The widely used genetic pleiotropic analysis of multiple phenotypes are often designed for examining the relationship between common variants and a few phenotypes. They are not suited for both high dimensional phenotypes and high dimensional genotype (next-generation sequencing) data. To overcome these limitations, we develop sparse structural equation models (SEMs) as a general framework for a new paradigm of genetic analysis of multiple phenotypes. To incorporate both common and rare variants into the analysis, we extend the traditional multivariate SEMs to sparse functional SEMs. To deal with high dimensional phenotype and genotype data, we employ functional data analysis and the alternative direction methods of multiplier (ADMM) techniques to reduce data dimension and improve computational efficiency. Using large scale simulations we showed that the proposed methods have higher power to detect true causal genetic pleiotropic structure than other existing methods. Simulations also demonstrate that the gene-based pleiotropic analysis has higher power than the single variant-based pleiotropic analysis. The proposed method is applied to exome sequence data from the NHLBI Exome Sequencing Project (ESP) with 11 phenotypes, which identifies a network with 137 genes connected to 11 phenotypes and 341 edges. Among them, 114 genes showed pleiotropic genetic effects and 45 genes were reported to be associated with phenotypes in the analysis or other cardiovascular disease (CVD) related phenotypes in the literature.

We present a very general geometrico-dynamical description of physical or more abstract entities, called the 'general tension-reduction' (GTR) model, where not only states, but also measurement-interactions can be represented, and the associated outcome probabilities calculated. Underlying the model is the hypothesis that indeterminism manifests as a consequence of unavoidable fluctuations in the experimental context, in accordance with the 'hidden-measurements interpretation' of quantum mechanics. When the structure of the state space is Hilbertian, and measurements are of the 'universal' kind, i.e., are the result of an average over all possible ways of selecting an outcome, the GTR-model provides the same predictions of the Born rule, and therefore provides a natural completed version of quantum mechanics. However, when the structure of the state space is non-Hilbertian and/or not all possible ways of selecting an outcome are available to be actualized, the predictions of the model generally differ from the quantum ones, especially when sequential measurements are considered. Some paradigmatic examples will be discussed, taken from physics and human cognition. Particular attention will be given to some known psychological effects, like question order effects and response replicability, which we show are able to generate non-Hilbertian statistics. We also suggest a realistic interpretation of the GTR-model, when applied to human cognition and decision, which we think could become the generally adopted interpretative framework in quantum cognition research.

Indoor localization is a supporting technology for a broadening range of pervasive wireless applications. One promis- ing approach is to locate users with radio frequency fingerprints. However, its wide adoption in real-world systems is challenged by the time- and manpower-consuming site survey process, which builds a fingerprint database a priori for localization. To address this problem, we visualize the 3-D RF fingerprint data as a function of locations (x-y) and indices of access points (fingerprint), as a tensor and use tensor algebraic methods for an adaptive tubal-sampling of this fingerprint space. In particular using a recently proposed tensor algebraic framework in [1] we capture the complexity of the fingerprint space as a low-dimensional tensor-column space. In this formulation the proposed scheme exploits adaptivity to identify reference points which are highly informative for learning this low-dimensional space. Further, under certain incoherency conditions we prove that the proposed scheme achieves bounded recovery error and near-optimal sampling complexity. In contrast to several existing work that rely on random sampling, this paper shows that adaptivity in sampling can lead to significant improvements in localization accuracy. The approach is validated on both data generated by the ray-tracing indoor model which accounts for the floor plan and the impact of walls and the real world data. Simulation results show that, while maintaining the same localization accuracy of existing approaches, the amount of samples can be cut down by 71% for the high SNR case and 55% for the low SNR case.

Martingales are indispensable in studying the temporal dynamics of stochastic processes arising in a multitude of fields [10, 14]. Particularly when such processes have complex long-range dependences, it is often of interest to concentrate martingales uniformly over time. On the theoretical side, a fundamental limit to such concentration is expressed by the law of the iterated logarithm (LIL). However, this only concerns asymptotic behavior. In many applications, it is more natural to instead consider concentration that holds uniformly over all finite times. This manuscript presents such bounds for the large classes of martingales which are addressed by Hoeffding [11] and Bernstein [8] inequalities. These new results are optimal within small constants, and can be viewed as finite-time generalizations of the upper half of the LIL.

Biclustering structures in data matrices were first formalized in a seminal paper by John Hartigan (1972) where one seeks to cluster cases and variables simultaneously. Such structures are also prevalent in block modeling of networks. In this paper, we develop a unified theory for the estimation and completion of matrices with biclustering structures, where the data is a partially observed and noise contaminated data matrix with a certain biclustering structure. In particular, we show that a constrained least squares estimator achieves minimax rate-optimal performance in several of the most important scenarios. To this end, we derive unified high probability upper bounds for all sub-Gaussian data and also provide matching minimax lower bounds in both Gaussian and binary cases. Due to the close connection of graphon to stochastic block models, an immediate consequence of our general results is a minimax rate-optimal estimator for sparse graphons.

We introduce the Locally Linear Latent Variable Model (LL-LVM), a probabilistic model for non-linear manifold discovery that describes a joint distribution over observations, their manifold coordinates and locally linear maps conditioned on a set of neighbourhood relationships. The model allows straightforward variational optimisation of the posterior distribution on coordinates and locally linear maps from the latent space to the observation space given the data. Thus, the LL-LVM encapsulates the local-geometry preserving intuitions that underlie non-probabilistic methods such as locally linear embedding (LLE). Its probabilistic semantics make it easy to evaluate the quality of hypothesised neighbourhood relationships, select the intrinsic dimensionality of the manifold, construct out-of-sample extensions and to combine the manifold model with additional probabilistic models that capture the structure of coordinates within the manifold.

The understanding of the type of inhibitory interaction plays an important role in drug design. Therefore, researchers are interested to know whether a drug has competitive or non-competitive interaction to particular protein targets. Method: to analyze the interaction types we propose factorization method Macau which allows us to combine different measurement types into a single tensor together with proteins and compounds. The compounds are characterized by high dimensional 2D ECFP fingerprints. The novelty of the proposed method is that using a specially designed noise injection MCMC sampler it can incorporate high dimensional side information, i.e., millions of unique 2D ECFP compound features, even for large scale datasets of millions of compounds. Without the side information, in this case, the tensor factorization would be practically futile. Results: using public IC50 and Ki data from ChEMBL we trained a model from where we can identify the latent subspace separating the two measurement types (IC50 and Ki). The results suggest the proposed method can detect the competitive inhibitory activity between compounds and proteins.

This paper proposes a distributionally robust approach to logistic regression. We use the Wasserstein distance to construct a ball in the space of probability distributions centered at the uniform distribution on the training samples. If the radius of this ball is chosen judiciously, we can guarantee that it contains the unknown data-generating distribution with high confidence. We then formulate a distributionally robust logistic regression model that minimizes a worst-case expected logloss function, where the worst case is taken over all distributions in the Wasserstein ball. We prove that this optimization problem admits a tractable reformulation and encapsulates the classical as well as the popular regularized logistic regression problems as special cases. We further propose a distributionally robust approach based on Wasserstein balls to compute upper and lower confidence bounds on the misclassification probability of the resulting classifier. These bounds are given by the optimal values of two highly tractable linear programs. We validate our theoretical out-of-sample guarantees through simulated and empirical experiments.

We consider the problem of minimizing a sum of $n$ functions over a convex parameter set $\mathcal{C} \subset \mathbb{R}^p$ where $n\gg p\gg 1$. In this regime, algorithms which utilize sub-sampling techniques are known to be effective. In this paper, we use sub-sampling techniques together with low-rank approximation to design a new randomized batch algorithm which possesses comparable convergence rate to Newton's method, yet has much smaller per-iteration cost. The proposed algorithm is robust in terms of starting point and step size, and enjoys a composite convergence rate, namely, quadratic convergence at start and linear convergence when the iterate is close to the minimizer. We develop its theoretical analysis which also allows us to select near-optimal algorithm parameters. Our theoretical results can be used to obtain convergence rates of previously proposed sub-sampling based algorithms as well. We demonstrate how our results apply to well-known machine learning problems. Lastly, we evaluate the performance of our algorithm on several datasets under various scenarios.

The Support Vector Machine (SVM) has been used in a wide variety of classification problems. The original SVM uses the hinge loss function, which is non-differentiable and makes the problem difficult to solve in particular for regularized SVMs, such as with $\ell_1$-regularization. This paper considers the Huberized SVM (HSVM), which uses a differentiable approximation of the hinge loss function. We first explore the use of the Proximal Gradient (PG) method to solving binary-class HSVM (B-HSVM) and then generalize it to multi-class HSVM (M-HSVM). Under strong convexity assumptions, we show that our algorithm converges linearly. In addition, we give a finite convergence result about the support of the solution, based on which we further accelerate the algorithm by a two-stage method. We present extensive numerical experiments on both synthetic and real datasets which demonstrate the superiority of our methods over some state-of-the-art methods for both binary- and multi-class SVMs.