Supervised learning in function spaces is an emerging area of machine learning research with applications to the prediction of complex physical systems such as fluid flows, solid mechanics, and climate modeling. By directly learning maps (operators) between infinite dimensional function spaces, these models are able to learn discretization invariant representations of target functions. A common approach is to represent such target functions as linear combinations of basis elements learned from data. However, there are simple scenarios where, even though the target functions form a low dimensional submanifold, a very large number of basis elements is needed for an accurate linear representation.

The principle of translation equivariance (if an input image is translated an output image should be translated by the same amount), led to the development of convolutional neural networks that revolutionized machine vision. Other symmetries, like rotations and reflections, play a similarly critical role, especially in biomedical image analysis, but exploiting these symmetries has not seen wide adoption. We hypothesize that this is partially due to the mathematical complexity of methods used to exploit these symmetries, which often rely on representation theory, a bespoke concept in differential geometry and group theory. In this work, we show that the same equivariance can be achieved using a simple form of convolution kernels that we call ``moment kernels,'' and prove that all equivariant kernels must take this form. These are a set of radially symmetric functions of a spatial position $x$, multiplied by powers of the components of $x$ or the identity matrix. We implement equivariant neural networks using standard convolution modules, and provide architectures to execute several biomedical image analysis tasks that depend on equivariance principles: classification (outputs are invariant under orthogonal transforms), 3D image registration (outputs transform like a vector), and cell segmentation (quadratic forms defining ellipses transform like a matrix).

Supervised learning in function spaces is an emerging area of machine learning research with applications to the prediction of complex physical systems such as fluid flows, solid mechanics, and climate modeling. By directly learning maps (operators) between infinite dimensional function spaces, these models are able to learn discretization invariant representations of target functions. A common approach is to represent such target functions as linear combinations of basis elements learned from data. However, there are simple scenarios where, even though the target functions form a low dimensional submanifold, a very large number of basis elements is needed for an accurate linear representation. We show this method is able to accurately learn low dimensional representations of solution manifolds to partial differential equations while outperforming linear models of larger size.

We show that many perception tasks, from visual recognition, semantic segmentation, optical flow, depth estimation to vocalization discrimination, are highly redundant functions of their input data. Images or spectrograms, projected into different subspaces, formed by orthogonal bases in pixel, Fourier or wavelet domains, can be used to solve these tasks remarkably well regardless of whether it is the top subspace where data varies the most, some intermediate subspace with moderate variability--or the bottom subspace where data varies the least. This phenomenon occurs because different subspaces have a large degree of redundant information relevant to the task.

This paper addresses the problem of accurately estimating a function on one domain when only its discrete samples are available on another domain. To answer this challenge, we utilize a neural network, which we train to incorporate prior knowledge of the function. In addition, by carefully analyzing the problem, we obtain a bound on the error over the extrapolation domain and define a condition number for this problem that quantifies the level of difficulty of the setup. Compared to other machine learning methods that provide time series prediction, such as transformers, our approach is suitable for setups where the interpolation and extrapolation regions are general subdomains and, in particular, manifolds. In addition, our construction leads to an improved loss function that helps us boost the accuracy and robustness of our neural network. We conduct comprehensive numerical tests and comparisons of our extrapolation versus standard methods. The results illustrate the effectiveness of our approach in various scenarios.

The aim of this paper is to present a mathematical framework for tensor PCA. The proposed approach is able to overcome the limitations of previous methods that extract a low dimensional subspace by iteratively solving an optimization problem. The core of the proposed approach is the derivation of a basis in tensor space from a real self-adjoint tensor operator, thus reducing the problem of deriving a basis to an eigenvalue problem. Three different cases have been studied to derive: i) a basis from a self-adjoint tensor operator; ii) a rank-1 basis; iii) a basis in a subspace. In particular, the equivalence between eigenvalue equation for a real self-adjoint tensor operator and standard matrix eigenvalue equation has been proven. For all the three cases considered, a subspace approach has been adopted to derive a tensor PCA. Experiments on image datasets validate the proposed mathematical framework.

We provide a full characterisation of all of the possible alternating group ($A_n$) equivariant neural networks whose layers are some tensor power of $\mathbb{R}^{n}$. In particular, we find a basis of matrices for the learnable, linear, $A_n$-equivariant layer functions between such tensor power spaces in the standard basis of $\mathbb{R}^{n}$. We also describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.