Any learner with the ability to predict the future of a structured time-varying signal must maintain a memory of the recent past. If the signal has a characteristic timescale relevant to future prediction, the memory can be a simple shift register---a moving window extending into the past, requiring storage resources that linearly grows with the timescale to be represented. However, an independent general purpose learner cannot a priori know the characteristic prediction-relevant timescale of the signal. Moreover, many naturally occurring signals show scale-free long range correlations implying that the natural prediction-relevant timescale is essentially unbounded. Hence the learner should maintain information from the longest possible timescale allowed by resource availability. Here we construct a fuzzy memory system that optimally sacrifices the temporal accuracy of information in a scale-free fashion in order to represent prediction-relevant information from exponentially long timescales. Using several illustrative examples, we demonstrate the advantage of the fuzzy memory system over a shift register in time series forecasting of natural signals. When the available storage resources are limited, we suggest that a general purpose learner would be better off committing to such a fuzzy memory system.

The kernel least mean squares (KLMS) algorithm is a computationally efficient nonlinear adaptive filtering method that "kernelizes" the celebrated (linear) least mean squares algorithm. We demonstrate that the least mean squares algorithm is closely related to the Kalman filtering, and thus, the KLMS can be interpreted as an approximate Bayesian filtering method. This allows us to systematically develop extensions of the KLMS by modifying the underlying state-space and observation models. The resulting extensions introduce many desirable properties such as "forgetting", and the ability to learn from discrete data, while retaining the computational simplicity and time complexity of the original algorithm.

Sequential decision-making problems with multiple objectives arise naturally in practice and pose unique challenges for research in decision-theoretic planning and learning, which has largely focused on single-objective settings. This article surveys algorithms designed for sequential decision-making problems with multiple objectives. Though there is a growing body of literature on this subject, little of it makes explicit under what circumstances special methods are needed to solve multi-objective problems. Therefore, we identify three distinct scenarios in which converting such a problem to a single-objective one is impossible, infeasible, or undesirable. Furthermore, we propose a taxonomy that classifies multi-objective methods according to the applicable scenario, the nature of the scalarization function (which projects multi-objective values to scalar ones), and the type of policies considered. We show how these factors determine the nature of an optimal solution, which can be a single policy, a convex hull, or a Pareto front. Using this taxonomy, we survey the literature on multi-objective methods for planning and learning. Finally, we discuss key applications of such methods and outline opportunities for future work.

We propose in this contribution a method for l one regularization in prototype based relevance learning vector quantization (LVQ) for sparse relevance profiles. Sparse relevance profiles in hyperspectral data analysis fade down those spectral bands which are not necessary for classification. In particular, we consider the sparsity in the relevance profile enforced by LASSO optimization. The latter one is obtained by a gradient learning scheme using a differentiable parametrized approximation of the $l_{1}$-norm, which has an upper error bound. We extend this regularization idea also to the matrix learning variant of LVQ as the natural generalization of relevance learning.

Feature extraction and dimensionality reduction are important tasks in many fields of science dealing with signal processing and analysis. The relevance of these techniques is increasing as current sensory devices are developed with ever higher resolution, and problems involving multimodal data sources become more common. A plethora of feature extraction methods are available in the literature collectively grouped under the field of Multivariate Analysis (MVA). This paper provides a uniform treatment of several methods: Principal Component Analysis (PCA), Partial Least Squares (PLS), Canonical Correlation Analysis (CCA) and Orthonormalized PLS (OPLS), as well as their non-linear extensions derived by means of the theory of reproducing kernel Hilbert spaces. We also review their connections to other methods for classification and statistical dependence estimation, and introduce some recent developments to deal with the extreme cases of large-scale and low-sized problems. To illustrate the wide applicability of these methods in both classification and regression problems, we analyze their performance in a benchmark of publicly available data sets, and pay special attention to specific real applications involving audio processing for music genre prediction and hyperspectral satellite images for Earth and climate monitoring.

The study of topological information of spatial objects has for a long time been a focus of research in disciplines like computational geometry, spatial reasoning, cognitive science, and robotics. While the majority of these researches emphasised the topological relations between spatial objects, this work studies the internal topological structure of bounded plane regions, which could consist of multiple pieces and/or have holes and islands to any finite level. The insufficiency of simple regions (regions homeomorphic to closed disks) to cope with the variety and complexity of spatial entities and phenomena has been widely acknowledged. Another significant drawback of simple regions is that they are not closed under set operations union, intersection, and difference. This paper considers bounded semi-algebraic regions, which are closed under set operations and can closely approximate most plane regions arising in practice.

Conventional SVM-based image coding methods are founded on independently restricting the distortion in every image coefficient at some particular image representation. Geometrically, this implies allowing arbitrary signal distortions in an $n$-dimensional rectangle defined by the $\varepsilon$-insensitivity zone in each dimension of the selected image representation domain. Unfortunately, not every image representation domain is well-suited for such a simple, scalar-wise, approach because statistical and/or perceptual interactions between the coefficients may exist. These interactions imply that scalar approaches may induce distortions that do not follow the image statistics and/or are perceptually annoying. Taking into account these relations would imply using non-rectangular $\varepsilon$-insensitivity regions (allowing coupled distortions in different coefficients), which is beyond the conventional SVM formulation. In this paper, we report a condition on the suitable domain for developing efficient SVM image coding schemes. We analytically demonstrate that no linear domain fulfills this condition because of the statistical and perceptual inter-coefficient relations that exist in these domains. This theoretical result is experimentally confirmed by comparing SVM learning in previously reported linear domains and in a recently proposed non-linear perceptual domain that simultaneously reduces the statistical and perceptual relations (so it is closer to fulfilling the proposed condition). These results highlight the relevance of an appropriate choice of the image representation before SVM learning.

Many problems in machine learning, statistics and signal processing may be cast as convex optimization problems. In large-scale situations, simple gradient-based algorithms with potentially many cheap iterations are often preferred over methods, such as Newton's method or interior-point methods, that rely on fewer but more expensive iterations. The choice of a first-order method depends on the structure of the problem, in particular (a) the smoothness and/or strong convexity of the objective function, and (b) the computational efficiency of certain operations related to the non-smooth parts of the objective function, when it is decomposable in a smooth and a non-smooth part. In this paper, we consider two classical algorithms, namely (a) subgradient descent and its mirror descent extension [29, 24, 4], and (b) conditional gradient algorithms, sometimes referred to as Frank-Wolfe algorithms [16, 13, 15, 14, 19]. Subgradient algorithms are adapted to non-smooth unstructured situations, and after t steps have a convergence rate of O(1/ t) in terms of objective values.

We propose a voted dual averaging method for online classification problems with explicit regularization. This method employs the update rule of the regularized dual averaging (RDA) method, but only on the subsequence of training examples where a classification error is made. We derive a bound on the number of mistakes made by this method on the training set, as well as its generalization error rate. We also introduce the concept of relative strength of regularization, and show how it affects the mistake bound and generalization performance. We experimented with the method using $\ell_1$ regularization on a large-scale natural language processing task, and obtained state-of-the-art classification performance with fairly sparse models.

In this paper, learning a Bayesian network structure that optimizes a scoring function for a given dataset is viewed as a shortest path problem in an implicit state-space search graph. This perspective highlights the importance of two research issues: the development of search strategies for solving the shortest path problem, and the design of heuristic functions for guiding the search. This paper introduces several techniques for addressing the issues. One is an A* search algorithm that learns an optimal Bayesian network structure by only searching the most promising part of the solution space. The others are mainly two heuristic functions. The first heuristic function represents a simple relaxation of the acyclicity constraint of a Bayesian network. Although admissible and consistent, the heuristic may introduce too much relaxation and result in a loose bound. The second heuristic function reduces the amount of relaxation by avoiding directed cycles within some groups of variables. Empirical results show that these methods constitute a promising approach to learning optimal Bayesian network structures.