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We define a new class of "implicit" deep learning prediction rules that generalize the recursive rules of feedforward neural networks. These models are based on the solution of a fixed-point equation involving a single a vector of hidden features, which is thus only implicitly defined. The new framework greatly simplifies the notation of deep learning, and opens up new possibilities, in terms of novel architectures and algorithms, robustness analysis and design, interpretability, sparsity, and network architecture optimization.

We describe a novel family of models of multilayer feedforward neural networks in which the activation functions are encoded via penalties in the training problem. Our approach is based on representing a non-decreasing activation function as the argmin of an appropriate convex optimization problem. The new framework allows for algorithms such as block-coordinate descent methods to be applied, in which each step is composed of a simple (no hidden layer) supervised learning problem that is parallelizable across data points and/or layers. Experiments indicate that the proposed models provide excellent initial guesses for weights for standard neural networks. In addition, the model provides avenues for interesting extensions, such as robustness against noisy inputs and optimizing over parameters in activation functions.

The paper evaluates the influence of the maximum vehicle acceleration and variable proportions of ACC/CACC vehicles on the throughput of an intersection. Two cases are studied: (1) free road downstream of the intersection; and (2) red light at some distance downstream of the intersection. Simulation of a 4-mile stretch of an arterial with 13 signalized intersections is used to evaluate the impact of (C)ACC vehicles on the mean and standard deviation of travel time as the proportion of (C)ACC vehicles is increased. The results suggest a very high urban mobility benefit of (C)ACC vehicles at little or no cost in infrastructure.

Outlier detection methods have become increasingly relevant in recent years due to increased security concerns and because of its vast application to different fields. Recently, Pauwels and Lasserre (2016) noticed that the sublevel sets of the inverse Christoffel function accurately depict the shape of a cloud of data using a sumof-squares polynomial and can be used to perform outlier detection. In this work, we propose a kernelized variant of the inverse Christoffel function that makes it computationally tractable for data sets with a large number of features. We compare our approach to current methods on 15 different data sets and achieve the best average area under the precision recall curve (AUPRC) score, the best average rank and the lowest root mean square deviation.