Procrustes Matching Problem

• Procrustes Matching Problem is an alignment task that seeks the optimal transformation (rotation, translation, reflection, and scaling) to minimize discrepancies between datasets.
• It encompasses several formulations, including rigid alignment, permutation recovery, and semidefinite constraints, often solved using SVD-based or convex relaxation techniques.
• Recent advancements extend the approach to robust, high-dimensional, and hyperbolic variants, enhancing its utility in statistical alignment and machine learning applications.

Searching arXiv for recent and foundational papers on the Procrustes matching problem and major variants. Searching "Procrustes matching problem". The Procrustes matching problem is a family of alignment problems in which one seeks a transformation of a dataset, or of one point set relative to another, that minimizes a prescribed discrepancy while respecting geometric or structural constraints. In its classical form, congruent Procrustes analysis aims to find the best matching between two point sets through rotation, reflection and translation; in matrix form, it is a matrix approximation problem of the type $\min_{A\in\mathcal S}\|AX-B\|_F^2$. The modern literature extends this template to rigid and similarity alignment, unknown correspondences and permutation recovery, robust power-$1$ objectives, hyperbolic isometries, positive semidefinite and non-symmetric semidefinite constraints, and high-dimensional embedding alignment (Tabaghi et al., 2021, Fulová et al., 2023, Amir et al., 2022, Even et al., 2024).

1. Canonical formulations

In Euclidean space, the classical point-set formulation seeks an isometry $T$ minimizing

$$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$

where $\mathcal T$ is the group of isometries, typically including rotations, translations, and possibly reflections and scalings (Tabaghi et al., 2021). A closely related matrix formulation asks, given matrices $X,B\in\mathbb R^{n\times m}$, for a matrix $A$ in a structured class $\mathcal S$ minimizing $\|AX-B\|_F^2$; changing $\mathcal S$ yields orthogonal, unitary, positive semidefinite, or non-symmetric positive semidefinite variants (Baghel et al., 2021).

When correspondences are known and the feasible set is orthogonal, the problem has a closed-form solution via singular value decomposition. In the orthogonal Procrustes problem, the optimal transformation is obtained from the SVD of the cross term, with the standard form $1$0 after decomposing the appropriate matrix product (Grave et al., 2018). This SVD-based solvability is one reason the Procrustes problem appears repeatedly as a subroutine in larger procedures, including sparse PCA on the Stiefel manifold and alternating methods for unsupervised alignment (Benidis et al., 2016).

Formulation	Representative objective	Decision variables
Orthogonal Procrustes	$1$1	orthogonal matrix
Procrustes matching	$1$2	orthogonal matrix, permutation matrix
PSDP / NSPSDP	$1$3 with $1$4 or $1$5	structured matrix
Procrustes-Wasserstein	$1$6	orthogonal matrix, permutation matrix

This range of formulations shows that “Procrustes” is not limited to labeled rigid point-set alignment. The term also covers constrained matrix approximation, inverse problems with algebraic structure, and unsupervised alignment in which the transformation and the correspondence are both unknown (Fulová et al., 2023).

2. Correspondence uncertainty and combinatorial matching

A central extension is the Procrustes matching problem in which the labeling itself must be recovered. For point clouds $1$7, the exact formulation is

$1$8

with $1$9 and $T$0 a permutation matrix (Dym et al., 2016). In this setting, the optimization is non-convex because the orthogonal and permutation constraints are coupled.

The exact decision version is computationally equivalent to the graph isomorphism problem (Dym et al., 2016). That equivalence places Procrustes matching within the same combinatorial landscape as graph matching and quadratic assignment. For exact PM problems, a convex semidefinite relaxation, PM-SDP, has been analyzed using a moment interpretation. For generic input shapes that are asymmetric or bilaterally symmetric, the relaxation returns a correct solution of PM. For symmetric shapes, however, PM has multiple solutions; the non-convex set of optimal PM solutions is strictly contained in the convex set of optimal PM-SDP solutions, and the true PM solutions are precisely the extreme points of the PM-SDP solution set (Dym et al., 2016). This resolves a common misconception that a convex relaxation must return a discrete matching whenever the objective value is exact.

Unsupervised alignment of embeddings produces a related formulation. Wasserstein Procrustes jointly estimates an orthogonal matrix and a permutation matrix through

$T$1

which is the squared Wasserstein distance between the aligned clouds under one-to-one matching (Grave et al., 2018). A later planted-model treatment writes the joint maximum-likelihood problem as

$T$2

and studies performance through Euclidean transport cost rather than only exact overlap (Even et al., 2024). In low dimension, that distinction is decisive: small transport cost may be achievable even when exact overlap is not.

Unknown correspondences also arise in contour registration. For two-dimensional statistical shape models, a dynamic time warping strategy was proposed for Procrustes registration without correspondences. The method alternates between order-preserving correspondence estimation and weighted Procrustes registration, handles open or closed contours, and extends generalized Procrustes analysis to ensembles without pre-assigned landmarks (Eguizabal et al., 2019). In that setting, the ordering along the contour is part of the structure and is enforced by the dynamic-programming constraint set.

3. Robust, probabilistic, and high-dimensional variants

The least-squares objective is sensitive to outliers, so a robust variant replaces the sum of squared distances with a power-$T$3 objective: $T$4 Unlike the least-squares problem, no closed-form solution is known for this robust objective (Amir et al., 2022). A symmetrized convex relaxation introduces

$T$5

then relaxes orthogonality, solves a convex program, and projects back to $T$6 (Amir et al., 2022). For $T$7, the resulting method has a $T$8-factor approximation guarantee, and under the dominance-of-inliers property or its affine analogue it exactly recovers the true transformation from correspondences contaminated by outliers (Amir et al., 2022).

In statistical alignment of high-dimensional matrices, Goodall’s perturbation model expresses each observation as

$T$9

with orthogonal $$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$0 and matrix-normal perturbations (Andreella et al., 2020). The ProMises model places a matrix von Mises–Fisher prior on $$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$1, yielding a conjugate posterior and a maximum a posteriori estimator obtained by SVD of $$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$2 (Andreella et al., 2020). This addresses the non-identifiability and interpretability problems of the original perturbation model. For the high-dimensional regime $$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$3, the Efficient ProMises model reduces the SVD from $$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$4 to $$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$5, lowering the computational complexity from $$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$6 to $$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$7 (Andreella et al., 2020).

The fitted Procrustes transformations can themselves be turned into similarity measures. Two distances have been proposed for between-matrices similarity: a residual-based distance,

$$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$8

and a rotational-based distance,

$$\widehat{T} = \arg\min_{T \in \mathcal{T}} \sum_{n=1}^{N} \|z_n - T(z'_n)\|_2^2,$$9

which distinguish differences remaining after alignment from differences in the rotations required to reach a common space (Andreella et al., 2023).

A related recent line studies interoperability between embedding models. If two sets of embeddings have approximately preserved Gram matrices, then there exists an orthogonal alignment with provably small Procrustes error; the alignment is again obtained by SVD and used as a Procrustes post-processing step (Maystre et al., 15 Oct 2025). This suggests that orthogonal alignment is not merely a fitting device but also a geometry-preserving interface between separately trained representation spaces.

4. Hyperbolic Procrustes analysis

The hyperbolic generalization replaces Euclidean geometry with the Lorentz, or hyperboloid, model. The ambient space is

$\mathcal T$0

with Lorentzian inner product

$\mathcal T$1

and hyperbolic distance

$\mathcal T$2

The hyperbolic Procrustes problem seeks the optimal hyperbolic isometry aligning two point sets in $\mathcal T$3 (Tabaghi et al., 2021).

In the hyperboloid model, any isometry can be written as

$\mathcal T$4

where $\mathcal T$5 and $\mathcal T$6 is a translation operator parameterized by $\mathcal T$7 (Tabaghi et al., 2021). The centering step is therefore more intricate than in Euclidean space. The hyperbolic centroid of $\mathcal T$8 is defined by

$\mathcal T$9

and translation by $X,B\in\mathbb R^{n\times m}$0 centers the point set in the sense that

$X,B\in\mathbb R^{n\times m}$1

(Tabaghi et al., 2021).

For noise-free matched data related by a hyperbolic isometry, the optimal map is obtained in two stages: center both sets by their hyperbolic centroids, then solve for the optimal rotation by SVD. If $X,B\in\mathbb R^{n\times m}$2 is the singular value decomposition of

$X,B\in\mathbb R^{n\times m}$3

then the optimal orthogonal factor is

$X,B\in\mathbb R^{n\times m}$4

and the full solution is

$X,B\in\mathbb R^{n\times m}$5

This reproduces the Euclidean “center, then rotate” pattern, but with Lorentzian centroid and hyperbolic isometries in place of Euclidean translations and rotations (Tabaghi et al., 2021).

Under measurement noise, the closed-form solution is approximate. The paper compares a closed-form estimator $X,B\in\mathbb R^{n\times m}$6, direct gradient descent $X,B\in\mathbb R^{n\times m}$7, and a hybrid $X,B\in\mathbb R^{n\times m}$8 initialized from the closed-form map. The discrepancy metric is

$X,B\in\mathbb R^{n\times m}$9

and the empirical analysis reports that the proposed method is fast and robust, with significantly fewer outlier cases than vanilla gradient descent, while the hybrid yields the best results (Tabaghi et al., 2021).

5. Constrained matrix variants

A major branch of the subject treats Procrustes problems as constrained matrix approximation. The positive semidefinite Procrustes problem (PSDP) is

$A$0

where $A$1 is the cone of real symmetric positive semidefinite matrices (Gillis et al., 2016). A key semi-analytical result reduces the problem, after an SVD of $A$2, to a smaller PSDP in dimension $A$3: $A$4 This reduction yields a complete characterization of the solution set, identifies when the infimum is not attained, and supports a fast gradient method with linear convergence on the reduced problem (Gillis et al., 2016).

The non-symmetric positive semidefinite Procrustes problem (NSPSDP) replaces symmetry by the condition $A$5: $A$6 As in the symmetric case, the problem can be reduced to a smaller diagonal full-rank instance, then solved by a fast gradient method whose main step is projection onto the NSPSD cone. The same framework extends to the complex case through an overparametrized real embedding (Baghel et al., 2021).

A more general conic-optimization view writes Procrustes problems as

$A$7

with $A$8 linear, $A$9 a feasible set defined by linear, quadratic, or semidefinite constraints, and the norm chosen from Frobenius, $\mathcal S$0, $\mathcal S$1, or $\mathcal S$2 (Fulová et al., 2023). This covers balanced and unbalanced weighted Procrustes problems, partially specified targets, oblique constraints $\mathcal S$3, projection problems such as $\mathcal S$4, and two-sided formulations. The reformulation produces a rank-constrained semidefinite program, so convex relaxations, trace or log-det heuristics, convex iteration, and bisection become available tools (Fulová et al., 2023).

The Procrustes subproblem also appears inside other non-convex optimizations. In orthogonal sparse PCA, an MM surrogate reduces each iteration to the rectangular Procrustes problem

$\mathcal S$5

whose solution is $\mathcal S$6 from the SVD of $\mathcal S$7 (Benidis et al., 2016). By contrast, in regularized multivariate analysis, replacing the second step of an alternating scheme by orthogonal Procrustes is not optimal from the perspective of the overall MVA method. An eigenvalue-based update preserves the uncorrelation of the extracted features and recovers the original unregularized solution as the regularization vanishes (Muñoz-Romero et al., 2016). This is a methodological caveat: a Procrustes step may be algebraically convenient without being statistically or geometrically appropriate for the larger problem.

Structured inverse eigenvalue problems furnish another specialized variant. For normal (skew) $\mathcal S$8-Hamiltonian and normal $\mathcal S$9-symplectic matrices, the Procrustes problem asks for the matrix in the structured solution set of $\|AX-B\|_F^2$0 closest in Frobenius norm to a prescribed $\|AX-B\|_F^2$1. The paper solves these problems by block-diagonalizing $\|AX-B\|_F^2$2, deriving explicit blockwise minimizers, and using the Cayley transform in the $\|AX-B\|_F^2$3-symplectic case (Gigola et al., 2024).

6. Statistical limits, Bayesian formulations, and large-scale applications

Recent work has made the statistical structure of Procrustes matching explicit. In a high-dimensional planted model,

$\|AX-B\|_F^2$4

with unknown permutation $\|AX-B\|_F^2$5, unknown orthogonal matrix $\|AX-B\|_F^2$6, and Gaussian noise, the task is exact recovery of $\|AX-B\|_F^2$7 from two sets of Gaussian vectors (Niu et al., 9 Jul 2026). A polynomial-time algorithm based on weighted counts of “wide” trees succeeds with high probability for $\|AX-B\|_F^2$8 whenever

$\|AX-B\|_F^2$9

where $\mathcal S$0 is Otter’s tree-counting constant (Niu et al., 9 Jul 2026). The same paper gives an information-theoretic guarantee for exact recovery when

$\mathcal S$1

and a low-degree analysis suggesting that the condition $\mathcal S$2 is necessary for tree-counting algorithms (Niu et al., 9 Jul 2026). This establishes a statistical-computational gap in the high-dimensional regime.

The Procrustes-Wasserstein literature studies a related planted model through the Euclidean transport cost. In the high-dimensional regime $\mathcal S$3, recovery of both the orthogonal map and the relabeling is information-theoretically possible as soon as $\mathcal S$4; in the low-dimensional regime $\mathcal S$5, small transport cost can still be achieved when exact overlap is much harder (Even et al., 2024). The Ping-Pong algorithm alternates a closed-form orthogonal Procrustes update with a linear assignment step, initialized by a Frank-Wolfe convex relaxation, and is analyzed for one-step recovery under sufficiently small noise (Even et al., 2024).

Probabilistic formulations also appear in applied reconstruction. In unposed 3D Gaussian Splatting, alignment of large submaps is formulated as a probabilistic Procrustes problem over similarity transformations: $\mathcal S$6 with soft correspondence weights $\mathcal S$7 and a soft dustbin mechanism for uncertain matches (Cheng et al., 24 Jul 2025). The method alternates a Kabsch-Umeyama closed-form initialization with entropy-regularized weight updates and gradient-based refinement of pose parameters. This suggests a convergence of Procrustes ideas with probabilistic coupling and large-scale scene reconstruction.

A Bayesian perspective predates many of these developments. For unlabelled point sets, Bayesian matching has been developed under two likelihoods: a Procrustes size-and-shape model, which optimizes out rotation and translation before modeling residuals, and a full configuration model, which treats rotation and translation as parameters and samples them by MCMC (Kenobi et al., 2010). An improved Procrustes algorithm introduces occasional large jumps during burn-in—nearness, rotation, translation, and flip moves—to improve convergence. The two models are linked by a Laplace approximation, under which the Procrustes posterior can be viewed as an approximation to the configuration posterior after marginalizing nuisance alignment parameters (Kenobi et al., 2010).

Across these lines of work, the main technical divide is between settings where closed-form SVD solves the entire problem and settings where SVD solves only a subproblem. Robust objectives, unknown correspondences, symmetry-induced multiplicity, and structured feasibility sets typically require convex relaxation, alternating minimization, first-order methods, semidefinite programming, or Bayesian computation (Amir et al., 2022, Dym et al., 2016, Gillis et al., 2016, Kenobi et al., 2010). A plausible implication is that the Procrustes matching problem is best understood not as a single algorithm, but as a unifying optimization paradigm for alignment under geometry, correspondence, and structure.