--- title: Symmetrized Robust Procrustes url: https://www.emergentmind.com/topics/symmetrized-robust-procrustes type: topic --- # Symmetrized Robust Procrustes 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 [2207.08592]. ## 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 $\ell_1$ cost maintains the integrity of dominant inlier correspondences [2207.08592]. ## 2. Symmetrized Convex Relaxation The SRP framework relaxes the orthogonality and translational constraints, considering a general affine map $A \in \mathbb{R}^{d \times d}$ and $t \in \mathbb{R}^d$, and introduces a symmetrized loss that simultaneously penalizes forward and adjoint residuals. For $p \in [1, \infty)$, $$ E_p(A, t, s) = \left\{\sum_{i=1}^n \|A p(i) + t - q(i)\|_2^p + \|A^T q(i) - p(i) + s\|_2^p\right\}^{1/p}, $$ with $s \in \mathbb{R}^d$ coupling the dual residuals. For $p=1$, the explicit objective is: $$ E_1(A, t, s) = \sum_{i=1}^n \|A p(i) + t - q(i)\|_2 + \|A^T q(i) - p(i) + s\|_2, $$ which is convex in $(A, t, s)$. In the orthogonal-only case (no translation), $t = s = 0$ and $E_p(A)$ penalizes only the symmetrized alignment residuals. To recover a feasible rigid motion, the SRP pipeline extracts $(A^*, t^*, s^*)$ from the convex program, then projects $A^*$ onto the orthogonal group via singular value decomposition (SVD): $A^* = U \Sigma V^T \implies R = U V^T$, and re-optimizes the translation $t$ in the $\ell_1$ Procrustes problem. All SRP objectives for $p=1,2,\infty$ can be formulated as second-order-cone programs (SOCPs) and solved to global optimality [2207.08592]. ## 3. Approximation Guarantee and Exact Recovery SRP provides a constant-factor guarantee for the robust ($\ell_1$) Procrustes objective. Denoting $E^*$ the global minimum of $E(R,t)$, any $(R,t)$ obtained from SRP satisfies: $$ E(R,t) \leq \sqrt{2} \cdot E_2(A^*, t^*, s^*) \leq \sqrt{2} \cdot E^*. $$ For the orthogonal-only formulation, the inequality $$ \sum_{i}\|R p(i) - q(i)\|_2 \leq \sqrt{2} \sum_{i} \|A^* p(i) - q(i)\|_2 \leq \sqrt{2} E^* $$ reflects that projection to the orthogonal group increases the $\ell_1$ objective by at most a factor $\sqrt{2}$. 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 [2207.08592]. Exact recovery is guaranteed under an affine dominance–in–projection (DIP) condition: if a subset $I \subset \{1\ldots n\}$ of inliers (so that $q(i) = R_0 p(i) + t_0$ for $i \in I$) dominates the outliers in every direction, i.e., for all $(u, a) \ne (0,0)$, $$ \sum_{i \in I} |u \cdot p(i) + a| > \sum_{i \notin I} |u \cdot p(i) + a| $$ (and symmetrically for $q$), then SRP returns the exact transformation $(R_0, t_0)$ for arbitrary outlier contamination [2207.08592]. ## 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 $O(n d^2)$ operations for residual and back-projection calculations, with convergence observed in tens of iterations. The projection to the orthogonal group incurs $O(d^3)$ floating-point operations for the SVD. Subsequently, optimizing translation $t$ given $R$ reduces to a convex SOCP (or a closed-form weighted median in the pure $\ell_1$ scenario), with cost $O(nd)$. Empirical results report that, for $n \sim 200$, $d \sim 100$, SRP runs in milliseconds per instance on standard hardware, matching a single IRLS iteration [2207.08592]. ## 5. Comparison with Iteratively Reweighted Least Squares The following table summarizes the salient distinctions between SRP and IRLS for Robust Procrustes: | Criterion | SRP$_2$ | IRLS | |------------------------------|----------------------------------|----------------| | Objective Quality | Lower bound cert. $\&\sqrt{2}$ approx. | No guarantee | | Initialization | Effective seed for IRLS | Variable | | Robustness to Dimension | $\sqrt{2}$, dimension-free | n/a | | Outlier Tolerance | Matches IRLS; provable recovery | Empirical only | | Extra Convex Regularization | Easily supported | Limited | SRP$_2$ consistently produces lower bound certificates on the true $\ell_1$-energy and remains competitive in recovery performance—oftentimes matching or slightly surpassed by local IRLS refinements. Seeding IRLS with SRP$_2$-obtained solutions improves final objective values compared to random or SVD-based initialization. Against RANSAC-type algorithms (with known exponential-in-$d$ approximation factors), SRP$_2$ is superior for $d$ up to 100 in both objective and recovery, while maintaining dimension-free guarantees. In synthetic tests with $200$ inliers and up to $1600$ outliers with $2\%$ noise in $d=3$ and $d=100$, SRP$_2$ exhibits robust outlier resistance and exact recovery upon DIP satisfaction [2207.08592]. ## 6. Extensions and Applications Two principal SRP extensions have been demonstrated: - **Semi-supervised Procrustes with Covariance Penalty**: When $n