Symmetrized Robust Procrustes

• Symmetrized Robust Procrustes is a convex relaxation framework for robust rigid alignment, achieving a universal √2-approximation of the ℓ1 objective.
• It employs a symmetrized loss penalizing both forward and adjoint residuals, enabling efficient global optimization via second-order cone programming.
• Under the affine DIP condition, SRP guarantees exact recovery even in high-dimensional settings with significant outlier contamination.

Symmetrized Robust Procrustes (SRP) is a convex relaxation framework for robust rigid alignment of point sets that achieves a universal $\sqrt{2}$-approximation of the $\ell_1$-based Robust Procrustes objective and possesses strong recovery guarantees under natural dominance conditions for outlier-corrupted correspondences. It enables efficient, certificate-backed solutions for high-dimensional and contaminated matching tasks, outperforming classical approaches such as Iteratively Reweighted Least Squares (IRLS) and providing a flexible foundation for further convex-regularized extensions in both geometric and linguistic domains (Amir et al., 2022).

1. The $\ell_1$-Robust Procrustes Problem

The robust Procrustes problem addresses the alignment of two ordered point sets $$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$, seeking the rigid transformation $(R, t)$ minimizing a fidelity criterion. While the classical Procrustes problem minimizes the squared error:

$$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$

the robust ($\ell_1$) variant adopts the sum of Euclidean norms for enhanced resistance to outliers:

$$(\mathrm{RP}_1) \qquad \min_{R \in O(d),\, t \in \mathbb{R}^d} E(R, t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2.$$

Replacing the $\ell_2^2$ cost with the $\ell_1$ cost improves robustness by limiting the influence of arbitrarily large misaligned points. In high dimensions or under point mismatches, the quadratic objective can be seriously degraded by outliers, whereas the $\ell_1$0 cost maintains the integrity of dominant inlier correspondences (Amir et al., 2022).

2. Symmetrized Convex Relaxation

The SRP framework relaxes the orthogonality and translational constraints, considering a general affine map $\ell_1$1 and $\ell_1$2, and introduces a symmetrized loss that simultaneously penalizes forward and adjoint residuals. For $\ell_1$3,

$\ell_1$4

with $\ell_1$5 coupling the dual residuals. For $\ell_1$6, the explicit objective is:

$\ell_1$7

which is convex in $\ell_1$8. In the orthogonal-only case (no translation), $\ell_1$9 and $\ell_1$0 penalizes only the symmetrized alignment residuals.

To recover a feasible rigid motion, the SRP pipeline extracts $\ell_1$1 from the convex program, then projects $\ell_1$2 onto the orthogonal group via singular value decomposition (SVD): $\ell_1$3, and re-optimizes the translation $\ell_1$4 in the $\ell_1$5 Procrustes problem. All SRP objectives for $\ell_1$6 can be formulated as second-order-cone programs (SOCPs) and solved to global optimality (Amir et al., 2022).

3. Approximation Guarantee and Exact Recovery

SRP provides a constant-factor guarantee for the robust ($\ell_1$7) Procrustes objective. Denoting $\ell_1$8 the global minimum of $\ell_1$9, any $$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$0 obtained from SRP satisfies:

$$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$1

For the orthogonal-only formulation, the inequality

$$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$2

reflects that projection to the orthogonal group increases the $$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$3 objective by at most a factor $$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$4. This factor is dimension-free, and the key inequality arises from the structure of the SVD and a coordinate-wise estimate over singular value components (Amir et al., 2022).

Exact recovery is guaranteed under an affine dominance–in–projection (DIP) condition: if a subset $$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$5 of inliers (so that $$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$6 for $$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$7) dominates the outliers in every direction, i.e., for all $$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$8,

$$P = [p(1),…,p(n)], Q = [q(1),…,q(n)] \subset \mathbb{R}^d$$9

(and symmetrically for $(R, t)$0), then SRP returns the exact transformation $(R, t)$1 for arbitrary outlier contamination (Amir et al., 2022).

4. Computational Aspects and Algorithmic Implementation

The SRP optimization is executed via a custom majorization-minimization solver for sums of Euclidean norms, substantially improving practical runtime over off-the-shelf convex software. Each iteration requires $(R, t)$2 operations for residual and back-projection calculations, with convergence observed in tens of iterations. The projection to the orthogonal group incurs $(R, t)$3 floating-point operations for the SVD. Subsequently, optimizing translation $(R, t)$4 given $(R, t)$5 reduces to a convex SOCP (or a closed-form weighted median in the pure $(R, t)$6 scenario), with cost $(R, t)$7. Empirical results report that, for $(R, t)$8, $(R, t)$9, SRP runs in milliseconds per instance on standard hardware, matching a single IRLS iteration (Amir et al., 2022).

5. Comparison with Iteratively Reweighted Least Squares

The following table summarizes the salient distinctions between SRP and IRLS for Robust Procrustes:

Criterion	SRP$$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$0	IRLS
Objective Quality	Lower bound cert. $$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$1 approx.	No guarantee
Initialization	Effective seed for IRLS	Variable
Robustness to Dimension	$$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$2, dimension-free	n/a
Outlier Tolerance	Matches IRLS; provable recovery	Empirical only
Extra Convex Regularization	Easily supported	Limited

SRP$$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$3 consistently produces lower bound certificates on the true $$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$4-energy and remains competitive in recovery performance—oftentimes matching or slightly surpassed by local IRLS refinements. Seeding IRLS with SRP$$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$5-obtained solutions improves final objective values compared to random or SVD-based initialization. Against RANSAC-type algorithms (with known exponential-in-$$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$6 approximation factors), SRP$$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$7 is superior for $$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$8 up to 100 in both objective and recovery, while maintaining dimension-free guarantees. In synthetic tests with $$\min_{R \in O(d),\, t \in \mathbb{R}^d} E_2(R,t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2^2,$$9 inliers and up to $\ell_1$0 outliers with $\ell_1$1 noise in $\ell_1$2 and $\ell_1$3, SRP$\ell_1$4 exhibits robust outlier resistance and exact recovery upon DIP satisfaction (Amir et al., 2022).

6. Extensions and Applications

Two principal SRP extensions have been demonstrated:

• Semi-supervised Procrustes with Covariance Penalty: When $\ell_1$5, incorporating a convex penalty enforcing covariance commutation

$\ell_1$6

regularizes mapping structure, and empirically improves both non-rigid shape alignment and functional-map correspondence accuracy in underdetermined and high-dimensional scenarios (Amir et al., 2022).

• Interlingual Word Translation: For multilingual Word2Vec embeddings ($\ell_1$7), and a small dictionary ($\ell_1$8 to $\ell_1$9), SRP$$(\mathrm{RP}_1) \qquad \min_{R \in O(d),\, t \in \mathbb{R}^d} E(R, t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2.$$0 with covariance penalty yields up to $$(\mathrm{RP}_1) \qquad \min_{R \in O(d),\, t \in \mathbb{R}^d} E(R, t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2.$$1 absolute accuracy improvement (CSLS measure) over vanilla Procrustes or IRLS when $$(\mathrm{RP}_1) \qquad \min_{R \in O(d),\, t \in \mathbb{R}^d} E(R, t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2.$$2. This application addresses extreme data scarcity and high-dimensional noisy alignment tasks (Amir et al., 2022).

SRP’s convex nature also facilitates the incorporation of other penalties (e.g., sparsity, Laplacians) in settings where IRLS and heuristic approaches are limited.

7. Summary and Broad Significance

Symmetrized Robust Procrustes delivers a convex “relax-and-project” strategy for the robust ($$(\mathrm{RP}_1) \qquad \min_{R \in O(d),\, t \in \mathbb{R}^d} E(R, t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2.$$3) Procrustes problem, offering:

• Dimension-free $$(\mathrm{RP}_1) \qquad \min_{R \in O(d),\, t \in \mathbb{R}^d} E(R, t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2.$$4-approximation for arbitrary outlier levels;
• Exact recovery under the affine DIP condition, regardless of outlier count;
• Objective lower bounds enabling solution certification and global branch-and-bound design;
• Flexibility for adding convex penalties crucial for high-dimensional and underdetermined matching;
• Competitive computational efficiency, suitable for moderately large $$(\mathrm{RP}_1) \qquad \min_{R \in O(d),\, t \in \mathbb{R}^d} E(R, t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2.$$5 and $$(\mathrm{RP}_1) \qquad \min_{R \in O(d),\, t \in \mathbb{R}^d} E(R, t) = \sum_{i=1}^n \|Rp(i) + t - q(i)\|_2.$$6.

These features position SRP as a rigorous, practical, and extensible foundation for point set alignment, robust shape analysis, and structured embedding transfer (Amir et al., 2022).