Do Log Factors Matter? On Optimal Wavelet Approximation and the Foundations of Compressed Sensing

Adcock, Ben; Brugiapaglia, Simone; King-Roskamp, Matthew

doi:10.1007/s10208-021-09501-3

Cited by 51 publications

(9 citation statements)

References 59 publications

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“…Finally, we test the proposed IHTL and CoSaMPL algorithms in the context of function approximation via compressed sensing. 14 We aim to approximate a function f : [0, 1] → R. In this context, it is convenient to adopt the terminology of decoders and encoders. An encoder is a linear mapping E m : L 2 ([0, 1]) → C m corresponding to the measurement phase.…”

Section: Function Approximation Via Compressed Sensingmentioning

confidence: 99%

“…Our goal is to find encoder-decoder pairs such that the approximation error f − fm L 2 decays at a rate as close as possible to the theoretically optimal O(m −α ). 14,30,31 Multilevel Fourier sampling sampling strategies have been recently showed to achieve a nearoptimal approximation rate O(log γ (m)/m α ) with γ = 13/4 + for any 0 < δ 1, when with Daubechies' wavelet approximation with a decoder based 1 minimization 14 (more precisely, socalled weighted square-root LASSO decoder 32 ). This near-optimal result heavily relies on the sparsity in levels structure.…”

Section: Function Approximation Via Compressed Sensingmentioning

confidence: 99%

“…This work was motivated by the question of optimal function approximation via compressed sensing. 14 Specifically, given a function class F for which it is known the best s-term approximation rate decays like s −α for some α > 0, can one design m compressed sensing measurements and a decoder so that the resulting approximation achieves an O(m −α ) error? Moreover, can this be achieved by a black box, with a decoder that has polynomial runtime in m?…”

Section: Introductionmentioning

confidence: 99%

“…The first question has been answered affirmatively for the class of piecewise α-Hölder functions of one variable. 14 Note that sparsity in level is crucial for obtaining the optimal error rate. However, the decoder is based on a weighted 1 minimization program, 14 and thus does not give an affirmative answer to the second question.…”

Section: Introductionmentioning

confidence: 99%

“…14 Note that sparsity in level is crucial for obtaining the optimal error rate. However, the decoder is based on a weighted 1 minimization program, 14 and thus does not give an affirmative answer to the second question. Striving for a true black box motivates one to consider non-optimization based approaches, such as the IHTL and CoSaMPL algorithms we introduce in this paper.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Iterative and greedy algorithms for the sparsity in levels model in compressed sensing

Adcock

Brugiapaglia

King-Roskamp

2019

Wavelets and Sparsity XVIII

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Motivated by the question of optimal functional approximation via compressed sensing, we propose generalizations of the Iterative Hard Thresholding and the Compressive Sampling Matching Pursuit algorithms able to promote sparse in levels signals. We show, by means of numerical experiments, that the proposed algorithms are successfully able to outperform their unstructured variants when the signal exhibits the sparsity structure of interest. Moreover, in the context of piecewise smooth function approximation, we numerically demonstrate that the structure promoting decoders outperform their unstructured variants and the basis pursuit program when the encoder is structure agnostic.

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Section: Function Approximation Via Compressed Sensingmentioning

confidence: 99%

Section: Function Approximation Via Compressed Sensingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Iterative and greedy algorithms for the sparsity in levels model in compressed sensing

Adcock

Brugiapaglia

King-Roskamp

2019

Wavelets and Sparsity XVIII

Self Cite

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Optimal approximation of infinite-dimensional holomorphic functions

Adcock,

Dexter,

Moraga

2024

Calcolo

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Compressive Imaging: Structure, Sampling, Learning

Adcock

Hansen

2021

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Images are everywhere. Whether on a social media page, a doctor's desk to aid diagnosis or a scientist's computer screen to help study a chemical, physical or biological process, the modern world is awash with digital images.All images are produced by acquiring measurements using a physical device. The list of different acquisition devices is long and varied. It includes simple devices found in most homes, such as digital cameras; specialist medical imaging equipment found in hospitals, such as a Magnetic Resonance Imaging (MRI) or X-ray Computed Tomography (CT) scanner; or scientific devices, such as electron microscopes, found in laboratories.The concern of this book is the task of image reconstruction. This is the algorithmic process of converting the raw data (the measurements) into the final image seen by the end user. The overarching aim of image reconstruction is to achieve the four following, and competing, objectives:Objective #1 (accuracy): to produce the highest-quality images.Accuracy is of course paramount. High-quality images are desirable in virtually all applications. However, in direct competition with this is:Objective #2 (sampling): to use as few measurements as possible.Acquiring more measurements usually comes at a cost. It could mean an additional outlay of time, power, monetary expense or risk, depending on the application at hand. Reducing the number of measurements is often the primary goal of image reconstruction. For example, in MRI, taking more measurements involves a longer scan time, which can be unpleasant and challenging for the patient -especially in paediatric MRI. It also makes the measurements acquired more susceptible to corruptions due, for instance, to patient motion. In X-ray CT, the number of measurements loosely corresponds to the amount of radiation to which the patient is exposed. Acquiring fewer measurements per scan opens the door for more frequent scans, which in turn allows for more effective treatment monitoring.Objective #3 (stability): to ensure that errors in the measurements or in the numerical computation do not significantly impact the quality of the recovered image.All imaging systems introduce error in the measurements, due to noise, corruptions or modelling assumptions. There are also round-off errors in the numerical computations performed by image reconstruction algorithms. It is vital that reconstruction algorithmsIn Fourier imaging the (noiseless) measurements y ∈ C m correspond to samplesis an important example of Fourier imaging. Key Point #10. Current deep learning strategies for compressive imaging are prone to instabilities, with certain small perturbations in the measurements leading to severe image artefacts. This arises because of the training procedure, which can cause the resulting network to over-perform. 1.7.4 Stable and Accurate Neural Networks for Compressive Imaging This situation is rather unsatisfactory. While neural networks have produced stunning results in various imaging tasks, their use in compressive imaging appears hampered 1.10...

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Do Log Factors Matter? On Optimal Wavelet Approximation and the Foundations of Compressed Sensing

Cited by 51 publications

References 59 publications

Iterative and greedy algorithms for the sparsity in levels model in compressed sensing

Iterative and greedy algorithms for the sparsity in levels model in compressed sensing

Optimal approximation of infinite-dimensional holomorphic functions

Compressive Imaging: Structure, Sampling, Learning

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