Predictable Feature Analysis

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

Understanding Slow Feature Analysis: A Mathematical Framework

Slow feature analysis is an algorithm for unsupervised learning of invariant representations from data with temporal correlations. Here, we present a mathematical analysis of slow feature analysis for the case where the input-output functions are not restricted in complexity. We show that the optimal functions obey a partial differential eigenvalue problem of a type that is common in theoretical physics. This analogy allows the transfer of mathematical techniques and intuitions from physics to concrete applications of slow feature analysis, thereby providing the means for analytical predictions and a better understanding of simulation results. We put particular emphasis on the situation where the input data are generated from a set of statistically independent sources.\ud The dependence of the optimal functions on the sources is calculated analytically for the cases where the sources have Gaussian or uniform distribution

An Extension of Slow Feature Analysis for Nonlinear Blind Source Separation

We present and test an extension of slow feature analysis as a novel approach to nonlinear blind source separation. The algorithm relies on temporal correlations and iteratively reconstructs a set of statistically independent sources from arbitrary nonlinear instantaneous mixtures. Simulations show that it is able to invert a complicated nonlinear mixture of two audio signals with a reliability of more than $90$\%. The algorithm is based on a mathematical analysis of slow feature analysis for the case of input data that are generated from statistically independent sources

Modular Toolkit for Data Processing (MDP): A Python Data Processing Framework

Modular toolkit for Data Processing (MDP) is a data processing framework written in Python. From the user's perspective, MDP is a collection of supervised and unsupervised learning algorithms and other data processing units that can be combined into data processing sequences and more complex feed-forward network architectures. Computations are performed efficiently in terms of speed and memory requirements. From the scientific developer's perspective, MDP is a modular framework, which can easily be expanded. The implementation of new algorithms is easy and intuitive. The new implemented units are then automatically integrated with the rest of the library. MDP has been written in the context of theoretical research in neuroscience, but it has been designed to be helpful in any context where trainable data processing algorithms are used. Its simplicity on the user's side, the variety of readily available algorithms, and the reusability of the implemented units make it also a useful educational tool

Independent slow feature analysis and nonlinear blind source separation

Abstract. We present independent slow feature analysis as a new method for nonlinear blind source separation. It circumvents the indeterminacy of nonlinear independent component analysis by combining the objectives of statistical independence and temporal slowness. The principle of temporal slowness is adopted from slow feature analysis, an unsupervised method to extract slowly varying features from a given observed vectorial signal. The performance of the algorithm is demonstrated on nonlinearly mixed speech data.