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Many neural networks for graphs are based on the graph convolution operator, proposed more than a decade ago. Since then, many alternative definitions have been proposed, that tend to add complexity (and non-linearity) to the model. In this…
We investigate spaces of operators which are invariant under translations or modulations by lattices in phase space. The natural connection to the Heisenberg module is considered, giving results on the characterisation of such operators as…
A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be…
In this work we begin a theoretical and numerical investigation on the spectra of evolution operators of neutral renewal equations, with the stability of equilibria and periodic orbits in mind. We start from the simplest form of linear…
Cellular automata (CA) are a class of computational models that exhibit rich dynamics emerging from the local interaction of cells arranged in a regular lattice. In this work we focus on a generalised version of typical CA, called graph…
A class of fermionic quantum field theories with interactions is shown to be equivalent to probabilistic cellular automata, namely cellular automata with a probability distribution for the initial states. Probabilistic cellular automata on…
The ``time-evolution operator'' in mechanics is a powerful tool which can be geometrically defined as a vector field along the Legendre map. It has been extensively used by several authors for studying the structure and properties of the…
Multiparameter persistent homology has been largely neglected as an input to machine learning algorithms. We consider the use of lattice-based convolutional neural network layers as a tool for the analysis of features arising from…
Cellular Automata are discrete--time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular Automata, are discrete…
Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the…
We apply the methods of lattice field theories to the quantization of cellular automata. We discuss the quantization of five main categories of cellular automata: bosonic, fermionic, supersymmetric, spin and quantum dot using path integral…
The finite-element approach to lattice field theory is both highly accurate (relative errors $\sim1/N^2$, where $N$ is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly…
Cellular automata, CA for short are continuous maps defined on the set of configurations over a finite alphabet A that commutes with the shift. They are characterized by the existence of local function which determine by local behavior the…
We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This analysis applies, through unitary equivalence, to…
Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an $L^2$-positive definite and zonal kernel on the unit sphere of $\mathbb{C}^q$ in order that the…
We consider operators $H_\mu$ of convolution with measures $\mu$ on locally compact groups. We characterize the spectrum of $H_\mu$ by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of $H_\mu$,…
A notion of convolution is presented in the context of formal power series together with lifting constructions characterising algebras of such series, which usually are quantales. A number of examples underpin the universality of these…
Unitarity of the global evolution is an extremely stringent condition on finite state models in discrete spacetime. Quantum cellular automata, in particular, are tightly constrained. In previous work we proved a simple No-go Theorem which…
We present a description of the function space and the smoothness class associated with a convolutional network using the machinery of reproducing kernel Hilbert spaces. We show that the mapping associated with a convolutional network…
One-dimensional cellular automata are discrete dynamical systems that operate on an infinite lattice of sites and are characterized by the locality and uniformity of their update rule. Permutations of the state set and isometric…