Beyond Procrustes distances: a multilinear Gromov-Wasserstein distance capturing chirality

Efficiently and robustly analyzing shape data is critical across many scientific disciplines. While chirality is a fundamental property in numerous applications - most notably in molecular science - existing shape analysis metrics fail to distinguish between a shape and its mirror image. To address this gap, we introduce a multilinear generalization of the Gromov-Wasserstein objective. Under mild assumptions, this objective yields a distance between shapes, represented as probability distributions quotiented by a symmetry group $G$. In particular, for $G = SO(d)$, we introduce the Chiral Gromov-Wasserstein ($\mathrm{CGW}$) distance, sensitive to chirality. We establish robustness properties for the multilinear Gromov-Wasserstein distances and develop efficient algorithms to compute them, reformulating the underlying optimization problem by projecting couplings onto a low-dimensional space. We derive algorithms for both local and approximate global solutions, yielding a fully polynomial-time approximation scheme for these problems. We validate the framework through numerical experiments that demonstrate the effectiveness of $\mathrm{CGW}$ as a shape metric for chiral objects.

Mixture of Geodesic Factor Analyzers on Riemannian Homogeneous Spaces

This paper introduces Mixtures of Geodesic Factor Analyzers (MGFA) on Riemannian homogeneous spaces. MGFA uses a geodesic factor model within each mixture component, providing greater expressiveness than mixtures of Riemannian radial distributions and enabling clustering of manifold-valued data with anisotropic subpopulations. We establish root-$n$ consistency for the MGFA maximum likelihood estimator (MLE), thereby filling a theoretical gap for mixtures of Riemannian radial distributions as a special case. We also propose an iterative estimation algorithm and implement it on spheres, shape spaces, and hyperbolic spaces. Numerical experiments show that MGFA substantially outperforms competing methods in well-specified regimes while remaining robust under model misspecification. Finally, case studies on corpus callosum and left hippocampus shape datasets demonstrate MGFA's effectiveness for both 2D contour and 3D shape analysis.

The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes

We introduce a mathematical framework for shape comparison based on mapping functions from the shape domain to a common reference domain. This Push-Forward Transform enables invariant and robust comparison of shapes, preserving intrinsic geometric information. Quantitatively comparing shapes and their temporal evolution is a fundamental challenge in image analysis. Meaningful shape comparison requires representations that are invariant to transformations that do not alter shape itself, such as translation, rotation, reflection, re-parametrization, and uniform scaling, while remaining sensitive to intrinsic geometric variation. Existing approaches often rely on sensitive parameterizations, landmark correspondence, or learned representations that are difficult to interpret and reproduce. We show that the Push-Forward Transform (PF-T) applied to Signed Distance Functions (SDFs) yields a continuous representation that captures both boundary and interior geometry. We derive an interpretable morphometric that quantifies shape similarity and reveals features such as skeletal topology and rotational symmetries. The push-forward transform applies consistently to two- and three-dimensional shapes, extends to time-evolving geometries, and supports the joint analysis of shape and additional scalar fields defined over shapes, such as intensity or molecular signals. We present the mathematical formulation, describe an efficient algorithm, and benchmark the approach on 2D, 3D, and temporal data sets.

Hi-TOPS: Hierarchical Topology-aware Scoring Prior for 3D Part Decomposition

Accurate 3D part decomposition requires separating shapes into structurally meaningful components with precise boundaries while preserving articulation seams and thin attachments. Existing approaches often suffer from a structural-scale mismatch: geometric evidence for separation is most reliable at the meso scale, yet many pipelines operate either too globally to respect joints or too locally to remain robust to noise. We propose Hi-TOPS, a Hierarchical Topology-aware Scoring Prior that aggregates complementary intrinsic cues into a multi-resolution Flow-Freeze field. Flow regions provide expandable support for primitive coverage, while Freeze regions restrict growth near articulations and thin structures. A TSDF-guided body-surface superquadric fitter then captures dominant cores and residual surface structures, followed by SQ-to-mesh assignment for connected, boundary-aligned parts. Across diverse benchmarks, Hi-TOPS delivers stable, editable decompositions without semantic supervision or 2D foundation priors.

Landmark shape spaces with induced metrics

We present a unification of Kendall's landmark shape spaces, where rigid motions are factored out and scale fixed on landmark configurations equipped with Euclidean geometry, with landmark configuration spaces carrying Riemannian metrics descending from right-invariant Sobolev metrics on the diffeomorphism group. The resulting new landmark shape spaces achieve the defining properties of both approaches: The regularity of the descending metric prevents landmarks from colliding, the metric is defined in the ambient space independent of the number of landmarks, local rigid transformations are preserved, global rigid motions are removed, and scale fixed. To achieve this, we define a particular Sobolev-type operator, the screened elasticity operator, whose null-space consists exactly of the rigid motions, we show how this operator descends to achieve the desired geometry, and we present approaches to solving matching problems and computing geodesics numerically. The resulting construction allows the use of landmark configuration spaces with sufficiently regular metrics in applications while retaining the shape invariances that are a hallmark of Kendall's shape spaces.

A Riemannian View on Active Subspaces

Active subspaces provide an explainable, eigenvalue-ordered principle for studying how scalar-valued quantities of interest change the most, on average, over a reduced basis of Euclidean domains. Composition with parallel transport generalizes this principle from Euclidean space to quantities of interest defined over Riemannian manifolds, and the resulting intrinsic formulation is contrasted with the extrinsic, embedding-based gradient average of manifold learning. Either strategy is studied in an intrinsically local sense, restricted to mean-centered geodesic-balls, and within that scope the two are not identical: on the central tangent space, eigenvalues agree to second order in the geodesic radius of the sampled domain, while dominant eigenspaces agree at the same order relative to the spectral gap. Extending activity beyond that central space then calls for either recomputed decompositions over changing tangent spaces or, intrinsically, parallel transport of a single central frame. Hyperspheres are emphasized throughout as a particular manifold of interest, motivated by applications over preshape spaces for statistical shape analysis. Numerical examples over the 2-sphere illustrate the formalism, including the derived ridge recovery at a curvature-limited quadratic rate.

Shape-Based Inductive Bias for Glioma Grading from Tumor Contours

Glioma grading from tumor contours is often treated as a pixel problem even when the signal of interest is shape. We align closed contours with a functional shape-alignment framework, separate global deformation from residual Fourier shape, and organize these quantities as frequency-ordered tokens. In five-fold patient-disjoint cross-validation on BraTS~2020 tumor contours, with model selection performed using grouped inner validation, a compact multilayer perceptron (MLP) achieves the highest mean balanced accuracy at 71.5\%, compared with 65.9\% for ResNet-18 and 63.3\% for ViT-Tiny. It also gives the highest mean low-grade glioma F1 at 54.9\%. Its pooled out-of-fold balanced accuracy is 72.4\% (patient-bootstrap 95\% CI: 66.4--77.8\%). The selected MLPs use 2.9k--117.3k parameters across folds, at least 46 times fewer than the pixel baselines. In a controlled noise-free simulation, shape-based models reach 56.3--71.5\% balanced accuracy while the pixel models remain at 50.0--52.5\%. This work demonstrates how incorporating a shape-based inductive bias at the representation level can improve interpretability and scalability while enabling substantial dimensionality reduction.

Intrinsic Riemannian Cross-covariance for Manifold-valued Random Objects

Covariance estimation yields a fundamental second-order statistic underlying representation learning, dimension reduction, and dependence modeling. While covariance has been well understood in Euclidean spaces, it is ill-defined for random objects residing on nonlinear Riemannian manifolds, which increasingly arise in modern machine learning applications involving shapes, symmetric positive definite (SPD) matrices, etc. This paper introduces an intrinsic Riemannian cross-covariance for manifold-valued random objects. Our approach defines covariance and correlation by transporting local variations to a common tangent space via parallel transport, yielding a second-order descriptor that is independent of arbitrary coordinate choices. We establish that the proposed covariance inherits desirable properties of its Euclidean counterparts and characterize its asymptotic behavior. Numerical studies on spheres and SPD manifolds, together with real-data experiments on heart valve shapes in Kendall's shape space, demonstrate the effectiveness of our estimators and verify the stated properties. Our results position the Riemannian covariance as a fundamental tool for second-order learning and analysis in non-Euclidean representation spaces.

Monte Carlo Steklov Operators for Large-Scale Geometry Processing in the Wild

Intrinsic methods fill the default toolbox for geometry processing on meshes. Intrinsic operators, in particular the Laplacian, underlie methods that require invariance to isometry and have hence been employed in many algorithms for shape analysis, learning, and editing. However, intrinsic methods are predicated on assumptions that quickly become brittle when working with in-the-wild geometry, where (i) mesh quality is not guaranteed, and (ii) many meshes are modeled with multiple connected components. In such settings, volumetric constructions are better-defined, since restrictions on surface topology can be relaxed. This paper presents a Monte Carlo method for estimating the Dirichlet-to-Neumann (DtN) operator -- a boundary-to-boundary volumetric operator -- and its associated Steklov eigenmodes. We build on recent developments in Monte Carlo geometry processing by casting this boundary operator itself as the subject of estimation. The DtN operator, defined through a volumetric stochastic process, is then generalized to the exterior domain, where it couples disconnected components through the surrounding ambient space. We show that our method is orders of magnitude faster than existing boundary-element approaches for computing Steklov spectra while remaining robust to poor triangulations, high-resolution meshes, and multi-component geometry. To demonstrate this scalability, we compute interior and exterior Steklov eigenspectra for approximately 450,000 shapes from the uncurated Objaverse dataset. We incorporate these operators into Steklov-CLIP, a mesh-based neural network that uses volumetric spectral operators for large-scale contrastive 3D representation learning. The resulting network learns semantically meaningful global and dense shape representations, illustrating that geometrically-principled volumetric operators can be made practical at the scale of modern 3D datasets.