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The structure of complex networks has been of interest in many scientific and engineering disciplines over the decades. A number of studies in the field have been focused on finding the common properties among different kinds of networks…

This article defines a partial order structure to study the relationship between levels and contents of conscious subjective experience in a single mathematical set-up. We understand phenomenal structure as extrapolated relationships among…

The estimation of functional networks through functional covariance and graphical models have recently attracted increasing attention in settings with high dimensional functional data, where the number of functional variables p is…

We show that MLP layers in transformer language models perform binary routing of continuous signals: the decision of whether a token needs nonlinear processing is well-captured by binary neuron activations, even though the signals being…

The human brain is a complex network comprised of functionally and anatomically interconnected brain regions. A growing number of studies have suggested that empirical estimates of brain networks may be useful for discovery of biomarkers of…

Inferring tie strengths (strong vs. weak) is a core task in network analysis, often guided by the Strong Triadic Closure (STC) principle. In multilayer networks, such as social platforms or biological systems, applying STC independently to…

In a function approximation with a neural network, an input dataset is mapped to an output index by optimizing the parameters of each hidden-layer unit. For a unary function, we present constraints on the parameters and its second…

This article investigates the probabilistic relationship between quantum classification of Boolean functions and their Hamming distance. By integrating concepts from quantum computing, information theory, and combinatorics, we explore how…

Characterizing judgments of similarity within a perceptual or semantic domain, and making inferences about the underlying structure of this domain from these judgments, has an increasingly important role in cognitive and systems…

Brain areas' functional repertoires are shaped by their incoming and outgoing structural connections. In empirically measured networks, most connections are short, reflecting spatial and energetic constraints. Nonetheless, a small number of…

This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hyperparameters are optimized,…

Functions correspond to one of the key concepts in mathematics and science, allowing the representation and modeling of several types of signals and systems. The present work develops an approach for characterizing the coverage and…

Supervised training of a convolutional network for object classification should make explicit any information related to the class of objects and disregard any auxiliary information associated with the capture of the image or the variation…

We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function $\sigma : \mathbb{C} \to…

Universal approximation theorem suggests that a shallow neural network can approximate any function. The input to neurons at each layer is a weighted sum of previous layer neurons and then an activation is applied. These activation…

One of the central questions in the theory of deep learning is to understand how neural networks learn hierarchical features. The ability of deep networks to extract salient features is crucial to both their outstanding generalization…

We study bisimulation and context equivalence in a probabilistic $\lambda$-calculus. The contributions of this paper are threefold. Firstly we show a technique for proving congruence of probabilistic applicative bisimilarity. While the…

Functional depth is used for ranking functional observations from most outlying to most typical. The ranks produced by functional depth have been proposed as the basis for functional classifiers, rank tests, and data visualization…

The rank of neural networks measures information flowing across layers. It is an instance of a key structural condition that applies across broad domains of machine learning. In particular, the assumption of low-rank feature representations…

We present an analysis of different techniques for selecting the connection be- tween layers of deep neural networks. Traditional deep neural networks use ran- dom connection tables between layers to keep the number of connections small and…