Contour Bootstrapping in Matrices & Shapes

• Contour bootstrapping is a dual-meaning method that combines resolvent-based spectral techniques for matrix perturbation with m-out-of-N bootstrapping for nonparametric shape inference.
• In matrix perturbation theory, it uses contour-integral representations to convert complex eigenvalue distributions into explicit bounds on projectors, low-rank approximants, and analytic matrix functionals under gap assumptions.
• In elastic shape analysis, it resamples image-derived contours to construct confidence regions and perform hypothesis tests for shape equivalence, invariant to translation, rotation, scaling, and reparameterization.

“Contour bootstrapping” designates two distinct technical constructions in recent arXiv literature. In matrix analysis, it is a resolvent-based method that converts contour-integral representations of spectral quantities into perturbation bounds for projectors, low-rank approximants, and analytic matrix functionals under a spectral gap assumption (Tran et al., 2024). In statistical shape analysis, it denotes a bootstrap-based inferential procedure for 2D contours extracted from images, implemented within Elastic Shape Analysis (ESA) through percentile contours, square-root velocity functions (SRVFs), elastic shape distance (ESD), and an $m$-out-of-$N$ bootstrap designed for non-smooth functionals (Glenn et al., 3 Jun 2026). The shared term is therefore terminologically ambiguous: in one setting the relevant contours lie in the complex plane, whereas in the other they are planar object boundaries.

1. Distinct meanings and mathematical settings

The matrix-analytic usage arises in the study of perturbations of real symmetric $n\times n$ matrices. Given a symmetric matrix $A$ with eigen-decomposition

$A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$

and a perturbation $E$, one studies $\widetilde A=A+E$ and seeks bounds on quantities such as the largest eigenvalue, spectral norm, leading eigenvector, eigenspace projectors, leading singular subspaces, best rank-$p$ approximants, and more general matrix functionals of the form

$f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$

The methodological core is spectral calculus via complex contours and resolvents (Tran et al., 2024).

The ESA usage arises in image-based shape inference. A 2D contour is modeled as an absolutely continuous parameterized closed planar curve

$c:\mathcal D\to \mathbb R^2,$

with $N$0 or $N$1, and in the closed case $N$2. Contours are extracted from images as percentile contours. If image intensities are $N$3 with empirical distribution $N$4, then for a percentile $N$5,

$N$6

That point set is then ordered and interpolated into a smooth closed curve. The inferential target is not Euclidean curve proximity but shape equivalence modulo translation, rotation, global scaling, and reparameterization (Glenn et al., 3 Jun 2026).

This terminological duality is central. The matrix method is an analytic perturbation technique; bootESA is a nonparametric hypothesis-testing framework. Any unified treatment must therefore begin by separating the two senses rather than conflating them.

2. Contour bootstrapping in matrix perturbation theory

For symmetric matrices, contour bootstrapping rests on the contour representation of spectral quantities. If $N$7 is a closed contour enclosing exactly the eigenvalues $N$8, then for analytic $N$9,

$n\times n$0

The same formula holds for $n\times n$1 provided $n\times n$2 encloses the corresponding perturbed eigenvalues and no others. Consequently,

$n\times n$3

The key algebraic input is the resolvent identity,

$n\times n$4

Tran–Vu then define

$n\times n$5

and obtain

$n\times n$6

A second expansion yields the self-referential inequality

$n\times n$7

where

$n\times n$8

and

$n\times n$9

The “bootstrapping” step is the absorption of the self-term. Under the gap condition

$A$0

one can choose $A$1 so that its distance to the spectrum is at least $A$2, implying $A$3. Therefore

$A$4

and the exact perturbation problem is reduced to estimating an integral involving only the resolvent of $A$5, the perturbation $A$6, and the contour geometry (Tran et al., 2024).

The method is carried out with rectangular contours whose vertical sides roughly bisect spectral gaps and whose horizontal sides are placed at imaginary height proportional to the relevant eigenvalue magnitude, for example $A$7. This contour design is not auxiliary: it encodes the separation needed both for spectral stability and for explicit logarithmic integral estimates.

3. Principal matrix bounds, refinements, and applications

The first major application is eigenspace perturbation. For the projector onto the top $A$8 eigenspace,

$A$9

let $A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$0, let $A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$1 be the smallest index such that $A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$2, and define

$A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$3

Under

$A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$4

Tran–Vu prove

$A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$5

This refines Davis–Kahan by separating a term controlled by $A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$6 from a term controlled by the finer directional quantity $A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$7, rather than only $A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$8 (Tran et al., 2024).

For best rank-$A=\sum_{i=1}^n \lambda_i u_i u_i^{\!*},$9 approximation, the paper derives several bounds. In the positive semi-definite case,

$E$0

under the corresponding gap assumption. For general analytic functionals

$E$1

the contour argument yields

$E$2

Projectors and rank-$E$3 approximants arise as the special cases $E$4 and $E$5.

The significance of these bounds lies in the structural split between global perturbation size and directional incoherence. The paper explicitly notes that for random noise models such as Wigner perturbations, $E$6 may be of order $E$7 while each bilinear form $E$8 may be only $E$9 with high probability. This opens regimes in which contour bootstrapping is substantially sharper than generic Davis–Kahan or Eckart–Young–Mirsky-based estimates.

Two applications are emphasized. In matrix sparsification, $\widetilde A=A+E$0 is often random and structured, and the resulting bounds quantify when spectral subspaces or low-rank surrogates of a sparse approximation remain close to those of the dense matrix. In differential privacy, one adds noise $\widetilde A=A+E$1 to a sensitive matrix $\widetilde A=A+E$2, and the contour-bootstrapping bounds convert the required noise scale into explicit spectral utility guarantees. The paper highlights that this analysis is not tied to Gaussian noise and can therefore support privacy mechanisms with more general noise distributions (Tran et al., 2024).

4. bootESA: contour bootstrapping in elastic shape analysis

The second usage, bootESA, belongs to statistical shape analysis of image-derived contours. Its geometric foundation is ESA, which compares shapes in a manner invariant to translation, rotation, global scaling, and reparameterization. The basic representation is the SRVF

$\widetilde A=A+E$3

defined for a parameterized curve $\widetilde A=A+E$4. Translation invariance is immediate because adding a constant vector to $\widetilde A=A+E$5 does not change $\widetilde A=A+E$6, and scale invariance is enforced by normalization (Glenn et al., 3 Jun 2026).

Shape comparison takes place in the quotient space

$\widetilde A=A+E$7

where $\widetilde A=A+E$8 is the reparameterization group and $\widetilde A=A+E$9 the rotation group. The elastic shape distance between two shape classes is defined by optimizing over rotations and reparameterizations and then applying a $p$0 geodesic map to the resulting $p$1 inner product. In practice, computation alternates between optimizing the rotation and optimizing the reparameterization, then evaluates the ESD after registration.

The inferential target is a one-sample shape test. From reconstructed images $p$2, one forms the learned mean image

$p$3

extracts the percentile contour $p$4, converts it to its SRVF $p$5, and compares it with a hypothesized true shape $p$6 having SRVF $p$7. The question is whether the underlying true shape has equivalence class $p$8 in elastic shape space.

A central obstacle is non-smoothness. The ESD functional is not Fréchet differentiable everywhere because the alignment optimizer can be non-unique and because the derivative of $p$9 diverges at $f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$0, exactly the regime corresponding to zero shape distance. The paper therefore does not use a standard delta-method bootstrap directly on ESD. Instead, it adopts an $f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$1-out-of-$f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$2 bootstrap tailored to non-smooth functionals (Glenn et al., 3 Jun 2026).

5. Bootstrap construction, confidence regions, and hypothesis testing

Given reconstructed images $f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$3, the bootstrap resamples in image space. For each bootstrap replicate $f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$4, sample indices $f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$5 with replacement from $f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$6 and define

$f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$7

From each $f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$8, extract the percentile contour $f_S(A)=\sum_{i\in S} f(\lambda_i)u_i u_i^{\!*}.$9, compute the SRVF $c:\mathcal D\to \mathbb R^2,$0, and evaluate the elastic shape distance to the reference mean shape $c:\mathcal D\to \mathbb R^2,$1. The bootstrap statistic is

$c:\mathcal D\to \mathbb R^2,$2

The resulting empirical distribution approximates the sampling law of $c:\mathcal D\to \mathbb R^2,$3 (Glenn et al., 3 Jun 2026).

The paper defines the unknown confidence region

$c:\mathcal D\to \mathbb R^2,$4

where $c:\mathcal D\to \mathbb R^2,$5 is the $c:\mathcal D\to \mathbb R^2,$6 quantile of the limiting distribution, and estimates the quantile by

$c:\mathcal D\to \mathbb R^2,$7

For one-sided intervals, the empirical confidence region for ESD is $c:\mathcal D\to \mathbb R^2,$8.

The one-sample test against a hypothesized shape $c:\mathcal D\to \mathbb R^2,$9 uses the observed statistic

$N$00

and the p-value

$N$01

One rejects the null at level $N$02 when the p-value is at most $N$03, equivalently when $N$04.

The $N$05-out-of-$N$06 device is the essential modification. The paper states that with the standard choice $N$07, bootstrap ESDs are often conservative because the non-smooth ESD overestimates variability near zero. It therefore recommends $N$08 with $N$09 and $N$10, while keeping the observed-data statistic scaled by $N$11. In simulations for ICF-like images, $N$12 provided reasonable type I error behavior for typical percentile levels $N$13. In real data examples, $N$14 bootstrap replicates were used (Glenn et al., 3 Jun 2026).

6. Empirical behavior, comparison with classical methods, and limitations

For the matrix method, the comparison class is classical perturbation theory. Davis–Kahan bounds eigenspace error in terms of $N$15, and Eckart–Young–Mirsky-based bounds control low-rank perturbation by $N$16. Contour bootstrapping differs by using a self-consistent resolvent inequality, by making the denominator depend on eigenvalue magnitude as well as spectral gap, and by admitting fine structure through quantities such as $N$17. Its limitations are also explicit: a nontrivial spectral gap is required, typically $N$18, and the treatment in the paper is restricted to symmetric or Hermitian matrices rather than non-normal ones (Tran et al., 2024).

For bootESA, the comparison class includes Legendre polynomial or spherical harmonic summaries, Euclidean or Procrustes approaches, and existing ESA tests. The paper argues that basis-coefficient methods are well suited to smooth near-spherical shapes but can miss localized asymmetries; Euclidean and Procrustes approaches generally require landmark correspondences and are sensitive to parameterization; and prior ESA-based tests were chiefly multi-sample procedures rather than one-sample tests against a specified shape. bootESA fills that one-sample inferential gap by working directly with contour-derived shapes and ESD (Glenn et al., 3 Jun 2026).

The simulation evidence reported for bootESA is consistent with its design. As sample size $N$19 increases, both empirical and bootstrap ESD distributions concentrate near zero. With $N$20, type I error is often below the nominal level and sometimes effectively zero; with $N$21, type I error is closer to the target $N$22 for many percentiles. Power is high for alternatives such as indented circles, ellipses with varying aspect ratio, and polygons with few sides, and it increases with both sample size and effect size. In two ICF experiments with $N$23 pinholes, some percentile contours were significantly different from circles while others were not; the second shot showed more contours, especially in middle and high percentile ranges, that were not significantly different from circles (Glenn et al., 3 Jun 2026).

A common misconception would be to treat “contour bootstrapping” as a single transferable recipe. Current usage indicates otherwise. In matrix perturbation theory, the contour is a complex-analytic integration path and the bootstrap is the absorption of a self-term in a resolvent inequality. In ESA, the contour is an image-derived planar boundary and the bootstrap is literal resampling for inference on a non-smooth distance functional. This suggests a family resemblance only at the level of indirect stabilization: both methods replace a difficult quantity by a more tractable surrogate under structural assumptions, but they do so in mathematically unrelated ways.

The principal open directions likewise diverge. The matrix paper points toward fuller treatments of sparsification, privacy mechanisms, structured random perturbations, and possible extensions to non-normal matrices. The ESA paper points toward 3D surfaces, joint inference across multiple percentile contours, symmetry measures beyond circularity, alternative shape metrics, and other bootstrap schemes such as wild bootstrap, residual bootstrap in reconstruction space, or subsampling approaches. Taken together, the two literatures establish “contour bootstrapping” as a technically rich but context-dependent term rather than a single canonical method (Tran et al., 2024, Glenn et al., 3 Jun 2026).