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We propose a tensor neural network ($t$-NN) framework that offers an exciting new paradigm for designing neural networks with multidimensional (tensor) data. Our network architecture is based on the $t$-product (Kilmer and Martin, 2011), an…

This paper establishes a nonlinear separation principle based on contraction theory and derives sharp stability conditions for recurrent neural networks (RNNs). First, we introduce a nonlinear separation principle that guarantees global…

It has been argued by Thom and Palm that sparsely-connected neural networks (SCNs) show improved performance over fully-connected networks (FCNs). Super-regular networks (SRNs) are neural networks composed of a set of stacked sparse layers…

Zero-one biochemical reaction networks play key roles in cell signalling such as signalling pathways regulated by protein phosphorylation. Multistability of reaction networks is a crucial dynamics feature enabling decision-making in cells.…

There is consensus in the current literature that stable states of asynchronous irregular spiking activity require (i) large networks of 10 000 or more neurons and (ii) external background activity or pacemaker neurons. Yet already in 1963,…

This paper studies the problem of selecting a minimum-size set of input nodes to guarantee stability of a networked system in the presence of uncertainties and time delays. Current approaches to input selection in networked dynamical…

Let $C$ be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps $f\colon C\to\bar{C}$. First we prove that if $f(C)$ is totally bounded, then it has an approximate…

We study the problem of change point localization in dynamic networks models. We assume that we observe a sequence of independent adjacency matrices of the same size, each corresponding to a realization of an unknown inhomogeneous Bernoulli…

We consider a PT-symmetric ladder-shaped optical array consisting of a chain of waveguides with gain coupled to a parallel chain of waveguides with loss. All waveguides have the focusing Kerr nonlinearity. The array supports two co-existing…

We study the problem of solving fixed-point equations for seminorm-contractive operators and establish foundational results on the non-asymptotic behavior of iterative algorithms in both deterministic and stochastic settings. Specifically,…

We study the computational problem of the stationarity test for the empirical loss of neural networks with ReLU activation functions. Our contributions are: Hardness: We show that checking a certain first-order approximate stationarity…

Despite the success of neural networks, the issue of classification robustness remains, particularly highlighted by adversarial examples. In this paper, we address this challenge by focusing on the continuum of functions implemented in…

We establish fixed-point theorems for Meir-Keeler-type contractions in b-metric spaces. While Lu et al. demonstrated via an explicit counterexample that classical Meir-Keeler contractions may fail to admit fixed points in this setting, we…

This paper concerns piecewise-smooth maps on $\mathbb{R}^d$ that are continuous but not differentiable on switching manifolds (where the functional form of the map changes). The stability of fixed points on switching manifolds is…

This paper analyzes stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy. The associated Markov jump processes are analyzed under a thermodynamic limit…

The widespread use of Multi-layer perceptrons (MLPs) often relies on a fixed activation function (e.g., ReLU, Sigmoid, Tanh) for all nodes within the hidden layers. While effective in many scenarios, this uniformity may limit the networks…

The question of what kind of convolutional neural network (CNN) structure performs well is fascinating. In this work, we move toward the answer with one more step by connecting zero stability and model performance. Specifically, we found…

We show NP-hardness of the minimum latency scheduling (MLS) problem under the physical model of wireless networking. In this model a transmission is received successfully if the Signal to Interference-plus-Noise Ratio (SINR), is above a…

The implicit bias induced by the training of neural networks has become a topic of rigorous study. In the limit of gradient flow and gradient descent with appropriate step size, it has been shown that when one trains a deep linear network…

Neural network controllers have become popular in control tasks thanks to their flexibility and expressivity. Stability is a crucial property for safety-critical dynamical systems, while stabilization of partially observed systems, in many…