Extended Predictable Feature Analysis (PFAx) [Richthofer and Wiskott, 2017] is an extension of PFA [Richthofer and Wiskott, 2015] that allows generating a goal-directed control signal of an agent whose dynamics has previously been learned during a training phase in an unsupervised manner. PFAx hardly requires assumptions or prior knowledge of the agent's sensor or control mechanics, or of the environment. It selects features from a high-dimensional input by intrinsic predictability and organizes them into a reasonably low-dimensional model. While PFA obtains a well predictable model, PFAx yields a model ideally suited for manipulations with predictable outcome. This allows for goal-directed manipulation of an agent and thus for local navigation, i.e. for reaching states where intermediate actions can be chosen by a permanent descent of distance to the goal. The approach is limited when it comes to global navigation, e.g. involving obstacles or multiple rooms. In this article, we extend theoretical results from [Sprekeler and Wiskott, 2008], enabling PFAx to perform stable global navigation. So far, the most widely exploited characteristic of Slow Feature Analysis (SFA) was that slowness yields invariances. We focus on another fundamental characteristics of slow signals: They tend to yield monotonicity and one significant property of monotonicity is that local optimization is sufficient to find a global optimum. We present an SFA-based algorithm that structures an environment such that navigation tasks hierarchically decompose into subgoals. Each of these can be efficiently achieved by PFAx, yielding an overall global solution of the task. The algorithm needs to explore and process an environment only once and can then perform all sorts of navigation tasks efficiently. We support this algorithm by mathematical theory and apply it to different problems.
In this paper we propose a novel method for learning a Mahalanobis distance measure to be used in the KNN classification algorithm. The algorithm directly maximizes a stochastic variant of the leave-one-out KNN score on the training set. It can also learn a low-dimensional linear embedding of labeled data that can be used for data visualization and fast classification. Unlike other methods, our classification model is non-parametric, making no assumptions about the shape of the class distributions or the boundaries between them. The performance of the method is demonstrated on several data sets, both for metric learning and linear dimensionality reduction. Abstract : This paper proposes a canonical transformation of a k dimensional stationary autoregressive process. The components of the transformed process are ordered from least predictable to most predictable. The least predictable components are often nearly white noise and the most predictable can be nearly nonstationary. Transformed variables which are white noise can reflect relationships which may be associated with or point to economic or physical laws. A 5-variate example is given. Introduce Forecastable Component Analysis (ForeCA), a novel dimension reduction technique for temporally dependent signals. Based on a new forecastability measure, ForeCA finds an optimal transformation to separate a multivariate time series into a forecastable and an orthogonal white noise space. I present a converging algorithm with a fast eigenvector solution. Applications to financial and macro-economic time series show that ForeCA can successfully discover informative structure, which can be used for forecasting as well as classification. The R package ForeCA accompanies this work and is publicly available on CRAN. Applying reinforcement learning algorithms in real-world domains is challenging because relevant state information is often embedded in a stream of high-dimensional sensor data. This paper describes a novel algorithm for learning task-relevant features through interactions with the environment. The key idea is that a feature is likely to be useful to the degree that its dynamics can be controlled by the actions of the agent. We describe an algorithm that can find such features and we demonstrate its effectiveness in an artificial domain. We propose graph-based predictable feature analysis (GPFA), a new method for unsupervised learning of predictable features from high-dimensional time series, where high predictability is understood very generically as low variance in the distribution of the next data point given the previous ones. We show how this measure of predictability can be understood in terms of graph embedding as well as how it relates to the information-theoretic measure of predictive information in special cases. We confirm the effectiveness of GPFA on different datasets, comparing it to three existing algorithms with similar objectives—namely slow feature analysis, forecastable component analysis, and predictable feature analysis—to which GPFA shows very competitive results. Every organism in an environment, whether biological, robotic or virtual, must be able to predict certain aspects of its environment in order to survive or perform whatever task is intended. It needs a model that is capable of estimating the consequences of possible actions, so that planning, control, and decision-making become feasible. For scientific purposes, such models are usually created in a problem specific manner using differential equations and other techniques from control- and system-theory. In contrast to that, we aim for an unsupervised approach that builds up the desired model in a self-organized fashion. Inspired by Slow Feature Analysis (SFA), our approach is to extract sub-signals from the input, that behave as predictable as possible. These "predictable features" are highly relevant for modeling, because predictability is a desired property of the needed consequence-estimating model by definition. In our approach, we measure predictability with respect to a certain prediction model. We focus here on the solution of the arising optimization problem and present a tractable algorithm based on algebraic methods which we call Predictable Feature Analysis (PFA). We prove that the algorithm finds the globally optimal signal, if this signal can be predicted with low error. To deal with cases where the optimal signal has a significant prediction error, we provide a robust, heuristically motivated variant of the algorithm and verify it empirically. Additionally, we give formal criteria a prediction-model must meet to be suitable for measuring predictability in the PFA setting and also provide a suitable default-model along with a formal proof that it meets these criteria. 