Information-theoretic Cost Function Research Articles

Topological belief space planning for active SLAM with pairwise Gaussian potentials and performance guarantees

Determining a globally optimal solution of belief space planning (BSP) in high-dimensional state spaces directly is computationally expensive, as it involves belief propagation and objective function evaluation for each candidate action. However, many problems of interest, such as active SLAM, exhibit structure that can be harnessed to expedite planning. Also, in order to choose an optimal action, an exact value of the objective function is not required as long as the actions can be sorted in the same way. In this paper we leverage these two key aspects and present the topological belief space planning (t-bsp) concept that uses topological signatures to perform this ranking for information-theoretic cost functions, considering only topologies of factor graphs that correspond to future posterior beliefs. In particular, we propose a highly efficient topological signature based on the von Neumann graph entropy that is a function of graph node degrees and supports an incremental update. We analyze it in the context of active pose SLAM and derive error bounds between the proposed topological signature and the original information-theoretic cost function. These bounds are then used to provide performance guarantees for t-bsp with respect to a given solver of the original information-theoretic BSP problem. Realistic and synthetic simulations demonstrate drastic speed-up of the proposed method with respect to the state-of-the-art methods while retaining the ability to select a near-optimal solution. A proof of concept of t-bsp is given in a small-scale real-world active SLAM experiment.

A discrepant information selection mechanism for cooperative sensors

Aiming at the discrepancy of information raised by different fields of view of cooperative sensors, this paper proposes a fusion rule based on a new information-theoretic cost function. Specifically, the optimal function is proposed based on the principle of Minimum Discrimination Information (MDI), where the fusion weights are the functions of the label set. The solution of the optimal function exhibits the information selection mechanism, i.e., the GLMB components of the global multitarget density are selected among the local GLMB components based on their weights. Besides, to evaluate the performance of the proposed method, we also propose the overlapping indicator which assesses the similarity among the FoVs of cooperative sensors. The simulation results demonstrate that the proposed method achieves robust performance under different values of the overlapping indicator.

Information-Theoretic Interactive Sensing and Inference for Autonomous Systems

This paper addresses an autonomous exploration problem in which a mobile sensor, placed in a previously unseen search area, utilizes an information-theoretic navigation cost function to dynamically select the next sensing action, i.e., location from which to take a measurement, to efficiently detect and classify objects of interest within the area. The information-theoretic cost function proposed in this paper consist of two <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">information gain</i> terms, one for detection and localization of objects and the other for sequential classification of the detected objects. We present a novel closed-form representation for the cost function, derived from the definition of mutual information. We evaluate three different policies for choosing the next sensing action: lawn mower, greedy, and non-greedy. For these three policies, we compare the results from our information-theoretic cost functions to the results of other information-theoretic inspired cost functions. Our simulation results show that search efficiency is greater using the proposed cost functions compared to those of the other methods.

A Generalized Framework For Kullback–Leibler Markov Aggregation

This paper proposes an information-theoretic cost function for aggregating a Markov chain via a (possibly stochastic) mapping. The cost function is motivated by two objectives: 1) The process obtained by observing the Markov chain through the mapping should be close to a Markov chain, and 2) the aggregated Markov chain should retain as much of the temporal dependence structure of the original Markov chain as possible. We discuss properties of this parameterized cost function and show that it contains the cost functions previously proposed by Deng et al., Xu et al., and Geiger et al. as special cases. We moreover discuss these special cases providing a better understanding and highlighting potential shortcomings: For example, the cost function proposed by Geiger et al. is tightly connected to approximate probabilistic bisimulation, but leads to trivial solutions if optimized without regularization. We furthermore propose a simple heuristic to optimize our cost function for deterministic aggregations and illustrate its performance on a set of synthetic examples.

Analysis of Agent Expertise in Ms. Pac-Man Using Value-of-Information-Based Policies

Conventional reinforcement learning methods for Markov decision processes rely on weakly-guided, stochastic searches to drive the learning process. It can therefore be difficult to predict what agent behaviors might emerge. In this paper, we consider an information-theoretic cost function for performing constrained stochastic searches that promote the formation of risk-averse to risk-favoring behaviors. This cost function is the value of information, which provides the optimal trade-off between the expected return of a policy and the policy's complexity; policy complexity is measured by number of bits and controlled by a single hyperparameter on the cost function. As the policy complexity is reduced, the agents will increasingly eschew risky actions. This reduces the potential for high accrued rewards. As the policy complexity increases, the agents will take actions, regardless of the risk, that can raise the long-term rewards. The obtainable reward depends on a single, tunable hyperparameter that regulates the degree of policy complexity. We evaluate the performance of value-of-information-based policies on a stochastic version of Ms. Pac-Man. A major component of this paper is the demonstration that ranges of policy complexity values yield different game-play styles and explaining why this occurs. We also show that our reinforcement-learning search mechanism is more efficient than the others we utilize. This result implies that the value of information theory is appropriate for framing the exploitation-exploration trade-off in reinforcement learning.

Co-Clustering via Information-Theoretic Markov Aggregation

We present an information-theoretic cost function for co-clustering, i.e., for simultaneous clustering of two sets based on similarities between their elements. By constructing a simple random walk on the corresponding bipartite graph, our cost function is derived from a recently proposed generalized framework for information-theoretic Markov chain aggregation. The goal of our cost function is to minimize relevant information loss, hence it connects to the information bottleneck formalism. Moreover, via the connection to Markov aggregation, our cost function is not ad hoc, but inherits its justification from the operational qualities associated with the corresponding Markov aggregation problem. We furthermore show that, for appropriate parameter settings, our cost function is identical to well-known approaches from the literature, such as Information-Theoretic Co-Clustering of Dhillon et al. Hence, understanding the influence of this parameter admits a deeper understanding of the relationship between previously proposed information-theoretic cost functions. We highlight some strengths and weaknesses of the cost function for different parameters. We also illustrate the performance of our cost function, optimized with a simple sequential heuristic, on several synthetic and real-world data sets, including the Newsgroup20 and the MovieLens100k data sets.

Adaptive Sampling for Learning Gaussian Processes Using Mobile Sensor Networks

This paper presents a novel class of self-organizing sensing agents that adaptively learn an anisotropic, spatio-temporal Gaussian process using noisy measurements and move in order to improve the quality of the estimated covariance function. This approach is based on a class of anisotropic covariance functions of Gaussian processes introduced to model a broad range of spatio-temporal physical phenomena. The covariance function is assumed to be unknown a priori. Hence, it is estimated by the maximum a posteriori probability (MAP) estimator. The prediction of the field of interest is then obtained based on the MAP estimate of the covariance function. An optimal sampling strategy is proposed to minimize the information-theoretic cost function of the Fisher Information Matrix. Simulation results demonstrate the effectiveness and the adaptability of the proposed scheme.

Information theoretic learning with adaptive kernels

This paper presents an online algorithm for adapting the kernel width that is a free parameter in information theoretic cost functions using Renyi's entropy. This kernel computes the interactions between the error samples and essentially controls the nature of the performance surface over which the parameters of the system adapt. Since the error in an adaptive system is non-stationary during training, a fixed value of the kernel width may affect the adaptation dynamics and even compromise the location of the global optimum in parameter space. The proposed online algorithm for adapting the kernel width is derived from first principles and minimizes the Kullback–Leibler divergence between the estimated error density and the true density. We characterize the performance of this novel approach with simulations of linear and nonlinear systems training, using the minimum error entropy criterion with the proposed adaptive kernel algorithm. We conclude that adapting the kernel width improves the rate of convergence of the parameters, and decouples the convergence rate and misadjustment of the filter weights.

Some Equivalences between Kernel Methods and Information Theoretic Methods

In this paper, we discuss some equivalences between two recently introduced statistical learning schemes, namely Mercer kernel methods and information theoretic methods. We show that Parzen window-based estimators for some information theoretic cost functions are also cost functions in a corresponding Mercer kernel space. The Mercer kernel is directly related to the Parzen window. Furthermore, we analyze a classification rule based on an information theoretic criterion, and show that this corresponds to a linear classifier in the kernel space. By introducing a weighted Parzen window density estimator, we also formulate the support vector machine in this information theoretic perspective.

An extension of statistical decision theory with information theoretic cost functions to decision fusion: Part II

This paper is the second of two papers that examine statistical decision theory for the case where the cost function is an information theoretic cost function. Part I reviewed the information theoretic decision rule that results from the information theoretic function as well as some of its unique characteristics. It was shown that information theoretic decisions are represented by characteristic probability distributions and that the optimal decision is the one with the smallest Kullbach–Leibler statistical distance from the measurement-based probability distribution. The information theoretic decision function is less sensitive to the assignment of prior probabilities and the abstraction levels of the decision and hypothesis sets when compared to Bayesian decision theory. This second part develops an entirely new decision fusion rule from the decision rule of Part I. The decision fusion rule is restructured to produce a recursive algorithm that has similarities with the Kalman filter. The performance characteristics of the recursive algorithm and a handful of extensions are compared to a recursive Bayesian decision fusion algorithm through the analysis of a simple decision problem.

An extension of statistical decision theory with information theoretic cost functions to decision fusion: Part I

A review of information theory and statistical decision theory has led to the recognition that decisions in statistical decision theory can be interpreted as being determined by the similarity between the distribution of probabilities obtained from measurements and characteristic distributions of probabilities representing the members of the set of decisions. The obscure interpretation was found during a review of statistical decision theory for the special case where the cost function of statistical decision theory is an information theoretic cost function. Additional research has found that the resulting information theoretic decision rule has a number of interesting characteristics that may have previously been recognized in terms of mathematical interest, but until now have not been recognized for their implications for information fusion. Bayesian probability theory has been criticized for problematic changes in decisions when hypotheses and decisions are reorganized to different levels of abstraction, weak justification for the selection of prior probabilities, and the need for all probability density functions to be defined. The characteristics of the information theoretic decision rule show that the decisions are less sensitive to changes in the reorganization of hypothesis and decision sets to different levels of abstraction in comparison to Bayesian probability theory. Extension of the information theoretic rule to a fusion rule (to be provided in a companion paper) will be shown to provide increased justification for the selection of prior probabilities through the adoption of Laplace’s principle of indifference. The criticism of the need for all probability density functions can be partially mitigated by arguing that the hypothesis abstraction levels can be selected so that all the probability density functions may be obtained. Further refutation of the third criticism will require that the assumption that the probability density functions are not definitively known but may be ambiguous as well and will not be pursued as a line of inquiry within the two companion papers.

Decoding neuronal spike trains: How important are correlations?

It has been known for >30 years that neuronal spike trains exhibit correlations, that is, the occurrence of a spike at one time is not independent of the occurrence of spikes at other times, both within spike trains from single neurons and across spike trains from multiple neurons. The presence of these correlations has led to the proposal that they might form a key element of the neural code. Specifically, they might act as an extra channel for information, carrying messages about events in the outside world that are not carried by other aspects of the spike trains, such as firing rate. Currently, there is no general consensus about whether this proposal applies to real spike trains in the nervous system. This is largely because it has been hard to separate information carried in correlations from that not carried in correlations. Here we propose a framework for performing this separation. Specifically, we derive an information-theoretic cost function that measures how much harder it is to decode neuronal responses when correlations are ignored than when they are taken into account. This cost function can be readily applied to real neuronal data.