Related papers: On Non-Linear operators for Geometric Deep Learnin…
There have been attempts to insert mathematical morphology (MM) operators into convolutional neural networks (CNN), and the most successful endeavor to date has been the morphological neural networks (MNN). Although MNN have performed…
In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful…
Classical spectral theory gives a complete description of a single normal operator, but it fails for noncommuting operators, where no canonical joint spectrum or simultaneous diagonalization exists. Existing approaches provide only partial…
The recent impressive results of deep learning-based methods on computer vision applications brought fresh air to the research and industrial community. This success is mainly due to the process that allows those methods to learn…
We study data-driven representations for three-dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and…
We consider the standard hypergeometric differential operator $D$ regarded as an operator on the complex plane $C$ and the complex conjugate operator $\overline D$. These operators formally commute and are formally adjoint one to another…
We define a new notion of fiber-wise linear differential operator on the total space of a vector bundle $E$. Our main result is that fiber-wise linear differential operators on $E$ are equivalent to (polynomial) derivations of an…
Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these methods are graph-based: they associate a vertex…
In this paper, we introduce the concepts of endomorphism operator, left averaging operator, differential operator and Rota-Baxter Operator, and we construct examples of these linear maps on associative algebras with a left identity, a…
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…
This paper introduces the class of grey-scale image stack operators as those that (a) map binary-images into binary-images and (b) commute on average with cross-sectioning. Equivalently, stack operators are 1-Lipchitz extensions of set…
Deep neural networks are an attractive alternative for simulating complex dynamical systems, as in comparison to traditional scientific computing methods, they offer reduced computational costs during inference and can be trained directly…
Despite the remarkable capabilities of deep neural networks in image recognition, the dependence on activation functions remains a largely unexplored area and has yet to be eliminated. On the other hand, Polynomial Networks is a class of…
This paper presents a novel framework for non-linear equivariant neural network layers on homogeneous spaces. The seminal work of Cohen et al. on equivariant $G$-CNNs on homogeneous spaces characterized the representation theory of such…
We construct a new class of operators that act on symmetric functions with two deformation parameters $q$ and $t$. Our combinatorial construction associates each operator with a specific lattice path, whose steps alternate between moving up…
Let F be a smooth real manifold with a linear connection in the tangent bundle. How can we extend the coefficients of the connection to bi-differential operators that incorporate the original structure at zero order? Take a constant mapping…
We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds $\mathcal{M}$ using…
By recursively summing node features over entire neighborhoods, spatial graph convolution operators have been heralded as key to the success of Graph Neural Networks (GNNs). Yet, despite the multiplication of GNN methods across tasks and…
We study a system of partial differential equations defined by commuting family of differential operators with regular singularities. We construct ideally analytic solutions depending on a holomorphic parameter. We give some explicit…
We investigate $\rho$-orthogonality and its local symmetry in the space of bounded linear operators. A characterization of Hilbert space operators with symmetric numerical range is established in terms of $\rho$-orthogonality. Further, we…