795 citations · 1.3k across the 32 of their papers we have counts for
35 papers
On Hallucinations in Inverse Problems: Fundamental Limits and Provable Assessment Methods
David Iagaru, Nina M. Gottschling, Anders C. Hansen +1
Artificial intelligence (AI) has transformed imaging inverse problems, from medical diagnostics to Earth observation. Yet deep neural networks can produce hallucinations, realistic…
Implicit regularization in AI meets generalized hardness of approximation in optimization -- Sharp results for diagonal linear networks
Johan S. Wind, Vegard Antun, Anders C. Hansen
Understanding the implicit regularization imposed by neural network architectures and gradient based optimization methods is a key challenge in deep learning and AI. In this work w…
Generalised hardness of approximation and the SCI hierarchy -- On determining the boundaries of training algorithms in AI
Luca Eva Gazdag, Alexander Bastounis, Anders C. Hansen
Generalised hardness of approximation (GHA) is the phenomenon that one can easily compute an -approximation to a solution of a computational problem for , but for $ε…
The extended Smale's 9th problem -- On computational barriers and paradoxes in estimation, regularisation, computer-assisted proofs and learning
Alexander Bastounis, Anders C Hansen, Verner Vlačić
Linear and semidefinite programming (LP, SDP), regularisation through basis pursuit (BP) and Lasso have seen great success in mathematics, statistics, data science, computer-assist…
The mathematics of adversarial attacks in AI -- Why deep learning is unstable despite the existence of stable neural networks
Alexander Bastounis, Anders C Hansen, Verner Vlačić
The unprecedented success of deep learning (DL) makes it unchallenged when it comes to classification problems. However, it is well established that the current DL methodology prod…
Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem
Matthew J. Colbrook, Vegard Antun, Anders C. Hansen
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…
Computing solutions of Schrödinger equations on unbounded domains- On the brink of numerical algorithms
Simon Becker, Anders Hansen
We address the open problem of determining which classes of time-dependent linear Schrödinger equations and focusing and defocusing cubic and quintic non-linear Schrödinger equatio…
The troublesome kernel -- On hallucinations, no free lunches and the accuracy-stability trade-off in inverse problems
Nina M. Gottschling, Vegard Antun, Anders C. Hansen +1
Methods inspired by Artificial Intelligence (AI) are starting to fundamentally change computational science and engineering through breakthrough performances on challenging problem…
Compressed Sensing MRI With Variable Density Averaging (CS-VDA) Outperforms Full Sampling At Low SNR
Jasper Schoormans, Gustav J. Strijkers, Anders C. Hansen +2
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-spa…
Non-uniform recovery guarantees for binary measurements and infinite-dimensional compressed sensing
Laura Thesing, Anders Christian Hansen
Due to the many applications in Magnetic Resonance Imaging (MRI), Nuclear Magnetic Resonance (NMR), radio interferometry, helium atom scattering etc., the theory of compressed sens…
The foundations of spectral computations via the Solvability Complexity Index hierarchy
Matthew J. Colbrook, Anders C. Hansen
The problem of computing spectra of operators is arguably one of the most investigated areas of computational mathematics. However, the problem of computing spectra of general boun…
On the stable sampling rate for binary measurements and wavelet reconstruction
Anders Christian Hansen, Laura Thesing
This paper is concerned with the problem of reconstructing an infinite-dimensional signal from a limited number of linear measurements. In particular, we show that for binary measu…
Linear reconstructions and the analysis of the stable sampling rate
Laura Thesing, Anders Christian Hansen
The theory of sampling and the reconstruction of data has a wide range of applications and a rich collection of techniques. For many methods a core problem is the estimation of the…
What do AI algorithms actually learn? - On false structures in deep learning
Laura Thesing, Vegard Antun, Anders C. Hansen
There are two big unsolved mathematical questions in artificial intelligence (AI): (1) Why is deep learning so successful in classification problems and (2) why are neural nets bas…
Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling
Ben Adcock, Vegard Antun, Anders C. Hansen
Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in the form of integral transforms such as the Fourier…
On instabilities of deep learning in image reconstruction - Does AI come at a cost?
Vegard Antun, Francesco Renna, Clarice Poon +2
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…
Analyzing the structure of multidimensional compressed sensing problems through coherence
Alex Jones, Ben Adcock, Anders Hansen
Recently it has been established that asymptotic incoherence can be used to facilitate subsampling, in order to optimize reconstruction quality, in a variety of continuous compress…
Complexity Issues in Computing Spectra, Pseudospectra and Resolvents
Anders C. Hansen, Olavi Nevanlinna
We display methods that allow for computations of spectra, pseudospectra and resolvents of linear operators on Hilbert spaces and also elements in unital Banach algebras. The paper…
Computing Spectra -- On the Solvability Complexity Index Hierarchy and Towers of Algorithms
Jonathan Ben-Artzi, Matthew J. Colbrook, Anders C. Hansen +2
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…
On the absence of the RIP in real-world applications of compressed sensing and the RIP in levels
Alexander Bastounis, Anders C. Hansen
The purpose of this paper is twofold. The first is to point out that the Restricted Isometry Property (RIP) does not hold in many applications where compressed sensing is successfu…
Density theorems for nonuniform sampling of bandlimited functions using derivatives or bunched measurements
Ben Adcock, Milana Gataric, Anders C. Hansen
We provide sufficient density condition for a set of nonuniform samples to give rise to a set of sampling for multivariate bandlimited functions when the measurements consist of po…
Recovering piecewise smooth functions from nonuniform Fourier measurements
Ben Adcock, Milana Gataric, Anders C. Hansen
In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of non…
On asymptotic structure in compressed sensing
Bogdan Roman, Anders Hansen, Ben Adcock
This paper demonstrates how new principles of compressed sensing, namely asymptotic incoherence, asymptotic sparsity and multilevel sampling, can be utilised to better understand u…
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
Ben Adcock, Milana Gataric, Anders C. Hansen
In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly.…