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Researcher profile

Anders C. Hansen

Anders C. Hansen contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.NAComputer VisionMachine LearningNumerical AnalysisComputational Complexityeess.IVInformation Theorymath-phmath.ITmath.LOmath.MPmath.OCmath.SPNeural and Evolutionary Computingphysics.med-ph

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Published work

8 published item(s)

preprint2026arXiv

On Hallucinations in Inverse Problems: Fundamental Limits and Provable Assessment Methods

Artificial intelligence (AI) has transformed imaging inverse problems, from medical diagnostics to Earth observation. Yet deep neural networks can produce hallucinations, realistic-looking but incorrect details, undermining their reliability, especially when ground truth data is unavailable. We develop a theoretical framework showing that such hallucinations are not merely artifacts of particular models, but can arise from the ill-posed nature of the inverse problem itself. We derive necessary and sufficient conditions for hallucinations, together with computable bounds on their magnitude that depend only on the forward model. Building on this theory, we introduce algorithms to: (1) estimate the minimum hallucination magnitude achievable by any reconstruction model for a given input; (2) assess the faithfulness of reconstructed details by a given reconstruction model. Experiments across three imaging tasks demonstrate that our approach applies broadly, including to modern generative models, and provides a principled way to quantify and evaluate AI hallucinations.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Generalised hardness of approximation and the SCI hierarchy -- On determining the boundaries of training algorithms in AI

Hardness of approximation (HA) -- the phenomenon that, assuming P $\neq$ NP, one can easily compute an $ε$-approximation to the solution of a discrete computational problem for $ε> ε_0 > 0$, but for $ε< ε_0$ it suddenly becomes intractable -- is a core phenomenon in the foundations of computations that has transformed computer science. In this paper we study the newly discovered phenomenon in the foundations of computational mathematics: generalised hardness of approximation (GHA) -- which in spirit is close to classical HA in computer science. However, GHA is typically independent of the P vs. NP question in many cases. Thus, it requires a new mathematical framework that we initiate in this paper. We demonstrate the hitherto undiscovered phenomenon that GHA happens when using AI techniques in order to train optimal neural networks (NNs). In particular, for any non-zero underdetermined linear problem the following phase transition may occur: One can prove the existence of optimal NNs for solving the problem but they can only be computed to a certain accuracy $ε_0 > 0$. Below the approximation threshold $ε_0$ -- not only does it become intractable to compute the NN -- it becomes impossible regardless of computing power, and no randomised algorithm can solve the problem with probability better than 1/2. In other cases, despite the existence of a stable optimal NN, any attempts of computing it below the approximation threshold $ε_0$ will yield an unstable NN. Our results use and extend the current mathematical framework of the Solvability Complexity Index (SCI) hierarchy and facilitate a program for detecting the GHA phenomenon throughout computational mathematics and AI.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale&#39;s 18th problem

Deep learning (DL) has had unprecedented success and is now entering scientific computing with full force. However, current DL methods typically suffer from instability, even when universal approximation properties guarantee the existence of stable neural networks (NNs). We address this paradox by demonstrating basic well-conditioned problems in scientific computing where one can prove the existence of NNs with great approximation qualities, however, there does not exist any algorithm, even randomised, that can train (or compute) such a NN. For any positive integers $K > 2$ and $L$, there are cases where simultaneously: (a) no randomised training algorithm can compute a NN correct to $K$ digits with probability greater than $1/2$, (b) there exists a deterministic training algorithm that computes a NN with $K-1$ correct digits, but any such (even randomised) algorithm needs arbitrarily many training data, (c) there exists a deterministic training algorithm that computes a NN with $K-2$ correct digits using no more than $L$ training samples. These results imply a classification theory describing conditions under which (stable) NNs with a given accuracy can be computed by an algorithm. We begin this theory by establishing sufficient conditions for the existence of algorithms that compute stable NNs in inverse problems. We introduce Fast Iterative REstarted NETworks (FIRENETs), which we both prove and numerically verify are stable. Moreover, we prove that only $\mathcal{O}(|\log(ε)|)$ layers are needed for an $ε$-accurate solution to the inverse problem.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Computing Spectra -- On the Solvability Complexity Index Hierarchy and Towers of Algorithms

This paper establishes some of the fundamental barriers in the theory of computations and finally settles the long-standing computational spectral problem. That is to determine the existence of algorithms that can compute spectra $\mathrm{sp}(A)$ of classes of bounded operators $A = \{a_{ij}\}_{i,j \in \mathbb{N}} \in \mathcal{B}(l^2(\mathbb{N}))$, given the matrix elements $\{a_{ij}\}_{i,j \in \mathbb{N}}$, that are sharp in the sense that they achieve the boundary of what a digital computer can achieve. Similarly, for a Schrödinger operator $H = -Δ+V$, determine the existence of algorithms that can compute the spectrum $\mathrm{sp}(H)$ given point samples of the potential function $V$. In order to solve these problems, we establish the Solvability Complexity Index (SCI) hierarchy and provide a collection of new algorithms that allow for problems that were previously out of reach. The SCI is the smallest number of limits needed in the computation, yielding a classification hierarchy for all types of problems in computational mathematics that determines the boundaries of what computers can achieve in scientific computing. In addition, the SCI hierarchy provides classifications of computational problems that can be used in computer-assisted proofs. The SCI hierarchy captures many key computational issues in the history of mathematics including the insolvability of the quintic, Smale&#39;s problem on the existence of iterative generally convergent algorithm for polynomial root finding, the computational spectral problem, inverse problems, optimisation etc.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Compressed Sensing MRI With Variable Density Averaging (CS-VDA) Outperforms Full Sampling At Low SNR

We investigated whether a combination of k-space undersampling and variable density averaging enhances image quality for low-SNR MRI acquisitions. We implemented 3D Cartesian k-space prospective undersampling with a variable number of averages for each k-line. The performance of this compressed sensing with variable-density averaging (CS-VDA) method was evaluated in retrospective analysis of fully sampled phantom MRI measurements, as well as for prospectively accelerated in vivo 3D brain and knee MRI scans. Both phantom and in vivo results showed that acquisitions using the CS-VDA approach resulted in better image quality as compared to full sampling of k-space in the same scan time. Specifically, CS-VDA with a higher number of averages in the center of k-space resulted in the best image quality, apparent from increased anatomical detail with preserved soft-tissue contrast. This novel approach will facilitate improved image quality of inherently low SNR data, such as those with high-resolution or specific contrast-weightings with low SNR efficiency.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

On instabilities of deep learning in image reconstruction - Does AI come at a cost?

Deep learning, due to its unprecedented success in tasks such as image classification, has emerged as a new tool in image reconstruction with potential to change the field. In this paper we demonstrate a crucial phenomenon: deep learning typically yields unstablemethods for image reconstruction. The instabilities usually occur in several forms: (1) tiny, almost undetectable perturbations, both in the image and sampling domain, may result in severe artefacts in the reconstruction, (2) a small structural change, for example a tumour, may not be captured in the reconstructed image and (3) (a counterintuitive type of instability) more samples may yield poorer performance. Our new stability test with algorithms and easy to use software detects the instability phenomena. The test is aimed at researchers to test their networks for instabilities and for government agencies, such as the Food and Drug Administration (FDA), to secure safe use of deep learning methods.

0 citations0 reviewsNo code linkedOpen access
preprint2010arXiv

A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases

We introduce a generalized framework for sampling and reconstruction in separable Hilbert spaces. Specifically, we establish that it is always possible to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently many of its samples in any other Riesz basis. This framework can be viewed as an extension of that of Eldar et al. However, whilst the latter imposes stringent assumptions on the reconstruction basis, and may in practice be unstable, our framework allows for recovery in any (Riesz) basis in a manner that is completely stable. Whilst the classical Shannon Sampling Theorem is a special case of our theorem, this framework allows us to exploit additional information about the approximated vector (or, in this case, function), for example sparsity or regularity, to design a reconstruction basis that is better suited. Examples are presented illustrating this procedure.

0 citations0 reviewsNo code linkedOpen access
preprint2010arXiv

Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon

We introduce a method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the number of samples, this procedure can be numerically implemented in a stable manner. Moreover, the accuracy of the resulting approximation is completely determined by the choice of reconstruction basis, meaning that the reconstruction vectors can be tailored to the particular problem at hand. An important example of this approach is the accurate recovery of a piecewise analytic function from its first few Fourier coefficients. Whilst the standard Fourier projection suffers from the Gibbs phenomenon, by reconstructing in a piecewise polynomial basis, we obtain an approximation with root exponential accuracy in terms of the number of Fourier samples and exponential accuracy in terms of the degree of the reconstruction function. Numerical examples illustrate the advantage of this approach over other existing methods.

0 citations0 reviewsNo code linkedOpen access