Nearest-Neighbor Interaction Systems in the Tensor-Train Format

Low-rank tensor approximation approaches have become an important tool in the scientific computing community. The aim is to enable the simulation and analysis of high-dimensional problems which cannot be solved using conventional methods anymore due to the so-called curse of dimensionality. This requires techniques to handle linear operators defined on extremely large state spaces and to solve the resulting systems of linear equations or eigenvalue problems. In this paper, we present a systematic tensor-train decomposition for nearest-neighbor interaction systems which is applicable to a host of different problems. With the aid of this decomposition, it is possible to reduce the memory consumption as well as the computational costs significantly. Furthermore, it can be shown that in some cases the rank of the tensor decomposition does not depend on the network size. The format is thus feasible even for high-dimensional systems. We will illustrate the results with several guiding examples such as the Ising model, a system of coupled oscillators, and a CO oxidation model

Active Perception by Interaction with Other Agents in a Predictive Coding Framework: Application to Internet of Things Environment

Predicting the state of an agent\u27s partially-observable environment is a problem of interest in many domains. Typically in the real world, the environment consists of multiple agents, not necessarily working towards a common goal. Though the goal and sensory observation for each agent is unique, one agent might have acquired some knowledge that may benefit the other. In essence, the knowledge base regarding the environment is distributed among the agents. An agent can sample this distributed knowledge base by communicating with other agents. Since an agent is not storing the entire knowledge base, its model can be small and its inference can be efficient and fault-tolerant. However, the agent needs to learn -- when, with whom and what -- to communicate (in general interact) under different situations.This dissertation presents an agent model that actively and selectively communicates with other agents to predict the state of its environment efficiently. Communication is a challenge when the internal models of other agents is unknown and unobservable. The proposed agent learns communication policies as mappings from its belief state to when, with whom and what to communicate. The policies are learned using predictive coding in an online manner, without any reinforcement. The proposed agent model is evaluated on widely-studied applications, such as human activity recognition from multimodal, multisource and heterogeneous sensor data, and transferring knowledge across sensor networks. In the applications, either each sensor or each sensor network is assumed to be monitored by an agent. The recognition accuracy on benchmark datasets is comparable to the state-of-the-art, even though our model has significantly fewer parameters and infers the state in a localized manner. The learned policy reduces number of communications. The agent is tolerant to communication failures and can recognize the reliability of each agent from its communication messages. To the best of our knowledge, this is the first work on learning communication policies by an agent for predicting the state of its environment

A literature survey of low-rank tensor approximation techniques

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors

Many body physics from a quantum information perspective

The quantum information approach to many body physics has been very successful in giving new insight and novel numerical methods. In these lecture notes we take a vertical view of the subject, starting from general concepts and at each step delving into applications or consequences of a particular topic. We first review some general quantum information concepts like entanglement and entanglement measures, which leads us to entanglement area laws. We then continue with one of the most famous examples of area-law abiding states: matrix product states, and tensor product states in general. Of these, we choose one example (classical superposition states) to introduce recent developments on a novel quantum many body approach: quantum kinetic Ising models. We conclude with a brief outlook of the field.Comment: Lectures from the Les Houches School on "Modern theories of correlated electron systems". Improved version new references adde

Low-rank tensor methods for large Markov chains and forward feature selection methods

In the first part of this thesis, we present and compare several approaches for the determination of the steady-state of large-scale Markov chains with an underlying low-rank tensor structure. Such structure is, in our context of interest, associated with the existence of interacting processes. The state space grows exponentially with the number of processes. This type of problems arises, for instance, in queueing theory, in chemical reaction networks, or in telecommunications. As the number of degrees of freedom of the problem grows exponentially with the number of processes, the so-called \textit{curse of dimensionality} severely impairs the use of standard methods for the numerical analysis of such Markov chains. We drastically reduce the number of degrees of freedom by assuming a low-rank tensor structure of the solution. We develop different approaches, all considering a formulation of the problem where all involved structures are considered in their low-rank representations in \textit{tensor train} format. The first approaches that we will consider are associated with iterative solvers, in particular focusing on solving a minimization problem that is equivalent to the original problem of finding the desired steady state. We later also consider tensorized multigrid techniques as main solvers, using different operators for restriction and interpolation. For instance, aggregation/disaggregation operators, which have been extensively used in this field, are applied. In the second part of this thesis, we focus on methods for feature selection. More concretely, since, among the various classes of methods, sequential feature selection methods based on mutual information have become very popular and are widely used in practice, we focus on this particular type of methods. This type of problems arises, for instance, in microarray analysis, in clinical prediction, or in text categorization. Comparative evaluations of these methods have been limited by being based on specific datasets and classifiers. We develop a theoretical framework that allows evaluating the methods based on their theoretical properties. Our framework is based on the properties of the target objective function that the methods try to approximate, and on a novel categorization of features, according to their contribution to the explanation of the class; we derive upper and lower bounds for the target objective function and relate these bounds with the feature types. Then, we characterize the types of approximations made by the methods, and analyse how these approximations cope with the good properties of the target objective function. We also develop a distributional setting designed to illustrate the various deficiencies of the methods, and provide several examples of wrong feature selections. In the context of this setting, we use the minimum Bayes risk as performance measure of the methods