Gaussian graphical models are widely used to represent conditional dependence among random variables. In this paper, we propose a novel estimator for data arising from a group of Gaussian graphical models that are themselves dependent. A motivating example is that of modeling gene expression collected on multiple tissues from the same individual: here the multivariate outcome is affected by dependencies acting not only at the level of the specific tissues, but also at the level of the whole body; existing methods that assume independence among graphs are not applicable in this case. To estimate multiple dependent graphs, we decompose the problem into two graphical layers: the systemic layer, which affects all outcomes and thereby induces cross- graph dependence, and the category-specific layer, which represents graph-specific variation. We propose a graphical EM technique that estimates both layers jointly, establish estimation consistency and selection sparsistency of the proposed estimator, and confirm by simulation that the EM method is superior to a simple one-step method. We apply our technique to mouse genomics data and obtain biologically plausible results.

An incremental/online state dynamic learning method is proposed for identification of the nonlinear Gaussian state space models. The method embeds the stochastic variational sparse Gaussian process as the probabilistic state dynamic model inside a particle filter framework. Model updating is done at measurement sample rate using stochastic gradient descent based optimization implemented in the state estimation filtering loop. The performance of the proposed method is compared with state-of-the-art Gaussian process based batch learning methods. Finally, it is shown that the state estimation performance significantly improves due to the online learning of state dynamics.

The Kalman filter (KF) is used in a variety of applications for computing the posterior distribution of latent states in a state space model. The model requires a linear relationship between states and observations. Extensions to the Kalman filter have been proposed that incorporate linear approximations to nonlinear models, such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). However, we argue that in cases where the dimensionality of observed variables greatly exceeds the dimensionality of state variables, a model for $p(\text{state}|\text{observation})$ proves both easier to learn and more accurate for latent space estimation. We derive and validate what we call the discriminative Kalman filter (DKF): a closed-form discriminative version of Bayesian filtering that readily incorporates off-the-shelf discriminative learning techniques. Further, we demonstrate that given mild assumptions, highly non-linear models for $p(\text{state}|\text{observation})$ can be specified. We motivate and validate on synthetic datasets and in neural decoding from non-human primates, showing substantial increases in decoding performance versus the standard Kalman filter.

We study the problem of determining the one-dimensional structure that best represents a given data set. More precisely, we take a variational approach to approximating a given measure (data) by curves. We consider an objective functional whose minimizers are a regularization of principal curves and introduce a new functional which allows for multiple curves. We prove existence of minimizers and investigate their properties. While both of the functionals used are non-convex, we show that enlarging the configuration space to allow for multiple curves leads to a simpler energy landscape with fewer undesirable (high-energy) local minima. We provide an efficient algorithm for approximating minimizers of the functional and demonstrate its performance on real and synthetic data. The numerical examples illustrate the effectiveness of the proposed approach in the presence of substantial noise, and the viability of the algorithm for high-dimensional data.

In part 1 I gave a brief introduction to classification. Just to recap, classification is the problem of identifying which group a piece of data belongs to. It's an example of supervised learning because the classifier predicts the classes based on the training data fed to it. An example of classification is finding out whether an email is spam or not. More formally, classification is about finding out a model that distinguishes one class of data from another so as to predict the class of data whose class is unknown.

The rate at which companies accumulate data continues to accelerate, and with it, so does the need to translate that data into meaningful, actionable insights to key decision makers and stakeholders. Where do our customers' interests lie today, tomorrow, and beyond? That demand has put increasing pressure on solution providers to deliver technology that addresses these questions, and Microsoft is responding to the call. Azure Machine Learning (Azure ML), one of Microsoft's cloud-based solutions, helps companies unlock key insights from its available data resources. Whereas traditional machine learning offerings focus on very rigid, tightly controlled processes and experiments, Azure ML's key benefit is its highly flexible machine learning model architecture.

Clustering high-dimensional data often requires some form of dimensionality reduction, where clustered variables are separated from "noise-looking" variables. We cast this problem as finding a low-dimensional projection of the data which is well-clustered. This yields a one-dimensional projection in the simplest situation with two clusters, and extends naturally to a multi-label scenario for more than two clusters. In this paper, (a) we first show that this joint clustering and dimension reduction formulation is equivalent to previously proposed discriminative clustering frameworks, thus leading to convex relaxations of the problem, (b) we propose a novel sparse extension, which is still cast as a convex relaxation and allows estimation in higher dimensions, (c) we propose a natural extension for the multi-label scenario, (d) we provide a new theoretical analysis of the performance of these formulations with a simple probabilistic model, leading to scalings over the form $d=O(\sqrt{n})$ for the affine invariant case and $d=O(n)$ for the sparse case, where $n$ is the number of examples and $d$ the ambient dimension, and finally, (e) we propose an efficient iterative algorithm with running-time complexity proportional to $O(nd^2)$, improving on earlier algorithms which had quadratic complexity in the number of examples.

Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace. Following the blueprint of classical Linear Discriminant Analysis (LDA), WDA selects the projection matrix that maximizes the ratio of two quantities: the dispersion of projected points coming from different classes, divided by the dispersion of projected points coming from the same class. To quantify dispersion, WDA uses regularized Wasserstein distances, rather than cross-variance measures which have been usually considered, notably in LDA. Thanks to the the underlying principles of optimal transport, WDA is able to capture both global (at distribution scale) and local (at samples scale) interactions between classes. Regularized Wasserstein distances can be computed using the Sinkhorn matrix scaling algorithm; We show that the optimization of WDA can be tackled using automatic differentiation of Sinkhorn iterations. Numerical experiments show promising results both in terms of prediction and visualization on toy examples and real life datasets such as MNIST and on deep features obtained from a subset of the Caltech dataset.

This paper introduces a novel technique to track structures in time varying graphs. The method uses a maximum a posteriori approach for adjusting a three-dimensional co-clustering of the source vertices, the destination vertices and the time, to the data under study, in a way that does not require any hyper-parameter tuning. The three dimensions are simultaneously segmented in order to build clusters of source vertices, destination vertices and time segments where the edge distributions across clusters of vertices follow the same evolution over the time segments. The main novelty of this approach lies in that the time segments are directly inferred from the evolution of the edge distribution between the vertices, thus not requiring the user to make any a priori quantization. Experiments conducted on artificial data illustrate the good behavior of the technique, and a study of a real-life data set shows the potential of the proposed approach for exploratory data analysis.

In high-dimensional data, many sparse regression methods have been proposed. However, they may not be robust against outliers. Recently, the use of density power weight has been studied for robust parameter estimation and the corresponding divergences have been discussed. One of such divergences is the $\gamma$-divergence and the robust estimator using the $\gamma$-divergence is known for having a strong robustness. In this paper, we consider the robust and sparse regression based on $\gamma$-divergence. We extend the $\gamma$-divergence to the regression problem and show that it has a strong robustness under heavy contamination even when outliers are heterogeneous. The loss function is constructed by an empirical estimate of the $\gamma$-divergence with sparse regularization and the parameter estimate is defined as the minimizer of the loss function. To obtain the robust and sparse estimate, we propose an efficient update algorithm which has a monotone decreasing property of the loss function. Particularly, we discuss a linear regression problem with $L_1$ regularization in detail. In numerical experiments and real data analyses, we see that the proposed method outperforms past robust and sparse methods.