ASPIRe: An Informative Trajectory Planner with Mutual Information Approximation for Target Search and Tracking

This paper proposes an informative trajectory planning approach, namely, \textit{adaptive particle filter tree with sigma point-based mutual information reward approximation} (ASPIRe), for mobile target search and tracking (SAT) in cluttered environments with limited sensing field of view. We develop a novel sigma point-based approximation to accurately estimate mutual information (MI) for general, non-Gaussian distributions utilizing particle representation of the belief state, while simultaneously maintaining high computational efficiency. Building upon the MI approximation, we develop the Adaptive Particle Filter Tree (APFT) approach with MI as the reward, which features belief state tree nodes for informative trajectory planning in continuous state and measurement spaces. An adaptive criterion is proposed in APFT to adjust the planning horizon based on the expected information gain. Simulations and physical experiments demonstrate that ASPIRe achieves real-time computation and outperforms benchmark methods in terms of both search efficiency and estimation accuracy.

SwarmPRM: Probabilistic Roadmap Motion Planning for Swarm Robotic Systems

Swarm robotic systems consisting of large-scale cooperative agents hold promise for performing autonomous tasks in diverse fields. However, existing planning strategies for swarm robotic systems often encounter a trade-off between scalability and solution quality. We introduce here SwarmPRM, a hierarchical, highly scalable, computationally efficient, and risk-aware sampling-based motion planning approach for swarm robotic systems, which is asymptotically optimal under mild assumptions. We employ probability density functions (PDFs) to represent the swarm's macroscopic state and utilize optimal mass transport (OMT) theory to measure the swarm's cost to go. A risk-aware Gaussian roadmap is constructed wherein each node encapsulates a distinct PDF and conditional-value-at-risk (CVaR) is employed to assess the collision risk, facilitating the generation of macroscopic PDFs in Wasserstein-GMM space. Extensive simulations demonstrate that the proposed approach outperforms state-of-the-art methods in terms of computational efficiency and the average travelling distance.

Sharp Analysis of Power Iteration for Tensor PCA

We investigate the power iteration algorithm for the tensor PCA model introduced in Richard and Montanari (2014). Previous work studying the properties of tensor power iteration is either limited to a constant number of iterations, or requires a non-trivial data-independent initialization. In this paper, we move beyond these limitations and analyze the dynamics of randomly initialized tensor power iteration up to polynomially many steps. Our contributions are threefold: First, we establish sharp bounds on the number of iterations required for power method to converge to the planted signal, for a broad range of the signal-to-noise ratios. Second, our analysis reveals that the actual algorithmic threshold for power iteration is smaller than the one conjectured in literature by a polylog(n) factor, where n is the ambient dimension. Finally, we propose a simple and effective stopping criterion for power iteration, which provably outputs a solution that is highly correlated with the true signal. Extensive numerical experiments verify our theoretical results.

Informative Path Planning of Autonomous Vehicle for Parking Occupancy Estimation

Parking occupancy estimation holds significant potential in facilitating parking resource management and mitigating traffic congestion. Existing approaches employ robotic systems to detect the occupancy status of individual parking spaces and primarily focus on enhancing detection accuracy through perception pipelines. However, these methods often overlook the crucial aspect of robot path planning, which can hinder the accurate estimation of the entire parking area. In light of these limitations, we introduce the problem of informative path planning for parking occupancy estimation using autonomous vehicles and formulate it as a Partially Observable Markov Decision Process (POMDP) task. Then, we develop an occupancy state transition model and introduce a Bayes filter to estimate occupancy based on noisy sensor measurements. Subsequently, we propose the Monte Carlo Bayes Filter Tree, a computationally efficient algorithm that leverages progressive widening to generate informative paths. We demonstrate that the proposed approach outperforms the benchmark methods in diverse simulation environments, effectively striking a balance between optimality and computational efficiency.

Probabilistic Visibility-Aware Trajectory Planning for Target Tracking in Cluttered Environments

Target tracking with a mobile robot has numerous significant applications in both civilian and military. Practical challenges such as limited field-of-view, obstacle occlusion, and system uncertainty may all adversely affect tracking performance, yet few existing works can simultaneously tackle these limitations. To bridge the gap, we introduce the concept of belief-space probability of detection (BPOD) to measure the predictive visibility of the target under stochastic robot and target states. An Extended Kalman Filter variant incorporating BPOD is developed to predict target belief state under uncertain visibility within the planning horizon. Furthermore, we propose a computationally efficient algorithm to uniformly calculate both BPOD and the chance-constrained collision risk by utilizing linearized signed distance function (SDF), and then design a two-stage strategy for lightweight calculation of SDF in sequential convex programming. Building upon these treatments, we develop a real-time, non-myopic trajectory planner for visibility-aware and safe target tracking in the presence of system uncertainty. The effectiveness of the proposed approach is verified by both simulations and real-world experiments.

Learning time-scales in two-layers neural networks

Gradient-based learning in multi-layer neural networks displays a number of striking features. In particular, the decrease rate of empirical risk is non-monotone even after averaging over large batches. Long plateaus in which one observes barely any progress alternate with intervals of rapid decrease. These successive phases of learning often take place on very different time scales. Finally, models learnt in an early phase are typically `simpler' or `easier to learn' although in a way that is difficult to formalize. Although theoretical explanations of these phenomena have been put forward, each of them captures at best certain specific regimes. In this paper, we study the gradient flow dynamics of a wide two-layer neural network in high-dimension, when data are distributed according to a single-index model (i.e., the target function depends on a one-dimensional projection of the covariates). Based on a mixture of new rigorous results, non-rigorous mathematical derivations, and numerical simulations, we propose a scenario for the learning dynamics in this setting. In particular, the proposed evolution exhibits separation of timescales and intermittency. These behaviors arise naturally because the population gradient flow can be recast as a singularly perturbed dynamical system.

Lower Bounds for the Convergence of Tensor Power Iteration on Random Overcomplete Models

Tensor decomposition serves as a powerful primitive in statistics and machine learning. In this paper, we focus on using power iteration to decompose an overcomplete random tensor. Past work studying the properties of tensor power iteration either requires a non-trivial data-independent initialization, or is restricted to the undercomplete regime. Moreover, several papers implicitly suggest that logarithmically many iterations (in terms of the input dimension) are sufficient for the power method to recover one of the tensor components. In this paper, we analyze the dynamics of tensor power iteration from random initialization in the overcomplete regime. Surprisingly, we show that polynomially many steps are necessary for convergence of tensor power iteration to any of the true component, which refutes the previous conjecture. On the other hand, our numerical experiments suggest that tensor power iteration successfully recovers tensor components for a broad range of parameters, despite that it takes at least polynomially many steps to converge. To further complement our empirical evidence, we prove that a popular objective function for tensor decomposition is strictly increasing along the power iteration path. Our proof is based on the Gaussian conditioning technique, which has been applied to analyze the approximate message passing (AMP) algorithm. The major ingredient of our argument is a conditioning lemma that allows us to generalize AMP-type analysis to non-proportional limit and polynomially many iterations of the power method.

Overparametrized linear dimensionality reductions: From projection pursuit to two-layer neural networks

Given a cloud of $n$ data points in $\mathbb{R}^d$, consider all projections onto $m$-dimensional subspaces of $\mathbb{R}^d$ and, for each such projection, the empirical distribution of the projected points. What does this collection of probability distributions look like when $n,d$ grow large? We consider this question under the null model in which the points are i.i.d. standard Gaussian vectors, focusing on the asymptotic regime in which $n,d\to\infty$, with $n/d\to\alpha\in (0,\infty)$, while $m$ is fixed. Denoting by $\mathscr{F}_{m, \alpha}$ the set of probability distributions in $\mathbb{R}^m$ that arise as low-dimensional projections in this limit, we establish new inner and outer bounds on $\mathscr{F}_{m, \alpha}$. In particular, we characterize the Wasserstein radius of $\mathscr{F}_{m,\alpha}$ up to logarithmic factors, and determine it exactly for $m=1$. We also prove sharp bounds in terms of Kullback-Leibler divergence and R\'{e}nyi information dimension. The previous question has application to unsupervised learning methods, such as projection pursuit and independent component analysis. We introduce a version of the same problem that is relevant for supervised learning, and prove a sharp Wasserstein radius bound. As an application, we establish an upper bound on the interpolation threshold of two-layers neural networks with $m$ hidden neurons.

Tractability from overparametrization: The example of the negative perceptron

In the negative perceptron problem we are given $n$ data points $({\boldsymbol x}_i,y_i)$, where ${\boldsymbol x}_i$ is a $d$-dimensional vector and $y_i\in\{+1,-1\}$ is a binary label. The data are not linearly separable and hence we content ourselves to find a linear classifier with the largest possible \emph{negative} margin. In other words, we want to find a unit norm vector ${\boldsymbol \theta}$ that maximizes $\min_{i\le n}y_i\langle {\boldsymbol \theta},{\boldsymbol x}_i\rangle$. This is a non-convex optimization problem (it is equivalent to finding a maximum norm vector in a polytope), and we study its typical properties under two random models for the data. We consider the proportional asymptotics in which $n,d\to \infty$ with $n/d\to\delta$, and prove upper and lower bounds on the maximum margin $\kappa_{\text{s}}(\delta)$ or -- equivalently -- on its inverse function $\delta_{\text{s}}(\kappa)$. In other words, $\delta_{\text{s}}(\kappa)$ is the overparametrization threshold: for $n/d\le \delta_{\text{s}}(\kappa)-\varepsilon$ a classifier achieving vanishing training error exists with high probability, while for $n/d\ge \delta_{\text{s}}(\kappa)+\varepsilon$ it does not. Our bounds on $\delta_{\text{s}}(\kappa)$ match to the leading order as $\kappa\to -\infty$. We then analyze a linear programming algorithm to find a solution, and characterize the corresponding threshold $\delta_{\text{lin}}(\kappa)$. We observe a gap between the interpolation threshold $\delta_{\text{s}}(\kappa)$ and the linear programming threshold $\delta_{\text{lin}}(\kappa)$, raising the question of the behavior of other algorithms.