Taking into account future risk is essential for an autonomously operating robot to find online not only the best but also a safe action to execute. In this paper, we build upon the recently introduced formulation of probabilistic belief-dependent constraints. In our methodology safety can be materialized with any general belief-dependent operator we call payoff. We present an anytime approach employing the Monte Carlo Tree Search (MCTS) method in continuous domains in terms of states, actions and observations and general-belief dependent reward and payoff operators. Unlike previous approaches, our method ensures safety anytime with respect to the currently expanded search tree without relying on the convergence of the search. We prove convergence in probability with an exponential rate of a version of our algorithms and study proposed techniques via extensive simulations. Even with a tiny number of tree queries, the best action found by our approach is much safer than the baseline. Moreover, our approach constantly yields better than the baseline action in terms of objective function. This is because we revise the values and statistics maintained in the search tree and remove from them the contribution of the pruned actions. We rigorously show that our cleaning routine is necessary. Without it, at the limit of convergence of MCTS, an infinite amount of sampled dangerous actions can be detrimental to the objective function.

Online decision making under uncertainty in partially observable domains, also known as Belief Space Planning, is a fundamental problem in Robotics and Artificial Intelligence. Due to an abundance of plausible future unravelings, calculating an optimal course of action inflicts an enormous computational burden on the agent. Moreover, in many scenarios, e.g., Information gathering, it is required to introduce a belief-dependent constraint. Prompted by this demand, in this article, we consider a recently introduced probabilistic belief-dependent constrained partially observable Markov decision process (POMDP). We present a technique to adaptively accept or discard a candidate action sequence with respect to a probabilistic belief-dependent constraint, before expanding a complete set of sampled future observations episodes and without any loss in accuracy. Moreover, using our proposed framework, we contribute an adaptive method to find a maximal feasible return (e.g., Information Gain) in terms of Value at Risk and a corresponding action sequence, given a set of candidate action sequences, with substantial acceleration. On top of that, we introduce anadaptive simplificationtechnique for a probabilistically constrained setting. Such an approach provably returns an identical-quality solution while dramatically accelerating the online decision making. Our universal framework applies to any belief-dependent constrained continuous POMDP with parameteric beliefs, as well as nonparameteric beliefs represented by particles. In the context of an information-theoretic constraint, our presented framework stochastically quantifies if a cumulative Information Gain along the planning horizon is sufficiently significant (for e.g., Information Gathering, active simultaneous localization and mapping (SLAM)). As a case study, we apply our method to two challenging problems of high dimensional belief space planning: active SLAM and sensor deployment. Extensive realistic simulations corroborate the superiority of our proposed ideas.

It is a long-standing objective to ease the computation burden incurred by the decision-making problem under partial observability. Identifying the sensitivity to simplification of various components of the original problem has tremendous ramifications. Yet, algorithms for decision-making under uncertainty usually lean on approximations or heuristics without quantifying their effect. Therefore, challenging scenarios could severely impair the performance of such methods. In this paper, we extend the decision-making mechanism to the whole by removing standard approximations and considering all previously suppressed stochastic sources of variability. On top of this extension, we scrutinize the distribution of the return. We begin from a return given a single candidate policy and continue to the pair of returns given a corresponding pair of candidate policies. Furthermore, we present novel stochastic bounds on the return and novel tools, Probabilistic Loss (PLoss) and its online accessible counterpart (PbLoss), to characterize the effect of a simplification.

We study how the topology of a data set $M = M_a \cup M_b \subseteq \mathbb{R}^d$, representing two classes $a$ and $b$ in a binary classification problem, changes as it passes through the layers of a well-trained neural network, i.e., one with perfect accuracy on training set and near-zero generalization error ($\approx 0.01\%$). The goal is to shed light on two mysteries in deep neural networks: (i) a nonsmooth activation function like ReLU outperforms a smooth one like hyperbolic tangent; (ii) successful neural network architectures rely on having many layers, even though a shallow network can approximate any function arbitrarily well. We performed extensive experiments on the persistent homology of a wide range of point cloud data sets, both real and simulated. The results consistently demonstrate the following: (1) Neural networks operate by changing topology, transforming a topologically complicated data set into a topologically simple one as it passes through the layers. No matter how complicated the topology of $M$ we begin with, when passed through a well-trained neural network $f : \mathbb{R}^d \to \mathbb{R}^p$, there is a vast reduction in the Betti numbers of both components $M_a$ and $M_b$; in fact they nearly always reduce to their lowest possible values: $\beta_k\bigl(f(M_i)\bigr) = 0$ for $k \ge 1$ and $\beta_0\bigl(f(M_i)\bigr) = 1$, $i =a, b$. (2) The reduction in Betti numbers is significantly faster for ReLU activation than for hyperbolic tangent activation as the former defines nonhomeomorphic maps that change topology, whereas the latter defines homeomorphic maps that preserve topology. (3) Shallow and deep networks transform data sets differently --- a shallow network operates mainly through changing geometry and changes topology only in its final layers, a deep one spreads topological changes more evenly across all layers.

In many areas of neuroscience and biological data analysis, it is desired to reveal common patterns among a group of subjects. Such analyses play important roles e.g., in detecting functional brain networks from fMRI scans and in identifying brain regions which show increased activity in response to certain stimuli. Group level techniques usually assume that all subjects in the group behave according to a single statistical model, or that deviations from the common model have simple parametric forms. Therefore, complex subject-specific deviations from the common model severely impair the performance of such methods. In this paper, we propose nonparametric algorithms for estimating the common covariance matrix and the common density function of several variables in a heterogeneous group of subjects. Our estimates converge to the true model as the number of subjects tends to infinity, under very mild conditions. We illustrate the effectiveness of our methods through extensive simulations as well as on real-data from fMRI scans and from arterial blood pressure and photoplethysmogram measurements.

Automated driving has become increasingly viable through the deployment of a number of sensing technologies on vehicles. These intelligent transportation systems (ITS) employ sensors such as radar, camera and lidars for collision avoidance and vehicle-to-everything (V2X) communication links for active environment sensing. Since the available spectrum for vehicular systems is limited, spectrum sharing in ITS sensors is a subject of active investigation. This paper presents a spectrum sharing technology that enables interference-free operation of an automotive radar and a V2X communication system within a common spectrum. Both systems dynamically share information with each other and optimize their spectral resources to the changing RF environment. The V2X system is a cognitive radio that is capable of blind sensing its spectrum using very low sampling and processing rates. The radar system is also modeled as a cognitive system that employs a Xampling-based sub-Nyquist receiver and transmits in several narrow bands that occupy a fraction of the conventional radar bandwidth. We present a hardware realization of this vehicular spectrum sharing technology and demonstrate spectral coexistence through real-time experiments.