For the published version of the article, see: Heck, D. W., Moshagen, M., & Erdfelder, E. (2014). Correspondence concerning this article should be addressed to Daniel W. Heck, Department of Psychology, School of Social Sciences, University of Mannheim, Schloss EO 254, D-68131 Mannheim, Germany. FISHER INFORMATION APPROXIMATION 2 Abstract The Fisher information approximation (FIA) is an implementation of the minimum description length principle for model selection. Unlike information criteria such as AIC or BIC, it has the advantage of taking the functional form of a model into account. Unfortunately, FIA can be misleading in finite samples, resulting in an inversion of the correct rank order of complexity terms for competing models in the worst case.

Matrix factorization is a key tool in data analysis; its applications include recommender systems, correlation analysis, signal processing, among others. Binary matrices are a particular case which has received significant attention for over thirty years, especially within the field of data mining. Dictionary learning refers to a family of methods for learning overcomplete basis (also called frames) in order to efficiently encode samples of a given type; this area, now also about twenty years old, was mostly developed within the signal processing field. In this work we propose two binary matrix factorization methods based on a binary adaptation of the dictionary learning paradigm to binary matrices. The proposed algorithms focus on speed and scalability; they work with binary factors combined with bit-wise operations and a few auxiliary integer ones. Furthermore, the methods are readily applicable to online binary matrix factorization. Another important issue in matrix factorization is the choice of rank for the factors; we address this model selection problem with an efficient method based on the Minimum Description Length principle. Our preliminary results show that the proposed methods are effective at producing interpretable factorizations of various data types of different nature.

We tackle the problem of penalty selection of regularization on the basis of the minimum description length (MDL) principle. In particular, we consider that the design space of the penalty function is high-dimensional. In this situation, the luckiness-normalized-maximum-likelihood(LNML)-minimization approach is favorable, because LNML quantifies the goodness of regularized models with any forms of penalty functions in view of the minimum description length principle, and guides us to a good penalty function through the high-dimensional space. However, the minimization of LNML entails two major challenges: 1) the computation of the normalizing factor of LNML and 2) its minimization in high-dimensional spaces. In this paper, we present a novel regularization selection method (MDL-RS), in which a tight upper bound of LNML (uLNML) is minimized with local convergence guarantee. Our main contribution is the derivation of uLNML, which is a uniform-gap upper bound of LNML in an analytic expression. This solves the above challenges in an approximate manner because it allows us to accurately approximate LNML and then efficiently minimize it. The experimental results show that MDL-RS improves the generalization performance of regularized estimates specifically when the model has redundant parameters.

We investigate the teaching of infinite concept classes through the effect of the learning bias (which is used by the learner to prefer some concepts over others and by the teacher to devise the teaching examples) and the sampling bias (which determines how the concepts are sampled from the class). We analyse two important classes: Turing machines and finite-state machines. We derive bounds for the biased teaching dimension when the learning bias is derived from a complexity measure (Kolmogorov complexity and minimal number of states respectively) and analyse the sampling distributions that lead to finite expected biased teaching dimensions. We highlight the existing trade-off between the bound and the representativeness of the sample, and its implications for the understanding of what teaching rich concepts to machines entails.

Segmenting temporal data sequences is an important problem which helps in understanding data dynamics in multiple applications such as epidemic surveillance, motion capture sequences, etc. In this paper, we give DASSA, the first self-guided and efficient algorithm to automatically find a segmentation that best detects the change of pattern in data sequences. To avoid introducing tuning parameters, we design DASSA to be a multi-level method which examines segments at each level of granularity via a compact data structure called the segment-graph. We build this data structure by carefully leveraging the information bottleneck method with the MDL principle to effectively represent each segment.Next, DASSA efficiently finds the optimal segmentation via a novel average-longest-path optimization on the segment-graph. Finally we show how the outputs from DASSA can be naturally interpreted to reveal meaningful patterns. We ran DASSA on multiple real datasets of varying sizes and it is very effective in finding the time-cut points of the segmentations (in some cases recovering the cut points perfectly) as well as in finding the corresponding changing patterns.

Telling cause from effect using observational data is a challenging problem, especially in the bivariate case. Contemporary methods often assume an independence between the cause and the generating mechanism of the effect given the cause. From this postulate, they derive asymmetries to uncover causal relationships. In this work, we propose such an approach, based on the link between Kolmogorov complexity and quantile scoring. We use a nonparametric conditional quantile estimator based on copulas to implement our procedure, thus avoiding restrictive assumptions about the joint distribution between cause and effect. In an extensive study on real and synthetic data, we show that quantile copula causal discovery (QCCD) compares favorably to state-of-the-art methods, while at the same time being computationally efficient and scalable.

Remarkably general method for bounding the statistical risk of penalized likelihood estimators comes from work on two-part coding, one of the minimum description length (MDL) approaches to statistical inference. Two-part coding MDL prescribes assigning codelengths to a model (or model class) then selecting the distribution that provides the most efficient description of one's data [1]. The total description length has two parts: the part that specifies a distribution within the model (as well as a model within the model class if necessary) and the part that specifies the data with reference to the specified distribution. If the codelengths are exactly Kraft-valid, this approach is equivalent to Bayesian maximum a posteriori (MAP) estimation, in that the two parts correspond to log reciprocal of prior and log reciprocal of likelihood respectively. More generally, one can call the part of the codelength specifying the distribution a penalty term; it is called the complexity in MDL literature. Let (Θ, L) denote a discrete set indexing distributions along with a complexity function. With X P, the (pointwise) redundancy of any θ Θ is its two-part codelength minus log(1/p(X)), the codelength one gets by using P as the coding distribution.

Given data over the joint distribution of two random variables $X$ and $Y$, we consider the problem of inferring the most likely causal direction between $X$ and $Y$. In particular, we consider the general case where both $X$ and $Y$ may be univariate or multivariate, and of the same or mixed data types. We take an information theoretic approach, based on Kolmogorov complexity, from which it follows that first describing the data over cause and then that of effect given cause is shorter than the reverse direction. The ideal score is not computable, but can be approximated through the Minimum Description Length (MDL) principle. Based on MDL, we propose two scores, one for when both $X$ and $Y$ are of the same single data type, and one for when they are mixed-type. We model dependencies between $X$ and $Y$ using classification and regression trees. As inferring the optimal model is NP-hard, we propose Crack, a fast greedy algorithm to determine the most likely causal direction directly from the data. Empirical evaluation on a wide range of data shows that Crack reliably, and with high accuracy, infers the correct causal direction on both univariate and multivariate cause-effect pairs over both single and mixed-type data.

Telling cause from effect from observational data is one of the fundamental problems in science [26], [18]. We consider the problem of inferring the most likely direction between two univariate numeric random variables X and Y. That is, we are interested in identifying whether X causes Y, whether Y causes X, or whether they are merely correlated. Traditional methods, that rely on conditional independence tests, cannot decide between the Markov equivalent classes of X Y and Y X [18]. Recently, it has been postulated however that if X Y, there exists an independence between the marginal distribution of the cause, P (X), and the conditional distribution of the effect given the cause, P (Y X) [25], [9]. The state of the art exploits this asymmetry in various ways, and overall obtain up to 70% accuracy on a well-known benchmark of cause-effect pairs [24], [8], [20], [10], [17]. In this paper we break this barrier, and give an elegant score that is computable in linear-time and obtains over 82% accuracy on the same benchmark.

Learning-based binary hashing has become a powerful paradigm for fast search and retrieval in massive databases. However, due to the requirement of discrete outputs for the hash functions, learning such functions is known to be very challenging. In addition, the objective functions adopted by existing hashing techniques are mostly chosen heuristically. In this paper, we propose a novel generative approach to learn hash functions through Minimum Description Length principle such that the learned hash codes maximally compress the dataset and can also be used to regenerate the inputs. We also develop an efficient learning algorithm based on the stochastic distributional gradient, which avoids the notorious difficulty caused by binary output constraints, to jointly optimize the parameters of the hash function and the associated generative model. Extensive experiments on a variety of large-scale datasets show that the proposed method achieves better retrieval results than the existing state-of-the-art methods.