Abstract. Diffuse optical tomography (DOT) is an imaging modality in which images of the optical properties of a biological tissue, specifically diffusion [Formula: see text] and absorption [Formula: see text], are estimated based on measurements of near-infrared light on the surface of the body. In practical applications, one often lacks exact knowledge of the measurement domain boundary. This poses a significant challenge, as inaccuracies in the boundary shape of the computational domain may result in substantial artifacts in the reconstructed images. In this study, the following two results are achieved in a two-dimensional setting: (i) when the measurement domain [Formula: see text] is known, it is shown that knowledge of the Robin-to-Neumann map for two modulation frequencies uniquely determines [Formula: see text] and [Formula: see text], both assumed to be isotropic. (ii) When the measurement domain [Formula: see text] is not exactly known, a method is proposed for simultaneous reconstruction of [Formula: see text] and [Formula: see text] as well as the boundary [Formula: see text]. For this, DOT measurements are needed for two modulation frequencies. This approach yields reconstructed coefficients that are a conformal deformation of the true coefficients in the exact domain. The new method is demonstrated using simulated noisy data.

Many numerical algorithms in scientific computing—particularly in areas like numerical linear algebra, PDE simulation, and inverse problems—produce outputs that can be represented by semialgebraic functions; that is, the graph of the computed function can be described by finitely many polynomial equalities and inequalities. In this work, we introduce Semialgebraic Neural Networks (SANNs), a neural network architecture capable of representing any bounded semialgebraic function, and computing such functions up to the accuracy of a numerical ODE solver chosen by the programmer. Conceptually, we encode the graph of the learned function as the kernel of a piecewise polynomial selected from a class of functions whose roots can be evaluated using a particular homotopy continuation method. We show by construction that the SANN architecture is able to execute this continuation method, thus evaluating the learned semialgebraic function. Furthermore, the architecture can exactly represent even discontinuous semialgebraic functions by executing a continuation method on each connected component of the target function. Lastly, we provide example applications of these networks and show they can be trained with traditional deep-learning techniques.

In this work we consider the problem of discretization of neural operators in a general setting. Using category theory, we give a no-go theorem that shows that diffeomorphisms between Hilbert spaces may not admit any continuous approximations by diffeomorphisms on finite spaces, even if the discretization is non-linear. This shows how infinite-dimensional Hilbert spaces and finite-dimensional vector spaces fundamentally differ. A key take-away is that to obtain discretization invariance, considerable effort is needed to ensure that finite-dimensional approximations of neural operator converge not only as sequences of functions, but that their representations converge in a suitable sense as well. With this perspective, we give several positive results. We first show that strongly monotone diffeomorphism operators always admit finite-dimensional strongly monotone diffeomorphisms. Next we show that bilipschitz neural operators may always be written via the repeated alternating composition of strongly monotone neural operators and invertible linear maps. We also show that such operators may be inverted locally via iteration provided that such inverse exists. Finally, we conclude by showing how our framework may be used `out of the box' to prove quantitative approximation results for discretization of neural operators.

Abstract. 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.

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 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 setting, is subtler in the infinite-rank setting and is proven 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 have many layers, 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 uncertainty quantification where injectivity enables likelihood estimation and in inverse problems where surjectivity and injectivity corresponds to existence and uniqueness of the solutions, respectively.

Abstract. We study the inverse problem of determining a finite weighted graph [Formula: see text] from the source-to-solution map on a vertex subset [Formula: see text] for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on [Formula: see text]. In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under a novel geometric condition (called the Two-Points Condition), the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on [Formula: see text], or from the observation on [Formula: see text] of one realization of the random walk.

We study approximation of probability measures supported on n-dimensional manifolds embedded in R^m by injective flows—neural networks composed of invertible flows and injective layers. We show that in general, injective flows between R^n and R^m universally approximate measures supported on images of extendable embeddings, which are a subset of standard embeddings: when the embedding dimension m is small, topological obstructions may preclude certain manifolds as admissible targets. When the embedding dimension is sufficiently large, m >= 3n+1, we use an argument from algebraic topology known as the clean trick to prove that the topological obstructions vanish and injective flows universally approximate any differentiable embedding. Along the way we show that the studied injective flows admit efficient projections on the range, and that their optimality can be established "in reverse," resolving a conjecture made in Brehmer & Cranmer 2020.

Injectivity plays an important role in generative models where it enables inference; in inverse problems and compressed sensing with generative priors it is a precursor to well posedness. We establish sharp characterizations of injectivity of fully-connected and convolutional ReLU layers and networks. First, through a layerwise analysis, we show that an expansivity factor of two is necessary and sufficient for injectivity by constructing appropriate weight matrices. We show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires larger expansivity between 3.4 and 10.5. We also characterize the stability of inverting an injective network via worst-case Lipschitz constants of the inverse. We then use arguments from differential topology to study injectivity of deep networks and prove that any Lipschitz map can be approximated by an injective ReLU network. Finally, using an argument based on random projections, we show that an end-to-end---rather than layerwise---doubling of the dimension suffices for injectivity. Our results establish a theoretical basis for the study of nonlinear inverse and inference problems using neural networks.

In this work, we consider the linear inverse problem $y=Ax+\varepsilon$, 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 $\varepsilon$, 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.

The objective of electrical impedance tomography (EIT) is to reconstruct the internal conductivity of a physical body based on current and voltage measurements at the boundary of the body. In many medical applications the exact shape of the domain boundary and contact impedances are not available. This is problematic as even small errors in the boundary shape of the computation domain or in the contact impedance values can produce large artifacts in the reconstructed images, which results in a loss of relevant information. A method is proposed that simultaneously reconstructs the conductivity, the contact impedances, and the boundary shape from EIT data. The approach consists of three steps: first, the unknown contact impedances and an anisotropic conductivity reproducing the measured EIT data in a model domain are computed. Second, using isothermal coordinates, a deformation is constructed that makes the conductivity isotropic. The final step minimizes the error of true and reconstructed known geometric properties (like the electrode lengths) using conformal deformations. The feasibility of the method is illustrated with experimental EIT data, with robust and accurate reconstructions of both conductivity and boundary shape.

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 us 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.

In the present work, we give a solution to the following question from manifold learning. Suppose data belonging to a high dimensional Euclidean space is drawn independently, identically distributed from a measure supported on a low dimensional twice differentiable embedded manifold $M$, and corrupted by a small amount of gaussian noise. How can we produce a manifold $M’$ whose Hausdorff distance to $M$ is small and whose reach is not much smaller than the reach of $M$?

A computational method is introduced for choosing the regularization parameter for total variation (TV) regularization. A partial understanding of the properties of the method is provided by rigorously proving that the TV norms of the reconstructions converge with any choice of regularization parameter. The computational approach is based on computing reconstructions at a few different resolutions and various values of regularization parameter. The chosen parameter is the smallest one resulting in approximately discretization-invariant TV norms of the reconstructions. The method is tested with simulated and experimental X-ray tomography data and compared to the S-curve method. The results are comparable to those of the S-curve method. However, the S-curve method needs quantitative a priori information about the expected sparsity (TV norm) of the unknown, while the proposed method does not require such input parameters.

In electrical impedance tomography (EIT) electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes at the boundary of the body. The resulting voltages are measured on the same electrodes and the objective is to reconstruct the unknown conductivity function inside the body based on these data. All the traditional approaches to the reconstruction problem assume that the boundary of the body and the electrode-skin contact impedances are known a priori. However, in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances, and therefore, approximate model domain and contact impedances have to be used in the image reconstruction. However, it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images, leading to loss of diagnostically relevant information. In a recent paper (Kolehmainen , 2006), we showed how in the 2-D case the errors induced by the inaccurately known boundary can be eliminated as part of the image reconstruction and introduced a novel method for finding a deformed image of the original isotropic conductivity using the theory of TeichmUller mappings. In this paper, the theory and reconstruction method are extended to include the estimation of unknown contact impedances. The method is implemented numerically and tested with experimental EIT data. The results show that the systematic errors caused by inaccurately known boundary and contact impedances can efficiently be eliminated by the reconstruction method.

Diagnostic and operational tasks based on dental radiology often require three-dimensional (3-D) information that is not available in a single X-ray projection image. Comprehensive 3-D information about tissues can be obtained by computerized tomography (CT) imaging. However, in dental imaging a conventional CT scan may not be available or practical because of high radiation dose, low-resolution or the cost of the CT scanner equipment. In this paper, we consider a novel type of 3-D imaging modality for dental radiology. We consider situations in which projection images of the teeth are taken from a few sparsely distributed projection directions using the dentist's regular (digital) X-ray equipment and the 3-D X-ray attenuation function is reconstructed. A complication in these experiments is that the reconstruction of the 3-D structure based on a few projection images becomes an ill-posed inverse problem. Bayesian inversion is a well suited framework for reconstruction from such incomplete data. In Bayesian inversion, the ill-posed reconstruction problem is formulated in a well-posed probabilistic form in which a priori information is used to compensate for the incomplete information of the projection data. In this paper we propose a Bayesian method for 3-D reconstruction in dental radiology. The method is partially based on Kolehmainen et al. 2003. The prior model for dental structures consist of a weighted l1 and total variation (TV)-prior together with the positivity prior. The inverse problem is stated as finding the maximum a posteriori (MAP) estimate. To make the 3-D reconstruction computationally feasible, a parallelized version of an optimization algorithm is implemented for a Beowulf cluster computer. The method is tested with projection data from dental specimens and patient data. Tomosynthetic reconstructions are given as reference for the proposed method.

The aim of X-ray tomography is to reconstruct an unknown physical body from a collection of projection images. When the projection images are only available from a limited angle of view, the reconstruction problem is a severely ill-posed inverse problem. Statistical inversion allows stable solution of the limited-angle tomography problem by complementing the measurement data by a priori information. In this work, the unknown attenuation distribution inside the body is represented as a wavelet expansion, and a Besov space prior distribution together with positivity constraint is used. The wavelet expansion is thresholded before reconstruction to reduce the dimension of the computational problem. Feasibility of the method is demonstrated by numerical examples using in vitro data from mammography and dental radiology.

We study the problem of gathering information about wind velocity from continuous-wave clear-air Doppler radar measurements. The radar is assumed to be a monostatic fixed-frequency Doppler radar, and the wind velocity as well as the reflectivity are modeled as random fields with statistical parameters depending on the altitude. We seek to reconstruct the hodograph curve of the wind profile, i.e., the projection of the wind velocity profile to the ground plane. We show that under certain assumptions of the wind field, the problem reduces to a well-known problem occurring in classical X-ray tomography. Numerical simulations based on the use of X-ray inversion methods are presented.