When a parameter space has a certain underlying structure, the ordinary gradient of a function does not represent its steepest direction but the natural gradient does. Information geometry is used for calculating the natural gradients in the parameter space of perceptrons, the space of matrices (for blind source separation) and the space of linear dynamical systems (for blind source deconvolution). The dynamical behavior of natural gradient on-line learning is analyzed and is proved to be Fisher efficient, implying that it has asymptotically the same performance as the optimal batch estimation of parameters. This suggests that the plateau phenomenon which appears in the backpropagation learning algorithm of multilayer perceptrons might disappear or might be not so serious when the natural gradient is used. An adaptive method of updating the learning rate is proposed and analyzed. 1 Introduction The stochastic gradient method (Widrow, 1963; Amari, 1967; Tsypkin, 1973; Rumelhart et al...

In this paper, on the one hand, we aim to give a review on literature dealing with the problem of supervised learning aided by additional unlabeled data. On the other hand, being a part of the author's first year PhD report, the paper serves as a frame to bundle related work by the author as well as numerous suggestions for potential future work. Therefore, this work contains more speculative and partly subjective material than the reader might expect from a literature review. We give a rigorous definition of the problem and relate it to supervised and unsupervised learning. The crucial role of prior knowledge is put forward, and we discuss the important notion of input-dependent regularization. We postulate a number of baseline methods, being algorithms or algorithmic schemes which can more or less straightforwardly be applied to the problem, without the need for genuinely new concepts. However, some of them might serve as basis for a genuine method. In the literature revi...

An exponential family or mixture family of probability distributions has a natural hierarchical structure. This paper gives an “orthogonal” decomposition of such a system based on information geometry. A typical example is the decomposition of stochastic dependency among a number of random variables. In general, they have a complex structure of dependencies. Pairwise dependency is easily represented by correlation, but it is more difficult to measure effects of pure triplewise or higher order interactions (dependencies) among these variables. Stochastic dependency is decomposed quantitatively into an “orthogonal” sum of pairwise, triplewise, and further higher order dependencies. This gives a new invariant decomposition of joint entropy. This problem is important for extracting intrinsic interactions in firing patterns of an ensemble of neurons and for estimating its functional connections. The orthogonal decomposition is given in a wide class of hierarchical structures including both exponential and mixture families. As an example, we decompose the dependency in a higher order Markov chain into a sum of those in various lower order Markov chains.

We introduce a method for deriving a metric, locally based on the Fisher information matrix, into the data space. A Self-Organizing Map is computed in the new metric to explore financial statements of enterprises. The metric measures local distances in terms of changes in the distribution of an auxiliary random variable that reflects what is important in the data. In this paper the variable indicates bankruptcy within the next few years. The conditional density of the auxiliary variable is first estimated, and the change in the estimate resulting from local displacements in the primary data space is measured using the Fisher information matrix. When a Self-Organizing Map is computed in the new metric it still visualizes the data space in a topology-preserving fashion, but represents the (local) directions in which the probability of bankruptcy changes the most.

i Abstract This thesis is concerned with the problem of Blind Source Separation. Specifically we considerthe Independent Component Analysis (ICA) model in which a set of observations are modelled by xt = Ast: (1) where A is an unknown mixing matrix and st is a vector of hidden source components attime t. The ICA problem is to find the sources given only a set of observations. In chapter 1, the blind source separation problem is introduced. In chapter 2 the methodof Ensemble Learning is explained. Chapter 3 applies Ensemble Learning to the ICA model and chapter 4 assesses the use of Ensemble Learning for model selection.Chapters 5-7 apply the Ensemble Learning ICA algorithm to data sets from physics (a medical imaging data set consisting of images of a tooth), biology (data sets from cDNAmicro-arrays) and astrophysics (Planck image separation and galaxy spectra separation).

This paper clarifies the relation between the learning curve and the algebraic geometrical structure of a non-identifiable learning machine such as a multilayer neural network whose true parameter set is an analytic set with singular points. By using a concept in algebraic analysis, we rigorously prove that the Bayesian stochastic complexity or the free energy is asymptotically equal to λ1 log n − (m1 − 1) log log n+constant, where n is the number of training samples and λ1 and m1 are the rational number and the natural number which are determined as the birational invariant values of the singularities in the parameter space. Also we show an algorithm to calculate λ1 and m1 based on the resolution of singularities in algebraic geometry. In regular statistical models, 2λ1 is equal to the number of parameters and m1 = 1, whereas in nonregular models such as multilayer networks, 2λ1 is not larger than the number of parameters and m1 ≥ 1. Since the increase of the stochastic complexity is equal to the learning curve or the generalization error, the non-identifiable learning machines are the better models than the regular ones if the Bayesian ensemble learning is applied. 1 1

The hypothesis testing problem of two quantum states is treated. We show a new inequality between the error of the first kind and the second kind, which complements the result of Hiai and Petz to establish the quantum version of Stein’s lemma. The inequality is also used to show a bound on the first kind error when the power exponent for the second kind error exceeds the quantum relative entropy, and the bound yields the strong converse in the quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi in the classical hypothesis testing.

In most algorithmic applications which compare two distributions, information theoretic distances are more natural than standard ℓp norms. In this paper we design streaming and sublinear time property testing algorithms for entropy and various information theoretic distances. Batu et al posed the problem of property testing with respect to the Jensen-Shannon distance. We present optimal algorithms for estimating bounded, symmetric f-divergences (including the Jensen-Shannon divergence and the Hellinger distance) between distributions in various property testing frameworks. Along the way, we close a (log n)/H gap between the upper and lower bounds for estimating entropy H, yielding an optimal algorithm over all values of the entropy. In a data stream setting (sublinear space), we give the first algorithm for estimating the entropy of a distribution. Our algorithm runs in polylogarithmic space and yields an asymptotic constant factor approximation scheme. An integral part of the algorithm is an interesting use of an F0 (the number of distinct elements in a set) estimation algorithm; we also provide other results along the space/time/approximation tradeoff curve. Our results have interesting structural implications that connect sublinear time and space constrained algorithms. The mediating model is the random order streaming model, which assumes the input is a random permutation of a multiset and was first considered by Munro and Paterson in 1980. We show that any property testing algorithm in the combined oracle model for calculating a permutation invariant functions can be simulated in the random order model in a single pass. This addresses a question raised by Feigenbaum et al regarding the relationship between property testing and stream algorithms. Further, we give a polylog-space PTAS for estimating the entropy of a one pass random order stream. This bound cannot be achieved in the combined oracle (generalized property testing) model. 1

In this paper we discus a wide class of loss (cost) functions for non-negative matrix factorization (NMF) and derive several novel algorithms with improved efficiency and robustness to noise and outliers. We review several approaches which allow us to obtain generalized forms of multiplicative NMF algorithms and unify some existing algorithms. We give also the flexible and relaxed form of the NMF algorithms to increase convergence speed and impose some desired constraints such as sparsity and smoothness of components. Moreover, the effects of various regularization terms and constraints are clearly shown. The scope of these results is vast since the proposed generalized divergence functions include quite large number of useful loss functions such as the squared Euclidean distance,Kulback-Leibler divergence, Itakura-Saito, Hellinger, Pearson’s chi-square, and Neyman’s chi-square distances, etc. We have applied successfully the developed algorithms to blind (or semi blind) source separation (BSS) where sources can be generally statistically dependent, however they satisfy some other conditions or additional constraints such as nonnegativity, sparsity and/or smoothness.

We propose a generic method for iteratively approximating various second-order gradient steps -- Newton, Gauss-Newton, Levenberg-Marquardt, and natural gradient -- in linear time per iteration, using special curvature matrix-vector products that can be computed in O(n). Two recent acceleration techniques for online learning, matrix momentum and stochastic meta-descent (SMD), in fact implement this approach. Since both were originally derived by very different routes, this o ers fresh insight into their operation, resulting in further improvements to SMD.