Stein showed that the multivariate sample mean is outperformed by "shrinking" to a constant target vector. Ledoit and Wolf extended this approach to the sample covariance matrix and proposed a multiple of the identity as shrinkage target. In a general framework, independent of a specific estimator, we extend the shrinkage concept by allowing simultaneous shrinkage to a set of targets. Application scenarios include settings with (A) additional data sets from potentially similar distributions, (B) non-stationarity, (C) a natural grouping of the data or (D) multiple alternative estimators which could serve as targets. We show that this Multi-Target Shrinkage can be translated into a quadratic program and derive conditions under which the estimation of the shrinkage intensities yields optimal expected squared error in the limit. For the sample mean and the sample covariance as specific instances, we derive conditions under which the optimality of MTS is applicable. We consider two asymptotic settings: the large dimensional limit (LDL), where the dimensionality and the number of observations go to infinity at the same rate, and the finite observations large dimensional limit (FOLDL), where only the dimensionality goes to infinity while the number of observations remains constant. We then show the effectiveness in extensive simulations and on real world data.

To estimate a sparse linear model from data with Gaussian noise, consilience from lasso and compressed sensing literatures is that thresholding estimators like lasso and the Dantzig selector have the ability in some situations to identify with high probability part of the significant covariates asymptotically, and are numerically tractable thanks to convexity. Yet, the selection of a threshold parameter $\lambda$ remains crucial in practice. To that aim we propose Quantile Universal Thresholding, a selection of $\lambda$ at the detection edge. We show with extensive simulations and real data that an excellent compromise between high true positive rate and low false discovery rate is achieved, leading also to good predictive risk.

Dzmitry Bahdanau Jacobs University Bremen, Germany Yoshua Bengio Université de Montréal CIFAR Senior Fellow We replace the Hidden Markov Model (HMM) which is traditionally used in in continuous speech recognition with a bidirectional recurrent neural network encoder coupled to a recurrent neural network decoder that directly emits a stream of phonemes. The alignment between the input and output sequences is established using an attention mechanism: the decoder emits each symbol based on a context created with a subset of input symbols selected by the attention mechanism. We report initial results demonstrating that this new approach achieves phoneme error rates that are comparable to the state-of-the-art HMM-based decoders, on the TIMIT dataset.

Diffusion-weighted MR imaging (DWI) is the only method we currently have to measure connections between different parts of the human brain in vivo. To elucidate the structure of these connections, algorithms for tracking bundles of axonal fibers through the subcortical white matter rely on local estimates of the fiber orientation distribution function (fODF) in different parts of the brain. These functions describe the relative abundance of populations of axonal fibers crossing each other in each location. Multiple models exist for estimating fODFs. The quality of the resulting estimates can be quantified by means of a suitable measure of distance on the space of fODFs. However, there are multiple distance metrics that can be applied for this purpose, including smoothed $L_p$ distances and the Wasserstein metrics. Here, we give four reasons for the use of the Earth Mover's Distance (EMD) equipped with the arc-length, as a distance metric. (continued)

Based on a new atomic norm, we propose a new convex formulation for sparse matrix factorization problems in which the number of nonzero elements of the factors is assumed fixed and known. The formulation counts sparse PCA with multiple factors, subspace clustering and low-rank sparse bilinear regression as potential applications. We compute slow rates and an upper bound on the statistical dimension of the suggested norm for rank 1 matrices, showing that its statistical dimension is an order of magnitude smaller than the usual $\ell\_1$-norm, trace norm and their combinations. Even though our convex formulation is in theory hard and does not lead to provably polynomial time algorithmic schemes, we propose an active set algorithm leveraging the structure of the convex problem to solve it and show promising numerical results.

Deep Gaussian processes provide a flexible approach to probabilistic modelling of data using either supervised or unsupervised learning. For tractable inference approximations to the marginal likelihood of the model must be made. The original approach to approximate inference in these models used variational compression to allow for approximate variational marginalization of the hidden variables leading to a lower bound on the marginal likelihood of the model [Damianou and Lawrence, 2013]. In this paper we extend this idea with a nested variational compression. The resulting lower bound on the likelihood can be easily parallelized or adapted for stochastic variational inference.

We expand the item response theory to study the case of "cheating students" for a set of exams, trying to detect them by applying a greedy algorithm of inference. This extended model is closely related to the Boltzmann machine learning. In this paper we aim to infer the correct biases and interactions of our model by considering a relatively small number of sets of training data. Nevertheless, the greedy algorithm that we employed in the present study exhibits good performance with a few number of training data. The key point is the sparseness of the interactions in our problem in the context of the Boltzmann machine learning: the existence of cheating students is expected to be very rare (possibly even in real world). We compare a standard approach to infer the sparse interactions in the Boltzmann machine learning to our greedy algorithm and we find the latter to be superior in several aspects.

Factor analysis provides linear factors that describe relationships between individual variables of a data set. We extend this classical formulation into linear factors that describe relationships between groups of variables, where each group represents either a set of related variables or a data set. The model also naturally extends canonical correlation analysis to more than two sets, in a way that is more flexible than previous extensions. Our solution is formulated as variational inference of a latent variable model with structural sparsity, and it consists of two hierarchical levels: The higher level models the relationships between the groups, whereas the lower models the observed variables given the higher level. We show that the resulting solution solves the group factor analysis problem accurately, outperforming alternative factor analysis based solutions as well as more straightforward implementations of group factor analysis. The method is demonstrated on two life science data sets, one on brain activation and the other on systems biology, illustrating its applicability to the analysis of different types of high-dimensional data sources.

An answer set is a plain set of literals which has no further structure that would explain why certain literals are part of it and why others are not. We show how argumentation theory can help to explain why a literal is or is not contained in a given answer set by defining two justification methods, both of which make use of the correspondence between answer sets of a logic program and stable extensions of the Assumption-Based Argumentation (ABA) framework constructed from the same logic program. Attack Trees justify a literal in argumentation-theoretic terms, i.e. using arguments and attacks between them, whereas ABA-Based Answer Set Justifications express the same justification structure in logic programming terms, that is using literals and their relationships. Interestingly, an ABA-Based Answer Set Justification corresponds to an admissible fragment of the answer set in question, and an Attack Tree corresponds to an admissible fragment of the stable extension corresponding to this answer set.

Human Motion Analysis (HMA) is currently one of the most popularly active research domains as such significant research interests are motivated by a number of real world applications such as video surveillance, sports analysis, healthcare monitoring and so on. However, most of these real world applications face high levels of uncertainties that can affect the operations of such applications. Hence, the fuzzy set theory has been applied and showed great success in the recent past. In this paper, we aim at reviewing the fuzzy set oriented approaches for HMA, individuating how the fuzzy set may improve the HMA, envisaging and delineating the future perspectives. To the best of our knowledge, there is not found a single survey in the current literature that has discussed and reviewed fuzzy approaches towards the HMA. For ease of understanding, we conceptually classify the human motion into three broad levels: Low-Level (LoL), Mid-Level (MiL), and High-Level (HiL) HMA.