We propose SPARFA-Trace, a new machine learning-based framework for time-varying learning and content analytics for education applications. We develop a novel message passing-based, blind, approximate Kalman filter for sparse factor analysis (SPARFA), that jointly (i) traces learner concept knowledge over time, (ii) analyzes learner concept knowledge state transitions (induced by interacting with learning resources, such as textbook sections, lecture videos, etc, or the forgetting effect), and (iii) estimates the content organization and intrinsic difficulty of the assessment questions. These quantities are estimated solely from binary-valued (correct/incorrect) graded learner response data and a summary of the specific actions each learner performs (e.g., answering a question or studying a learning resource) at each time instance. Experimental results on two online course datasets demonstrate that SPARFA-Trace is capable of tracing each learner's concept knowledge evolution over time, as well as analyzing the quality and content organization of learning resources, the question-concept associations, and the question intrinsic difficulties. Moreover, we show that SPARFA-Trace achieves comparable or better performance in predicting unobserved learner responses than existing collaborative filtering and knowledge tracing approaches for personalized education.

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.

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.

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.

We study the representation of systems S of linear equations over the two-element field (aka xor- or parity-constraints) via conjunctive normal forms F (boolean clause-sets). First we consider the problem of finding an "arc-consistent" representation ("AC"), meaning that unit-clause propagation will fix all forced assignments for all possible instantiations of the xor-variables. Our main negative result is that there is no polysize AC-representation in general. On the positive side we show that finding such an AC-representation is fixed-parameter tractable (fpt) in the number of equations. Then we turn to a stronger criterion of representation, namely propagation completeness ("PC") --- while AC only covers the variables of S, now all the variables in F (the variables in S plus auxiliary variables) are considered for PC. We show that the standard translation actually yields a PC representation for one equation, but fails so for two equations (in fact arbitrarily badly). We show that with a more intelligent translation we can also easily compute a translation to PC for two equations. We conjecture that computing a representation in PC is fpt in the number of equations.