussian spectral clustering, both based on

d_{\mathrm{SK}}

, achieve mean adjusted Rand indices (ARI) of

0.756

and

0.800

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

0.750

obtained by average linkage on

W_2

. The Gaussian

d_{\mathrm{SK}}

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

Sebastien Tchitchek, Julien Tierny

CNRS and Sorbonne Université

Wasserstein Distance Persistent Homology Kernel Method Clustering Diagrams Approximation

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