Charts are an excellent way to convey patterns and trends in data, but they do not facilitate further modeling of the data or close inspection of individual data points. We present a fully automated system for extracting the numerical values of data points from images of scatter plots. We use deep learning techniques to identify the key components of the chart, and optical character recognition together with robust regression to map from pixels to the coordinate system of the chart. We focus on scatter plots with linear scales, which already have several interesting challenges. Previous work has done fully automatic extraction for other types of charts, but to our knowledge this is the first approach that is fully automatic for scatter plots. Our method performs well, achieving successful data extraction on 89% of the plots in our test set.

Estimating the state of a dynamical system from a series of noise-corrupted observations is fundamental in many areas of science and engineering. The most well-known method, the Kalman smoother (and the related Kalman filter), relies on assumptions of linearity and Gaussianity that are rarely met in practice. In this paper, we introduced a new dynamical smoothing method that exploits the remarkable capabilities of convolutional neural networks to approximate complex non-linear functions. The main idea is to generate a training set composed of both latent states and observations from an ensemble of simulators and to train the deep network to recover the former from the latter. Importantly, this method only requires the availability of the simulators and can therefore be applied in situations in which either the latent dynamical model or the observation model cannot be easily expressed in closed form. In our simulation studies, we show that the resulting ConvNet smoother has almost optimal performance in the Gaussian case even when the parameters are unknown. Furthermore, the method can be successfully applied to extremely non-linear and non-Gaussian systems. Finally, we empirically validate our approach via the analysis of measured brain signals.

This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a local-entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the local entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.

The minimization of non-differentiable convex cost functions is a critical step in the solution of many design problems [3]-[5], including the design of sparse-aware (LASSO) solutions [6], [7], support-vector machine (SVM) learners [8]-[12], or total-variation-based image denoising solutions [13], [14]. Several powerful techniques have been proposed in the literature to deal with the non-differentiability aspect of the problem formulation, including methods that employ sub-gradient iterations [3]-[5], cutting-plane techniques [15], or proximal iterations [16], [17]. This work focuses on the class of sub-gradient methods for the reasons explained in the sequel. The sub-gradient technique is closely related to the traditional gradient-descent method [3], [4] where the actual gradient is replaced by a sub-gradient at points of nondifferentiability. It is one of the simplest methods in current practice but is known to suffer from slow convergence. For instance, it is shown in [5] that, for convex cost functions, the optimal convergence rate that can be delivered by sub-gradient methods in deterministic optimization problems cannot be faster than O(1/ i) under worst case conditions, where i is the iteration index.

Graphs are a central tool in machine learning and information processing as they allow to conveniently capture the structure of complex datasets. In this context, it is of high importance to develop flexible models of signals defined over graphs or networks. In this paper, we generalize the traditional concept of wide sense stationarity to signals defined over the vertices of arbitrary weighted undirected graphs. We show that stationarity is expressed through the graph localization operator reminiscent of translation. We prove that stationary graph signals are characterized by a well-defined Power Spectral Density that can be efficiently estimated even for large graphs. We leverage this new concept to derive Wiener-type estimation procedures of noisy and partially observed signals and illustrate the performance of this new model for denoising and regression.

Training convolutional networks (CNN's) that fit on a single GPU with minibatch stochastic gradient descent has become effective in practice. However, there is still no effective method for training large CNN's that do not fit in the memory of a few GPU cards, or for parallelizing CNN training. In this work we show that a simple hard mixture of experts model can be efficiently trained to good effect on large scale hashtag (multilabel) prediction tasks. Mixture of experts models are not new [7, 3], but in the past, researchers have had to devise sophisticated methods to deal with data fragmentation. We show empirically that modern weakly supervised data sets are large enough to support naive partitioning schemes where each data point is assigned to a single expert. Because the experts are independent, training them in parallel is easy, and evaluation is cheap for the size of the model. Furthermore, we show that we can use a single decoding layer for all the experts, allowing a unified feature embedding space. We demonstrate that it is feasible (and in fact relatively painless) to train far larger models than could be practically trained with standard CNN architectures, and that the extra capacity can be well used on current datasets.

The analysis in Part I revealed interesting properties for subgradient learning algorithms in the context of stochastic optimization when gradient noise is present. These algorithms are used when the risk functions are non-smooth and involve non-differentiable components. They have been long recognized as being slow converging methods. However, it was revealed in Part I that the rate of convergence becomes linear for stochastic optimization problems, with the error iterate converging at an exponential rate $\alpha^i$ to within an $O(\mu)-$neighborhood of the optimizer, for some $\alpha \in (0,1)$ and small step-size $\mu$. The conclusion was established under weaker assumptions than the prior literature and, moreover, several important problems (such as LASSO, SVM, and Total Variation) were shown to satisfy these weaker assumptions automatically (but not the previously used conditions from the literature). These results revealed that sub-gradient learning methods have more favorable behavior than originally thought when used to enable continuous adaptation and learning. The results of Part I were exclusive to single-agent adaptation. The purpose of the current Part II is to examine the implications of these discoveries when a collection of networked agents employs subgradient learning as their cooperative mechanism. The analysis will show that, despite the coupled dynamics that arises in a networked scenario, the agents are still able to attain linear convergence in the stochastic case; they are also able to reach agreement within $O(\mu)$ of the optimizer.

This paper presents new algorithms for distributed statistical estimation that can take advantage of the divide-and-conquer approach. We show that one of the key benefits attained by an appropriate divide-and-conquer strategy is robustness, an important characteristic of large distributed systems. We introduce a class of algorithms that are based on the properties of the geometric median, establish connections between performance of these distributed algorithms and rates of convergence in normal approximation, and provide tight deviations guarantees for resulting estimators in the form of exponential concentration inequalities. Our techniques are illustrated through several examples: in particular, we obtain new results for the median-of-means estimator, as well as provide performance guarantees for robust distributed maximum likelihood estimation.

Spectral dimensionality reduction algorithms are widely used in numerous domains, including for recognition, segmentation, tracking and visualization. However, despite their popularity, these algorithms suffer from a major limitation known as the "repeated Eigen-directions" phenomenon. That is, many of the embedding coordinates they produce typically capture the same direction along the data manifold. This leads to redundant and inefficient representations that do not reveal the true intrinsic dimensionality of the data. In this paper, we propose a general method for avoiding redundancy in spectral algorithms. Our approach relies on replacing the orthogonality constraints underlying those methods by unpredictability constraints. Specifically, we require that each embedding coordinate be unpredictable (in the statistical sense) from all previous ones. We prove that these constraints necessarily prevent redundancy, and provide a simple technique to incorporate them into existing methods. As we illustrate on challenging high-dimensional scenarios, our approach produces significantly more informative and compact representations, which improve visualization and classification tasks.

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