This section provides basic theoretical details on the log-Sinkhorn operator and its convergence results. Then, we define the functional F(Al): C2(M)/R R, as follows: F(Al) = Z Sl A(x)dµθt(x)+ Z Hϵµ[Sl A](y)dνt(y). The log-Sinkhorn iteration S has the a point in C2(M)/R. This fixed point is determined up to an additive constant, and minimizes the functional F uniformly: F(S Al) F(Al+1) F(Al). Then, the function Al L is approximated to the d2/2-Legendre transformation (11) of the function Bm M. [Al L] If A is a fixed point of the log-Sinkhorn operator S on C2(M)/R, 36th Conference on Neural Information Processing Systems (NeurIPS 2022).

Unlike distance metric learning where the subsequent tasks utilizing the estimated distance metric is the usual focus, the proposal focuses on the estimated metric characterizing the geometry structure. Despite the illustrated taxi and MNIST examples, it is still open to finding more compelling applications that target the data space geometry. Interpreting mathematical concepts such as Riemannian metric and geodesic in the context of potential application (e.g., cognition and perception research where similarity measures are common) could be inspiring. Our proposal requires sufficiently dense data, which could be demanding, especially for high-dimensional data due to the curse of dimensionality. Dimensional reduction (e.g., manifold embedding as in the MNIST example) can substantially alleviate the curse of dimensionality, and the dense data requirement will more likely hold true.

The key quantity of interest here is the Riemannian metric, which characterizes the Riemannian geometry and defines straight lines and derivatives on the manifold.

The key quantity of interest here is the Riemannian metric, which characterizes the Riemannian geometry and defines straight lines and derivatives on the manifold.

We introduce Einstein Fields, a neural representation that is designed to compress computationally intensive four-dimensional numerical relativity simulations into compact implicit neural network weights. By modeling the \emph{metric}, which is the core tensor field of general relativity, Einstein Fields enable the derivation of physical quantities via automatic differentiation. However, unlike conventional neural fields (e.g., signed distance, occupancy, or radiance fields), Einstein Fields are \emph{Neural Tensor Fields} with the key difference that when encoding the spacetime geometry of general relativity into neural field representations, dynamics emerge naturally as a byproduct. Einstein Fields show remarkable potential, including continuum modeling of 4D spacetime, mesh-agnosticity, storage efficiency, derivative accuracy, and ease of use. We address these challenges across several canonical test beds of general relativity and release an open source JAX-based library, paving the way for more scalable and expressive approaches to numerical relativity. Code is made available at https://github.com/AndreiB137/EinFields

Geometrical interpretations of deep learning models offer insightful perspectives into their underlying mathematical structures. In this work, we introduce a novel approach that leverages differential geometry, particularly concepts from Riemannian geometry, to integrate multiple models into a unified transfer learning framework. By aligning the Ricci curvature of latent space of individual models, we construct an interrelated architecture, namely Geometric Embedding Alignment via cuRvature matching in transfer learning (GEAR), which ensures comprehensive geometric representation across datapoints. This framework enables the effective aggregation of knowledge from diverse sources, thereby improving performance on target tasks. We evaluate our model on 23 molecular task pairs sourced from various domains and demonstrate significant performance gains over existing benchmark model under both random (14.4%) and scaffold (8.3%) data splits.

In the realm of robotics, the concept of "controller design in the physical domain" (Sharon et al., 1989, 1991) and the associated methodology of "control by interconnection" (Stramigioli, 2001; van der Schaft, 2016) emphasize that robot controllers should be more than mere signal processors. Instead, they should have a physical interpretation (Lachner, 2022), which is especially important for robots that physically interact with the environment (Hogan, 1988; Dietrich and Hogan, 2022; Hogan, 2022). This paper delves into impedance control (Hogan, 1984) during physical interaction, specifically focusing on the symmetry of the stiffness matrix and its role in ensuring passive physical equivalent robot controllers. Passivity is a fundamental property for ensuring coupled stability when interacting with arbitrary passive objects (Colgate and Hogan, 1988a). Stability in robotics can be achieved by monitoring and controlling the energy supplied by the controller (Colgate and Hogan, 1987, 1988b,a; Stramigioli, 2015). In impedance-controlled robots, this monitoring is particularly straightforward, as energy is stored in virtual springs (potential energy) and transferred into kinetic energy during movement (Lachner et al., 2021). During physical interaction, stiffness plays a crucial role, as it defines how energy is stored and exchanged between the robot and its environment. Task-space stiffness determines interaction forces due to contact, which is especially important at low frequencies (e.g., steady-state). 1

Traditional Markov Chain Monte Carlo sampling methods often struggle with sharp curvatures, intricate geometries, and multimodal distributions. Slice sampling can resolve local exploration inefficiency issues and Riemannian geometries help with sharp curvatures. Recent extensions enable slice sampling on Riemannian manifolds, but they are restricted to cases where geodesics are available in closed form. We propose a method that generalizes Hit-and-Run slice sampling to more general geometries tailored to the target distribution, by approximating geodesics as solutions to differential equations. Our approach enables exploration of regions with strong curvature and rapid transitions between modes in multimodal distributions. We demonstrate the advantages of the approach over challenging sampling problems.