We consider a point cloud $X_n := \{ x_1, \dots, x_n \}$ uniformly distributed on the flat torus $\mathbb{T}^d : = \mathbb{R}^d / \mathbb{Z}^d $, and construct a geometric graph on the cloud by connecting points that are within distance $\epsilon$ of each other. We let $\mathcal{P}(X_n)$ be the space of probability measures on $X_n$ and endow it with a discrete Wasserstein distance $W_n$ as introduced independently by Maas and Zhou et al. for general finite Markov chains. We show that as long as $\epsilon= \epsilon_n$ decays towards zero slower than an explicit rate depending on the level of uniformity of $X_n$, then the space $(\mathcal{P}(X_n), W_n)$ converges in the Gromov-Hausdorff sense towards the space of probability measures on $\mathbb{T}^d$ endowed with the Wasserstein distance.

Hidden Markov Models (HMMs) are a ubiquitous tool to model time series data, and have been widely used in two main tasks of Automatic Music Transcription (AMT): note segmentation, i.e. identifying the played notes after a multi-pitch estimation, and sequential post-processing, i.e. correcting note segmentation using training data. In this paper, we employ the multi-pitch estimation method called Probabilistic Latent Component Analysis (PLCA), and develop AMT systems by integrating different HMM-based modules in this framework. For note segmentation, we use two different twostate on/o? HMMs, including a higher-order one for duration modeling. For sequential post-processing, we focused on a musicological modeling of polyphonic harmonic transitions, using a first- and second-order HMMs whose states are defined through candidate note mixtures. These different PLCA plus HMM systems have been evaluated comparatively on two different instrument repertoires, namely the piano (using the MAPS database) and the marovany zither. Our results show that the use of HMMs could bring noticeable improvements to transcription results, depending on the instrument repertoire.

We introduce interacting particle Markov chain Monte Carlo (iPMCMC), a PMCMC method based on an interacting pool of standard and conditional sequential Monte Carlo samplers. Like related methods, iPMCMC is a Markov chain Monte Carlo sampler on an extended space. We present empirical results that show significant improvements in mixing rates relative to both non-interacting PMCMC samplers, and a single PMCMC sampler with an equivalent memory and computational budget. An additional advantage of the iPMCMC method is that it is suitable for distributed and multi-core architectures.

This paper presents sampling-based speech parameter generation using moment-matching networks for Deep Neural Network (DNN)-based speech synthesis. Although people never produce exactly the same speech even if we try to express the same linguistic and para-linguistic information, typical statistical speech synthesis produces completely the same speech, i.e., there is no inter-utterance variation in synthetic speech. To give synthetic speech natural inter-utterance variation, this paper builds DNN acoustic models that make it possible to randomly sample speech parameters. The DNNs are trained so that they make the moments of generated speech parameters close to those of natural speech parameters. Since the variation of speech parameters is compressed into a low-dimensional simple prior noise vector, our algorithm has lower computation cost than direct sampling of speech parameters. As the first step towards generating synthetic speech that has natural inter-utterance variation, this paper investigates whether or not the proposed sampling-based generation deteriorates synthetic speech quality. In evaluation, we compare speech quality of conventional maximum likelihood-based generation and proposed sampling-based generation. The result demonstrates the proposed generation causes no degradation in speech quality.

Personalized predictive medicine necessitates the modeling of patient illness and care processes, which inherently have long-term temporal dependencies. Healthcare observations, recorded in electronic medical records, are episodic and irregular in time. We introduce DeepCare, an end-to-end deep dynamic neural network that reads medical records, stores previous illness history, infers current illness states and predicts future medical outcomes. At the data level, DeepCare represents care episodes as vectors in space, models patient health state trajectories through explicit memory of historical records. Built on Long Short-Term Memory (LSTM), DeepCare introduces time parameterizations to handle irregular timed events by moderating the forgetting and consolidation of memory cells. DeepCare also incorporates medical interventions that change the course of illness and shape future medical risk. Moving up to the health state level, historical and present health states are then aggregated through multiscale temporal pooling, before passing through a neural network that estimates future outcomes. We demonstrate the efficacy of DeepCare for disease progression modeling, intervention recommendation, and future risk prediction. On two important cohorts with heavy social and economic burden -- diabetes and mental health -- the results show improved modeling and risk prediction accuracy.

It is common to subsample Markov chain output to reduce the storage burden. Geyer (1992) shows that discarding $k-1$ out of every $k$ observations will not improve statistical efficiency, as quantified through variance in a given computational budget. That observation is often taken to mean that thinning MCMC output cannot improve statistical efficiency. Here we suppose that it costs one unit of time to advance a Markov chain and then $\theta>0$ units of time to compute a sampled quantity of interest. For a thinned process, that cost $\theta$ is incurred less often, so it can be advanced through more stages. Here we provide examples to show that thinning will improve statistical efficiency if $\theta$ is large and the sample autocorrelations decay slowly enough. If the lag $\ell\ge1$ autocorrelations of a scalar measurement satisfy $\rho_\ell\ge\rho_{\ell+1}\ge0$, then there is always a $\theta<\infty$ at which thinning becomes more efficient for averages of that scalar. Many sample autocorrelation functions resemble first order AR(1) processes with $\rho_\ell =\rho^{|\ell|}$ for some $-1<\rho<1$. For an AR(1) process it is possible to compute the most efficient subsampling frequency $k$. The optimal $k$ grows rapidly as $\rho$ increases towards $1$. The resulting efficiency gain depends primarily on $\theta$, not $\rho$. Taking $k=1$ (no thinning) is optimal when $\rho\le0$. For $\rho>0$ it is optimal if and only if $\theta \le (1-\rho)^2/(2\rho)$. This efficiency gain never exceeds $1+\theta$. This paper also gives efficiency bounds for autocorrelations bounded between those of two AR(1) processes.

Reinforcement learning is considered to be a strong AI paradigm which can be used to teach machines through interaction with the environment and learning from their mistakes. Despite its perceived utility, it has not yet been successfully applied in automotive applications. Motivated by the successful demonstrations of learning of Atari games and Go by Google DeepMind, we propose a framework for autonomous driving using deep reinforcement learning. This is of particular relevance as it is difficult to pose autonomous driving as a supervised learning problem due to strong interactions with the environment including other vehicles, pedestrians and roadworks. As it is a relatively new area of research for autonomous driving, we provide a short overview of deep reinforcement learning and then describe our proposed framework. It incorporates Recurrent Neural Networks for information integration, enabling the car to handle partially observable scenarios. It also integrates the recent work on attention models to focus on relevant information, thereby reducing the computational complexity for deployment on embedded hardware. The framework was tested in an open source 3D car racing simulator called TORCS. Our simulation results demonstrate learning of autonomous maneuvering in a scenario of complex road curvatures and simple interaction of other vehicles.

The pseudo-likelihood method is one of the most popular algorithms for learning sparse binary pairwise Markov networks. In this paper, we formulate the $L_1$ regularized pseudo-likelihood problem as a sparse multiple logistic regression problem. In this way, many insights and optimization procedures for sparse logistic regression can be applied to the learning of discrete Markov networks. Specifically, we use the coordinate descent algorithm for generalized linear models with convex penalties, combined with strong screening rules, to solve the pseudo-likelihood problem with $L_1$ regularization. Therefore a substantial speedup without losing any accuracy can be achieved. Furthermore, this method is more stable than the node-wise logistic regression approach on unbalanced high-dimensional data when penalized by small regularization parameters. Thorough numerical experiments on simulated data and real world data demonstrate the advantages of the proposed method.

The Gibbs sampler is a particularly popular Markov chain used for learning and inference problems in Graphical Models (GMs). These tasks are computationally intractable in general, and the Gibbs sampler often suffers from slow mixing. In this paper, we study the Swendsen-Wang dynamics which is a more sophisticated Markov chain designed to overcome bottlenecks that impede the Gibbs sampler. We prove O(\log n) mixing time for attractive binary pairwise GMs (i.e., ferromagnetic Ising models) on stochastic partitioned graphs having n vertices, under some mild conditions, including low temperature regions where the Gibbs sampler provably mixes exponentially slow. Our experiments also confirm that the Swendsen-Wang sampler significantly outperforms the Gibbs sampler when they are used for learning parameters of attractive GMs.

In many sequential decision-making problems one is interested in minimizing an expected cumulative cost while taking into account \emph{risk}, i.e., increased awareness of events of small probability and high consequences. Accordingly, the objective of this paper is to present efficient reinforcement learning algorithms for risk-constrained Markov decision processes (MDPs), where risk is represented via a chance constraint or a constraint on the conditional value-at-risk (CVaR) of the cumulative cost. We collectively refer to such problems as percentile risk-constrained MDPs. Specifically, we first derive a formula for computing the gradient of the Lagrangian function for percentile risk-constrained MDPs. Then, we devise policy gradient and actor-critic algorithms that (1) estimate such gradient, (2) update the policy in the descent direction, and (3) update the Lagrange multiplier in the ascent direction. For these algorithms we prove convergence to locally optimal policies. Finally, we demonstrate the effectiveness of our algorithms in an optimal stopping problem and an online marketing application.