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The topology of many real complex networks has been conjectured to be embedded in hidden metric spaces, where distances between nodes encode their likelihood of being connected. Besides of providing a natural geometrical interpretation of…
Deep neural networks often contain far more parameters than training examples, yet they still manage to generalize well in practice. Classical complexity measures such as VC-dimension or PAC-Bayes bounds usually become vacuous in this…
Compressive sampling has great potential for making wideband spectrum sensing possible at sub-Nyquist sampling rates. As a result, there have recently been research efforts that leverage compressive sampling to enable efficient wideband…
Generative mechanisms which lead to empirically observed structure of networked systems from diverse fields like biology, technology and social sciences form a very important part of study of complex networks. The structure of many…
In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control systems, we provide a…
Deep networks have recently been shown to be vulnerable to universal perturbations: there exist very small image-agnostic perturbations that cause most natural images to be misclassified by such classifiers. In this paper, we propose the…
This article describes a gradient complex network model whose weights are proportional to the difference between uniformly distributed ``fitness'' values assigned to the nodes. It is shown analytically and experimentally that the strength…
This paper considers the growth in the length of one-dimensional trajectories as they are passed through deep ReLU neural networks, which, among other things, is one measure of the expressivity of deep networks. We generalise existing…
In the present study, we investigate a universality of neural networks, which concerns a density of the set of two-layer neural networks in a function spaces. There are many works that handle the convergence over compact sets. In the…
As network research becomes more sophisticated, it is more common than ever for researchers to find themselves not studying a single network but needing to analyze sets of networks. An important task when working with sets of networks is…
Various iterative reconstruction algorithms for inverse problems can be unfolded as neural networks. Empirically, this approach has often led to improved results, but theoretical guarantees are still scarce. While some progress on…
Neural networks are nowadays highly successful despite strong hardness results. The existing hardness results focus on the network architecture, and assume that the network's weights are arbitrary. A natural approach to settle the…
We derive upper bounds on the complexity of ReLU neural networks approximating the solution of a linear system given the matrix and the right-hand side. We focus on matrices which are symmetric positive definite and sparse, as they appear…
The learnability of different neural architectures can be characterized directly by computable measures of data complexity. In this paper, we reframe the problem of architecture selection as understanding how data determines the most…
We show generalisation error bounds for deep learning with two main improvements over the state of the art. (1) Our bounds have no explicit dependence on the number of classes except for logarithmic factors. This holds even when formulating…
Recent advances in deep learning have achieved impressive gains in classification accuracy on a variety of types of data, including images and text. Despite these gains, however, concerns have been raised about the calibration, robustness,…
The abundance of models of complex networks and the current insufficient validation standards make it difficult to judge which models are strongly supported by data and which are not. We focus here on likelihood maximization methods for…
The widespread relevance of complex networks is a valuable tool in the analysis of a broad range of systems. There is a demand for tools which enable the extraction of meaningful information and allow the comparison between different…
The weight of links in a network is often related to the similarity of the nodes. Here, we introduce a simple tunable measure for analysing the similarity of nodes across different link weights. In particular, we use the measure to analyze…
This paper presents a general framework for norm-based capacity control for $L_{p,q}$ weight normalized deep neural networks. We establish the upper bound on the Rademacher complexities of this family. With an $L_{p,q}$ normalization where…