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Noise Stability and Rotational Equivariance of Image-Induced Hamiltonian Spectra for Robust Unsupervised Segmentation

Soumic Sarkar

arXiv:2609.30280v1eess.IV

TL;DR

The paper addresses the lack of theoretical noise-stability and geometric-consistency analyses for image-induced Hamiltonian segmentation. It derives noise bounds for the graph Hamiltonian and analyzes its behavior under planar rigid motions. The reported results include exact equivariance at 0° and 90° in experiments and a trade-off that prevents simultaneous certification of spectral-gap and segmentation-margin conditions by parameter scaling alone.

  • Problem

    The noise robustness of image-induced Hamiltonian segmentation had been supported empirically or qualitatively, with prior rigorous analysis limited to a 1D spectral quantity, and equivariance had not been analyzed for this operator family.

    Source: §I · p. 1

  • Method

    The paper extends perturbation analysis to a discrete graph Hamiltonian, bounds noise-induced eigenvalue and ground-state drift, and characterizes equivariance under planar rigid motions.

    Source: §I · p. 1, §VIII · p. 9

  • Results

    Experiments report exact zero equivariance gap at 0° and 90° and a statistically significant anisotropic-versus-isotropic difference; the analysis also identifies a non-vanishing gap for the axis-restricted construction.

    Source: D. Discussion of Results · p. 8, §VIII · p. 9

  • Takeaways & Limitations

    Scaling potential parameters alone cannot certify both the spectral-gap and segmentation-margin conditions with fixed, non-vanishing margins.

    Source: §I · p. 1, D. Discussion of Results · p. 8

    The noise model fixes graph topology, so sufficiently large noise that perturbs edge weights is outside the analysis; eigenvector and mask guarantees cover only the single ground-state readout.

    Source: B. Limitations and Relation to… · p. 9

Abstract

from arXiv · show

Unsupervised segmentation methods that embed an image as the potential of a Schr\"{o}dinger-type Hamiltonian and read out object structure from its low-lying eigenstates have demonstrated strong empirical performance across natural and volumetric imagery, yet the robustness of this spectral representation to sensor noise and geometric transformation has never been established theoretically. This paper closes that gap. A discrete graph-Hamiltonian formulation is adopted, consistent with existing quantum-inspired segmentation frameworks, and an explicit finite-sample perturbation bound is derived that links the drift of its eigenvalues and ground-state eigenvector under additive noise of known variance to the operator's spectral gap, extending a one-dimensional noise-error analysis originally developed for semiclassical signal analysis to the two-dimensional graph setting used in image segmentation. The eigenvector-level bound is further propagated to the induced binary segmentation mask, giving, for the first time, a quantitative guarantee on segmentation stability under noise rather than an empirical observation of it. A complementary equivariance analysis characterizes the conditions under which the operator's spectrum transforms exactly under the planar rigid-motion group, and quantifies the equivariance gap introduced by anisotropic potential terms commonly used in practice. Theoretical predictions are validated numerically on natural-image and noise-perturbed benchmarks, showing close agreement between the derived bounds and observed spectral and segmentation-mask deviation, and offering practitioners a principled criterion for selecting operator parameters under known noise conditions.

I. INTRODUCTION

The introduction identifies unresolved questions about noise stability and rotational consistency in image-induced Hamiltonian segmentation, then situates them among quantum-inspired segmentation, semiclassical signal analysis, graph perturbation theory, and equivariant vision models.

A. Quantum-Inspired and Spectral Segmentation

  • Quantum Cuts reads segmentation masks from Hamiltonian eigenstates, while related methods extend this spectral approach to volumetric imagery and crack segmentation.The Quantum Cuts masks were reported as competitive with supervised alternatives without labeled training data.Source: §I · p. 1, A. Quantum-Inspired and Spectral… · pp. 1–2

I. INTRODUCTION

  • The literature lacked a 2D noise-stability bound reaching the segmentation mask and an equivariance analysis for image-induced Hamiltonians, motivating both contributions.The paper addresses these gaps in the discrete graph-Hamiltonian setting and reports numerical validation on real imagery.Source: §I · p. 1

B. Semiclassical Signal and Image Analysis

  • Semiclassical signal analysis reconstructs signals from Schrödinger-operator spectra, but its rigorous noise bound covers only the one-dimensional case.Its 2D extension uses row/column tensor-product decomposition.Source: B. Semiclassical Signal and Image… · p. 2

C. Spectral Graph Methods and Perturbation Theory

  • Existing spectral-clustering stability work studies large-sample consistency, not finite-sample robustness of a fixed-image segmentation mask to pixel noise.Weyl’s inequality and Davis–Kahan are standard tools, but their application to image-induced Hamiltonians had no identified precedent.Source: C. Spectral Graph Methods and… · p. 2

D. Equivariance in Vision Models

  • Learned vision models have established group-equivariant and steerable architectures, while equivariance of the non-learned image-induced Hamiltonian family remained unexamined.The paper asks whether these operators are equivariant and quantifies failure when they are not.Source: D. Equivariance in Vision Models · p. 2

III. PRELIMINARIES

The preliminaries define the image-induced graph Hamiltonian and its ground-state segmentation readout, specify an additive-noise model, and recall the perturbation results used to analyze stability.

A. Discrete Graph-Hamiltonian Formulation

  • The image is represented as a weighted graph, and its Hamiltonian is the graph Laplacian plus a diagonal potential derived from local image statistics.The ground-state eigenvector is squared and thresholded at its mean to produce the fixed segmentation readout; the spectral gap separates the ground state from the next eigenvalue.Source: A. Discrete Graph-Hamiltonian… · p. 2

B. Noise Model

  • The noise model adds i.i.d. pixel noise with mean µ and variance σ² while holding graph topology, bandwidth, and potential construction fixed.This isolates noise to the potential term.Source: B. Noise Model · p. 2

C. Classical Perturbation Results

  • Weyl’s inequality bounds eigenvalue drift by the perturbation norm, while Davis–Kahan bounds ground-state eigenvector drift when the perturbation is small relative to the spectral gap.The paper applies these results to the image-induced operator to obtain eigenvalue-, eigenvector-, and mask-level noise bounds.Source: C. Classical Perturbation Results · p. 3

D. The Planar Rigid-Motion Group

The paper defines spectral equivariance under planar rigid motions and develops a noise-based eigenvalue bound for the graph Hamiltonian, accounting for the nonlinear image-derived potential.

D. The Planar Rigid-Motion Group

  • For planar rigid motions, equivariance means the transformed operator has matching spectrum and eigenvectors transformed by the induced pixel-grid action.The passage points to conditions on the potential for exact equivariance.Source: D. The Planar Rigid-Motion Group · p. 3

A. Eigenvalue Stability

  • With graph topology fixed, noise perturbs the diagonal potential, whose gradient-derived term requires bounding nonlinear changes in local image differences.The bound uses bounded finite-difference operators and a bounded clean-image gradient; its derivation applies a Chebyshev noise tail and Weyl’s inequality. The extension differs from the 1D analysis by treating a 2D graph operator and accounting for the quadratic noise term.Source: A. Eigenvalue Stability · p. 3
  • The eigenvalue perturbation theorem bounds drift for every eigenvalue under the stated assumptions, while Gaussian noise permits a tighter exponential concentration bound at the cost of a distribution-specific assumption.The general bound retains a distribution-free guarantee.Source: A. Eigenvalue Stability · p. 3

B. Eigenvector Stability

The ground-state eigenvector bound makes segmentation-relevant stability depend explicitly on the spectral gap and the noise-induced operator perturbation.

B. Eigenvector Stability

  • Eigenvalue stability alone cannot ensure stable segmentation because eigenvectors near degenerate eigenvalues can rotate substantially under small perturbations.The ground state is the eigenvector used by the Quantum Cuts readout.Source: B. Eigenvector Stability · p. 3
  • When the clean spectral gap exceeds twice the eigenvalue-perturbation bound, the ground-state eigenvector drift is bounded with the same probability as the eigenvalue result.Davis–Kahan makes the bound depend on the perturbation relative to the gap.Source: B. Eigenvector Stability · pp. 3–4
  • The result bounds noise-induced drift of the eigenvector used to produce a segmentation mask, rather than only an intermediate spectral quantity.The passage characterizes this as the first explicit bound of this kind for the Quantum Cuts family.Source: B. Eigenvector Stability · p. 4

C. Propagation to the Segmentation Mask

The section carries ground-state eigenvector stability through the thresholding step to a condition guaranteeing an unchanged binary mask. The guarantee requires a margin separating every clean-image mask value from the threshold.

C. Propagation to the Segmentation Mask

  • The binary mask thresholds each squared ground-state amplitude at the mean of those amplitudes.The continuous quantity y = ψ0 ◦ ψ0 is converted into the segmentation mask by thresholding at ȳ.Source: C. Propagation to the Segmentation Mask · p. 4
  • A margin δ requires every clean squared ground-state amplitude to lie at least δ from the mean threshold.This excludes nodes exactly at the threshold, where an arbitrarily small perturbation could flip a label.Source: C. Propagation to the Segmentation Mask · p. 4
  • Under Assumptions 1–4, if 2∥ψ0∥∞·∥ψ̃0−ψ0∥2 < δ, the clean and noisy segmentation masks are identical on Theorem 2’s event.The sufficient condition can be expressed through the noise variance, spectral gap, and operator constants.Source: C. Propagation to the Segmentation Mask · p. 4
  • If each squared-amplitude perturbation is smaller than δ, no node crosses the threshold, so every mask label is preserved.The proof bounds the per-node change using the ground-state eigenvector drift and its infinity norm.Source: C. Propagation to the Segmentation Mask · p. 4

D. The Ground-State Degeneracy Mechanism and the Gap– Margin Trade-off

The section identifies a structural tension between noise and mask stability as the potential scale shrinks. The spectral gap approaches the graph’s algebraic connectivity, while the ground-state density’s threshold margin vanishes, preventing scale alone from uniformly certifying both conditions.

D. The Ground-State Degeneracy Mechanism and the Gap– Margin Trade-off

  • Reducing potential weights widens the noise-stability range but drives the mask margin toward zero, so scaling alone cannot maintain both conditions with a fixed non-vanishing margin.The paper proposes changing the potential parameterization or relaxing the uniform-margin mask guarantee as possible routes to joint certification.Source: D. The Ground-State Degeneracy… · pp. 4–5
  • The mechanism uses H(I) = L + tΦ, with L the bare graph Laplacian and µFiedler(L) its positive second eigenvalue for a connected graph.The stated connectivity and eigenvalue-separation assumptions support the first-order perturbation analysis.Source: D. The Ground-State Degeneracy… · p. 4
  • As t → 0, the spectral gap approaches µFiedler(L), while the ground-state density margin vanishes linearly in t for pixels with c(i) ≠ 0.The ground state tends to the constant vector, whose associated density has zero margin.Source: D. The Ground-State Degeneracy… · pp. 4–5
  • For fixed noise level, sufficiently small t can satisfy the gap condition, but a fixed target margin requires t bounded away from zero.When the margin’s lower-bound scale exceeds the noise condition’s upper-bound scale, no single t satisfies both.Source: D. The Ground-State Degeneracy… · p. 5

V. EQUIVARIANCE ANALYSIS

The analysis characterizes rigid-motion equivariance for the image-induced Hamiltonian, separating the continuum operator result from discretization effects. In the continuum, the isotropic potential yields exact SE(2) equivariance, invariant eigenvalues, and covariant eigenfunctions.

V. EQUIVARIANCE ANALYSIS

  • The analysis separates potential-construction effects from finite-grid sampling because these are independent sources of departure from exact equivariance.The continuum result is established before the discrete pixel-lattice case is considered.Source: A. Exact Equivariance for Isotropic… · p. 5, §V · p. 5

A. Exact Equivariance for Isotropic Potentials

  • For every planar rigid motion, the continuum Hamiltonian with the isotropic potential is SE(2)-equivariant, so its eigenvalues are invariant and its eigenfunctions transform covariantly.The potential uses gradient magnitude and the Laplacian, whose invariance under rotations and translations gives the operator identity.Source: A. Exact Equivariance for Isotropic… · p. 5
  • Under the continuum isotropic construction, rotating or translating the input transforms the ground state and its segmentation mask by the same rigid motion, with no approximation error.This is exact equivariance by construction, with a mechanism distinct from steerable or circular-harmonic convolutional filters.Source: A. Exact Equivariance for Isotropic… · pp. 5–6

B. Discretization Effects

Discretization breaks exact equivariance of the isotropic continuum operator at general rotation angles. Under a bounded-stencil-error assumption, the discrete operator’s equivariance gap is bounded in proportion to pixel spacing and |sin θ|.

B. Discretization Effects

  • The continuum theorem does not give exact equivariance for the discrete pixel-lattice operator at general rotation angles.Finite-difference approximations of the gradient and Laplacian, together with the square lattice, limit exact rotation symmetry to multiples of 90°.Source: B. Discretization Effects · p. 6
  • The bound vanishes at 0, π/2, π, and 3π/2, where the square pixel lattice is rotation-symmetric, and is largest near π/4.Thus the isotropic construction is exactly lattice-equivariant at those four angles, with an error that vanishes linearly in h otherwise.Source: B. Discretization Effects · p. 6
  • The discrete isotropic Hamiltonian’s equivariance gap is bounded by c3(κ1, κ2, λ1, λ2) h |sin θ| under Assumption 6.The constant depends on the stencil and potential weights, but not on the image or h.Source: B. Discretization Effects · p. 6

C. The Equivariance Gap for Anisotropic Potentials

Axis-aligned nearest-neighbor kinetic terms break rotational consistency beyond the 90°-rotation symmetry of their lattice stencil. Their equivariance gap remains bounded away from zero as pixel spacing shrinks, unlike the discretization gap of the isotropic construction.

C. The Equivariance Gap for Anisotropic Potentials

  • For an axis-aligned four-neighbor Hamiltonian, some rotation and image yield an equivariance gap of at least c4 > 0, independent of pixel spacing h.The result concerns the construction used in.Source: C. The Equivariance Gap for… · p. 6
  • At 45°, the rotated operator couples diagonal neighbors, while recomputing with the same axis-aligned stencil retains only horizontal and vertical couplings.A ramp oriented at 45° makes the difference persist under grid refinement because one operator has nonzero diagonal couplings where the other has none.Source: C. The Equivariance Gap for… · p. 6
  • The axis-aligned stencil is invariant under 90° rotations, but not under arbitrary planar rotations; the isotropic construction’s residual gap vanishes as h → 0.This separates a resolution-limited discretization gap from the non-vanishing anisotropic gap.Source: C. The Equivariance Gap for… · p. 6

VI. EXPERIMENTAL VALIDATION

On 15 randomly selected BSDS500 validation images, the experiments compare operating and certified parameter regimes under Gaussian noise. Lower potential weights reduce spectral drift and sometimes satisfy the spectral-gap condition, but mask stability remains uncertified because the margin condition fails.

VI. EXPERIMENTAL VALIDATION

  • Fifteen randomly selected validation images were grayscale center-cropped to 48 × 48 pixels, and tested in operating and certified regimes with six Gaussian-noise trials per σ.The regimes use λ1 = λ2 = 0.15 and 0.01, respectively; the lower weights were chosen to widen the range satisfying Assumption 3.Source: §VI · p. 7, A. Noise Stability: Empirical… · p. 7

A. Noise Stability: Empirical Verification

B. The Certified Regime and the Gap–Margin Trade-off

  • At λ1 = λ2 = 0.01, eigenvector drift at σ = 0.08 falls from 0.188 to 0.014, yet mask Hamming distance stays near 0.28 and the mask guarantee remains empty.The margin-fragile share at δ = 10^-4 rises from 36.6% in the operating regime to 100% in the certified regime; the spectral-gap condition is satisfied for 27% of images at σ = 0.005, 13% at σ = 0.01, and none beyond.Source: B. The Certified Regime and the… · pp. 7–8, Table II · p. 8

C. Equivariance: Empirical Verification

Across 15 BSDS500 images, equivariance gaps were exactly zero at 0° and 90° for both constructions, while the anisotropic construction had larger gaps away from those angles. The discussion also reports that the margin condition was rarely certified in practice and that scaling potential weights alone cannot preserve both stability conditions.

C. Equivariance: Empirical Verification

  • Anisotropic gaps exceeded isotropic gaps in 76% of 75 non-trivial image-angle pairs (t = 5.50, p ≈10−6), while both gaps were zero at 0°/90° for every image.Away from those angles, the isotropic gap peaked at 0.256 near 45° and the anisotropic gap at 0.371.Source: C. Equivariance: Empirical Verification · p. 8
  • The certified regime’s margin diagnostic reached 100% of pixels below δ = 10−4, consistent with the predicted margin collapse as t →0.The figure reports fractions averaged over 15 images.Source: Figure 5 · p. 8

D. Discussion of Results

  • Potential-weight scaling alone cannot maintain both the spectral-gap and fixed-margin conditions, and the discussion says neither condition is certified with practical frequency.The discussion also reports that Tables I–II’s monotonic trends support the qualitative content of Theorems 1–2 beyond Corollary 1’s sufficient condition.Source: D. Discussion of Results · p. 8

VII. DISCUSSION

The discussion emphasizes that the bounds apply to existing Quantum Cuts operators but that the noise bound can be conservative and parameter choices must balance spectral-gap and margin conditions. The conclusion reports validation on BSDS500 and identifies fixed-topology noise, single-eigenvector mask stability, and limited transformation scope as boundaries.

VII. DISCUSSION

  • The bounds apply without modification to the existing Quantum Cuts operator and direct descendants; the segmentation rule itself is unchanged..Source: §VII · p. 8

A. Bound Looseness, Parameter Selection, and the Trade-off

  • Weyl’s inequality discards perturbation structure beyond its operator norm, making the distribution-free eigenvalue bound loose but conservative.A Bernstein-type concentration argument could tighten it at the cost of the current distribution-free guarantee.Source: A. Bound Looseness, Parameter… · pp. 8–9
  • Parameter selection must balance the spectral-gap condition against the margin condition, since reducing potential weights can widen the certified-noise range while collapsing the margin.The discussion recommends choosing parameters jointly rather than optimizing the spectral gap alone.Source: A. Bound Looseness, Parameter… · p. 9

B. Limitations and Relation to Semiclassical Signal Analysis

  • Noise analysis fixes graph topology, mask-stability bounds cover only the single ground-state eigenvector, and equivariance proofs cover planar rotations and translations rather than affine or projective transformations.A unified stability theory spanning this segmentation objective and semiclassical signal reconstruction remains open.Source: B. Limitations and Relation to… · p. 9

VIII. CONCLUSION

  • On BSDS500, the predicted gap–margin trade-off was observed, exact equivariance at lattice-symmetric angles held without exception, and anisotropic gaps were significantly larger than isotropic gaps.The conclusion also reports that the spectral-gap and margin conditions cannot both be certified with fixed, non-vanishing margins by potential-weight scaling alone.Source: §VIII · p. 9

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