Geometry, Conditioning, and Parity in One-Sided Directional Curvature Recovery, with a Controlled Comparison of Estimators

Technical Summary: Geometry, Conditioning, and Parity in One-Sided Directional Curvature Recovery

Problem Statement
The paper addresses the fundamental limits of recovering a Hessian matrix (second-order derivative information) from function values sampled along rays emanating from a single point. This scenario arises in derivative-free optimization, finite-difference approximations near domain boundaries, and inverse problems where the complement of a domain is inaccessible. In these settings, the admissible directions are constrained by geometry (e.g., active inequality constraints, thin structures, or boundaries), preventing the formation of symmetric stencils. The central question is not how to design a specific estimator, but rather what is theoretically possible given these geometric constraints: which quantities are identifiable, how conditioning degrades with restricted access, and what order of accuracy is attainable.

Methodology and Framework
The study adopts a geometric and analytical approach rather than proposing a new numerical estimator. The core methodology involves:

1. Measurement Model: Using a one-sided directional probe defined by three points on a ray: the center $x$, an interior node $x + R h u$, and an endpoint $x + h u$. The probe forms a second divided difference to estimate the directional curvature $u^T H u$.
2. Geometric Analysis: Analyzing the identifiability of the Hessian based on the distribution of sampled directions. The author utilizes the space of symmetric $3 \times 3$ matrices ($\text{Sym}_3(\mathbb{R})$) and spherical harmonics to decompose the measurement operator.
3. Parity Decomposition: Exploiting the parity (even/odd symmetry) of homogeneous polynomials on the sphere. The probe expansion is split into even parts (carrying the Hessian) and odd parts (carrying the leading truncation error).
4. Controlled Comparison: A rigorous benchmarking protocol comparing eight estimators (including quadratic regression, moving least squares, and various directional schemes) under a "fairness protocol." Every method is granted its own oracle-optimal sampling radius, and all share identical noise realizations and feasibility constraints.

Key Contributions and Results

1. Identifiability Bounds (Theorem 4, Proposition 5):

   • Evaluation Floor: In $\mathbb{R}^3$, without smoothness priors, at least 10 point evaluations in general position are necessary and sufficient to uniquely determine a Hessian.
   • Directional Uniqueness: For directional measurements, the condition for uniqueness is not merely that directions span $\mathbb{R}^3$, but that the directions do not lie on a common quadric cone.
   • Cost Implication: A directional scheme using $m$ directions requires $1 + 2m$ evaluations. Since $m \ge 6$ is required for uniqueness, the minimum cost is 13 evaluations, exceeding the 10-evaluation floor for general point sets.

2. Conditioning under Restricted Access (Theorem 10):

   • When admissible directions are confined to a spherical cap of half-angle $\theta$, the recovery operator's singular values scale as $\{1, \theta, \theta, \theta^2, \theta^2, \theta^2\}$.
   • Consequently, the condition number grows as $\kappa(\theta) \asymp \theta^{-2}$. The paper derives exact constants for this growth (e.g., $\kappa \theta^2 \to \sqrt{24} \approx 4.899$ for surface-uniform sampling).
   • Interpretation: Narrow cones measure normal curvature well but mixed and tangential curvatures poorly (at orders $\theta$ and $\theta^2$ respectively). A smooth boundary (hemisphere, $\theta \approx 90^\circ$) is benign; conditioning issues arise only at corners, cusps, or highly constrained active sets.

3. Parity Decomposition and Truncation (Propositions 13, 14, Corollary 15):

   • The one-sided probe expands into even (Hessian-related) and odd (truncation-related) spherical harmonic components.
   • Hessian-Free Measurement: The odd projection isolates the third-order directional derivative ($D^3 f$) independent of the Hessian, accurate to $O(h^2)$. This allows for the measurement of the truncation term without prior knowledge of the Hessian.
   • Attainable Order (Proposition 17): Second-order accuracy in the aggregated Hessian estimate is attainable if and only if the feasible direction set is antipodally symmetric ($u \in U \implies -u \in U$). If the set is genuinely one-sided (no antipodal pairs), the error remains $O(h)$ regardless of the number of directions or weighting schemes.

4. Noise and Pilot Design (Propositions 19, 21, 22):

   • The paper quantifies the variance of the probe and the resulting lower bound on the usable step size (span). The optimal span scales as $\sigma^{1/3}$, consistent with standard finite difference theory.
   • Extracting the truncation constant via the odd parity projection requires a "pilot" phase. The paper notes this pilot is expensive, requiring at least 18 directions to be stable, which may exceed the cost of the Hessian estimate itself.

5. Controlled Comparison of Estimators:

   • Under the fairness protocol, directional schemes did not outperform standard quadratic regression or $\Lambda$-poised interpolation on the same feasible points.
   • Where symmetric stencils are feasible, central differences are most accurate. Where they are not, quadratic regression and moving least squares match or exceed directional schemes.
   • The failure of directional schemes to gain an advantage is attributed to the fact that they cannot overcome the $O(h)$ truncation error inherent to one-sided access (Proposition 17) and suffer from the same geometric conditioning limits as other methods.

Significance and Claims
The paper explicitly frames its contributions as structural bounds on the measurement geometry rather than improvements to specific algorithms.

• Constraints on Optimality: The results define the "impossible" region: no estimator can achieve second-order accuracy without antipodal symmetry, and no estimator can bypass the $\theta^{-2}$ conditioning penalty of narrow cones.
• Unexploited Structure: The identification of the odd spherical-harmonic component as a Hessian-free measure of the truncation term is highlighted as a structural insight, though the paper cautions that extracting it is computationally expensive.
• Methodological Rigor: A significant portion of the paper is dedicated to documenting nine specific experimental artifacts (e.g., parameter propagation errors, geometry generation bugs, and flawed fairness assumptions) that had to be removed to produce trustworthy results. The author argues that the negative result—that directional schemes offer no advantage over regression under fair conditions—is more valuable than a potentially spurious positive result.

Modesty and Limitations
The author is careful to distinguish between proven theorems and numerical observations. They do not claim that one-sided directional sampling is superior near boundaries; in fact, their results and the comparison suggest it is not. They acknowledge that their parity decomposition results may be instances of existing Slepian theory or restricted Veronese embedding literature, framing these as "open positioning questions" rather than novel discoveries. The paper concludes that while the geometry of one-sided recovery is well-characterized, the practical utility of directional schemes is limited by the fundamental trade-offs between geometry, truncation, and noise.