Reliable diagnosis of depressive disorder is essential for both optimal treatment and prevention of fatal outcomes. In this study, we aimed to elucidate the effectiveness of two non-linear measures, Higuchi Fractal Dimension (HFD) and Sample Entropy (SampEn), in detecting depressive disorders when applied on EEG. HFD and SampEn of EEG signals were used as features for seven machine learning algorithms including Multilayer Perceptron, Logistic Regression, Support Vector Machines with the linear and polynomial kernel, Decision Tree, Random Forest, and Naive Bayes classifier, discriminating EEG between healthy control subjects and patients diagnosed with depression. We confirmed earlier observations that both non-linear measures can discriminate EEG signals of patients from healthy control subjects. The results suggest that good classification is possible even with a small number of principal components. Average accuracy among classifiers ranged from 90.24% to 97.56%. Among the two measures, SampEn had better performance. Using HFD and SampEn and a variety of machine learning techniques we can accurately discriminate patients diagnosed with depression vs controls which can serve as a highly sensitive, clinically relevant marker for the diagnosis of depressive disorders.

Scientific fields such as insider-threat detection and highway-safety planning often lack sufficient amounts of time-series data to estimate statistical models for the purpose of scientific discovery. Moreover, the available limited data are quite noisy. This presents a major challenge when estimating time-series models that are robust to overfitting and have well-calibrated uncertainty estimates. Most of the current literature in these fields involve visualizing the time-series for noticeable structure and hard coding them into pre-specified parametric functions. This approach is associated with two limitations. First, given that such trends may not be easily noticeable in small data, it is difficult to explicitly incorporate expressive structure into the models during formulation. Second, it is difficult to know $\textit{a priori}$ the most appropriate functional form to use. To address these limitations, a nonparametric Bayesian approach was proposed to implicitly capture hidden structure from time series having limited data. The proposed model, a Gaussian process with a spectral mixture kernel, precludes the need to pre-specify a functional form and hard code trends, is robust to overfitting and has well-calibrated uncertainty estimates.

Subspace clustering is a useful technique for many computer vision applications in which the intrinsic dimension of high-dimensional data is often smaller than the ambient dimension. Spectral clustering, as one of the main approaches to subspace clustering, often takes on a sparse representation or a low-rank representation to learn a block diagonal self-representation matrix for subspace generation. However, existing methods require solving a large scale convex optimization problem with a large set of data, with computational complexity reaches O(N^3) for N data points. Therefore, the efficiency and scalability of traditional spectral clustering methods can not be guaranteed for large scale datasets. In this paper, we propose a subspace clustering model based on the Kronecker product. Due to the property that the Kronecker product of a block diagonal matrix with any other matrix is still a block diagonal matrix, we can efficiently learn the representation matrix which is formed by the Kronecker product of k smaller matrices. By doing so, our model significantly reduces the computational complexity to O(kN^{3/k}). Furthermore, our model is general in nature, and can be adapted to different regularization based subspace clustering methods. Experimental results on two public datasets show that our model significantly improves the efficiency compared with several state-of-the-art methods. Moreover, we have conducted experiments on synthetic data to verify the scalability of our model for large scale datasets.

Variational inference relies on flexible approximate posterior distributions. Normalizing flows provide a general recipe to construct flexible variational posteriors. We introduce Sylvester normalizing flows, which can be seen as a generalization of planar flows. Sylvester normalizing flows remove the well-known single-unit bottleneck from planar flows, making a single transformation much more flexible. We compare the performance of Sylvester normalizing flows against planar flows and inverse autoregressive flows and demonstrate that they compare favorably on several datasets.

Distributed sparse learning with a cluster of multiple machines has attracted much attention in machine learning, especially for large-scale applications with high-dimensional data. One popular way to implement sparse learning is to use $L_1$ regularization. In this paper, we propose a novel method, called proximal \mbox{SCOPE}~(\mbox{pSCOPE}), for distributed sparse learning with $L_1$ regularization. pSCOPE is based on a \underline{c}ooperative \underline{a}utonomous \underline{l}ocal \underline{l}earning~(\mbox{CALL}) framework. In the \mbox{CALL} framework of \mbox{pSCOPE}, we find that the data partition affects the convergence of the learning procedure, and subsequently we define a metric to measure the goodness of a data partition. Based on the defined metric, we theoretically prove that pSCOPE is convergent with a linear convergence rate if the data partition is good enough. We also prove that better data partition implies faster convergence rate. Furthermore, pSCOPE is also communication efficient. Experimental results on real data sets show that pSCOPE can outperform other state-of-the-art distributed methods for sparse learning.

We present a formulation of deep learning that aims at producing a large margin classifier. The notion of margin, minimum distance to a decision boundary, has served as the foundation of several theoretically profound and empirically successful results for both classification and regression tasks. However, most large margin algorithms are applicable only to shallow models with a preset feature representation; and conventional margin methods for neural networks only enforce margin at the output layer. Such methods are therefore not well suited for deep networks. In this work, we propose a novel loss function to impose a margin on any chosen set of layers of a deep network (including input and hidden layers). Our formulation allows choosing any norm on the metric measuring the margin. We demonstrate that the decision boundary obtained by our loss has nice properties compared to standard classification loss functions. Specifically, we show improved empirical results on the MNIST, CIFAR-10 and ImageNet datasets on multiple tasks: generalization from small training sets, corrupted labels, and robustness against adversarial perturbations. The resulting loss is general and complementary to existing data augmentation (such as random/adversarial input transform) and regularization techniques (such as weight decay, dropout, and batch norm).

Momentum based stochastic gradient methods such as heavy ball (HB) and Nesterov's accelerated gradient descent (NAG) method are widely used in practice for training deep networks and other supervised learning models, as they often provide significant improvements over stochastic gradient descent (SGD). Rigorously speaking, "fast gradient" methods have provable improvements over gradient descent only for the deterministic case, where the gradients are exact. In the stochastic case, the popular explanations for their wide applicability is that when these fast gradient methods are applied in the stochastic case, they partially mimic their exact gradient counterparts, resulting in some practical gain. This work provides a counterpoint to this belief by proving that there exist simple problem instances where these methods cannot outperform SGD despite the best setting of its parameters. These negative problem instances are, in an informal sense, generic; they do not look like carefully constructed pathological instances. These results suggest (along with empirical evidence) that HB or NAG's practical performance gains are a by-product of mini-batching. Furthermore, this work provides a viable (and provable) alternative, which, on the same set of problem instances, significantly improves over HB, NAG, and SGD's performance. This algorithm, referred to as Accelerated Stochastic Gradient Descent (ASGD), is a simple to implement stochastic algorithm, based on a relatively less popular variant of Nesterov's Acceleration. Extensive empirical results in this paper show that ASGD has performance gains over HB, NAG, and SGD.

Recent efforts on combining deep models with probabilistic graphical models are promising in providing flexible models that are also easy to interpret. We propose a variational message-passing algorithm for variational inference in such models. First, we propose structured inference networks that incorporate the structure of the graphical model in the inference network of variational auto-encoders (V AE). Second, we establish conditions under which such inference networks enable fast amortized inference similar to V AE. Finally, we derive a variational message passing algorithm to perform efficient natural-gradient inference while retaining the efficiency of the amortized inference. By simultaneously enabling structured, amortized, and natural-gradient inference for deep structured models, our method simplifies and generalizes existing methods. To analyze real-world data, machine learning relies on models that can extract useful patterns. Deep Neural Networks (DNNs) are a popular choice for this purpose because they can learn flexible representations. Another popular choice are probabilistic graphical models (PGMs) which can find interpretable structures in the data. Recent work on combining these two types of models hopes to exploit their complimentary strengths and provide powerful models that are also easy to interpret (Johnson et al., 2016; Krishnan et al., 2015; Archer et al., 2015; Fraccaro et al., 2016). To apply such hybrid models to real-world problems, we need efficient algorithms that can extract useful structure from the data. For deep learning, stochastic-gradient methods are the most popular choice, e.g., those based on back-propagation.

Distance metric learning is a successful way to enhance the performance of the nearest neighbor classifier. In most cases, however, the distribution of data does not obey a regular form and may change in different parts of the feature space. Regarding that, this paper proposes a novel local distance metric learning method, namely Local Mahalanobis Distance Learning (LMDL), in order to enhance the performance of the nearest neighbor classifier. LMDL considers the neighborhood influence and learns multiple distance metrics for a reduced set of input samples. The reduced set is called as prototypes which try to preserve local discriminative information as much as possible. The proposed LMDL can be kernelized very easily, which is significantly desirable in the case of highly nonlinear data. The quality as well as the efficiency of the proposed method assesses through a set of different experiments on various datasets and the obtained results show that LDML as well as the kernelized version is superior to the other related state-of-the-art methods.

Tick is a statistical learning library for Python~3, with a particular emphasis on time-dependent models, such as point processes, and tools for generalized linear models and survival analysis. The core of the library is an optimization module providing model computational classes, solvers and proximal operators for regularization. tick relies on a C++ implementation and state-of-the-art optimization algorithms to provide very fast computations in a single node multi-core setting. Source code and documentation can be downloaded from https://github.com/X-DataInitiative/tick