Optimal Transport (OT) provides a rigorous mathematical framework for comparing empirical distributions, such as point clouds. However, standard formulations based on the Wasserstein distances are highly sensitive to global rigid transformations in the feature space. Unlike other approaches being computationally prohibitive (e.g. Gromov-Wasserstein) the Procrustes-Wasserstein (PW) distance addresses this limitation in a more scalable way by directly incorporating alignment into the optimization problem. However, current approaches remain confined to discrete settings and lacks generalization to unseen samples. In this work, we extend the PW framework to the continuous domain by formally deriving its dual and semi-dual formulations. Leveraging results in both convex analysis and, more recently, neural optimal transport, we introduce NeuralPW, a framework that jointly learns rigid alignments and continuous transport maps. We empirically demonstrate that NeuralPW accurately recovers geometric transformations, explicitly decoupling global pose alignment from local non-rigid deformation. Finally, we validate our approach on a real-world case study in archaeozoology, demonstrating its ability to robustly match high-density point clouds where discrete primal solvers are computationally intractable.

Kernel methods have been widely used in machine learning as they are a powerful tool for implicitly mapping data into high-dimensional spaces, enabling the discovery of complex patterns that might be challenging to capture in the original feature space. Although some classification and regression problems can be successfully addressed with a single kernel, sometimes real-world scenarios exhibit complex structures, and it is desirable to employ several kernel types, one for each notion of similarity that one aims to take into account. This is where multiple kernel learning (MKL) comes into play. This paper revisits multi-kernel classification with a specific focus on kernel(s) selection in the light of recent developments in Monte Carlo (importance weighted) variational inference (MCVI). In the framework of kernelized logistic regression (KLR), we consider positive semidefinite linear combinations of kernels and treat the kernel weights as random variables. Proper choices of prior distributions coupled with the explicit derivation of the importance-weighted lower bound (IW-ELBO), generalization of the traditional variational lower bound (ELBO), allow us to both perform kernel selection via shrinking and to perform posterior inference on the kernel weights, without needing MCMC sampling. Unlike pure optimization-based approaches to MKL, our optimization problem does not require explicit constraints and can be optimized by standard stochastic gradient descent.

The faunal remains from numerous Holocene archaeological sites across southwest Asia frequently include the bones of several wild and domestic ungulates, such as sheep, goats, ibexes, roe deer and gazelles. These assemblages may provide insight into hunting and animal husbandry strategies and offer palaeoecological information on ancient human societies. However, the skeletons of these taxa are highly similar in appearance, which presents a challenge for accurate identification based on their bones. This paper presents a case study to test the potential of topological data analysis (TDA) and multiple kernel learning (MKL) for inter-specific identification of 150 3D astragali belonging to modern and archaeological specimens. The joint application of TDA and MKL demonstrated remarkable efficacy in accurately identifying wild species, with a correct identification rate of approximately 90%. In contrast, the identification of domestic species exhibited a lower success rate, at approximately 60%. The misidentification of sheep and goat species is attributed to the morphological variability of domestic breeds. Moreover, while these methods assist in clearly identifying wild taxa from one another, they also highlight their morphological diversity. In this context, TDA and MKL could be invaluable for investigating intra-specific variability in domestic and wild animals. These methods offer a means of expanding our understanding of past domestic animal selection practices and techniques. They also facilitate an investigation into the morphological evolution of wild animal populations over time.

Due to its invariance to rigid transformations such as rotations and reflections, Procrustes-Wasserstein (PW) was introduced in the literature as an optimal transport (OT) distance, alternative to Wasserstein and more suited to tasks such as the alignment and comparison of point clouds. Having that application in mind, we carefully build a space of discrete probability measures and show that over that space PW actually is a distance. Algorithms to solve the PW problems already exist, however we extend the PW framework by discussing and testing several initialization strategies. We then introduce the notion of PW barycenter and detail an algorithm to estimate it from the data. The result is a new method to compute representative shapes from a collection of point clouds. We benchmark our method against existing OT approaches, demonstrating superior performance in scenarios requiring precise alignment and shape preservation. We finally show the usefulness of the PW barycenters in an archaeological context. Our results highlight the potential of PW in boosting 2D and 3D point cloud analysis for machine learning and computational geometry applications.