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Classical neural network approximation results take the form: for every function $f$ and every error tolerance $\epsilon > 0$, one constructs a neural network whose architecture and weights depend on $\epsilon$. This paper introduces a…
The quantum approximate optimization algorithm (QAOA) is a leading iterative variational quantum algorithm for heuristically solving combinatorial optimization problems. A large portion of the computational effort in QAOA is spent by the…
The first decade of this century has seen the nascency of the first mathematical theory of general artificial intelligence. This theory of Universal Artificial Intelligence (UAI) has made significant contributions to many theoretical,…
Recently, versions of neural networks with infinite-dimensional affine operators inside the computational units (``neural operator'' networks) have been applied to learn solutions to differential equations. To enable practical computations,…
We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family to map the input…
We investigate the concept of Best Approximation for Feedforward Neural Networks (FNN) and explore their convergence properties through the lens of Random Projection (RPNNs). RPNNs have predetermined and fixed, once and for all, internal…
In this work we consider a model problem of deep neural learning, namely the learning of a given function when it is assumed that we have access to its point values on a finite set of points. The deep neural network interpolant is the the…
We review several of the most widely used techniques for training recurrent neural networks to approximate dynamical systems, then describe a novel algorithm for this task. The algorithm is based on an earlier theoretical result that…
Infinitely wide or deep neural networks (NNs) with independent and identically distributed (i.i.d.) parameters have been shown to be equivalent to Gaussian processes. Because of the favorable properties of Gaussian processes, this…
The learning speed of feed-forward neural networks is notoriously slow and has presented a bottleneck in deep learning applications for several decades. For instance, gradient-based learning algorithms, which are used extensively to train…
We present $\phi-$DeepONet, a physics-informed neural operator designed to learn mappings between function spaces that may contain discontinuities or exhibit non-smooth behavior. Classical neural operators are based on the universal…
We establish universal approximation theorems for infinite-dimensional geometric rough paths, i.e., we show that continuous functions on the space of infinite-dimensional weakly geometric H\"older continuous rough paths can be approximated…
The mean field algorithm is a widely used approximate inference algorithm for graphical models whose exact inference is intractable. In each iteration of mean field, the approximate marginals for each variable are updated by getting…
Modern mobile neural networks with a reduced number of weights and parameters do a good job with image classification tasks, but even they may be too complex to be implemented in an FPGA for video processing tasks. The article proposes…
Function regression/approximation is a fundamental application of machine learning. Neural networks (NNs) can be easily trained for function regression using a sufficient number of neurons and epochs. The forward-forward learning algorithm…
We study the approximation of shift-invariant or equivariant functions by deep fully convolutional networks from the dynamical systems perspective. We prove that deep residual fully convolutional networks and their continuous-layer…
Many approaches have been proposed to estimate camera poses by directly minimizing photometric error. However, due to the non-convex property of direct alignment, proper initialization is still required for these methods. Many robust norms…
This paper addresses the growing need to process non-Euclidean data, by introducing a geometric deep learning (GDL) framework for building universal feedforward-type models compatible with differentiable manifold geometries. We show that…
This paper extends the proof of density of neural networks in the space of continuous (or even measurable) functions on Euclidean spaces to functions on compact sets of probability measures. By doing so the work parallels a more then a…
Approximation Fixpoint Theory (AFT) is a powerful theory covering various semantics of non-monotonic reasoning formalisms in knowledge representation such as Logic Programming and Answer Set Programming. Many semantics of such non-monotonic…