Networks are a fundamental model of complex systems throughout the sciences, and network datasets are typically analyzed through lower-order connectivity patterns described at the level of individual nodes and edges. However, higher-order connectivity patterns captured by small subgraphs, also called network motifs, describe the fundamental structures that control and mediate the behavior of many complex systems. We develop three tools for network analysis that use higher-order connectivity patterns to gain new insights into network datasets: (1) a framework to cluster nodes into modules based on joint participation in network motifs; (2) a generalization of the clustering coefficient measurement to investigate higher-order closure patterns; and (3) a definition of network motifs for temporal networks and fast algorithms for counting them. Using these tools, we analyze data from biology, ecology, economics, neuroscience, online social networks, scientific collaborations, telecommunications, transportation, and the World Wide Web.

We show that the error probability of reconstructing kernel matrices from Random Fourier Features for the Gaussian kernel function is at most $\mathcal{O}(R^{2/3} \exp(-D))$, where $D$ is the number of random features and $R$ is the diameter of the data domain. We also provide an information-theoretic method-independent lower bound of $\Omega(\exp(-D))$ for $R>2.1$. Compared to prior work, we are the first to show that the error probability for random Fourier features is independent of the dimensionality of data points. As applications of our theory, we obtain dimension-independent bounds for kernel ridge regression and support vector machines.

In this paper we propose strategies for estimating performance of a classifier when labels cannot be obtained for the whole test set. The number of test instances which can be labeled is very small compared to the whole test data size. The goal then is to obtain a precise estimate of classifier performance using as little labeling resource as possible. Specifically, we try to answer, how to select a subset of the large test set for labeling such that the performance of a classifier estimated on this subset is as close as possible to the one on the whole test set. We propose strategies based on stratified sampling for selecting this subset. We show that these strategies can reduce the variance in estimation of classifier accuracy by a significant amount compared to simple random sampling (over 65% in several cases). Hence, our proposed methods are much more precise compared to random sampling for accuracy estimation under restricted labeling resources. The reduction in number of samples required (compared to random sampling) to estimate the classifier accuracy with only 1% error is high as 60% in some cases.

Learning to recognize touch gestures: recurrent vs. convolutional features and dynamic sampling Abstract-- We propose a fully automatic method for learning gestures on big touch devices in a potentially multi-user context. The goal is to learn general models capable of adapting to different gestures, user styles and hardware variations (e.g. Based on deep neural networks, our method features a novel dynamic sampling and temporal normalization component, transforming variable length gestures into fixed length representations while preserving finger/surface contact transitions, that is, the topology of the signal. This sequential representation is then processed with a convolutional model capable, unlike recurrent networks, of learning hierarchical representations with different levels of abstraction. To demonstrate the interest of the proposed method, we introduce a new touch gestures dataset with 6591 gestures performed by 27 people, which is, up to our knowledge, the first of its kind: a publicly available multi-touch gesture dataset for interaction. We also tested our method on a standard dataset of symbolic touch gesture recognition, the MMG dataset, outperforming the state of the art and reporting close to perfect performance. I. INTRODUCTION Touch screen technology has been widely integrated into many different devices for about a decade, becoming a major interface with different use cases ranging from smartphones to big touch tables. Starting with simple interactions, such as taps or single touch gestures, we are now using these interfaces to perform more and more complex actions, involving multiple touches and/or multiple users. If simple interactions do not require complicated engineering to perform well, advanced manipulations such as navigating through a 3D modelisation or designing a document in parallel with different users still craves for easier and better interactions. As of today, different methods and frameworks for touch gesture recognition were developed (see for instance [15], [28] and [7] for reviews). These methods define a specific model for the class, and it is up to the user to execute the correct protocol.

Time series are one of the most common data types encountered in daily life. Financial prices, weather, home energy usage, and even weight are all examples of data that can be collected at regular intervals. Almost every data scientist will encounter time series in their daily work and learning how to model them is an important skill in the data science toolbox. One powerful yet simple method for analyzing and predicting periodic data is the additive model. The idea is straightforward: represent a time-series as a combination of patterns at different scales such as daily, weekly, seasonally, and yearly, along with an overall trend.

Most people in the field of analytics can remember writing their own analytical code. Today, our Data Scientists in the MSiA program at Northwestern, can produce analytical models from regression, decision trees, support vector machines (and more) – all with more or less one simple execution. The manual step is minor. In fact, the manual step is being removed as analytics moves into automation and artificial intelligence. Career Take Away: Develop skills in many model types.

Throughout its history, Machine Learning (ML) has coexisted with Statistics uneasily, like an ex-boyfriend accidentally seated with the groom's family at a wedding reception: both uncertain where to lead the conversation, but painfully aware of the potential for awkwardness. This is caused in part by the fact that Machine Learning has adopted many of Statistics' methods, but was never intended to replace statistics, or even to have a statistical basis originally. Nevertheless, Statisticians and ML practitioners have often ended up working together, or working on similar tasks, and wondering what each was about. The question, "What's the difference between Machine Learning and Statistics?" has been asked now for decades. Machine Learning is largely a hybrid field, taking its inspiration and techniques from all manner of sources. It has changed directions throughout its history and often seemed like an enigma to those outside of it.1

We analyse a linear regression problem with nonconvex regularization called smoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basis for Gaussian random data. We propose an approximate message passing (AMP) algorithm considering nonconvex regularization, namely SCAD-AMP, and analytically show that the stability condition corresponds to the de Almeida--Thouless condition in spin glass literature. Through asymptotic analysis, we show the correspondence between the density evolution of SCAD-AMP and the replica symmetric solution. Numerical experiments confirm that for a sufficiently large system size, SCAD-AMP achieves the optimal performance predicted by the replica method. Through replica analysis, a phase transition between replica symmetric (RS) and replica symmetry breaking (RSB) region is found in the parameter space of SCAD. The appearance of the RS region for a nonconvex penalty is a significant advantage that indicates the region of smooth landscape of the optimization problem. Furthermore, we analytically show that the statistical representation performance of the SCAD penalty is better than that of L1-based methods, and the minimum representation error under RS assumption is obtained at the edge of the RS/RSB phase. The correspondence between the convergence of the existing coordinate descent algorithm and RS/RSB transition is also indicated.

We study the detailed path-wise behavior of the discrete-time Langevin algorithm for non-convex Empirical Risk Minimization (ERM) through the lens of metastability, adopting some techniques from Berglund and Gentz. For a particular local optimum of the empirical risk, with an arbitrary initialization, we show that, with high probability, one of the two mutually exclusive events will occur: either the Langevin trajectory ends up somewhere outside the $\varepsilon$-neighborhood of this particular optimum within a short recurrence time; or it enters this $\varepsilon$-neighborhood by the recurrence time and stays there until an exponentially long escape time. We call this phenomenon empirical metastability. This two-timescale characterization aligns nicely with the existing literature in the following two senses. First, the recurrence time is dimension-independent, and resembles the convergence time of deterministic Gradient Descent (GD). However unlike GD, the Langevin algorithm does not require strong conditions on local initialization, and has the possibility of eventually visiting all optima. Second, the scaling of the escape time is consistent with the Eyring-Kramers law, which states that the Langevin scheme will eventually visit all local minima, but it will take an exponentially long time to transit among them. We apply this path-wise concentration result in the context of statistical learning to examine local notions of generalization and optimality.

We provide a new computationally-efficient class of estimators for risk minimization. We show that these estimators are robust for general statistical models: in the classical Huber epsilon-contamination model and in heavy-tailed settings. Our workhorse is a novel robust variant of gradient descent, and we provide conditions under which our gradient descent variant provides accurate estimators in a general convex risk minimization problem. We provide specific consequences of our theory for linear regression, logistic regression and for estimation of the canonical parameters in an exponential family. These results provide some of the first computationally tractable and provably robust estimators for these canonical statistical models. Finally, we study the empirical performance of our proposed methods on synthetic and real datasets, and find that our methods convincingly outperform a variety of baselines.