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This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…
Deep convolutional networks often append additive constant ("bias") terms to their convolution operations, enabling a richer repertoire of functional mappings. Biases are also used to facilitate training, by subtracting mean response over…
Proposing an effective and flexible matrix to represent a graph is a fundamental challenge that has been explored from multiple perspectives, e.g., filtering in Graph Fourier Transforms. In this work, we develop a novel and general…
The increasing availability of geometric data has motivated the need for information processing over non-Euclidean domains modeled as manifolds. The building block for information processing architectures with desirable theoretical…
In two-dimensional noncommutive space for the case of both position - position and momentum - momentum noncommuting, the consistent deformed bosonic algebra at the non-perturbation level described by the deformed annihilation and creation…
We develop a theoretical analysis for special neural network architectures, termed operator recurrent neural networks, for approximating nonlinear functions whose inputs are linear operators. Such functions commonly arise in solution…
A very first step to develop non-commutative algebraic geometry is the arithmetic of polynomials in non-commuting variables over a commutative field, that is, the study of elements in free associative algebras. This investigation is…
Imposing orthogonality on the layers of neural networks is known to facilitate the learning by limiting the exploding/vanishing of the gradient; decorrelate the features; improve the robustness. This paper studies the theoretical properties…
We introduce a new algebraic structure for multi-dimensional compositional embeddings, built on directional non-commutative monoidal operators. The core contribution of this work is this novel framework, which exhibits appealing theoretical…
Wiatowski and B\"olcskei, 2015, proved that deformation stability and vertical translation invariance of deep convolutional neural network-based feature extractors are guaranteed by the network structure per se rather than the specific…
In this paper, we develop a finite mixture of convolutional distributions, a statistical model to analyze continuous data distributed approximately on a mixture of low-dimensional affine subspaces. The observations are assumed independent…
Predicting the evolution of a representative sample of a material with microstructure is a fundamental problem in homogenization. In this work we propose a graph convolutional neural network that utilizes the discretized representation of…
Neural network weights are typically initialized at random from univariate distributions, controlling just the variance of individual weights even in highly-structured operations like convolutions. Recent ViT-inspired convolutional networks…
We introduce the filtered *-bialgebra which is a noncommutative analog of the *-bialgebra of multivariate polynomials with the canonical coproduct and counit. We study the associated filtered convolutions, random walks and random variables.…
Convolutional neural networks (CNNs) have achieved remarkable performance in various fields, particularly in the domain of computer vision. However, why this architecture works well remains to be a mystery. In this work we move a small step…
Subspace preserving quantum circuits are a class of quantum algorithms that, relying on some symmetries in the computation, can offer theoretical guarantees for their training. Those algorithms have gained extensive interest as they can…
Convolutional neural networks (CNNs) based solutions have achieved state-of-the-art performances for many computer vision tasks, including classification and super-resolution of images. Usually the success of these methods comes with a cost…
In this paper, we present a novel transformer-based architecture for end-to-end image compression. Our architecture incorporates blocks that effectively capture local dependencies between tokens, eliminating the need for positional encoding…
We make an attempt to develop "noncommutative algebraic geometry" in which noncommutative affine schemes are in one-to-one correspondence with associative algebras. In the first part we discuss various aspects of smoothness in affine…
Deep learning has become an area of interest in most scientific areas, including physical sciences. Modern networks apply real-valued transformations on the data. Particularly, convolutions in convolutional neural networks discard phase…