Projection-Onto-Linear-Variety (PLV) Algorithm

• The PLV algorithm is a rigorously defined method that computes optimal projection operators by minimizing integral error functionals in linear systems.
• It leverages spectral information and convex geometry to yield explicit closed-form projectors and minimum-energy solutions for both model reduction and signal recovery.
• Its construction via spectral decomposition and Gramian inversion underpins advanced applications in reducing transient dynamics and calibrating continuous angular power spectra.

The Projection-Onto-Linear-Variety (PLV) algorithm constitutes a rigorously defined, uniquely optimal projection method in both finite- and infinite-dimensional settings. It arises from a variational principle minimizing a natural integral error functional: either the cumulative in-time squared error between full and reduced dynamics for stable linear systems, or minimum-norm spectrum estimation subject to covariance constraints for signal recovery. PLV leverages spectral information and convex geometry to yield explicit closed-form projectors and minimum-energy solutions. In recent research, it underpins advanced spectral model reduction (Dynamically Optimal Projection, DOP) and exact recovery of continuous angular power spectra by affine projection in weighted Hilbert spaces (Kogelbauer et al., 23 Mar 2025, Luo et al., 29 Dec 2025).

1. Mathematical Foundations and Variational Principle

PLV is rooted in the minimization of integral error functionals associated with constrained linear estimation. For stable linear ODEs, consider: $$\dot{x} = A x, \quad x \in \mathbb{R}^N \text{ or } \mathbb{C}^N,$$ and select a slow subspace $$\mathcal{V}_{\rm slow} = \mathrm{span}\{\hat{v}_1, ..., \hat{v}_n\}$$ comprising $n$ linearly independent eigenvectors with eigenvalues satisfying $$\Re \lambda_1 \geq \cdots \geq \Re \lambda_n > \max_{k > n} \Re \lambda_k$$. The task is to find an initial condition on $\mathcal{V}_{\rm slow}$, $x_{\rm slow}(\xi) = \sum_{j=1}^n \xi_j \hat{v}_j$, minimizing the cumulative squared error: $$J(x_0, \xi) = \frac12 \int_0^\infty \|e^{tA} x_0 - e^{tA} x_{\rm slow}(\xi)\|^2 \, dt,$$ which is strictly convex in $\xi$ if $A$ is Hurwitz $(\Re\,\text{spec}\,A < 0)$. The minimizer $\xi^*(x_0)$ parameterizes the dynamically optimal projection of $x_0$ onto the slow manifold (Kogelbauer et al., 23 Mar 2025).

In signal recovery contexts (continuous angular power spectrum estimation): $$R = \int_{- \pi / 2}^{\pi / 2} \rho(\theta) a(\theta) a(\theta)^\mathrm{H} d\theta,$$ where $R$ is the measured covariance and $\rho(\theta)$ is the unknown spectrum, PLV selects among all spectra reproducing $R$ the one with minimum $L^2$-norm. This is realized as the orthogonal projection of the zero function onto the affine variety defined by covariance constraints (Luo et al., 29 Dec 2025).

2. Explicit Construction: Spectrally Weighted Gramian and Projector

For linear systems, the optimal projector $P^*$ is constructed via spectral decomposition:

• Compute the $n \times n$ spectrally weighted Gramian:

$$G_{ij} = \frac{\langle \hat{v}_i, \hat{v}_j \rangle}{\lambda_i + \overline{\lambda}_j}, \quad 1 \leq i,j \leq n,$$

where $\langle \cdot, \cdot \rangle$ denotes inner product, and for $\Re \lambda_i < 0$, $G$ is negative-definite.

• The interaction vector is:

$$I_j(x_0) = \left\langle (A + \overline{\lambda}_j I)^{-1} x_0, \hat{v}_j \right\rangle.$$

• Solve $G^T \xi = I(x_0)$ for the optimal $\xi$.
• The projector $P^*$ is then:

$$P^* x = \sum_{i=1}^n \sum_{j=1}^n \hat{v}_i \left[(G^T)^{-1}\right]_{ij} \left\langle (A + \overline{\lambda}_j I)^{-1} x, \hat{v}_j \right\rangle.$$

$P^*$ is idempotent ($P^{*2} = P^*$) and projects onto $\mathcal{V}_{\rm slow}$; for normal $A$, this reduces to the standard orthogonal projector (Kogelbauer et al., 23 Mar 2025).

3. Algorithmic Procedure and Computational Complexity

The implementation involves the following steps:

1. Spectral Data: Compute or receive the slow eigenpairs $(\lambda_j, \hat{v}_j)$.
2. Gramian Assembly: Form $G_{ij}$ using eigenvectors and eigenvalues.
3. Matrix Inversion: Invert $G^T$ (complexity $O(n^3)$).
4. Projector Construction: Use LU factorization or spectral formulas to compute $(A + \overline{\lambda}_j I)^{-1}$, then construct $P^*$ from rank-$n$ representation.
5. Application: Apply $P^*$ either as a full matrix (moderate $N$) or as a rank-$n$ operator (large $N$).

Complexity breakdown:

• Eigen-decomposition: $O(N^3)$ or $O(N^2 n)$ (iterative)
• Gramian assembly: $O(n^2 N)$
• Matrix inversion: $O(n^3)$
• Factorization/solves: $O(n N^3)$ or $O(n N^2)$ (iterative)
• Projector application: $O(nN + n\,\mathrm{cost\_solve})$ (Kogelbauer et al., 23 Mar 2025).

4. Infinite-Dimensional Affine Projection in Weighted Fourier Domains

In continuous APS recovery, the PLV problem is posed as projection onto an affine subspace $\mathcal{V}_w$ of weighted $L^2$-functions, with covariance constraints expressed as weighted Fourier moments: $$r_m = \left\langle g,\, e^{i\kappa_m x} \right\rangle_w, \quad m = 0, \dots, M-1,$$ where $g(x) = \rho(\arcsin x)$ and $w(x) = 1 / \sqrt{1 - x^2}$. $\mathcal{V}_w$ is an affine flat of infinite co-dimension, whose direction $\mathcal{N}$ consists of functions orthogonal to all measured moments.

The orthogonal complement $\mathcal{N}^\perp$ is the finite-dimensional trigonometric polynomial space

$$\mathcal{N}^\perp = \mathrm{span} \left\{ 1, \cos(\kappa_m x), \sin(\kappa_m x): m = 1, \dots, M-1 \right\}.$$

Thus, the PLV solution lies in the intersection $\mathcal{V}_w \cap \mathcal{N}^\perp$, representable as

$$g_{\mathrm{plv}}(x) = b_0 + \sum_{m=1}^{M-1} b_m \cos(\kappa_m x) + \sum_{m=1}^{M-1} b_{M-1 + m} \sin(\kappa_m x),$$

with $b$ given by the solution to the real linear system $G b = y$, where $G$ is the Gram matrix of basis functions and $y$ encodes the measured covariance moments. $G \succ 0$ ensures uniqueness, and the closed-form solution is $b = G^{-1} y$ (Luo et al., 29 Dec 2025).

5. Error and Resolution Characteristics

PLV yields an exact Hilbert-space energy identity for APS estimation: $$\|g_{\mathrm{plv}} - g_*\|_w^2 = \|g_*\|_w^2 - y^T G^{-1} y \geq 0,$$ where $g_*$ is the ground-truth spectrum. PLV achieves perfect recovery ($\|g_{\mathrm{plv}} - g_*\|_w^2 = 0$) if and only if $g_*$ lies in the trigonometric polynomial subspace $\mathcal{N}^\perp$. Any component orthogonal to $\mathcal{N}^\perp$ is unrecoverable given finite measurement aperture; thus, the resolution limit is set by the maximal spatial frequency $\kappa_{M-1}$ of the measurement model (Luo et al., 29 Dec 2025). This provides a sharp identifiability characterization.

6. Application to Model Reduction and Signal Processing

PLV has direct application to model reduction in linear system dynamics, where it guarantees optimal trajectory fitting for both transient and asymptotic behavior. For instance, in the three-component Grad moment system, PLV recovers pressure oscillations in decaying dynamics with much higher fidelity compared to orthogonal projection, achieving pointwise error reduction of $5$--$10\times$ for oscillatory modes (Kogelbauer et al., 23 Mar 2025). In continuous APS estimation using uniform linear arrays, PLV reconstructs the spectrum as a minimum-energy trigonometric polynomial consistent with measured covariances and exposes intrinsic resolution limits in spectral recovery (Luo et al., 29 Dec 2025).

7. Assumptions, Limitations, and Geometric Interpretation

Key assumptions include linearity, spectral separation (the slow eigenvalues must be isolated for matrix invertibility), absence of Jordan blocks (though blocks can be handled with Gramian modification), and stability ($\Re\,\text{spec}\,A < 0$ for integral convergence). The requirement for full spectral knowledge of slow modes may be burdensome in very high-dimensional problems (Kogelbauer et al., 23 Mar 2025).

Geometrically, PLV performs minimum-norm affine projection in Hilbert space: for dynamics, it projects the full trajectory onto the slow manifold to best match temporal behavior; for APS estimation, it projects onto the unique finite-dimensional subspace consistent with measurement, quantifying the attainable resolution via convex geometry and spectral bounds.

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References:

• "Dynamically Optimal Projection onto Slow Spectral Manifolds for Linear Systems" (Kogelbauer et al., 23 Mar 2025)
• "Affine-Projection Recovery of Continuous Angular Power Spectrum: Geometry and Resolution" (Luo et al., 29 Dec 2025)