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For an arbitrary evolution family, we consider the notion of a polynomial dichotomy with respect to a family of norms and characterize it in terms of the admissibility property, that is, the existence of a unique bounded solution for each…
The success of neural networks across most machine learning tasks and the persistence of adversarial examples have made the verification of such models an important quest. Several techniques have been successfully developed to verify…
We propose a hierarchical training algorithm for standard feed-forward neural networks that adaptively extends the network architecture as soon as the optimization reaches a stationary point. By solving small (low-dimensional) optimization…
Equivariance of linear neural network layers is well studied. In this work, we relax the equivariance condition to only be true in a projective sense. We propose a way to construct a projectively equivariant neural network through building…
We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the…
This paper presents a transformative framework for artificial neural networks over graded vector spaces, tailored to model hierarchical and structured data in fields like algebraic geometry and physics. By exploiting the algebraic…
Even when neural networks are widely used in a large number of applications, they are still considered as black boxes and present some difficulties for dimensioning or evaluating their prediction error. This has led to an increasing…
The theory of Chebyshev (uniform) approximation for univariate polynomial and piecewise polynomial functions has been studied for decades. The optimality conditions are based on the notion of alternating sequence. However, the extension the…
We develop a framework that systematically casts the solvability and uniqueness conditions of linearized geometric boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the…
There is growing interest in understanding how the structural interconnections among brain regions change with the occurrence of neurological diseases. Diffusion weighted MRI imaging has allowed researchers to non-invasively estimate a…
Biological nervous systems consist of networks of diverse, sophisticated information processors in the form of neurons of different classes. In most artificial neural networks (ANNs), neural computation is abstracted to an activation…
We investigate the complexity of the reachability problem for (deep) neural networks: does it compute valid output given some valid input? It was recently claimed that the problem is NP-complete for general neural networks and…
Diversity is a concept of prime importance in almost all disciplines based on information processing. In telecommunications, for example, spatial, temporal, and frequency diversity, as well as redundant coding, are fundamental concepts that…
A theory of simultaneous resolution of singularities for families of embedded varieties (over a field of characteristic zero) parametrized by the spectrum of a suitable artinian ring, and compatible with a given algorithm of resolution, is…
Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but…
The notion of a core is generalized to multiway networks. To determine the multiway cores, we adapted already-known algorithms for determining the generalized cores in one-mode and two-mode networks. A new node property, node diversity has…
We define a neural network in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fr\'echet space $\X$ into a Banach space $\Y$. The…
We train an artificial neural network which distinguishes chaotic and regular dynamics of the two-dimensional Chirikov standard map. We use finite length trajectories and compare the performance with traditional numerical methods which need…
In this paper we discuss the stability properties of convolutional neural networks. Convolutional neural networks are widely used in machine learning. In classification they are mainly used as feature extractors. Ideally, we expect similar…
Hypergraphs provide a natural framework to model higher-order interactions in scientific, social, and biological systems. Hypergraph neural networks (HGNNs) aim to learn from such data, yet it remains unclear which higher-order structures…