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).

Determining the binding pose of a ligand to a protein, known as molecular docking, is a fundamental task in drug discovery. Generative approaches promise faster, improved, and more diverse pose sampling than physics-based methods, but are often hindered by chemically implausible outputs, poor generalisability, and high computational cost. To address these challenges, we introduce a novel fragmentation scheme, leveraging inductive biases from structural chemistry, to decompose ligands into rigid-body fragments. Building on this decomposition, we present SigmaDock, an SE(3) Riemannian diffusion model that generates poses by learning to reassemble these rigid bodies within the binding pocket. By operating at the level of fragments in SE(3), SigmaDock exploits well-established geometric priors while avoiding overly complex diffusion processes and unstable training dynamics. Experimentally, we show SigmaDock achieves state-of-the-art performance, reaching Top-1 success rates (RMSD<2 & PB-valid) above 79.9% on the PoseBusters set, compared to 12.7-30.8% reported by recent deep learning approaches, whilst demonstrating consistent generalisation to unseen proteins. SigmaDock is the first deep learning approach to surpass classical physics-based docking under the PB train-test split, marking a significant leap forward in the reliability and feasibility of deep learning for molecular modelling.

This paper proposes a principled extension of the traditional single-layer flat sparse coding scheme, where a two-layer coding scheme is derived based on theoretical analysis of nonlinear functional approximation that extends recent results for local coordinate coding. The two-layer approach can be easily generalized to deeper structures in a hierarchical multiple-layer manner. Empirically, it is shown that the deep coding approach yields improved performance in benchmark datasets.

Many training methods, such as Adam(W) and Shampoo, learn a positive-definite curvature matrix and apply an inverse root before preconditioning. Recently, non-diagonal training methods, such as Shampoo, have gained significant attention; however, they remain computationally inefficient and are limited to specific types of curvature information due to the costly matrix root computation via matrix decomposition. To address this, we propose a Riemannian optimization approach that dynamically adapts spectral-factorized positive-definite curvature estimates, enabling the efficient application of arbitrary matrix roots and generic curvature learning. We demonstrate the efficacy and versatility of our approach in positive-definite matrix optimization and covariance adaptation for gradient-free optimization, as well as its efficiency in curvature learning for neural net training.

In this technical report we present TrafficBots V1.5, a baseline method for the closed-loop simulation of traffic agents. TrafficBots V1.5 achieves baseline-level performance and a 3rd place ranking in the Waymo Open Sim Agents Challenge (WOSAC) 2024. It is a simple baseline that combines TrafficBots, a CVAE-based multi-agent policy conditioned on each agent's individual destination and personality, and HPTR, the heterogeneous polyline transformer with relative pose encoding. To improve the performance on the WOSAC leaderboard, we apply scheduled teacher-forcing at the training time and we filter the sampled scenarios at the inference time. The code is available at https://github.com/zhejz/TrafficBotsV1.5.

Gradient descent, or negative gradient flow, is a standard technique in optimization to find minima of functions. Many implementations of gradient descent rely on discretized versions, i.e., moving in the gradient direction for a set step size, recomputing the gradient, and continuing. In this paper, we present an approach to manifold learning where gradient descent takes place in the infinite dimensional space $\mathcal{E} = {\rm Emb}(M,\mathbb{R}^N)$ of smooth embeddings $\phi$ of a manifold $M$ into $\mathbb{R}^N$. Implementing a discretized version of gradient descent for $P:\mathcal{E}\to {\mathbb R}$, a penalty function that scores an embedding $\phi \in \mathcal{E}$, requires estimating how far we can move in a fixed direction -- the direction of one gradient step -- before leaving the space of smooth embeddings. Our main result is to give an explicit lower bound for this step length in terms of the Riemannian geometry of $\phi(M)$. In particular, we consider the case when the gradient of $P$ is pointwise normal to the embedded manifold $\phi(M)$. We prove this case arises when $P$ is invariant under diffeomorphisms of $M$, a natural condition in manifold learning.

Dynamical Systems (DS) are an effective and powerful means of shaping high-level policies for robotics control. They provide robust and reactive control while ensuring the stability of the driving vector field. The increasing complexity of real-world scenarios necessitates DS with a higher degree of non-linearity, along with the ability to adapt to potential changes in environmental conditions, such as obstacles. Current learning strategies for DSs often involve a trade-off, sacrificing either stability guarantees or offline computational efficiency in order to enhance the capabilities of the learned DS. Online local adaptation to environmental changes is either not taken into consideration or treated as a separate problem. In this paper, our objective is to introduce a method that enhances the complexity of the learned DS without compromising efficiency during training or stability guarantees. Furthermore, we aim to provide a unified approach for seamlessly integrating the initially learned DS's non-linearity with any local non-linearities that may arise due to changes in the environment. We propose a geometrical approach to learn asymptotically stable non-linear DS for robotics control. Each DS is modeled as a harmonic damped oscillator on a latent manifold. By learning the manifold's Euclidean embedded representation, our approach encodes the non-linearity of the DS within the curvature of the space. Having an explicit embedded representation of the manifold allows us to showcase obstacle avoidance by directly inducing local deformations of the space. We demonstrate the effectiveness of our methodology through two scenarios: first, the 2D learning of synthetic vector fields, and second, the learning of 3D robotic end-effector motions in real-world settings.