Trust-Region Optimization for Smooth Potential-Interaction Energies in Wasserstein Space

• Nonconvex potential–interaction energies call for methods that use second-order information while controlling local-model reliability.Source: §1 · p. 2
• Applications include Gaussian-core soft-particle models and MMD approximation, where half the squared discrepancy equals the energy up to a target-dependent constant under stated kernel conditions [6].Source: §1 · p. 2
• The trust-region method uses quadratic pushforward models and an L2(ρ) radius for finite-second-moment measures, while its analysis controls changing Hilbert spaces and atomic iterates uniformly.Source: §1 · pp. 2–3
• Under the stated assumptions, Wasserstein-gradient norms converge to zero and total outer trials, including rejected trials, are bounded by O(ε^-2).Source: §1 · p. 3
• For empirical measures, complexity constants are independent of particle number under uniform initial-gap bounds and fixed algorithmic parameters; experiments cover soft-particle and distributional-approximation problems.Source: §1 · p. 3
• The guarantees concern first-order stationarity, not convergence to a local or global minimizer or to a single measure limit, and assume exact evaluations.Source: §1 · p. 3

2.1 Problem

• For the potential–interaction energy, the Wasserstein gradient is ∇V + ∇W ∗ρ, which supplies a stationarity criterion and second-variation curvature information.Source: §2.1 · p. 3

2.2 Preliminaries

• Pushforward updates preserve finite second moments, bound Wasserstein displacement by the L2(ρ) step norm, and match the model's linear term to the first variation.Source: §2.2 · p. 4
• The ambient L2(ρ) second-variation operator is self-adjoint, and its projection onto the tangent space gives the intrinsic Hessian on tangent directions.Source: §2.2 · p. 4

2.3 Wasserstein trust-region algorithm

• Each iteration builds a quadratic model, computes a feasible Steihaug–Toint step with at least Cauchy decrease, accepts by the actual-to-predicted decrease ratio, and updates the radius.Source: §2.3 · p. 5

3.1 Assumptions and preparatory estimates

• Under the smoothness and lower-bound assumptions, uniformly bounded Hessians control the second variation, model operator, and quadratic-model error for every finite-second-moment measure.Source: §3.1 · p. 6

3.2 Complexity in the total number of outer trust-region trials

• The objective is nonincreasing, and an ε-stationary iterate is reached within O(ε^-2) outer trials, counting accepted and rejected trials.Source: §3.2 · pp. 7–8

3.3 Convergence of the gradient norms

• Wasserstein-gradient norms converge to zero; with quadratic coercivity, every weak accumulation point is stationary.Source: §3.3 · pp. 9–10

3.4 A tangent-space variant

• Projecting each model-operator application onto the tangent space preserves convergence and complexity guarantees with the same constants; from finite-support starts, tangent and ambient algorithms coincide.Source: §3.4 · pp. 10–11

4.1 The empirical reduction

• Empirical updates are finite-dimensional in L2(ρN), with N-independent trial bounds under uniform initial-gap bounds and fixed parameters; per-trial all-pairs work can still scale as O(N^2d).For generic dense pairwise Hessians, model-operator applications can cost O(N^2d^2), or O(JN^2d^2) for J truncated-CG applications.Source: §4.1 · p. 12

4.2 A Gaussian-core soft-particle model for fourth-generation dendrimers in an isotropic harmonic trap

• Trust-region optimization reduced median energy from 0.4743 to 0.1696 in 20 trials, versus fixed-step Wasserstein gradient descent’s terminal median energy of 0.1921 after 120 trials.Across eight seeds, trust-region runs used a median of 14 accepted trials and had a 19–24 outer-trial range.Source: §4.2 · p. 13, Figure 1 · p. 13

4.3 Maximum-mean-discrepancy fitting and component ablations

• Full method median final holdout losses were 1.419 × 10^-3 (banana), 1.674 × 10^-3 (two-moons), and 8.349 × 10^-4 (ring), 4.1–5.8% below Cauchy-only; all 24 Cauchy-only runs hit the cap.Disabling radius growth changed final medians by at most 0.09%, while increasing median trial counts by 1–6.Source: §4.3 · p. 14, Figure 2 · p. 15

4.4 Scaling study

• Across particle-count sweeps, median total trials ranged from 13 at N = 10 to 59 at N = 10^7; across dimension sweeps, they ranged from 5 at d = 1 to 92 at d = 10^4.Each of the 21 grid points per sweep used twelve seeds; Figure 3 shows medians and interquartile ranges.Source: §4.4 · p. 16, Figure 3 · p. 16

4.5 Full Gaussian-interaction runtime comparison

• The runtime comparison holds the full objective and gradient fixed, adds model-operator work for the trust-region method, and times each method from setup through stopping checks.Separate objective, gradient, and model-action counts are workload diagnostics and do not enter the time-to-tolerance summaries.Source: §4.5 · p. 17, Table 1 · p. 18

5 Conclusions and future directions

• The method uses quadratic models along pushforward curves and applies to every probability measure with finite second moment, including empirical measures.Under the stated assumptions, the objective is nonincreasing, Wasserstein-gradient norms converge to zero, and an ε-stationary iterate is reached in O(ε^-2) outer trials, including rejected trials.Source: §5 · p. 18
• With quadratic coercivity of the potential, every weak accumulation point is stationary; empirical-measure complexity constants are particle-number independent under uniform initial-gap bounds and fixed parameters.Source: §5 · p. 19
• Experiments evaluate a reduced-unit Gaussian-core soft-particle model and MMD minimization for non-Gaussian targets, including component ablations and scaling studies.Source: §5 · p. 19
• Open questions include curvature-dependent local rates, finite-particle approximation error, inexact or stochastic evaluations with fast interaction summation, and the joint effects of kernel choice, target-sample size, and optimization accuracy.Source: §5 · p. 19

Appendix A Continuity of interaction fields under weak convergence

• For weakly convergent measures with uniformly bounded second moments, convolution with a globally Lipschitz kernel converges locally uniformly.The uniform second-moment bound controls the kernel’s linear growth, while lower semicontinuity supplies the corresponding bound for the weak limit.Source: Appendix A · pp. 19–21
• The proof splits the convolution into a bounded-region part and tails, with the tails vanishing uniformly over compact sets.A finite net controls the bounded-region convergence; the truncation and tail estimates then give local uniform convergence.Source: Appendix A · p. 20
• Applying the continuity result with K = ∇W gives local uniform convergence of the interaction fields.Source: Appendix A · p. 21

Appendix B The Steihaug–Toint truncated-CG subsolver

• The subsolver treats a Hilbert-space quadratic model with a bounded self-adjoint operator that need not be positive semidefinite, using operator products as its oracle.For the Wasserstein application, the space is TρP2 and the operator is Hρ.Source: Appendix B · pp. 21–22
• Under exact arithmetic, the returned Steihaug–Toint step is feasible and achieves a model decrease bounded below by a constant fraction of the Cauchy decrease.The proposition assumes a bounded self-adjoint model operator and at least one inner iteration.Source: Appendix B · p. 22
• The truncated-CG iterates do not increase the model beyond the Cauchy point, including when the method exits on negative curvature or at the trust-region boundary.Source: Appendix B · pp. 22–23
• For empirical measures, the implementation uses particle coordinates and the inherited inner product; the Gaussian-core, MMD, and scaling runs use different inner-iteration limits and tolerances.The Cauchy-only ablation sets B = 0 and bypasses conjugate gradients.Source: Appendix B · p. 23