We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough ($L^\infty$) coefficients with rigorous a-priori accuracy and performance estimates. The method is discovered through a decision/game theory formulation of the problems of (1) identifying restriction and interpolation operators (2) recovering a signal from incomplete measurements based on norm constraints on its image under a linear operator (3) gambling on the value of the solution of the PDE based on a hierarchy of nested measurements of its solution or source term. The resulting elementary gambles form a hierarchy of (deterministic) basis functions of $H^1_0(\Omega)$ (gamblets) that (1) are orthogonal across subscales/subbands with respect to the scalar product induced by the energy norm of the PDE (2) enable sparse compression of the solution space in $H^1_0(\Omega)$ (3) induce an orthogonal multiresolution operator decomposition. The operating diagram of the multigrid method is that of an inverted pyramid in which gamblets are computed locally (by virtue of their exponential decay), hierarchically (from fine to coarse scales) and the PDE is decomposed into a hierarchy of independent linear systems with uniformly bounded condition numbers. The resulting algorithm is parallelizable both in space (via localization) and in bandwith/subscale (subscales can be computed independently from each other). Although the method is deterministic it has a natural Bayesian interpretation under the measure of probability emerging (as a mixed strategy) from the information game formulation and multiresolution approximations form a martingale with respect to the filtration induced by the hierarchy of nested measurements.

This paper studies the cooperative training of two probabilistic models of signals such as images. Both models are parametrized by convolutional neural networks (ConvNets). The first network is a descriptor network, which is an exponential family model or an energy-based model, whose feature statistics or energy function are defined by a bottom-up ConvNet, which maps the observed signal to the feature statistics. The second network is a generator network, which is a non-linear version of factor analysis. It is defined by a top-down ConvNet, which maps the latent factors to the observed signal. The maximum likelihood training algorithms of both the descriptor net and the generator net are in the form of alternating back-propagation, and both algorithms involve Langevin sampling. We observe that the two training algorithms can cooperate with each other by jumpstarting each other's Langevin sampling, and they can be naturally and seamlessly interwoven into a CoopNets algorithm that can train both nets simultaneously.

Bayesian inference is a way to get sharper predictions from your data. It's particularly useful when you don't have as much data as you would like and want to juice every last bit of predictive strength from it. Although it is sometimes described with reverence, Bayesian inference isn't magic or mystical. And even though the math under the hood can get dense, the concepts behind it are completely accessible. In brief, Bayesian inference lets you draw stronger conclusions from your data by folding in what you already know about the answer. Bayesian inference is based on the ideas of Thomas Bayes, a nonconformist Presbyterian minister in London about 300 years ago. He wrote two books, one on theology, and one on probability. His work included his now famous Bayes Theorem in raw form, which has since been applied to the problem of inference, the technical term for educated guessing. The popularity of Bayes' ideas was aided immeasurably by another minister, Richard Price. He saw their significance, refined them and published them. It would be more accurate and historically just to call Bayes' Theorem the Bayes-Price Rule.

We consider the problem of estimating the underlying graph associated with a Markov random field, with the added twist that the decoding algorithm can iteratively choose which subsets of nodes to sample based on the previous samples, resulting in an active learning setting. Considering both Ising and Gaussian models, we provide algorithm-independent lower bounds for high-probability recovery within the class of degree-bounded graphs. Our main results are minimax lower bounds for the active setting that match the best known lower bounds for the passive setting, which in turn are known to be tight in several cases of interest. Our analysis is based on Fano's inequality, along with novel mutual information bounds for the active learning setting, and the application of restricted graph ensembles. While we consider ensembles that are similar or identical to those used in the passive setting, we require different analysis techniques, with a key challenge being bounding a mutual information quantity associated with observed subsets of nodes, as opposed to full observations.

Point patterns are sets or multi-sets of unordered elements that can be found in numerous data sources. However, in data analysis tasks such as classification and novelty detection, appropriate statistical models for point pattern data have not received much attention. This paper proposes the modelling of point pattern data via random finite sets (RFS). In particular, we propose appropriate likelihood functions, and a maximum likelihood estimator for learning a tractable family of RFS models. In novelty detection, we propose novel ranking functions based on RFS models, which substantially improve performance.

This paper proposes a hierarchical feature extractor for non-stationary streaming time series based on the concept of switching observable Markov chain models. The slow time-scale non-stationary behaviors are considered to be a mixture of quasi-stationary fast time-scale segments that are exhibited by complex dynamical systems. The idea is to model each unique stationary characteristic without a priori knowledge (e.g., number of possible unique characteristics) at a lower logical level, and capture the transitions from one low-level model to another at a higher level. In this context, the concepts in the recently developed Symbolic Dynamic Filtering (SDF) is extended, to build an online algorithm suited for handling quasi-stationary data at a lower level and a non-stationary behavior at a higher level without a priori knowledge. A key observation made in this study is that the rate of change of data likelihood seems to be a better indicator of change in data characteristics compared to the traditional methods that mostly consider data likelihood for change detection. The algorithm minimizes model complexity and captures data likelihood. Efficacy demonstration and comparative evaluation of the proposed algorithm are performed using time series data simulated from systems that exhibit nonlinear dynamics. We discuss results that show that the proposed hierarchical SDF algorithm can identify underlying features with significantly high degree of accuracy, even under very noisy conditions. Algorithm is demonstrated to perform better than the baseline Hierarchical Dirichlet Process-Hidden Markov Models (HDP-HMM). The low computational complexity of algorithm makes it suitable for on-board, real time operations.

The use of Bayesian methods in large-scale data settings is attractive because of the rich hierarchical models, uncertainty quantification, and prior specification they provide. Standard Bayesian inference algorithms are computationally expensive, however, making their direct application to large datasets difficult or infeasible. Recent work on scaling Bayesian inference has focused on modifying the underlying algorithms to, for example, use only a random data subsample at each iteration. We leverage the insight that data is often redundant to instead obtain a weighted subset of the data (called a coreset) that is much smaller than the original dataset. We can then use this small coreset in any number of existing posterior inference algorithms without modification. In this paper, we develop an efficient coreset construction algorithm for Bayesian logistic regression models. We provide theoretical guarantees on the size and approximation quality of the coreset -- both for fixed, known datasets, and in expectation for a wide class of data generative models. Crucially, the proposed approach also permits efficient construction of the coreset in both streaming and parallel settings, with minimal additional effort. We demonstrate the efficacy of our approach on a number of synthetic and real-world datasets, and find that, in practice, the size of the coreset is independent of the original dataset size. Furthermore, constructing the coreset takes a negligible amount of time compared to that required to run MCMC on it.

Consumer Demand Response (DR) is an important research and industry problem, which seeks to categorize, predict and modify consumer's energy consumption. Unfortunately, traditional clustering methods have resulted in many hundreds of clusters, with a given consumer often associated with several clusters, making it difficult to classify consumers into stable representative groups and to predict individual energy consumption patterns. In this paper, we present a shape-based approach that better classifies and predicts consumer energy consumption behavior at the household level. The method is based on Dynamic Time Warping. DTW seeks an optimal alignment between energy consumption patterns reflecting the effect of hidden patterns of regular consumer behavior. Using real consumer 24-hour load curves from Opower Corporation, our method results in a 50% reduction in the number of representative groups and an improvement in prediction accuracy measured under DTW distance. We extend the approach to estimate which electrical devices will be used and in which hours.

Includes profile, demo downloads, and job openings. Developer of software systems that solve resource optimization, planning, scheduling, and deployment problems for the air transportation, gaming, healthcare, hospitality, and security industries. Source for neural network based data modeling, prediction, forecasting and optimization solutions. Areas of focus includes: Banking and Finance, Manufacturing, Marketing, Medical. Uses artificial-intelligence technologies to prevent fraud in transaction environments such as finance, e-commerce, telecommunications, and insurance.

Weighted model counting (WMC) has recently emerged as an effective and general approach to probabilistic inference, offering a computational framework for encoding a variety of formalisms, such as factor graphs and Bayesian networks.The advent of large-scale probabilistic knowledge bases has generated further interest in relational probabilistic representations, obtained by according weights to first-order formulas, whose semantics is given in terms of the ground theory, and solved by WMC. A fundamental limitation is that the domain of quantification, by construction and design, is assumed to be finite, which is at odds with areas such as vision and language understanding, where the existence of objects must be inferred from raw data. Dropping the finite-domain assumption has been known to improve the expressiveness of a first-order language for open-universe purposes, but these languages, so far, have eluded WMC approaches. In this paper, we revisit relational probabilistic models over an infinite domain, and establish a number of results that permit effective algorithms. We demonstrate this language on a number of examples, including a parameterized version of Pearl's Burglary-Earthquake-Alarm Bayesian network.