Reeb spaces, as well as their discretized versions called Mappers, are common descriptors used in Topological Data Analysis, with plenty of applications in various fields of science, such as computational biology and data visualization, among others. The stability and quantification of the rate of convergence of the Mapper to the Reeb space has been studied a lot in recent works~\cite{Brown2019, Carriere2018a, Carriere2018, Munch2016}, focusing on the case where a scalar-valued filter is used for the computation of Mapper. On the other hand, much less is known in the multivariate case, where the domain of the filter is in $\mathbb R^d$ instead of $\mathbb R$. The only available result in this setting~\cite{Munch2016} only works for topological spaces and cannot be used as is for finite metric spaces representing data, such as point clouds and distance matrices. In this article, we present an approximation result for the Reeb space in the multivariate case using a Mapper-based estimator, which is a slight modification of the usual Mapper construction. Moreover, our approximation is stated with respect to a pseudometric that is an extension of the usual {\em interleaving distance} between persistence modules~\cite{Chazal2016}. Finally, we apply our results to the case where the filter function used to compute the Mapper is estimated from the data. We provide applications of this setting in statistics and machine learning and probability for different kinds of target filters, as well as numerical experiments that demonstrate the relevance of our approach.

Evaluation of Bayesian deep learning (BDL) methods is challenging. We often seek to evaluate the methods' robustness and scalability, assessing whether new tools give `better' uncertainty estimates than old ones. These evaluations are paramount for practitioners when choosing BDL tools on-top of which they build their applications. Current popular evaluations of BDL methods, such as the UCI experiments, are lacking: Methods that excel with these experiments often fail when used in application such as medical or automotive, suggesting a pertinent need for new benchmarks in the field. We propose a new BDL benchmark with a diverse set of tasks, inspired by a real-world medical imaging application on \emph{diabetic retinopathy diagnosis}. Visual inputs (512x512 RGB images of retinas) are considered, where model uncertainty is used for medical pre-screening---i.e. to refer patients to an expert when model diagnosis is uncertain. Methods are then ranked according to metrics derived from expert-domain to reflect real-world use of model uncertainty in automated diagnosis. We develop multiple tasks that fall under this application, including out-of-distribution detection and robustness to distribution shift. We then perform a systematic comparison of well-tuned BDL techniques on the various tasks. From our comparison we conclude that some current techniques which solve benchmarks such as UCI `overfit' their uncertainty to the dataset---when evaluated on our benchmark these underperform in comparison to simpler baselines. The code for the benchmark, its baselines, and a simple API for evaluating new BDL tools are made available at https://github.com/oatml/bdl-benchmarks.

With the growth of using cell phones and the increase in diversity of smart mobile devices, a massive volume of data is generated continuously in the process of using these devices. Among these data, Call Detail Records, CDR, is highly remarkable. Since CDR contains both temporal and spatial labels, mobility analysis of CDR is one of the favorite subjects of study among the researchers. The user next location prediction is one of the main problems in the field of human mobility analysis. In this paper, we propose a data processing framework to predict user next location. We propose domain-specific data processing strategies and design a deep neural network model which is based on recurrent neurons and perform regression tasks. Using this prediction framework, the error of the prediction decreases from 74% to 55% in comparison to the worst and best performing traditional models. Methods, strategies, the framework and the results of this paper can be helpful in many applications such as urban planning and digital marketing.

Consider a Bayesian inference problem where a variable of interest does not take values in a Euclidean space. These "non-standard" data structures are in reality fairly common. They are frequently used in problems involving latent discrete factor models, networks, and domain specific problems such as sequence alignments and reconstructions, pedigrees, and phylogenies. In principle, Bayesian inference should be particularly well-suited in such scenarios, as the Bayesian paradigm provides a principled way to obtain confidence assessment for random variables of any type. However, much of the recent work on making Bayesian analysis more accessible and computationally efficient has focused on inference in Euclidean spaces. In this paper, we introduce Blang, a domain specific language (DSL) and library aimed at bridging this gap. Blang allows users to perform Bayesian analysis on arbitrary data types while using a declarative syntax similar to BUGS. Blang is augmented with intuitive language additions to invent data types of the user's choosing. To perform inference at scale on such arbitrary state spaces, Blang leverages recent advances in parallelizable, non-reversible Markov chain Monte Carlo methods.

Reinforcement learning (RL) algorithms have been successfully applied to a range of challenging sequential decision making and control tasks. In this paper, we classify RL into direct and indirect methods according to how they seek optimal policy of the Markov Decision Process (MDP) problem. The former solves optimal policy by directly maximizing an objective function using gradient descent method, in which the objective function is usually the expectation of accumulative future rewards. The latter indirectly finds the optimal policy by solving the Bellman equation, which is the sufficient and necessary condition from Bellman's principle of optimality. We take vanilla policy gradient and approximate policy iteration to study their internal relationship, and reveal that both direct and indirect methods can be unified in actor-critic architecture and are equivalent if we always choose stationary state distribution of current policy as initial state distribution of MDP. Finally, we classify the current mainstream RL algorithms and compare the differences between other criteria including value-based and policy-based, model-based and model-free.

Model-based reinforcement learning algorithms make decisions by building and utilizing a model of the environment. However, none of the existing algorithms attempts to infer the dynamics of any state-action pair from known state-action pairs before meeting it for sufficient times. We propose a new model-based method called Greedy Inference Model (GIM) that infers the unknown dynamics from known dynamics based on the internal spectral properties of the environment. In other words, GIM can "learn by analogy". We further introduce a new exploration strategy which ensures that the agent rapidly and evenly visits unknown state-action pairs. GIM is much more computationally efficient than state-of-the-art model-based algorithms, as the number of dynamic programming operations is independent of the environment size. Lower sample complexity could also be achieved under mild conditions compared against methods without inferring. Experimental results demonstrate the effectiveness and efficiency of GIM in a variety of real-world tasks.

To simultaneously capture syntax and global semantics from a text corpus, we propose a new larger-context recurrent neural network (RNN) based language model, which extracts recurrent hierarchical semantic structure via a dynamic deep topic model to guide natural language generation. Moving beyond a conventional RNN based language model that ignores long-range word dependencies and sentence order, the proposed model captures not only intra-sentence word dependencies, but also temporal transitions between sentences and inter-sentence topic dependences. For inference, we develop a hybrid of stochastic-gradient MCMC and recurrent autoencoding variational Bayes. Experimental results on a variety of real-world text corpora demonstrate that the proposed model not only outperforms state-of-the-art larger-context RNN-based language models, but also learns interpretable recurrent multilayer topics and generates diverse sentences and paragraphs that are syntactically correct and semantically coherent.

Markov Decision Process (MDP) problems can be solved using Dynamic Programming (DP) methods which suffer from the curse of dimensionality and the curse of modeling. To overcome these issues, Reinforcement Learning (RL) methods are adopted in practice. In this paper, we aim to obtain the optimal admission control policy in a system where different classes of customers are present. Using DP techniques, we prove that it is optimal to admit the $i$ th class of customers only upto a threshold $\tau(i)$ which is a non-increasing function of $i$. Contrary to traditional RL algorithms which do not take into account the structural properties of the optimal policy while learning, we propose a structure-aware learning algorithm which exploits the threshold structure of the optimal policy. We prove the asymptotic convergence of the proposed algorithm to the optimal policy. Due to the reduction in the policy space, the structure-aware learning algorithm provides remarkable improvements in storage and computational complexities over classical RL algorithms. Simulation results also establish the gain in the convergence rate of the proposed algorithm over other RL algorithms. The techniques presented in the paper can be applied to any general MDP problem covering various applications such as inventory management, financial planning and communication networking.

We give a complete characterization of the sampling complexity of best Markovian arm identification in one-parameter Markovian bandit models. We derive instance specific nonasymptotic and asymptotic lower bounds which generalize those of the IID setting. We analyze the Track-and-Stop strategy, initially proposed for the IID setting, and we prove that asymptotically it is at most a factor of four apart from the lower bound. Our one-parameter Markovian bandit model is based on the notion of an exponential family of stochastic matrices for which we establish many useful properties. For the analysis of the Track-and-Stop strategy we derive a novel concentration inequality for Markov chains that may be of interest in its own right.

New methodologies will be needed to ensure the airspace remains safe and efficient as traffic densities rise to accommodate new unmanned operations. This paper explores how unmanned free-flight traffic may operate in dense airspace. We develop and analyze autonomous collision avoidance systems for aircraft operating in dense airspace where traditional collision avoidance systems fail. We propose a metric for quantifying the decision burden on a collision avoidance system as well as a metric for measuring the impact of the collision avoidance system on airspace. We use deep reinforcement learning to compute corrections for an existing collision avoidance approach to account for dense airspace. The results show that a corrected collision avoidance system can operate more efficiently than traditional methods in dense airspace while maintaining high levels of safety.