A distinctive property of human and animal intelligence is the ability to form abstractions by neglecting irrelevant information which allows to separate structure from noise. From an information theoretic point of view abstractions are desirable because they allow for very efficient information processing. In artificial systems abstractions are often implemented through computationally costly formations of groups or clusters. In this work we establish the relation between the free-energy framework for decision making and rate-distortion theory and demonstrate how the application of rate-distortion for decision-making leads to the emergence of abstractions. We argue that abstractions are induced due to a limit in information processing capacity.

This note provides a simple example demonstrating that, if exact computations are allowed, the number of iterations required for the value iteration algorithm to find an optimal policy for discounted dynamic programming problems may grow arbitrarily quickly with the size of the problem. In particular, the number of iterations can be exponential in the number of actions. Thus, unlike policy iterations, the value iteration algorithm is not strongly polynomial for discounted dynamic programming. Keywords: Markov Decision Process, value iteration, strongly polynomial, policy, algorithm 1. Introduction Value iterations, policy iterations, and linear programming are three major methods for computing optimal policies for Markov Decision Processes (MDPs) with expected total discounted rewards [1], [3, Chapter 6], also known under the name of discounted dynamic programming. As is well-known, policy iterations can be viewed as implementations of the simplex method applied to one of the two major linear programs used to solve MDPs; see e.g.

Unfortunately, real-world datasets often include only samples of observed values mixed with many missing or unknown elements. Missing values may occur due to human impatience, human error during data entry, data loss, faulty sensory equipment, changes in data collection methods, inability to decipher handwriting, privacy issues, legal requirements, and a variety of other practical factors. Thus, improvements to methods for imputing missing values can have far-reaching impact on improving the effectiveness of existing learning algorithms for operating on real-world data. We present a method for imputation called Unsupervised Backpropagation (UBP), which trains a multilayer perceptron (MLP) to fit to the manifold represented by the known features in a dataset. We demonstrate this algorithm with the task of imputing missing values, and we show that it is significantly more effective than other methods for imputation. Backpropagation has long been a popular method for training neural networks (Rumelhart et al., 1986; Werbos, 1990).

Probabilistic models play a central role in modern machine learning. They offer a powerful framework for learning from data, and they have found applications in a variety of scientific fields beyond machine learning. A longstanding goal in machine learning and statistics is to achieve a separation between modeling and inference, so that users of these tools may focus on specifying models without having to implement new inference algorithms every time the models change. Recently, work in probabilistic programming has taken up this challenge, seeking to unify probabilistic modeling with computer programming in order to dramatically increase the effectiveness of machine learning experts (DARPA, 2013) and to equip non-experts with effective tools for specifying models and performing inference. We anticipate that continued success toward these goals will decrease the reliance of machine learning practitioners on tried-and-true models and will shift the community toward a paradigm grounded in flexible tools for rapidly prototyping and designing new models (Bishop, 2013).

Systematic, or market, risk is one of the most studied risk models not only in financial engineering, but also in actuarial sciences, in business and corporate management, and in several other domains. It is associated to the beta (β) coefficient, which is familiar in the investment industry since Sharpe's capital asset pricing model (CAPM) [30]. The pitfalls and shortcomings of β have been detailed by a number of excellent authors.

Hypergraphs allow one to encode higher-order relationships in data and are thus a very flexible modeling tool. Current learning methods are either based on approximations of the hypergraphs via graphs or on tensor methods which are only applicable under special conditions. In this paper, we present a new learning framework on hypergraphs which fully uses the hypergraph structure. The key element is a family of regularization functionals based on the total variation on hypergraphs.

Non-Negative Matrix Factorization, NMF, attempts to find a number of archetypal response profiles, or parts, such that any sample profile in the dataset can be approximated by a close profile among these archetypes or a linear combination of these profiles. The non-negativity constraint is imposed while estimating archetypal profiles, due to the non-negative nature of the observed signal. Apart from non negativity, a volume constraint can be applied on the Score matrix W to enhance the ability of learning parts of NMF. In this report, we describe a very simple algorithm, which in effect achieves volume minimization, although indirectly.

Functional Classification, i.e. the binary classification problem when the input observation X (X(t)) is of the form of a (possibly sampled) random curve/function and the output variable Y { 1, 1} is a binary label, has been the subject of a good deal of attention in the machine-learning literature in the past few years, see [1] or [2]. In contrast, Bipartite Ranking, termed Nonparametric Scoring sometimes, has never been tackled in a functional framework, except from the restrictive angle of Functional Logistic Regression, see [3] or [4] for instance. This global learning task consists in ordering all possible input observations X so that positive ones appear on top of the list with highest probability. This predictive problem, which can be cast in terms of ROC curve optimization (see [5]), covers a wide variety of applications, ranging from anomaly detection in signal processing to automatic design of diagnosis tools in medicine through creditscoring in mathematical finance or the conception of search engines in information retrieval. Functional versions of many popular approaches for classification have been developed, relying in general on a preliminary finite dimensional representation/projection of the input data.

The goal of supervised statistical learning is to develop prediction rules taking the values of predictor variables as input and returning a predicted value of the response variable. A prediction rule is typically learnt by applying a learning algorithm M to a so-called learning data set. A typical example in biomedical research is the prediction of patient outcome (e.g. The practitioners are usually interested in the accuracy of the prediction rule learnt from their data set to predict future patients, while methodological researchers rather want to know whether the learning algorithm is good at learning accurate prediction rules for different data sets drawn from a distribution of interest. The first perspective is called "conditional" (since referring to a specific data set) while the latter, which we take in this paper, is denoted as "unconditional". If the data set is very large, one can observe independent realizations of estimators of the unconditional error rates and use them for a paired t-test (see Section 2.3). In practise, however, huge data sets are rarely available. Prediction errors are thus usually estimated by resampling procedures consisting of splitting the available data set into learning and test sets a large number of times and averaging the estimated error over these iterations.

In this paper we explore the applicability of the unsupervised machine learning technique of Self Organizing Maps (SOM) to estimate galaxy photometric redshift probability density functions (PDFs). This technique takes a spectroscopic training set, and maps the photometric attributes, but not the redshifts, to a two dimensional surface by using a process of competitive learning where neurons compete to more closely resemble the training data multidimensional space. The key feature of a SOM is that it retains the topology of the input set, revealing correlations between the attributes that are not easily identified. We test three different 2D topological mapping: rectangular, hexagonal, and spherical, by using data from the DEEP2 survey. We also explore different implementations and boundary conditions on the map and also introduce the idea of a random atlas where a large number of different maps are created and their individual predictions are aggregated to produce a more robust photometric redshift PDF. We also introduced a new metric, the $I$-score, which efficiently incorporates different metrics, making it easier to compare different results (from different parameters or different photometric redshift codes). We find that by using a spherical topology mapping we obtain a better representation of the underlying multidimensional topology, which provides more accurate results that are comparable to other, state-of-the-art machine learning algorithms. Our results illustrate that unsupervised approaches have great potential for many astronomical problems, and in particular for the computation of photometric redshifts.