Learning of continuous exponential family distributions with unbounded support remains an important area of research for both theory and applications in high-dimensional statistics. In recent years, score matching has become a widely used method for learning exponential families with continuous variables due to its computational ease when compared against maximum likelihood estimation. However, theoretical understanding of the statistical properties of score matching is still lacking. In this work, we provide a non-asymptotic sample complexity analysis for learning the structure of exponential families of polynomials with score matching. The derived sample bounds show a polynomial dependence on the model dimension. These bounds are the first of its kind, as all prior work has shown only asymptotic bounds on the sample complexity.

As quantum computers become available to the general public, the need has arisen to train a cohort of quantum programmers, many of whom have been developing classical computer programs for most of their careers. While currently available quantum computers have less than 100 qubits, quantum computing hardware is widely expected to grow in terms of qubit count, quality, and connectivity. This review aims at explaining the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional. We give an introduction to quantum computing algorithms and their implementation on real quantum hardware. We survey 20 different quantum algorithms, attempting to describe each in a succinct and self-contained fashion. We show how these algorithms can be implemented on IBM’s quantum computer, and in each case, we discuss the results of the implementation with respect to differences between the simulator and the actual hardware runs. This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations.

The usual setting for learning the structure and parameters of a graphical model assumes the availability of independent samples produced from the corresponding multivariate probability distribution. However, for many models the mixing time of the respective Markov chain can be very large and i.i.d. samples may not be obtained. We study the problem of reconstructing binary graphical models from correlated samples produced by a dynamical process, which is natural in many applications. We analyze the sample complexity of two estimators that are based on the interaction screening objective and the conditional likelihood loss. We observe that for samples coming from a dynamical process far from equilibrium, the sample complexity reduces exponentially compared to a dynamical process that mixes quickly.

Spreading processes play an increasingly important role in modeling for diffusion networks, information propagation, marketing and opinion setting. We address the problem of learning of a spreading model such that the predictions generated from this model are accurate and could be subsequently used for the optimization, and control of diffusion dynamics. We focus on a challenging setting where full observations of the dynamics are not available, and standard approaches such as maximum likelihood quickly become intractable for large network instances. We introduce a computationally efficient algorithm, based on a scalable dynamic message-passing approach, which is able to learn parameters of the effective spreading model given only limited information on the activation times of nodes in the network. The popular Independent Cascade model is used to illustrate our approach. We show that tractable inference from the learned model generates a better prediction of marginal probabilities compared to the original model. We develop a systematic procedure for learning a mixture of models which further improves the prediction quality.

What is the optimal number of independent observations from which a sparse Gaussian Graphical Model can be correctly recovered? Information-theoretic arguments provide a lower bound on the minimum number of samples necessary to perfectly identify the support of any multivariate normal distribution as a function of model parameters. For a model defined on a sparse graph with $p$ nodes, a maximum degree $d$ and minimum normalized edge strength $\kappa$, this necessary number of samples scales at least as $d \log p/\kappa^2$. The sample complexity requirements of existing methods for perfect graph reconstruction exhibit dependency on additional parameters that do not enter in the lower bound. The question of whether the lower bound is tight and achievable by a polynomial time algorithm remains open. In this paper, we constructively answer this question and propose an algorithm, termed DICE, whose sample complexity matches the information-theoretic lower bound up to a universal constant factor. We also propose a related algorithm SLICE that has a slightly higher sample complexity, but can be implemented as a mixed integer quadratic program which makes it attractive in practice. Importantly, SLICE retains a critical advantage of DICE in that its sample complexity only depends on quantities present in the information theoretic lower bound. We anticipate that this result will stimulate future search of computationally efficient sample-optimal algorithms.

Graphical models are widely used in science to represent joint probability distributions with an underlying conditional dependence structure. The inverse problem of learning a discrete graphical model given i.i.d samples from its joint distribution can be solved with near-optimal sample complexity using a convex optimization method known as Generalized Regularized Interaction Screening Estimator (GRISE). But the computational cost of GRISE becomes prohibitive when the energy function of the true graphical model has higher order terms. We introduce NeurISE, a neural net based algorithm for graphical model learning, to tackle this limitation of GRISE. We use neural nets as function approximators in an Interaction Screening objective function. The optimization of this objective then produces a neural-net representation for the conditionals of the graphical model. NeurISE algorithm is seen to be a better alternative to GRISE when the energy function of the true model has a high order with a high degree of symmetry. In these cases NeurISE is able to find the correct parsimonious representation for the conditionals without being fed any prior information about the true model. NeurISE can also be used to learn the underlying structure of the true model with some simple modifications to its training procedure. In addition, we also show a variant of NeurISE that can be used to learn a neural net representation for the full energy function of the true model.

Graphical models are useful tools for describing structured high-dimensional probability distributions. Development of efficient algorithms for learning graphical models with least amount of data remains an active research topic. Reconstruction of graphical models that describe the statistics of discrete variables is a particularly challenging problem, for which the maximum likelihood approach is intractable. In this work, we provide the first sample-efficient method based on the Interaction Screening framework that allows one to provably learn fully general discrete factor models with node-specific discrete alphabets and multi-body interactions, specified in an arbitrary basis. We identify a single condition related to model parametrization that leads to rigorous guarantees on the recovery of model structure and parameters in any error norm, and is readily verifiable for a large class of models. Importantly, our bounds make explicit distinction between parameters that are proper to the model and priors used as an input to the algorithm. Finally, we show that the Interaction Screening framework includes all models previously considered in the literature as special cases, and for which our analysis shows a systematic improvement in sample complexity.

Spreading processes are often modelled as a stochastic dynamics occurring on top of a given network with edge weights corresponding to the transmission probabilities. Knowledge of veracious transmission probabilities is essential for prediction, optimization, and control of diffusion dynamics. Unfortunately, in most cases the transmission rates are unknown and need to be reconstructed from the spreading data. Moreover, in realistic settings it is impossible to monitor the state of each node at every time, and thus the data is highly incomplete. We introduce an efficient dynamic message-passing algorithm, which is able to reconstruct parameters of the spreading model given only partial information on the activation times of nodes in the network. The method is generalizable to a large class of dynamic models, as well to the case of temporal graphs.

We consider the problem of learning the underlying graph of an unknown Ising model on p spins from a collection of i.i.d. samples generated from the model. We suggest a new estimator that is computationally efficient and requires a number of samples that is near-optimal with respect to previously established information theoretic lower-bound. Our statistical estimator has a physical interpretation in terms of "interaction screening". The estimator is consistent and is efficiently implemented using convex optimization. We prove that with appropriate regularization, the estimator recovers the underlying graph using a number of samples that is logarithmic in the system size p and exponential in the maximum coupling-intensity and maximum node-degree.