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Researcher profile

Ben Adcock

Ben Adcock contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.NANumerical AnalysisMachine LearningInformation Theorymath.ITComputer Vision

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Published work

15 published item(s)

preprint2026arXiv

Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributions

State estimation is a critical task in scientific, engineering and control applications. Since the reliability of reconstructions depends on the number and position of sensors, optimal sensor placement (OSP) is essential in scenarios where measurements are sparse and expensive. Classical OSP approaches rely on Gaussian assumptions and are consequently unable to account for the complex distributions encountered in many real-world systems. Generative-model-based reconstruction using sensor guided diffusion posterior sampling (DPS) has emerged as a promising technique for reconstructing states from highly complex distributions. However, existing sensor-selection methods either require unrealistically many sensors or emulate classical OSP, creating a mismatch between modern recovery models with classical OSP tools motivating the need for fundamentally new ideas towards OSP that match the recent advances made in powerful recovery models. We introduce a distribution-free sensor placement framework based on the Christoffel function: a mathematical formulation of optimal sampling and recovery guarantees for posterior sampling with arbitrary sensors and signal distributions, from which we derive a new OSP strategy with non-asymptotic bounds on the number of sensors needed for recovery. We develop Christoffel-DPS, with offline and online variants, instantiating Christoffel sampling for generative models. Christoffel-DPS outperforms Gaussian OSP baselines and existing generative-model placement methods, validating that distribution-free sensing is both theoretically principled and practically superior. The framework is model-agnostic; we demonstrate its application to a range of unconditional DPS and flow-matching models on structurally non-Gaussian benchmarks, showing the efficacy of Christoffel-DPS in low sensor budget regimes.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains

Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular discretisations. However, practical implementations have struggled to fully realise these benefits. Existing backbones like Fourier neural operators are often biased towards regular grids and fail to generalise to complex domain topologies. We propose a novel architecture for function-space diffusion models that represents generalised graph convolutional kernels as finite element functions, enabling the model to naturally handle unstructured meshes and complex geometries. We demonstrate the efficacy of our network architecture through a series of unconditional and conditional sampling experiments across diverse geometries, including non-convex and multiply-connected domains. Our results show that the proposed method maintains resolution invariance and achieves high fidelity in capturing functional distributions on non-trivial geometries.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

CAS4DL: Christoffel Adaptive Sampling for function approximation via Deep Learning

The problem of approximating smooth, multivariate functions from sample points arises in many applications in scientific computing, e.g., in computational Uncertainty Quantification (UQ) for science and engineering. In these applications, the target function may represent a desired quantity of interest of a parameterized Partial Differential Equation (PDE). Due to the large cost of solving such problems, where each sample is computed by solving a PDE, sample efficiency is a key concerning these applications. Recently, there has been increasing focus on the use of Deep Neural Networks (DNN) and Deep Learning (DL) for learning such functions from data. In this work, we propose an adaptive sampling strategy, CAS4DL (Christoffel Adaptive Sampling for Deep Learning) to increase the sample efficiency of DL for multivariate function approximation. Our novel approach is based on interpreting the second to last layer of a DNN as a dictionary of functions defined by the nodes on that layer. With this viewpoint, we then define an adaptive sampling strategy motivated by adaptive sampling schemes recently proposed for linear approximation schemes, wherein samples are drawn randomly with respect to the Christoffel function of the subspace spanned by this dictionary. We present numerical experiments comparing CAS4DL with standard Monte Carlo (MC) sampling. Our results demonstrate that CAS4DL often yields substantial savings in the number of samples required to achieve a given accuracy, particularly in the case of smooth activation functions, and it shows a better stability in comparison to MC. These results therefore are a promising step towards fully adapting DL towards scientific computing applications.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Fast and stable approximation of analytic functions from equispaced samples via polynomial frames

We consider approximating analytic functions on the interval $[-1,1]$ from their values at a set of $m+1$ equispaced nodes. A result of Platte, Trefethen \& Kuijlaars states that fast and stable approximation from equispaced samples is generally impossible. In particular, any method that converges exponentially fast must also be exponentially ill-conditioned. We prove a positive counterpart to this `impossibility' theorem. Our `possibility' theorem shows that there is a well-conditioned method that provides exponential decay of the error down to a finite, but user-controlled tolerance $ε> 0$, which in practice can be chosen close to machine epsilon. The method is known as \textit{polynomial frame} approximation or \textit{polynomial extensions}. It uses algebraic polynomials of degree $n$ on an extended interval $[-γ,γ]$, $γ> 1$, to construct an approximation on $[-1,1]$ via a SVD-regularized least-squares fit. A key step in the proof of our main theorem is a new result on the maximal behaviour of a polynomial of degree $n$ on $[-1,1]$ that is simultaneously bounded by one at a set of $m+1$ equispaced nodes in $[-1,1]$ and $1/ε$ on the extended interval $[-γ,γ]$. We show that linear oversampling, i.e., $m = c n \log(1/ε) / \sqrt{γ^2-1}$, is sufficient for uniform boundedness of any such polynomial on $[-1,1]$. This result aside, we also prove an extended impossibility theorem, which shows that such a possibility theorem (and consequently the method of polynomial frame approximation) is essentially optimal.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Towards optimal sampling for learning sparse approximation in high dimensions

In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector- or even Hilbert space-valued. Our main objective is to study how the sampling strategy affects the sample complexity -- that is, the number of samples that suffice for accurate and stable recovery -- and to use this insight to obtain optimal or near-optimal sampling procedures. We consider two settings. First, when a target sparse representation is known, in which case we present a near-complete answer based on drawing independent random samples from carefully-designed probability measures. Second, we consider the more challenging scenario when such representation is unknown. In this case, while not giving a full answer, we describe a general construction of sampling measures that improves over standard Monte Carlo sampling. We present examples using algebraic and trigonometric polynomials, and for the former, we also introduce a new procedure for function approximation on irregular (i.e., nontensorial) domains. The effectiveness of this procedure is shown through numerical examples. Finally, we discuss a number of structured sparsity models, and how they may lead to better approximations.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Data

Accurate approximation of scalar-valued functions from sample points is a key task in computational science. Recently, machine learning with Deep Neural Networks (DNNs) has emerged as a promising tool for scientific computing, with impressive results achieved on problems where the dimension of the data or problem domain is large. This work broadens this perspective, focusing on approximating functions that are Hilbert-valued, i.e. take values in a separable, but typically infinite-dimensional, Hilbert space. This arises in science and engineering problems, in particular those involving solution of parametric Partial Differential Equations (PDEs). Such problems are challenging: 1) pointwise samples are expensive to acquire, 2) the function domain is high dimensional, and 3) the range lies in a Hilbert space. Our contributions are twofold. First, we present a novel result on DNN training for holomorphic functions with so-called hidden anisotropy. This result introduces a DNN training procedure and full theoretical analysis with explicit guarantees on error and sample complexity. The error bound is explicit in three key errors occurring in the approximation procedure: the best approximation, measurement, and physical discretization errors. Our result shows that there exists a procedure (albeit non-standard) for learning Hilbert-valued functions via DNNs that performs as well as, but no better than current best-in-class schemes. It gives a benchmark lower bound for how well DNNs can perform on such problems. Second, we examine whether better performance can be achieved in practice through different types of architectures and training. We provide preliminary numerical results illustrating practical performance of DNNs on parametric PDEs. We consider different parameters, modifying the DNN architecture to achieve better and competitive results, comparing these to current best-in-class schemes.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Do log factors matter? On optimal wavelet approximation and the foundations of compressed sensing

A signature result in compressed sensing is that Gaussian random sampling achieves stable and robust recovery of sparse vectors under optimal conditions on the number of measurements. However, in the context of image reconstruction, it has been extensively documented that sampling strategies based on Fourier measurements outperform this purportedly optimal approach. Motivated by this seeming paradox, we investigate the problem of optimal sampling for compressed sensing. Rigorously combining the theories of wavelet approximation and infinite-dimensional compressed sensing, our analysis leads to new error bounds in terms of the total number of measurements $m$ for the approximation of piecewise $α$-Hölder functions. Our theoretical findings suggest that Fourier sampling outperforms random Gaussian sampling when the Hölder exponent $α$ is large enough. Moreover, we establish a provably optimal sampling strategy. This work is an important first step towards the resolution of the claimed paradox, and provides a clear theoretical justification for the practical success of compressed sensing techniques in imaging problems.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

The gap between theory and practice in function approximation with deep neural networks

Deep learning (DL) is transforming industry as decision-making processes are being automated by deep neural networks (DNNs) trained on real-world data. Driven partly by rapidly-expanding literature on DNN approximation theory showing they can approximate a rich variety of functions, such tools are increasingly being considered for problems in scientific computing. Yet, unlike traditional algorithms in this field, little is known about DNNs from the principles of numerical analysis, e.g., stability, accuracy, computational efficiency and sample complexity. In this paper we introduce a computational framework for examining DNNs in practice, and use it to study empirical performance with regard to these issues. We study performance of DNNs of different widths & depths on test functions in various dimensions, including smooth and piecewise smooth functions. We also compare DL against best-in-class methods for smooth function approx. based on compressed sensing (CS). Our main conclusion from these experiments is that there is a crucial gap between the approximation theory of DNNs and their practical performance, with trained DNNs performing relatively poorly on functions for which there are strong approximation results (e.g. smooth functions), yet performing well in comparison to best-in-class methods for other functions. To analyze this gap further, we provide some theoretical insights. We establish a practical existence theorem, asserting existence of a DNN architecture and training procedure that offers the same performance as CS. This establishes a key theoretical benchmark, showing the gap can be closed, albeit via a strategy guaranteed to perform as well as, but no better than, current best-in-class schemes. Nevertheless, it demonstrates the promise of practical DNN approx., by highlighting potential for better schemes through careful design of DNN architectures and training strategies.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Approximating smooth, multivariate functions on irregular domains

In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the corresponding \textit{Nikolskii inequality} for the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of $d$, for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Frame approximation with bounded coefficients

Due to their flexibility, frames of Hilbert spaces are attractive alternatives to bases in approximation schemes for problems where identifying a basis is not straightforward or even feasible. Computing a best approximation using frames, however, can be challenging since it requires solving an ill-conditioned linear system. One consequence of this ill-conditioning is that the coefficients of such a frame approximation can grow large. In this paper we resolve this issue by introducing two methods for frame approximation that possess bounded coefficients. As we show, these methods typically lead to little or no deterioration in the approximation accuracy, but successfully avoid the large coefficients inherent to previous approaches, thus making them attractive in situations where large coefficients are undesirable. We also present theoretical analysis to support these conclusions.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Frames and numerical approximation II: generalized sampling

In a previous paper [Adcock & Huybrechs, 2019] we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but ill-conditioning often prevents the numerical computation of best approximations. We showed that, in spite of said ill-conditioning, approximations with regularization may still provide accuracy up to order $\sqrtε$, where $ε$ is a small truncation threshold. When using frames, i.e. complete systems that are generally redundant but which provide infinite representations with coefficients of bounded norm, this accuracy can actually be achieved for all functions in a space. Here, we generalize that setting in two ways. We assume information or samples from $f$ from a wide class of linear operators acting on $f$, rather than inner products associated with the best approximation projection. This enables the analysis of fully discrete approximations based, for instance, on function values only. Next, we allow oversampling, leading to least-squares approximations. We show that this leads to much improved accuracy on the order of $ε$ rather than $\sqrtε$. Overall, we demonstrate that numerical function approximation using redundant representations may lead to highly accurate approximations in spite of having to solve ill-conditioned systems of equations.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Improved recovery guarantees and sampling strategies for TV minimization in compressive imaging

In this paper, we consider the use of Total Variation (TV) minimization for compressive imaging; that is, image reconstruction from subsampled measurements. Focusing on two important imaging modalities -- namely, Fourier imaging and structured binary imaging via the Walsh--Hadamard transform -- we derive uniform recovery guarantees asserting stable and robust recovery for arbitrary random sampling strategies. Using this, we then derive a class of theoretically-optimal sampling strategies. For Fourier sampling, we show recovery of an image with approximately $s$-sparse gradient from $m \gtrsim_d s \cdot \log^2(s) \cdot \log^4(N)$ measurements, in $d \geq 1$ dimensions. When $d = 2$, this improves the current state-of-the-art result by a factor of $\log(s) \cdot \log(N)$. It also extends it to arbitrary dimensions $d \geq 2$. For Walsh sampling, we prove that $m \gtrsim_d s \cdot \log^2(s) \cdot \log^2(N/s) \cdot \log^3(N) $ measurements suffice in $d \geq 2$ dimensions. To the best of our knowledge, this is the first recovery guarantee for structured binary sampling with TV minimization.

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
preprint2019arXiv

On oracle-type local recovery guarantees in compressed sensing

We present improved sampling complexity bounds for stable and robust sparse recovery in compressed sensing. Our unified analysis based on l1 minimization encompasses the case where (i) the measurements are block-structured samples in order to reflect the structured acquisition that is often encountered in applications; (ii) the signal has an arbitrary structured sparsity, by results depending on its support S. Within this framework and under a random sign assumption, the number of measurements needed by l1 minimization can be shown to be of the same order than the one required by an oracle least-squares estimator. Moreover, these bounds can be minimized by adapting the variable density sampling to a given prior on the signal support and to the coherence of the measurements. We illustrate both numerically and analytically that our results can be successfully applied to recover Haar wavelet coefficients that are sparse in levels from random Fourier measurements in dimension one and two, which can be of particular interest in imaging problems. Finally, a preliminary numerical investigation shows the potential of this theory for devising adaptive sampling strategies in sparse polynomial approximation.

0 citations0 reviewsNo code linkedOpen access
preprint2017arXiv

Computing reconstructions from nonuniform Fourier samples: Universality of stability barriers and stable sampling rates

We study the problem of recovering an unknown compactly-supported multivariate function from samples of its Fourier transform that are acquired nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction problems of this kind arise in various imaging applications, where Fourier samples are taken along radial lines or spirals for example. Specifically, we consider finite-dimensional reconstructions, where a limited number of samples is available, and investigate the rate of convergence of such approximate solutions and their numerical stability. We show that the proportion of Fourier samples that allow for stable approximations of a given numerical accuracy is independent of the specific sampling geometry and is therefore universal for different sampling scenarios. This allows us to relate both sufficient and necessary conditions for different sampling setups and to exploit several results that were previously available only for very specific sampling geometries. The results are obtained by developing: (i) a transference argument for different measures of the concentration of the Fourier transform and Fourier samples; (ii) frame bounds valid up to the critical sampling density, which depend explicitly on the sampling set and the spectrum. As an application, we identify sufficient and necessary conditions for stable and accurate reconstruction of algebraic polynomials or wavelet coefficients from nonuniform Fourier data.

0 citations0 reviewsNo code linkedOpen access