E3: Riemannian Optimization for ETKDG

Module: sci_form::experimental::riemannianFeature flag: experimental-riemannian

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Overview

Replaces the Euclidean BFGS optimizer in the embedding step with Riemannian L-BFGS over the manifold of fixed-rank positive semidefinite (PSD) matrices. Negative eigenvalues are eliminated by design, reducing the retry loop failure rate to zero.

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Theory

PSD Manifold

The metric matrix $G$ must be PSD for valid 3D coordinates. Instead of optimizing in ${\mathbb{R}}^{N\times N}$ and projecting, we optimize directly on:

$$\mathcal{M}=\{X\in {\mathbb{R}}^{N\times N}:X⪰0,\text{rank}(X)=r\}$$

Retraction

The retraction maps a tangent vector back to the manifold:

$${\text{Retr}}_X(\xi )=P_{+}(X+\xi +\frac{1}{2}\xi X^{-1}\xi )$$

where $P_{+}$ projects onto the PSD cone by zeroing negative eigenvalues.

Riemannian Gradient

The Euclidean gradient is projected to the tangent space:

$${\text{grad}}_{R}f=P_{T_X}\mathrm{\nabla }f$$

For the Stiefel manifold: $P_{T_X}(\xi )=\xi -X\cdot \text{sym}(X^T\xi )$

L-BFGS on the Manifold

• Store $m=5$ curvature pairs $(s_k,y_k)$ in the tangent space
• Two-loop recursion with vector transport along geodesics
• Line search with Armijo condition along the manifold geodesic

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API

use sci_form::experimental::riemannian::*;

// PSD projection
let (projected, n_neg) = project_psd(&matrix);

// Retraction
let retracted = retract_psd(&x, &tangent_vector, epsilon);

// Riemannian gradient
let rgrad = riemannian_gradient(&x, &euclidean_grad);

// Full L-BFGS optimization
let config = RiemannianConfig {
    max_iter: 500,
    tol: 1e-6,
    memory: 5,
    ..Default::default()
};
let result = riemannian_lbfgs(&x0, &objective, &gradient, &config);
// result.x, result.iterations, result.converged, result.final_gradient_norm

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Convergence

• Primary: $\parallel {\text{grad}}_{R}f\parallel <{10}^{-6}$
• Secondary: Maximum distance violation $<0.01$ Å
• Fallback: Reverts to Euclidean BFGS + PSD projection if not converged in 500 iterations

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Tests

cargo test --features experimental-riemannian --test regression -- test_riemannian

Covers: PSD projection correctness, retraction stays on manifold, Riemannian gradient orthogonality, L-BFGS convergence on Rosenbrock and distance geometry objectives.