Geometric Framework for Optimal Experiment Design

• The topic is a geometric framework for experimental design, representing designs as probability measures, convex polytopes, and point configurations with structured information matrices.
• It links traditional criteria like D-, A-, and E-optimality with spectral and geometric insights, unifying classical design theory with Bayesian and decision-theoretic methods.
• It advances design selection through convex relaxations, greedy and exchange algorithms, and specialized optimization techniques applicable in sensor placement and environmental monitoring.

A geometric framework for optimal experimental design (OED) formalizes the selection of experiments as an intrinsically geometric problem, wherein designs are treated as mathematical objects—probability measures, convex polytopes, or point configurations—in spaces endowed with structure derived from statistical, algebraic, or analytic considerations. This perspective connects classical design theory, modern Bayesian and decision-theoretic OED, and computational advances through a common language emphasizing spectra, symmetry, and geometry.

1. Designs as Measures, Spectra, and Polytopes

In geometric OED, a design is represented as a probability measure ξ over a compact design space $\mathcal{X} \subseteq \mathbb{R}^d$, or equivalently, as a weight vector over candidate points or combinations thereof (Huan et al., 23 Jul 2024). For linear models with uncorrelated homoscedastic noise, the (Fisher) information matrix induced by ξ is

$$F(\xi) = \int_{\mathcal{X}} f(x) f(x)^\top d\xi(x),$$

where $f(x) \in \mathbb{R}^p$ is the regression basis at $x$. The set of all possible ξ forms a convex set; Carathéodory’s theorem ensures there exists an optimal design supported on at most $p(p+1)/2$ points for classical (linear, Gaussian) criteria.

Recent work has demonstrated that, for any fixed information matrix $M_*$, the set of all ξ attaining $M_*$ is a convex polytope in weight space. The vertices of this polytope—vertex optimal designs (VODs)—have minimal support and crucial combinatorial significance. Every other optimal design is a convex combination of VODs, endowing the class of optimal designs with rich structure and enabling secondary optimization (e.g., for cost, robustness, or sparsity) (Harman et al., 2023).

2. Spectral and Geometric Design Criteria

Optimality criteria have geometric interpretations in terms of the spectrum (eigenvalues) of the information matrix:

• D-optimality: $\max \log\det(F(\xi))$—minimizes the volume of the confidence ellipsoid for the parameter vector (Huan et al., 23 Jul 2024, Castro et al., 2017, Henrion et al., 6 Sep 2024).
• A-optimality: $\min \operatorname{trace}(F(\xi)^{-1})$—minimizes the average parameter variance.
• E-optimality: $\max \lambda_{\min}(F(\xi))$—minimizes the maximal variance direction.
• Elementary symmetric polynomials (ESPs): Interpolates between trace and determinant, allowing for partial-volume or spectrum-sensitive design objectives (Mariet et al., 2017).

The geometry of these criteria is prominent: maximizing determinant corresponds to maximizing total “spread” (volume) in parameter space, while trace and spectral variants control partial axes. The Christoffel function for polynomial regression and the dual polynomial arising in SDP relaxations localize the support of optimal designs to points where certain polynomials vanish, giving geometric certificates for optimality (Castro et al., 2017).

3. Geometric Allocation and Chebyshev Systems

For inverse quadratic regression and other classes of nonlinear problems, the gradient vectors of the regression function may form a Chebyshev system on the design region. In such cases, a geometric allocation rule selects support points at the Chebyshev (equioscillating) points, often with weights computable through analytic formulas involving Lagrange multipliers or the structure of the regression (0809.4938). When the experiment region is sufficiently broad, these points form a geometric progression linked to system symmetry; explicit formulas

$$u_0 = \frac{1}{\rho} \frac{\theta_0}{\theta_2}, \, u_1 = \frac{\theta_0}{\theta_2}, \, u_2 = \rho \frac{\theta_0}{\theta_2}$$

describe support for $c$-, $D$-, and $E$-optimal designs.

Chebyshev polynomial systems guarantee unique extremal points for variance functions and support the unification of geometric approaches to multiple optimality criteria. This supports analytic design selection in applications such as agriculture and chemistry and underpins shortening of confidence intervals compared to uniform or ad hoc designs.

4. Geometry in Bayesian and Decision-Theoretic OED

Modern OED frequently adopts a Bayesian or decision-theoretic viewpoint, treating parameters as random, observations as uncertain, and the utility of a design as the expected information gain or reduction in decision-theoretic loss (Huan et al., 23 Jul 2024). The mathematical object of central interest is often mutual information,

$$U(\xi) = \mathbb{E}_{Y, \theta | \xi}[\log \frac{p(\theta | Y, \xi)}{p(\theta)}] = I(Y; \theta \mid \xi),$$

which is equivalent to the expected decrease in “uncertainty volume” (the determinant of the posterior covariance) in linear–Gaussian settings. Here, geometry appears both explicitly, via ellipsoid volume, and spectrally, via the generalized eigenproblem for Fisher information. In high dimensions, optimal design leverages the low-rank structure of likelihood-informed subspaces for computational tractability (Wu et al., 2020, Alexanderian et al., 2017).

For nonlinear, non-Gaussian, or implicit models, geometric approaches extend to expected utility estimation via transport maps, normalizing flows, or variational bounds. When physical cost or practical constraints enter, as in environmental or sensor placement applications, geometric optimization is conducted either over the space of weighted designs (measures), via sequential design policies (MDPs), or with explicit polyhedral constraints (costs, resource limits) (Huan et al., 23 Jul 2024, Kuriki et al., 2017, Aquino-López et al., 10 Jul 2025).

5. Optimization Techniques and Geometric Algorithms

Optimizing geometric OED criteria requires specialized algorithms:

• Convex relaxations and semidefinite programming (SDP): The moment-SOS hierarchy and interior-point SDP allow the recovery of optimal design measures from convex relaxations. Duality theory and the Christoffel-Darboux polynomial localize the support points (Castro et al., 2017).
• Greedy and exchange algorithms: Fedorov exchange, swapping greedy, and volume-sampling–inspired procedures are widely used for D-, A-, and ESP-optimality, with theoretical guarantees linked to convex geometry and submodularity (Mariet et al., 2017, Allen-Zhu et al., 2017, Wu et al., 2020).
• Branch-and-bound and projected Newton methods: Recent advances in exact design exploit second-order self-concordant algorithms and vertex exchange strategies within a branch-and-bound tree, accelerating convergence to integer-valued optimal designs (Liang et al., 27 Sep 2024, Ahipasaoglu et al., 3 Jul 2025).
• Grid exploration and neighborhood search: Grid and star-set based algorithms exploit the geometry of the design space (grid structure, neighborhoods) to efficiently locate promising regions for high-dimensional multifactor designs (Harman et al., 2021).

For continuous design spaces, frameworks based on Wasserstein gradient flow on the space of probability measures (equipped with the $W_2$ metric) are used to derive optimal designs as stationary points in measure space, integrating geometric measure theory with variational OED objectives (Jin et al., 15 Jan 2024).

6. Beyond Shannon Information: Transport-Based and Geometric-Dependence Criteria

Classical OED maximizes information-theoretic quantities (e.g., mutual information), which are invariant under injective transformations and indifferent to underlying loss metrics. Recent advances propose criteria grounded in optimal transport theory. The mutual transport dependence (MTD) replaces the Kullback–Leibler divergence with an optimal transport cost: $T(d) = OT_c[p(\theta, y | d), p(\theta)p(y|d)]$ for a cost function $c(\cdot)$ that reflects a geometric error metric (e.g., squared Euclidean distance on outcome–parameter pairs). By adjusting the cost or applying transformations to the spaces, OED can be tailored directly to the performance criteria of downstream tasks, breaking the inflexibility of MI-based metrics. Sample-based computation of MTD, coupled with differentiability through reparameterization, enables scalable optimization in high-dimensional or implicit models (Kerrigan et al., 16 Oct 2025). This approach extends geometric OED to settings where custom loss, region-specific weighting, or other task-aligned geometry is central.

Alternative geometric criteria have also been introduced for data-consistent inversion (DCI) via singular value decompositions of the experimental Jacobian: the expected scaling effect (ESE) and expected skewness effect (ESK). These directly quantify “inverse volume” and “shape” of parameter pre-images, yielding closed-form surrogates for design utility and supporting computationally efficient OED without repeated inversion (Butler et al., 13 Jun 2025).

7. Applications, Impact, and Limitations

The geometric framework for OED has had significant impact across regression, signal recovery, sensor placement, environmental monitoring, and the design of materials and metamaterials (0809.4938, Zianni, 15 Aug 2025). In polynomial regression, designs constructed via equilibrium measures (from pluripotential theory) admit explicit characterization in terms of moment sequences, and atomic approximations converge to the equilibrium measure as design complexity increases (Henrion et al., 6 Sep 2024). In fields such as microplastics monitoring, formally Bayesian (with variance-based loss) and geometric criteria guide resource-constrained sampling and analysis decisions (Aquino-López et al., 10 Jul 2025).

Limitations arise in settings with nonconvex, nonlinear, or implicit models where information matrices or expected utilities cannot be computed tractably. In such cases, variational estimators, transport-based surrogates, or alternative geometric proxies may be required, but assurances of optimality or interpretability may be weaker. Polytopal structure and VODs may not extend straightforwardly to non-linear, nonparametric, or sequential OED.

Nonetheless, the geometric approach provides an integrated and universal perspective, fusing measure-theoretic, spectral, and algebraic tools with computational optimization, and enabling systematic tailoring of design objectives to both theoretical metrics and practical estimation goals. The framework’s flexibility, especially in enabling alignment of statistical design with physical constraints, cost, and downstream error metrics, continues to drive broad development in OED (Huan et al., 23 Jul 2024, Harman et al., 2023, Kerrigan et al., 16 Oct 2025).