The classical $\textit{Procrustes}$ problem is to find a rigid motion (orthogonal transformation and translation) that best aligns two given point-sets in the least-squares sense. The $\textit{Robust Procrustes}$ problem is an important variant, in which a power-1 objective is used instead of least squares to improve robustness to outliers. While the optimal solution of the least-squares problem can be easily computed in closed form, dating back to Sch\"onemann (1966), no such solution is known for the power-1 problem. In this paper we propose a novel convex relaxation for the Robust Procrustes problem. Our relaxation enjoys several theoretical and practical advantages: Theoretically, we prove that our method provides a $\sqrt{2}$-factor approximation to the Robust Procrustes problem, and that, under appropriate assumptions, it exactly recovers the true rigid motion from point correspondences contaminated by outliers. In practice, we find in numerical experiments on both synthetic and real robust Procrustes problems, that our method performs similarly to the standard Iteratively Reweighted Least Squares (IRLS). However the convexity of our algorithm allows incorporating additional convex penalties, which are not readily amenable to IRLS. This turns out to be a substantial advantage, leading to improved results in high-dimensional problems, including non-rigid shape alignment and semi-supervised interlingual word translation.

In this paper, we aim to improve the Quality-of-Service (QoS) of Ultra-Reliability and Low-Latency Communications (URLLC) in interference-limited wireless networks. To obtain time diversity within the channel coherence time, we first put forward a random repetition scheme that randomizes the interference power. Then, we optimize the number of reserved slots and the number of repetitions for each packet to minimize the QoS violation probability, defined as the percentage of users that cannot achieve URLLC. We build a cascaded Random Edge Graph Neural Network (REGNN) to represent the repetition scheme and develop a model-free unsupervised learning method to train it. We analyze the QoS violation probability using stochastic geometry in a symmetric scenario and apply a modelbased Exhaustive Search (ES) method to find the optimal solution. Simulation results show that in the symmetric scenario, the QoS violation probabilities achieved by the model-free learning method and the model-based ES method are nearly the same. In more general scenarios, the cascaded REGNN generalizes very well in wireless networks with different scales, network topologies, cell densities, and frequency reuse factors. It outperforms the model-based ES method in the presence of the model mismatch. Yuhong Liu, Changyang She, Wibowo Hardjawana and Branka Vucetic are with School of Electrical and Information Engineering, The University of Sydney, Sydney, Australia. Yi Zhong is with School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan, P. R. China.

Sampling-based model predictive control (MPC) optimization methods, such as Model Predictive Path Integral (MPPI), have recently shown promising results in various robotic tasks. However, it might produce an infeasible trajectory when the distributions of all sampled trajectories are concentrated within high-cost even infeasible regions. In this study, we propose a new method called log-MPPI equipped with a more effective trajectory sampling distribution policy which significantly improves the trajectory feasibility in terms of satisfying system constraints. The key point is to draw the trajectory samples from the normal log-normal (NLN) mixture distribution, rather than from Gaussian distribution. Furthermore, this work presents a method for collision-free navigation in unknown cluttered environments by incorporating the 2D occupancy grid map into the optimization problem of the sampling-based MPC algorithm. We first validate the efficiency and robustness of our proposed control strategy through extensive simulations of 2D autonomous navigation in different types of cluttered environments as well as the cartpole swing-up task. We further demonstrate, through real-world experiments, the applicability of log-MPPI for performing a 2D grid-based collision-free navigation in an unknown cluttered environment, showing its superiority to be utilized with the local costmap without adding additional complexity to the optimization problem. A video demonstrating the real-world and simulation results is available at https://youtu.be/_uGWQEFJSN0.

The action governor is an add-on scheme to a nominal control loop that monitors and adjusts the control actions to enforce safety specifications expressed as pointwise-in-time state and control constraints. In this paper, we introduce the Robust Action Governor (RAG) for systems the dynamics of which can be represented using discrete-time Piecewise Affine (PWA) models with both parametric and additive uncertainties and subject to non-convex constraints. We develop the theoretical properties and computational approaches for the RAG. After that, we introduce the use of the RAG for realizing safe Reinforcement Learning (RL), i.e., ensuring all-time constraint satisfaction during online RL exploration-and-exploitation process. This development enables safe real-time evolution of the control policy and adaptation to changes in the operating environment and system parameters (due to aging, damage, etc.). We illustrate the effectiveness of the RAG in constraint enforcement and safe RL using the RAG by considering their applications to a soft-landing problem of a mass-spring-damper system.

Cloth simulation has wide applications in computer animation, garment design, and robot-assisted dressing. This work presents a differentiable cloth simulator whose additional gradient information facilitates cloth-related applications. Our differentiable simulator extends a state-of-the-art cloth simulator based on Projective Dynamics (PD) and with dry frictional contact. We draw inspiration from previous work to propose a fast and novel method for deriving gradients in PD-based cloth simulation with dry frictional contact. Furthermore, we conduct a comprehensive analysis and evaluation of the usefulness of gradients in contact-rich cloth simulation. Finally, we demonstrate the efficacy of our simulator in a number of downstream applications, including system identification, trajectory optimization for assisted dressing, closed-loop control, inverse design, and real-to-sim transfer. We observe a substantial speedup obtained from using our gradient information in solving most of these applications.

The travelling salesperson problem (TSP) is a classic resource allocation problem used to find an optimal order of doing a set of tasks while minimizing (or maximizing) an associated objective function. It is widely used in robotics for applications such as planning and scheduling. In this work, we solve TSP for two objectives using reinforcement learning (RL). Often in multi-objective optimization problems, the associated objective functions can be conflicting in nature. In such cases, the optimality is defined in terms of Pareto optimality. A set of these Pareto optimal solutions in the objective space form a Pareto front (or frontier). Each solution has its trade-off. We present the Pareto frontier approximation network (PA-Net), a network that generates good approximations of the Pareto front for the bi-objective travelling salesperson problem (BTSP). Firstly, BTSP is converted into a constrained optimization problem. We then train our network to solve this constrained problem using the Lagrangian relaxation and policy gradient. With PA-Net we improve the performance over an existing deep RL-based method. The average improvement in the hypervolume metric, which is used to measure the optimality of the Pareto front, is 2.3%. At the same time, PA-Net has 4.5x faster inference time. Finally, we present the application of PA-Net to find optimal visiting order in a robotic navigation task/coverage planning. Our code is available on the project website.

The theoretical analysis of spectral clustering mainly focuses on consistency, while there is relatively little research on its generalization performance. In this paper, we study the excess risk bounds of the popular spectral clustering algorithms: \emph{relaxed} RatioCut and \emph{relaxed} NCut. Firstly, we show that their excess risk bounds between the empirical continuous optimal solution and the population-level continuous optimal solution have a $\mathcal{O}(1/\sqrt{n})$ convergence rate, where $n$ is the sample size. Secondly, we show the fundamental quantity in influencing the excess risk between the empirical discrete optimal solution and the population-level discrete optimal solution. At the empirical level, algorithms can be designed to reduce this quantity. Based on our theoretical analysis, we propose two novel algorithms that can not only penalize this quantity, but also cluster the out-of-sample data without re-eigendecomposition on the overall sample. Experiments verify the effectiveness of the proposed algorithms.

We consider the problem of designing uniformly stable first-order optimization algorithms for empirical risk minimization. Uniform stability is often used to obtain generalization error bounds for optimization algorithms, and we are interested in a general approach to achieve it. For Euclidean geometry, we suggest a black-box conversion which given a smooth optimization algorithm, produces a uniformly stable version of the algorithm while maintaining its convergence rate up to logarithmic factors. Using this reduction we obtain a (nearly) optimal algorithm for smooth optimization with convergence rate $\widetilde{O}(1/T^2)$ and uniform stability $O(T^2/n)$, resolving an open problem of Chen et al. (2018); Attia and Koren (2021). For more general geometries, we develop a variant of Mirror Descent for smooth optimization with convergence rate $\widetilde{O}(1/T)$ and uniform stability $O(T/n)$, leaving open the question of devising a general conversion method as in the Euclidean case.

Recently the "SP" (Stochastic Polyak step size) method has emerged as a competitive adaptive method for setting the step sizes of SGD. SP can be interpreted as a method specialized to interpolated models, since it solves the interpolation equations. SP solves these equation by using local linearizations of the model. We take a step further and develop a method for solving the interpolation equations that uses the local second-order approximation of the model. Our resulting method SP2 uses Hessian-vector products to speed-up the convergence of SP. Furthermore, and rather uniquely among second-order methods, the design of SP2 in no way relies on positive definite Hessian matrices or convexity of the objective function. We show SP2 is very competitive on matrix completion, non-convex test problems and logistic regression. We also provide a convergence theory on sums-of-quadratics.

In this paper, we tackle a significant challenge in PCA: heterogeneity. When data are collected from different sources with heterogeneous trends while still sharing some congruency, it is critical to extract shared knowledge while retaining unique features of each source. To this end, we propose personalized PCA (PerPCA), which uses mutually orthogonal global and local principal components to encode both unique and shared features. We show that, under mild conditions, both unique and shared features can be identified and recovered by a constrained optimization problem, even if the covariance matrices are immensely different. Also, we design a fully federated algorithm inspired by distributed Stiefel gradient descent to solve the problem. The algorithm introduces a new group of operations called generalized retractions to handle orthogonality constraints, and only requires global PCs to be shared across sources. We prove the linear convergence of the algorithm under suitable assumptions. Comprehensive numerical experiments highlight PerPCA's superior performance in feature extraction and prediction from heterogeneous datasets. As a systematic approach to decouple shared and unique features from heterogeneous datasets, PerPCA finds applications in several tasks including video segmentation, topic extraction, and distributed clustering.