Deep learning, due to its unprecedented success in tasks such as image classification, has emerged a new tool reconstruction with potential change the field. In this paper, we demonstrate crucial phenomenon: learning typically yields unstable methods for reconstruction. The instabilities usually occur several forms: 1) Certain tiny, almost undetectable perturbations, both and sampling domain, may result severe artefacts reconstruction; 2) small structural change, example, tumor, not be...

This paper presents a framework for compressed sensing that bridges gap between existing theory and the current use of in many real-world applications. In doing so, it also introduces new sampling method yields substantially improved recovery over techniques. applications sensing, including medical imaging, standard principles incoherence sparsity are lacking. Whilst is often used successfully such applications, done largely without mathematical explanation. The introduced this provides...

Significance Instability is the Achilles’ heel of modern artificial intelligence (AI) and a paradox, with training algorithms finding unstable neural networks (NNs) despite existence stable ones. This foundational issue relates to Smale’s 18th mathematical problem for 21st century on limits AI. By expanding methodologies initiated by Gödel Turing, we demonstrate limitations (even randomized) computing NNs. Despite numerous results NNs great approximation properties, only in specific cases do...

We show that it is possible to compute spectra and pseudospectra of linear operators on separable Hilbert spaces given their matrix elements. The core in the theory pseudospectral analysis particular the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>-pseudospectrum residual pseudospectrum....

Generalized sampling is a recently developed linear framework for and reconstruction in separable Hilbert spaces. It allows one to recover any element finite-dimensional subspace given finitely many of its samples with respect an arbitrary basis or frame. Unlike more common approaches this problem, such as the consistent technique Eldar others, it leads numerical methods possessing both guaranteed stability accuracy. The purpose paper twofold. First, we give complete formal analysis...

This paper demonstrates how new principles of compressed sensing, namely asymptotic incoherence, sparsity and multilevel sampling, can be utilised to better understand underlying phenomena in practical sensing improve results real-world applications. The contribution the is fourfold: First, it explains sampling strategy depends not only on signal but also its structure, shows design effective strategies utilising this. Second, that optimal efficiency resolution problem, this phenomenon...

We prove that any stable method for resolving the Gibbs phenomenon---that is, recovering high-order accuracy from first $m$ Fourier coefficients of an analytic and nonperiodic function---can converge at best root-exponentially fast in $m$. Any with faster convergence must also be unstable, particular, exponential implies ill-conditioning. This result is analogous to a recent theorem Platte, Trefethen, Kuijlaars concerning recovery pointwise function values on equispaced $m$-grid. The main...

We consider the problem of recovering a compactly supported function from finite collection pointwise samples its Fourier transform taken nonuniformly. First, we show that under suitable conditions on sampling frequencies---specifically, their density and bandwidth---it is possible to recover any such $f$ in stable accurate manner given finite-dimensional subspace, particular, one which well suited for approximating $f$. In practice, this carried out using so-called nonuniform generalized...

Abstract Compressed Sensing (CS) techniques are used to measure and reconstruct surface dynamical processes with a helium spin-echo spectrometer for the first time. Helium atom scattering is well established method examining structure dynamics of materials at atomic sized resolution technique opens up possibility compressing data acquisition process. CS methods demonstrating compressibility spectra presented several measurements. Recent developments on structured multilevel sampling that...

We consider signal recovery from Fourier measurements using compressed sensing (CS) with wavelets. For discrete signals structured sparse Haar wavelet coefficients, we give the first proof of near-optimal samples taken according to an appropriate variable density sampling scheme. Crucially, in taking into account such sparsity-known as sparsity levels-as opposed just sparsity, this result yields guarantees that agree empirically observed properties CS setting. This complements a recent...

The purpose of this paper is twofold. first to point out that the property uniform recovery, meaning all sparse vectors are recovered, does not hold in many applications where compressed sensing successfully used. This includes fields like magnetic resonance imaging (MRI), nuclear computerized tomography, electron radio interferometry, helium atom scattering, and fluorescence microscopy. We demonstrate for natural matrices involving a level based reconstruction basis (e.g., wavelets), number...

Computing the spectra of operators is a fundamental problem in sciences, with wide-ranging applications condensed-matter physics, quantum mechanics and chemistry, statistical mechanics, etc. While there are algorithms that certain cases converge to spectrum, no general procedure known (a) always converges, (b) provides bounds on errors approximation, (c) approximate eigenvectors. This may lead incorrect simulations. It has been an open since 1950s decide whether such reliable methods exist...

This paper provides an extension of compressed sensing which bridges a substantial gap between existing theory and its current use in real-world applications. It introduces mathematical framework that generalizes the three standard pillars - namely, sparsity, incoherence uniform random subsampling to new concepts: asymptotic multilevel sampling. The theorems show is also possible, reveals several advantages, under these substantially relaxed conditions. importance this threefold. First,...