Recovering Weighted Tangent Geometry from a Single-Scale Score Field

Ziqi Zhao; Qingjian Ni

stat.MLAug 23, 2026

Recovering Weighted Tangent Geometry from a Single-Scale Score Field

Authors: Ziqi Zhao, Qingjian Ni

Organizations: School of Computer Science and Engineering Southeast University Nanjing, China

Abstract

Near a smooth data manifold, one tangent space summarizes local geometry. At a branch point, the corresponding first-order object is instead a measure over tangent directions, whose normalized masses record the local share of each branch under the chosen data measure. We ask whether a score field at one noise level determines this weighted tangent geometry when the branch center and homogeneity degree $d$ are unknown. In this tangent-measure model, $d$ is the local measure dimension. Gaussian smoothing of a homogeneous tangent measure satisfies an Ornstein--Uhlenbeck eigenfunction equation. Its weak form turns score values---without score derivatives---into a linear system for the center and homogeneity degree, with an explicit rank condition and perturbation bound. After this calibration, the tangential score on one sphere is the spherical log-gradient of a scalar Gaussian--cone transform. Integration recovers that transform up to scale, and all its spherical-harmonic multipliers are positive. Thus one exact shell identifies the normalized angular measure in every ambient dimension $D\geq2$ . For at most $K$ positive rays, moments through degree $2K-1$ constructively recover count, directions, and weights in arbitrary dimension. Any fixed observation scheme needs at least $KD-1$ scalar tangential components. In the plane, degree $K$ is both sufficient and necessary, and we give quantitative finite-query certificates. For finite planar $C^{1,β}$ branches with positive $C^{0,β}$ densities, we prove $O(σ^β)$ convergence from the finite-noise score to its tangent model. In controlled experiments, 50k-step training lowers validation normalized-score error across four geometries yet raises angular-moment error, separating ordinary score fit from geometry recovery.

Figures & tables

Figure 1: One-point collision and shell conditioning. (a) For

D=2,d=1

, regular three- and four-ray measures share the vertex score Hessian

-I_{2}/2

but have different

R=2

shell densities and tangential scores. (b) Inverse harmonic multipliers grow rapidly with degree at small radii. Marker and line style identify radii; shading marks degrees

1{:}4

used by the planar

K=4

experiments.

Task and field	$D$	Runs	Center error	$\|\widehat{d}-d\|$	Full geometry
Exact weak calibration	2	20	$4.4\times 10^{-4}$	$8.7\times 10^{-5}$	–
Exact weak calibration	3	20	$8.5\times 10^{-4}$	$2.2\times 10^{-4}$	–
Exact weak calibration	5	20	$4.3\times 10^{-4}$	$9.7\times 10^{-5}$	–
Score perturbation ( $\mathrm{RMS}=0.01$ )	2	20	$0.018$	$0.004$	–
Score perturbation ( $\mathrm{RMS}=0.01$ )	3	20	$0.017$	$0.006$	–
Score perturbation ( $\mathrm{RMS}=0.01$ )	5	20	$0.016$	$0.005$	–

Table 1: Single-scale calibration and higher-dimensional reconstruction. Center error is normalized by

\sigma

. Exact-calibration rows report the maximum numerical quadrature error; perturbed and learned rows report the median. “Full geometry” counts trials meeting the generated count, direction, and weight criteria. For the three-dimensional score-shell rows, these criteria are correct count, maximum angular error

0.1^{\circ}

, and maximum weight error

10^{-3}

. Every entry is generated from seed-level rows.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 2: Finite-noise tangent-bias sweep. The panels report the maximum denominator error, numerator error, and normalized-score ratio error over a fixed normalized query set. Each curve is computed by Gaussian quadrature; the

\beta=1/2

\beta=3/4

, and

\beta=1

cases exhibit their predicted log–log exponents.

Stage	Declared class information	Sufficient test	Output on success / failure
Count	Minimum positive Toeplitz eigenvalue	Signal gap exceeds twice $E_{K}$	Branch count / abstain
Directions	Count, angular separation, and root margin	Appendix Rouché inequality	Angles within the declared tolerance / no directions
Weights	Direction bound and minimum Vandermonde singular value	Appendix perturbed-solve inequality	Normalized weights within tolerance / no weights

Appendix

Table 2: The conditional certificate proceeds from count to directions to weights. Every row uses class information fixed before the field is inspected. Failure stops the procedure before the next output.

Role	Information beyond the queried field	Interpretation
Blind candidate	$R$ , $K$ , query grid, and numerical rank rule	Uncertified count, directions, and weights
Oracle diagnostic	True synthetic signal gap or geometry	Controlled diagnosis only; not deployable
Certified output	Valid moment-error budget and class bounds fixed in advance	Stagewise guarantee, otherwise abstention

Appendix

Table 3: Information permissions for the three reported inference roles. Only a certified output carries a sufficient guarantee, and only when its external error budget and declared class bounds are valid.

Moment-error bound	Count	Directions	Full geometry
$1e-09$	48/48	48/48	48/48
$1e-08$	48/48	48/48	48/48
$1e-07$	48/48	48/48	48/48
$1e-06$	48/48	48/48	24/48
$1e-04$	48/48	24/48	0/48
$1e-03$	48/48	0/48	0/48

Appendix

Table 4: Execution of the pre-specified count–direction–weight certificate. Entries are positive reports out of 48 bounded perturbations at each error level; unreported cases abstain at that stage.

Figure 3: Finite-query recovery separates conditioning, query, noise, and center-offset failure modes. Rates aggregate 12 seeds per configuration; success requires correct count, maximum angular error $5^{\circ}$ , and maximum weight error $0.05$ . Top: an isolated close pair and a four-ray cluster at matched minimum separation and weight. Bottom: query/noise sensitivity and center-offset sensitivity by shell radius. Cell numbers, markers, and line styles make the rates readable without relying on color.

Figure 4: Empirical KDE recovery approaches the population field as local coverage grows. Across 6,528 trials, local coverage is the expected effective number of nearby samples receiving appreciable Gaussian weight at a query. Left: oracle-rank and blind recovery, with fixed-template and one-point Hessian/PCA checks; the rank gate uses the true synthetic signal gap. Middle: normalized-score RMS, maximum moment error, and an inverse-square-root reference slope. Right: blind recovery on named, held-out, imbalanced, and near-resolution subsets. Markers and line styles identify every curve without relying on color.

Figure 5: Longer training can restore count without full geometry. The study uses 5 seeds per configuration; its local-coverage proxy is $N\sigma/L$ . Left: full-geometry success versus coverage. Middle: estimated count, with the true count marked by a dashed line. Right: undercount rate versus noise scale. Markers and line styles distinguish empirical, time-conditioned, and single-noise fields and their training budgets. The external legend identifies the six middle-panel fields; the other panels use local legends.

Geometry	Angles (radians)	Weights
regular Y	$0.20,2.2944,4.3888$	$1/3,1/3,1/3$
weak Y	$0.25,2.15,4.55$	$0.05,0.40,0.55$
clustered four	$0.10,0.75,1.40,4.20$	$0.25,0.25,0.25,0.25$
weak four	$0.05,1.18,3.02,5.10$	$0.05,0.25,0.30,0.40$

Appendix

Table 5: Synthetic geometries used in the fixed cross-geometry learned-score evaluation.

Geometry	Network	Emp.	Count 1k $\to$ 5k	Full 1k $\to$ 5k	Angle 5k	Weight 5k
clustered four	plain	4/5	3/5 $\to$ 5/5	0/5 $\to$ 1/5	6.501	0.055
clustered four	residual	4/5	3/5 $\to$ 5/5	0/5 $\to$ 0/5	5.344	0.068
regular Y	plain	5/5	5/5 $\to$ 5/5	2/5 $\to$ 2/5	1.853	0.065
regular Y	residual	5/5	5/5 $\to$ 5/5	2/5 $\to$ 1/5	2.221	0.092
weak four	plain	5/5	3/5 $\to$ 5/5	1/5 $\to$ 0/5	2.505	0.099
weak four	residual	5/5	2/5 $\to$ 4/5	0/5 $\to$ 0/5	3.083	0.111

Appendix

Table 6: Fixed learned-score evaluation (successes out of 5 seeds). “Emp.” is exact empirical KDE full recovery; count and full columns show learned fields at 1k

\to

5k updates. Angle (degrees) and weight columns are medians over count-correct 5k seeds only.

Figure 6: Fixed evaluation across four geometries and two parameter-matched multilayer-perceptron (MLP) parameterizations. Left: count recovery; right: full-geometry recovery. Hatching distinguishes 1k from 5k updates, and error bars are 95% Wilson intervals over 5 seeds. C4, Y, W4, and WY denote clustered four, regular Y, weak four, and weak Y; P and R denote plain and residual MLPs.

Geometry	Arch.	Updates	Score RMS	Moment err.	Count	Full
clustered four	plain	50,000	0.0824	0.105	10/10	2/10
regular Y	plain	50,000	0.0993	0.133	10/10	2/10
weak four	plain	50,000	0.104	0.132	10/10	4/10
weak Y	plain	50,000	0.12	0.165	6/10	2/10

Appendix

Table 7: Validation-selected train-to-convergence results at

\sigma=0.1

(ten seeds per row). “Score RMS” is the root-mean-square error of the normalized score against its population value; moment error is the maximum error through order

K

. Full recovery requires correct count, maximum angle error at most

5^{\circ}

, and maximum weight error at most

0.05

Figure 7: Train-to-convergence learned-score diagnostics. (a) Validation normalized-score RMS across optimizer updates. (b) Test normalized-score RMS versus maximum angular-moment error at saved checkpoints. (c) Seed-level count and full-geometry outcomes versus moment error. (d) Validation-selected comparison of parameter-matched plain/residual MLPs with a larger residual MLP on three geometries. Curves and intervals aggregate ten seeds; checkpoint selection uses validation score error only.

Angle tolerance	$0.02$	$0.05$	$0.10$
$2^{\circ}$	0/40	2/40	7/40
$5^{\circ}$	1/40	6/40	19/40
$10^{\circ}$	1/40	8/40	23/40

Appendix

Table 8: Pooled full-geometry successes over the released long-run learned fields. Rows give the maximum angle error and columns the maximum weight error; each entry is successes out of 40 fields. These fields span four geometries, two parameterizations, and 5 seeds, so the pooled entries are descriptive reaggregations rather than additional independent replications.

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Sep 1, 2026J/K move · Enter open · S save

Matched Queries for Curvature and Density at Branching Junctions

Ziqi Zhao, Qingjian Ni

School of Computer Science and Engineering Southeast University Nanjing, China

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At a junction, a score field can reveal weighted tangent rays, yet these first-order quantities do not determine how individual branches bend or how their densities change away from the center. Recovering this missing information is necessary for describing local continuation beyond a single point, but finite observations must separate branchwise second-order effects while allowing error in the estimated center. We address this inverse problem using matched score queries at noise scales

σ

and

λσ

. For a finite union of

C^{2,α}

half-branches in

\mathbb{R}^D

, the normalized score has the expansion

F_σ=F_0+σG+O(σ^{1+α})

. Matched subtraction cancels the tangent contribution and exposes

G

, which depends linearly on branchwise curvature and log-density slope. Given tangent directions and weights on distinct rays,

G

uniquely identifies all

sD

branch parameters, and

sD

scalar component observations are necessary. An

O(σ^2)

center error introduces

D

translation modes, leading to

(s+1)D

observations under full-rank calibration, except for a translation-invariant full line. We also establish a perturbation bound and a conditional kernel-density-estimation rate. Experiments reproduce the predicted population and

N^{-1/5}

trends and remain full rank up to

D=20

with 16 supplied branches. In end-to-end tests for

D=3

5

, a known-count first-order frontend yields full rank in all 135 population systems and a median relative jet error of 0.132. With strong first-order error, matched responses reduce median parameter error by a factor of 49.4 relative to naive tangent subtraction.

Abstract

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