Matched Queries for Curvature and Density at Branching Junctions

Ziqi Zhao; Qingjian Ni

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Matched Queries for Curvature and Density at Branching Junctions

Ziqi Zhao, Qingjian Ni

arXiv:2609.01319v1Sep 1, 2026stat.MLcs.LG

TL;DR

Weighted tangent rays do not determine how junction branches bend or how density changes along them, leaving local continuation unresolved. The paper uses matched score queries across two noise scales to isolate the first correction and jointly handles center error; experiments show unique recovery with stated information counts and strong empirical performance.

Problem
Weighted tangent geometry omits branchwise curvature and outward density change, so the paper asks whether finite score observations can recover these quantities despite center-estimation error.
Method
Matched score queries at scales σ and λσ cancel the tangent contribution, exposing a correction linear in branch curvature and logarithmic density slope.
Results
49.4 times more accurate than naive subtraction under strong population first-order error, while all 135 population systems are full rank with median relative jet error 0.132.
Takeaways & Limitations
The first score correction uniquely determines branch jets on supplied distinct rays, with scalar-information complexity sD or (s + 1)D when center bias is jointly recovered.
Takeaways & Limitations
The theory assumes C2,α half-branches, positive C1,α densities, Gaussian smoothing, two known scales, branch correspondence, and supplied first-order geometry.

Abstract

from arXiv · show

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.

1 Introduction

Tangent geometry captures weighted branch directions but misses curvature and outward density change. Matched score queries cancel the tangent field, enabling recovery of branch jets and center bias from finite scalar observations.

Tangent blow-up preserves branch directions and weights but removes curvature and outward density change.
The inverse problem recovers how branches continue away from a junction despite identical leading tangent fields.The missing quantities appear in the next-order correction and require cross-scale isolation, branchwise assignment, and center-error control.
Matched queries at scales σ and λσ cancel the shared tangent field and expose the correction containing curvature and density change.Using the same normalized location and provisional center also leaves the center correction visible.
sD scalar observations are necessary and sufficient for branch-jet recovery, while center bias increases the requirement to (s + 1)D observations.The augmented system excludes the translation-invariant complete-line case.
The superposed correction uniquely identifies every branch jet on supplied distinct rays without requiring fixed angular separation.Stability is governed separately by conditioning, including ray collisions and vanishing weights.
1 Introduction: Across 180 population fits, the median remainder exponent is 0.998, and matched responses are 49.4 times more accurate than naive subtraction under strong first-order error.Scale-up remains full rank through D = 20 and 16 supplied branches.
1 Introduction: All 135 population systems are full rank in end-to-end D = 3–5 tests, with median relative jet error 0.132.KDE scores have median relative jet error 0.957 under a fixed 131,072-sample budget, improving to 0.688–0.810 at σ = 0.08.

2 Model and Second-Order Target

The model describes a finite junction as weighted, distinct C2,α half-branches with positive densities. Its second-order target is each branch’s logarithmic density slope and normal curvature, assembled into a branch jet.

The model assumes a finite union of one-dimensional half-branches near a junction center x0 in R^D.Branches are parameterized by arc length and include a finite remainder supported away from the center.
Distinct branches have distinct tangent directions vj, and arc-length parameterization makes each curvature vector kj orthogonal to vj.
The regularity assumptions use C2,α branch behavior and uniform jet remainders with 0 < α ≤ 1.
The weighted tangent measure contains the first-order geometry, with weights normalized because a score cannot identify total scale.The known first-order quantities are the directions and normalized weights.
The branch jet J2(µ, x0) consists of (aj, kj), where aj measures outward log-density change and kj measures bending.
At order σ, density slope changes mass along a tangent ray while curvature displaces the branch normally away from it.These are distinct signatures within the first correction to the tangent field.

3 Why Second-Order Geometry Appears Across Scale

At physical radius r = σu, curvature and density variation both enter the normalized score at order σ. A matched query at scales σ and λσ cancels the tangent field F0 and exposes the correction G.

Second-order mechanism: At physical radius r = σu, density change and normal curvature displacement both contribute at order σ after normalization.Gaussian smoothing places both effects in the same correction term G.
Second-order mechanism: The second-order expansion separates the tangent field F0 from the branch-jet correction G, whose forward superposition is later inverted.F0 records tangent rays, while G contains the branch jets.
Matched scale queries: Evaluating the same normalized location at x0 + σz and x0 + λσz enables matched subtraction across the two noise scales.The shared query location is essential to comparing the two score responses.
Matched scale queries: Matched subtraction removes the common tangent field F0 and exposes G without differentiating a noisy score.If the exact tangent score were available, direct subtraction would have the same limit, but matched responses cancel the leading field in the observations.

4 Recovering Branch Jets

With known distinct tangent directions and weights, the matched correction becomes a finite-dimensional linear inverse problem for branch jets. Exactly sD scalar component evaluations are sufficient and necessary for identifying all branch parameters.

Jet identifiability: Theorem 4.1 establishes injectivity of the map from branch jets to the superposed correction for distinct directions and positive weights.Gaussian convolution injectivity and narrowing-tube tests isolate each ray's curvature and density contributions.
Observation complexity: sD scalar component evaluations are sufficient and necessary to identify every branch jet under the theorem's conditions.This is a scalar-information count; one network evaluation can return all D coordinates simultaneously.
Jet parameterization: Each branch jet contains one logarithmic density slope and D − 1 curvature coordinates, giving sD unknown scalar parameters across s branches.The branch basis uses an orthonormal basis for each direction's normal space.
Observation complexity: Stability depends on conditioning, with ray collisions and vanishing weights degrading the inverse problem even when uniqueness holds.Well-conditioned rows can be selected from a larger candidate matrix using rank-revealing QR.

5 Recovering Jets with an Imperfect Center

An imperfect center contributes translation-shaped modes at the same normalized order as the branch-jet signal, but these modes can be jointly estimated under a non-translation-invariant junction. This raises the scalar requirement to (s + 1)D and refines the center when calibration is full rank.

Center-bias expansion: A score-only first stage produces a nearby center whose physical O(σ^2) error becomes order σ after normalization, matching the branch correction.The shared center is reused at both scales so the mismatch has a known translation shape.
Center-robust identifiability: Appending D translation fields makes the joint inverse problem have (s + 1)D coordinates.These modes are generated by the Jacobian of the tangent score.
Center-robust identifiability: Except for a translation-invariant complete line, (s + 1)D fixed scalar observations are sufficient and fewer cannot identify every parameter vector in an open set.A center shift cannot imitate branchwise curvature and density changes when the junction lacks nonzero translation invariance.
Center refinement: Estimating the translation coefficient refines the physical center to O(σ^{2+α}) when the coefficient error is O(σ^α).This refinement follows directly from the center expansion.

6 Error Propagation and Finite Samples

The paper bounds recovery error under perturbed designs, responses, score estimates, and center localization, then derives a conditional KDE rate. Cross-scale consistency, design conditioning, and local sampling jointly control second-order recovery.

Error model: For known-center recovery, p = sD scalar evaluations are required; joint center recovery uses p = (s + 1)D parameters.The query system stacks scalar evaluations into a design matrix B and two-scale response y.
Error model: If the implemented design perturbation satisfies ∥E∥op < γ, the least-squares estimator remains controlled by the smallest singular value γ = σmin(B).Finite-scale bias and response noise are amplified by the inverse smallest singular value.
Error model: Scale differencing divides normalized-score noise by σ, while an unmodeled physical center remainder is divided by σ2.These two amplification mechanisms are separated in the deterministic perturbation decomposition.
Finite samples: The KDE rate balances deterministic remainder σα against scale-differenced sampling fluctuation and incorporates score errors at both noise scales.Only Nσ samples contribute locally, while the scale difference adds another factor 1/σ to the variance term.
Finite samples: Cross-scale consistency, rather than accuracy at either scale alone, controls second-order recovery.The rate assumes a fixed query design, positive local tangent mass, bounded branch jets, and a comparably accurate first stage.

7 Experiments

Experiments validate the predicted finite-scale and sampling behavior, show strong dependence on geometry and conditioning, and test matched responses, frontend recovery, and learned scores across increasingly difficult settings.

Finite-scale expansion: 0.998 was the median remainder exponent across 180 population fits in D ∈ {2, 3, 5}, matching the predicted O(σ) order.The 5th–95th percentile was 0.957–1.03, and exact- and O(σ2)-biased-center runs followed the predicted order.
Scalar information versus stable designs: 0.00552 was the median relative error for D-optimal row selection, 2.89 times lower than random selection at response-noise RMS 10−3.Overdetermination and conditioning-aware rows increased the smallest singular value.
Geometry and scale: 184-fold separated the phase-map median errors 0.698 and 0.00379, while condition numbers changed from 5.61 × 104 to 24.6.All 200 scale-up systems remained full rank through D = 20, 16 branches, 340 unknowns, and 680 queries.
Matched responses: 49.4 times more accurate were matched responses than naive subtraction under strong population first-order error.The corresponding population error ratios under mild, moderate, and strong perturbations were 7.75, 26.6, and 49.4.
Finite-sample KDE: N−1/5 was contained by all seed-bootstrap 95% intervals across four planar junctions and twenty seeds, with error reductions of 52.1–66%.Bandwidth comparisons identified a broad stable operating region across geometries.
Estimated first-order frontends: 0.132 was the median relative jet error for population runs with a known-count frontend in D = 3–5, and all 270 second-stage systems were full rank.The frontend achieved 100% blind count accuracy on population data and 98.8% on KDE scores.
Learned scores: 0.676 was the final median jet error in the hard learned-score study despite score RMS improving to 0.0143.The controlled experiment isolates cross-scale consistency as the mechanism used by the inverse.

8 Related Work

Prior work recovers tangent, curvature, and forward score corrections, whereas this paper addresses the converse problem of decomposing superposed corrections into branchwise second-order coefficients on supplied rays.

Scope relative to prior work: Classical and score-based methods recover manifolds, tangent spaces, curvature, and weighted singular tangents, while this work targets branchwise second-order continuation at singular junctions.The target is continuation along supplied rays rather than only first-order junction geometry.
Inverse formulation: Forward asymptotics map known geometry to density and curvature corrections; this paper instead uniquely decomposes a superposed correction using finite scalar information and center translation.Unlike Prony and super-resolution, the unknowns are coefficients on supplied rays, not atom locations.

9 Limitations and Conclusion

The theory establishes unique branch-jet recovery and matched-response stability under supplied geometric conditions, while experiments identify the scope boundaries and practical regimes of the result.

Limitations: The theory assumes zero-thickness C2,α half-branches, positive C1,α densities, Gaussian smoothing, two known scales, branch correspondence, and supplied first-order geometry.Conditioning captures ray collisions and vanishing weights, while the KDE rate is conditioned on first-stage accuracy.
Limitations: Experiments cover known-count full-pipeline recovery in D = 3–5, blind planar counts, and supplied-geometry scaling through D = 20 with 16 branches.The full-line case retains intrinsic translation symmetry, and learned-score experiments use controlled synthetic junctions.
Conclusion: The first score correction uniquely determines branch bending and density change on supplied distinct rays, while matched subtraction exposes it with sharp scalar-information complexity.Translation modes accommodate imperfect localization, and experiments carry score-estimated first-order geometry through the inverse in D = 3–5.
Conclusion: The stability map, matched-response comparison, and learned-score trajectories identify geometric and statistical conditions governing recovery accuracy.The paper positions branch jets as a test of whether score fields retain junction continuation.

B Distributional proof of jet identifiability

The proof shows that the branchwise curvature and density-slope fields have no nontrivial nullspace after Gaussian convolution, establishing jet identifiability.

Gaussian convolution is injective because its Fourier multiplier is everywhere positive.Thus vanishing convolved correction implies the underlying distributional correction is zero.
Localized test functions with arbitrary normal gradients isolate each branch and force its curvature vector to vanish.The test support excludes other rays and the origin, while the test function vanishes on the selected ray.
Repeating the argument over all branches proves the nullspace is trivial and establishes uniqueness for two parameter collections.
On an interior ray segment, the remaining density relation implies a_j u = C at every point.Because the interval contains more than one point, this forces a_j = 0 and C = 0.

C Finite evaluations and the query lower bound

Finite point-coordinate evaluations recover linearly independent vector fields, while invariance of domain proves that fewer than sD scalar outputs cannot continuously identify all branch parameters.

Linearly independent vector fields admit p point-coordinate evaluations whose evaluation matrix is nonsingular.The coordinate evaluation functionals separate points of the span and contain a basis of its dual.
The sD jet fields are linearly independent, so Theorem C.1 supplies an upper bound of sD scalar observations.
Any continuous injective scheme using M < sD scalar outputs would embed an open parameter box into a lower-dimensional space, contradicting invariance of domain.

D.1 Why weak calibration has a physical O(σ2) error

Weak calibration produces an O(σ^2) physical center error, whose translation modes can be jointly identified with branch jets under full-rank designs; perturbation and KDE analyses quantify stability and sampling effects.

D.1 Why weak calibration has a physical O(σ2) error: Matched two-scale expansion exposes the center bias as a translation term alongside the first score correction.The expansion contains σ∇F0(z)b + σG(z), while quadratic bias interactions enter the remainder.
D.1 Why weak calibration has a physical O(σ2) error: Except for translation lineality, the augmented branch-jet and center-bias fields remain identifiable from the calibrated design.Translation lineality for a finite positive ray junction can occur only along a complete line with opposite rays and matching density.
E. Stability and sampling: The least-squares perturbation is controlled by the pseudoinverse applied to finite-scale, response, and design errors.Weyl’s inequality lower-bounds the perturbed smallest singular value by σ_min(B) − ||E||op.
E. Stability and sampling: The KDE rate balances the deterministic remainder σ^α against scale-differenced sampling fluctuation, with first-stage errors entering separately.At the selected bandwidth, the N^-1σ^-2 term is lower order for every α > 0.
E. Stability and sampling: Cross-scale consistency is required for learned-score second-order recovery, because errors at the two noise scales must be aligned.
F. Experiments: The end-to-end experiments use matched responses, exact-first-order diagnostics, and crossed designs to separate first-stage, row-selection, and coefficient-basis effects.The crossed E4 diagnostic reports a population estimated-basis/exact-basis ratio of 0.611 as finite-scale bias cancellation.