Mixture of Experts Soften the Curse of Dimensionality in Operator Learning

In this paper, we construct a mixture of neural operators (MoNOs) between function spaces whose complexity is distributed over a network of expert neural operators (NOs), with each NO satisfying parameter scaling restrictions. Our main result is a \textit{distributed} universal approximation theorem guaranteeing that any Lipschitz non-linear operator between $L^2([0,1]^d)$ spaces can be approximated uniformly over the Sobolev unit ball therein, to any given $\varepsilon>0$ accuracy, by an MoNO while satisfying the constraint that: each expert NO has a depth, width, and rank of $\mathcal{O}(\varepsilon^{-1})$. Naturally, our result implies that the required number of experts must be large, however, each NO is guaranteed to be small enough to be loadable into the active memory of most computers for reasonable accuracies $\varepsilon$. During our analysis, we also obtain new quantitative expression rates for classical NOs approximating uniformly continuous non-linear operators uniformly on compact subsets of $L^2([0,1]^d)$.

TILT: topological interface recovery in limited-angle tomography

A novel reconstruction method is introduced for the severely ill-posed inverse problem of limited-angle tomography. It is well known that, depending on the available measurement, angles specify a subset of the wavefront set of the unknown target, while some oriented singularities remain invisible in the data. Topological Interface recovery for Limited-angle Tomography, or TILT, is based on lifting the visible part of the wavefront set under a universal covering map. In the space provided, it is possible to connect the appropriate pieces of the lifted wavefront set correctly using dual-tree complex wavelets, a dedicated metric, and persistent homology. The result is not only a suggested invisible boundary but also a computational representation for all interfaces in the target.

Globally injective and bijective neural operators

Recently there has been great interest in operator learning, where networks learn operators between function spaces from an essentially infinite-dimensional perspective. In this work we present results for when the operators learned by these networks are injective and surjective. As a warmup, we combine prior work in both the finite-dimensional ReLU and operator learning setting by giving sharp conditions under which ReLU layers with linear neural operators are injective. We then consider the case the case when the activation function is pointwise bijective and obtain sufficient conditions for the layer to be injective. We remark that this question, while trivial in the finite-rank case, is subtler in the infinite-rank case and is proved using tools from Fredholm theory. Next, we prove that our supplied injective neural operators are universal approximators and that their implementation, with finite-rank neural networks, are still injective. This ensures that injectivity is not `lost' in the transcription from analytical operators to their finite-rank implementation with networks. Finally, we conclude with an increase in abstraction and consider general conditions when subnetworks, which may be many layers deep, are injective and surjective and provide an exact inversion from a `linearization.' This section uses general arguments from Fredholm theory and Leray-Schauder degree theory for non-linear integral equations to analyze the mapping properties of neural operators in function spaces. These results apply to subnetworks formed from the layers considered in this work, under natural conditions. We believe that our work has applications in Bayesian UQ where injectivity enables likelihood estimation and in inverse problems where surjectivity and injectivity corresponds to existence and uniqueness, respectively.

A Transfer Principle: Universal Approximators Between Metric Spaces From Euclidean Universal Approximators

We build universal approximators of continuous maps between arbitrary Polish metric spaces $\mathcal{X}$ and $\mathcal{Y}$ using universal approximators between Euclidean spaces as building blocks. Earlier results assume that the output space $\mathcal{Y}$ is a topological vector space. We overcome this limitation by "randomization": our approximators output discrete probability measures over $\mathcal{Y}$. When $\mathcal{X}$ and $\mathcal{Y}$ are Polish without additional structure, we prove very general qualitative guarantees; when they have suitable combinatorial structure, we prove quantitative guarantees for H\"older-like maps, including maps between finite graphs, solution operators to rough differential equations between certain Carnot groups, and continuous non-linear operators between Banach spaces arising in inverse problems. In particular, we show that the required number of Dirac measures is determined by the combinatorial structure of $\mathcal{X}$ and $\mathcal{Y}$. For barycentric $\mathcal{Y}$, including Banach spaces, $\mathbb{R}$-trees, Hadamard manifolds, or Wasserstein spaces on Polish metric spaces, our approximators reduce to $\mathcal{Y}$-valued functions. When the Euclidean approximators are neural networks, our constructions generalize transformer networks, providing a new probabilistic viewpoint of geometric deep learning.

Deep Invertible Approximation of Topologically Rich Maps between Manifolds

How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? The answer is with a coordinate projection. Neural networks based on topological data analysis (TDA) use tools such as persistent homology to learn topological signatures of data and stabilize training but may not be universal approximators or have stable inverses. Other architectures universally approximate data distributions on submanifolds but only when the latter are given by a single chart, making them unable to learn maps that change topology. By exploiting the topological parallels between locally bilipschitz maps, covering spaces, and local homeomorphisms, and by using universal approximation arguments from machine learning, we find that a novel network of the form $\mathcal{T} \circ p \circ \mathcal{E}$, where $\mathcal{E}$ is an injective network, $p$ a fixed coordinate projection, and $\mathcal{T}$ a bijective network, is a universal approximator of local diffeomorphisms between compact smooth submanifolds embedded in $\mathbb{R}^n$. We emphasize the case when the target map changes topology. Further, we find that by constraining the projection $p$, multivalued inversions of our networks can be computed without sacrificing universality. As an application, we show that learning a group invariant function with unknown group action naturally reduces to the question of learning local diffeomorphisms for finite groups. Our theory permits us to recover orbits of the group action. We also outline possible extensions of our architecture to address molecular imaging of molecules with symmetries. Finally, our analysis informs the choice of topologically expressive starting spaces in generative problems.

Learning a microlocal priorfor limited-angle tomography

Digital breast tomosynthesis is an ill posed inverse problem. In this paper, we provide a try to overcome the problem of stretching artefacts of DBT with the help of learning from the microlocal priors.

Universal Joint Approximation of Manifolds and Densities by Simple Injective Flows

We analyze neural networks composed of bijective flows and injective expansive elements. We find that such networks universally approximate a large class of manifolds simultaneously with densities supported on them. Among others, our results apply to the well-known coupling and autoregressive flows. We build on the work of Teshima et al. 2020 on bijective flows and study injective architectures proposed in Brehmer et al. 2020 and Kothari et al. 2021. Our results leverage a new theoretical device called the embedding gap, which measures how far one continuous manifold is from embedding another. We relate the embedding gap to a relaxation of universally we call the manifold embedding property, capturing the geometric part of universality. Our proof also establishes that optimality of a network can be established in reverse, resolving a conjecture made in Brehmer et al. 2020 and opening the door for simple layer-wise training schemes. Finally, we show that the studied networks admit an exact layer-wise projection result, Bayesian uncertainty quantification, and black-box recovery of network weights.

Learning the optimal regularizer for inverse problems

In this work, we consider the linear inverse problem $y=Ax+\epsilon$, where $A\colon X\to Y$ is a known linear operator between the separable Hilbert spaces $X$ and $Y$, $x$ is a random variable in $X$ and $\epsilon$ is a zero-mean random process in $Y$. This setting covers several inverse problems in imaging including denoising, deblurring, and X-ray tomography. Within the classical framework of regularization, we focus on the case where the regularization functional is not given a priori but learned from data. Our first result is a characterization of the optimal generalized Tikhonov regularizer, with respect to the mean squared error. We find that it is completely independent of the forward operator $A$ and depends only on the mean and covariance of $x$. Then, we consider the problem of learning the regularizer from a finite training set in two different frameworks: one supervised, based on samples of both $x$ and $y$, and one unsupervised, based only on samples of $x$. In both cases, we prove generalization bounds, under some weak assumptions on the distribution of $x$ and $\epsilon$, including the case of sub-Gaussian variables. Our bounds hold in infinite-dimensional spaces, thereby showing that finer and finer discretizations do not make this learning problem harder. The results are validated through numerical simulations.

Globally Injective ReLU Networks

We study injective ReLU neural networks. Injectivity plays an important role in generative models where it facilitates inference; in inverse problems with generative priors it is a precursor to well posedness. We establish sharp conditions for injectivity of ReLU layers and networks, both fully connected and convolutional. We make no architectural assumptions beyond the ReLU activations so our results apply to a very general class of neural networks. We show through a layer-wise analysis that an expansivity factor of two is necessary for injectivity; we also show sufficiency by constructing weight matrices which guarantee injectivity. Further, we show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires considerably larger expansivity which might seem counterintuitive. We then derive the inverse Lipschitz constants and study the approximation-theoretic properties of injective neural networks. Using arguments from differential topology we prove that, under mild technical conditions, any Lipschitz map can be approximated by an injective neural network. This justifies the use of injective neural networks in problems which a priori do not require injectivity. Our results establish a theoretical basis for the study of nonlinear inverse and inference problems using neural networks.

Deep neural networks for inverse problems with pseudodifferential operators: an application to limited-angle tomography

We propose a novel convolutional neural network (CNN), called $\Psi$DONet, designed for learning pseudodifferential operators ($\Psi$DOs) in the context of linear inverse problems. Our starting point is the Iterative Soft Thresholding Algorithm (ISTA), a well-known algorithm to solve sparsity-promoting minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling and convolution, which characterize our $\Psi$DONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited angle X-ray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of $\Psi$DONet on simulated data from limited-angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are $\Psi$DOs or Fourier integral operators.