arXiv subjects

Matteo Santacesaria

Publications and source records attributed to Matteo Santacesaria.

At least 19 recordsLinked to original sources

arXiv2026

Analytic inverse problems with finitely many random measurements

While infinite-dimensional inverse problems are traditionally analyzed assuming continuous data, practical applications rely on finitely many discrete measurements. Recent deterministic approaches establish that unknowns belonging to a $d$-dimensional model class can be stably recovered from finitely many measurements. However, for severely ill-posed problems, such as the Calder\'on problem and inverse scattering, the known constructions may require a number of measurements that is exponential in $d$. We show that random sampling reduces this count dramatically if one asks only for exact identifiability. By exploiting the analytic geometry of the forward maps, we prove that, whenever the infinite-data problem is injective on the model class, $2d+1$ random scalar measurements determine the unknown uniquely, almost surely. Applications are given to the Calder\'on problem, with both infinite- and finite-dimensional boundary sampling, and to inverse medium scattering from randomly sampled far-field values.

math.AP↗

arXiv2026

PIKS: Universal Physics-Informed Kernel Methods

Physics-informed machine learning incorporates physical principles --often expressed via differential operators-- into data-driven models. While physics-informed neural networks (PINNs) dominate empirical applications, the complexity of neural network architectures and optimization landscapes hinders the development of a corresponding learning theory. In turn, kernel methods offer an appealing alternative with closed-form solutions and analytical tractability, yet existing guarantees primarily cover the well-specified setting where the target belongs to the native Reproducing Kernel Hilbert Space (RKHS). This imposes unrealistic regularity assumptions that physical targets often fail to satisfy. In this paper, we introduce and analyze Physics-Informed Kernel methodS (PIKS). We establish the universal consistency of PIKS for linear differential constraints, proving that for universal kernels (such as Gaussian or Mat\'ern), the estimator asymptotically learns the target while satisfying physical constraints. We further derive finite-sample bounds under suitable source conditions. Our analysis is based on extending classical operator-theoretic analysis of kernel methods to physics-informed machine learning. Numerical experiments demonstrate that PIKS can be competitive with PINNs and traditional finite element methods.

stat.ML↗

arXiv2026

Stochastic Generalized Sampling

Reconstructing an infinite-dimensional signal from a finite set of measurements is a fundamental problem in approximation theory and signal processing. While the generalized sampling (GS) framework provides a robust methodology for recovering elements in arbitrary separable Hilbert spaces, deterministic approaches suffer from severe basis-dependent dimensionality constraints, often requiring a quadratic sample complexity $m \gtrsim n^2$ to avoid numerical instability. In this paper, we introduce a fully stochastic framework for GS that natively overcomes these deterministic barriers. By drawing measurements according to an optimal leverage-score probability distribution, we prove that stable recovery is guaranteed with high probability at a near-linear sample complexity of $m \gtrsim n\log n$. Crucially, this optimal rate is universal-independent of the specific choice of measurement and reconstruction bases-and holds even when the sensing system is a highly redundant frame. To establish these guarantees, we derive a novel matrix Bernstein inequality for random rectangular operators, allowing us to rigorously control the aliasing error governed by the empirical cross-term. Finally, we demonstrate the practical efficacy of our approach on the classical problem of recovering analytic functions from continuous Fourier measurements via Legendre polynomials, where our randomized method achieve near-exponential convergence rates.

math.FA↗

arXiv2026

Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomography

Deep generative models have emerged as state-of-the-art for solving inverse problems, but applying them to inverse problems for PDEs, like electrical impedance tomography (EIT) remains challenging. Because physical domains are naturally discretized as unstructured meshes rather than regular grids, standard convolutional architectures are often inadequate. In this paper, we propose a novel framework that extends diffusion posterior sampling (DPS) to graph-structured data. We develop an unconditional score-based diffusion model directly on a 2D triangular mesh to learn an accurate prior over the physical solution space. Furthermore, we introduce a regularized variant, RDPS, which incorporates explicit regularization terms, such as total variation and generalized Tikhonov, to complement the implicit diffusion prior and mitigate severe ill-posedness. Extensive experiments on synthetic and real 2D EIT datasets demonstrate that RDPS produces stable, physically plausible reconstructions. Our approach generalizes well to out-of-distribution inclusion geometries, is highly robust to measurement noise, and outperforms current state-of-the-art solvers (e.g., GPnP-BM3D, DP-SGS) in reconstruction accuracy and artifact reduction.

eess.IV↗

arXiv2026

Infinite dimensional generative sensing

Deep generative models have become a standard for modeling priors for inverse problems, going beyond classical sparsity-based methods. However, existing theoretical guarantees are mostly confined to finite-dimensional vector spaces, creating a gap when the physical signals are modeled as functions in Hilbert spaces. This work presents a rigorous framework for generative compressed sensing in Hilbert spaces. We extend the notion of local coherence in an infinite-dimensional setting, to derive optimal, resolution-independent sampling distributions. Thanks to a generalization of the Restricted Isometry Property, we show that stable recovery holds when the number of measurements is proportional to the prior's intrinsic dimension (up to logarithmic factors), independent of the ambient dimension. Finally, numerical experiments on the Darcy flow equation validate our theoretical findings and demonstrate that in severely undersampled regimes, employing lower-resolution generators acts as an implicit regularizer, improving reconstruction stability.

math.NA↗

arXiv2026

Inverse problems for quasi-linear elliptic systems modeling electrolysers

We investigate the electrochemical processes within an electrolyser cell, which are modelled by a coupled system of second-order quasi-linear elliptic PDEs. In this context, we study an inverse problem aiming to reconstruct both the non-linear diffusion coefficients and the phenomenological relation defining the electric potential. Our main results state that boundary measurements alone are not enough to reconstruct these non-linear quantities. However, we show that a combination of boundary and interior measurements allow for their unique reconstruction. To achieve this result we generalise a linearisation result in the context of the scalar quasi-linear Calder\'{o}n problem, [Sun, Math. Z. 221 (1996)], to the setting of a system of PDEs with non-local nonlinearities. In contrast to the Calder\'{o}n case, the generalised linearisation does not "freeze" the coefficients. We show that interior measurements are precisely what is required to achieve this freezing and thus enable the unique reconstruction.

math.AP↗

arXiv2025

MAD: Manifold Attracted Diffusion

Score-based diffusion models are a highly effective method for generating samples from a distribution of images. We consider scenarios where the training data comes from a noisy version of the target distribution, and present an efficiently implementable modification of the inference procedure to generate noiseless samples. Our approach is motivated by the manifold hypothesis, according to which meaningful data is concentrated around some low-dimensional manifold of a high-dimensional ambient space. The central idea is that noise manifests as low magnitude variation in off-manifold directions in contrast to the relevant variation of the desired distribution which is mostly confined to on-manifold directions. We introduce the notion of an extended score and show that, in a simplified setting, it can be used to reduce small variations to zero, while leaving large variations mostly unchanged. We describe how its approximation can be computed efficiently from an approximation to the standard score and demonstrate its efficacy on toy problems, synthetic data, and real data.

stat.ML↗

arXiv2025

Deep Unfolding Network for Nonlinear Multi-Frequency Electrical Impedance Tomography

Multi-frequency Electrical Impedance Tomography (mfEIT) represents a promising biomedical imaging modality that enables the estimation of tissue conductivities across a range of frequencies. Addressing this challenge, we present a novel variational network, a model-based learning paradigm that strategically merges the advantages and interpretability of classical iterative reconstruction with the power of deep learning. This approach integrates graph neural networks (GNNs) within the iterative Proximal Regularized Gauss Newton (PRGN) framework. By unrolling the PRGN algorithm, where each iteration corresponds to a network layer, we leverage the physical insights of nonlinear model fitting alongside the GNN's capacity to capture inter-frequency correlations. Notably, the GNN architecture preserves the irregular triangular mesh structure used in the solution of the nonlinear forward model, enabling accurate reconstruction of overlapping tissue fraction concentrations.

math.NA↗

arXiv2025

On spatial systems of cities

Are there multiple equilibria in the spatial economy? This paper develops a unified framework that integrates systems of cities and regional models to address this question within a general geographic space. A key feature is the endogenous formation of commuting areas linking a continuum of residential locations to a finite set of potential business districts. Using tools from computational geometry and shape optimization, we derive sufficient conditions for the existence and uniqueness of spatial equilibria. For plausible parameter values, urban location is indeterminate, but, conditional on an urban system, city sizes are uniquely determined. The framework reconciles seemingly conflicting empirical findings on the role of geography and scale economies in shaping the spatial economy.

econ.TH↗

arXiv2025

DIMA: DIffusing Motion Artifacts for unsupervised correction in brain MRI images

Motion artifacts remain a significant challenge in Magnetic Resonance Imaging (MRI), compromising diagnostic quality and potentially leading to misdiagnosis or repeated scans. Existing deep learning approaches for motion artifact correction typically require paired motion-free and motion-affected images for training, which are rarely available in clinical settings. To overcome this requirement, we present DIMA (DIffusing Motion Artifacts), a novel framework that leverages diffusion models to enable unsupervised motion artifact correction in brain MRI. Our two-phase approach first trains a diffusion model on unpaired motion-affected images to learn the distribution of motion artifacts. This model then generates realistic motion artifacts on clean images, creating paired datasets suitable for supervised training of correction networks. Unlike existing methods, DIMA operates without requiring k-space manipulation or detailed knowledge of MRI sequence parameters, making it adaptable across different scanning protocols and hardware. Comprehensive evaluations across multiple datasets and anatomical planes demonstrate that our method achieves comparable performance to state-of-the-art supervised approaches while offering superior generalizability to real clinical data. DIMA represents a significant advancement in making motion artifact correction more accessible for routine clinical use, potentially reducing the need for repeat scans and improving diagnostic accuracy.

eess.IV↗

arXiv2025

TomoSelfDEQ: Self-Supervised Deep Equilibrium Learning for Sparse-Angle CT Reconstruction

Deep learning has emerged as a powerful tool for solving inverse problems in imaging, including computed tomography (CT). However, most approaches require paired training data with ground truth images, which can be difficult to obtain, e.g., in medical applications. We present TomoSelfDEQ, a self-supervised Deep Equilibrium (DEQ) framework for sparse-angle CT reconstruction that trains directly on undersampled measurements. We establish theoretical guarantees showing that, under suitable assumptions, our self-supervised updates match those of fully-supervised training with a loss including the (possibly non-unitary) forward operator like the CT forward map. Numerical experiments on sparse-angle CT data confirm this finding, also demonstrating that TomoSelfDEQ outperforms existing self-supervised methods, achieving state-of-the-art results with as few as 16 projection angles.

eess.IV↗

arXiv2025

Assessing the use of Diffusion models for motion artifact correction in brain MRI

Magnetic Resonance Imaging generally requires long exposure times, while being sensitive to patient motion, resulting in artifacts in the acquired images, which may hinder their diagnostic relevance. Despite research efforts to decrease the acquisition time, and designing efficient acquisition sequences, motion artifacts are still a persistent problem, pushing toward the need for the development of automatic motion artifact correction techniques. Recently, diffusion models have been proposed as a solution for the task at hand. While diffusion models can produce high-quality reconstructions, they are also susceptible to hallucination, which poses risks in diagnostic applications. In this study, we critically evaluate the use of diffusion models for correcting motion artifacts in 2D brain MRI scans. Using a popular benchmark dataset, we compare a diffusion model-based approach with state-of-the-art methods consisting of Unets trained in a supervised fashion on motion-affected images to reconstruct ground truth motion-free images. Our findings reveal mixed results: diffusion models can produce accurate predictions or generate harmful hallucinations in this context, depending on data heterogeneity and the acquisition planes considered as input.

eess.IV↗

arXiv2025

Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI

This paper extends the sample complexity theory for ill-posed inverse problems developed in a recent work by the authors [`Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform', J. Eur. Math. Soc., to appear], which was originally focused on the sparse Radon transform. We demonstrate that the underlying abstract framework, based on infinite-dimensional compressed sensing and generalized sampling techniques, can effectively handle a variety of practical applications. Specifically, we analyze three case studies: (1) The reconstruction of a sparse signal from a finite number of pointwise blurred samples; (2) The recovery of the (sparse) source term of an elliptic partial differential equation from finite samples of the solution; and (3) A moderately ill-posed variation of the classical sensing problem of recovering a wavelet-sparse signal from finite Fourier samples, motivated by magnetic resonance imaging. For each application, we establish rigorous recovery guarantees by verifying the key theoretical requirements, including quasi-diagonalization and coherence bounds. Our analysis reveals that careful consideration of balancing properties and optimized sampling strategies can lead to improved reconstruction performance. The results provide a unified theoretical foundation for compressed sensing approaches to inverse problems while yielding practical insights for specific applications.

math.FA↗

arXiv2024

Learning sparsity-promoting regularizers for linear inverse problems

This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as $B$, which regularizes the inverse problem while promoting sparsity in the solution. The method leverages statistical properties of the underlying data and incorporates prior knowledge through the choice of $B$. We establish the well-posedness of the optimization problem, provide theoretical guarantees for the learning process, and present sample complexity bounds. The approach is demonstrated through theoretical infinite-dimensional examples, including compact perturbations of a known operator and the problem of learning the mother wavelet, and through extensive numerical simulations. This work extends previous efforts in Tikhonov regularization by addressing non-differentiable norms and proposing a data-driven approach for sparse regularization in infinite dimensions.

stat.ML↗

arXiv2024

Is in-domain data beneficial in transfer learning for landmarks detection in x-ray images?

In recent years, deep learning has emerged as a promising technique for medical image analysis. However, this application domain is likely to suffer from a limited availability of large public datasets and annotations. A common solution to these challenges in deep learning is the usage of a transfer learning framework, typically with a fine-tuning protocol, where a large-scale source dataset is used to pre-train a model, further fine-tuned on the target dataset. In this paper, we present a systematic study analyzing whether the usage of small-scale in-domain x-ray image datasets may provide any improvement for landmark detection over models pre-trained on large natural image datasets only. We focus on the multi-landmark localization task for three datasets, including chest, head, and hand x-ray images. Our results show that using in-domain source datasets brings marginal or no benefit with respect to an ImageNet out-of-domain pre-training. Our findings can provide an indication for the development of robust landmark detection systems in medical images when no large annotated dataset is available.

cs.CV↗

arXiv2024

Localization of point scatterers via sparse optimization on measures

We consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities. In the Foldy-Lax model, the estimation of the scatterers' locations and intensities from far field measurements can be recast as the recovery of a discrete measure from nonlinear observations. We propose a "linearize and locally optimize" approach to perform this reconstruction. We first solve a convex program in the space of measures (known as the Beurling LASSO), which involves a linearization of the forward operator (the far field pattern in the Born approximation). Then, we locally minimize a second functional involving the nonlinear forward map, using the output of the first step as initialization. We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated. We also provide numerical evidence that the second step still allows for accurate recovery in settings that are more involved.

math.NA↗

arXiv2024

Learning a Gaussian Mixture for Sparsity Regularization in Inverse Problems

In inverse problems, it is widely recognized that the incorporation of a sparsity prior yields a regularization effect on the solution. This approach is grounded on the a priori assumption that the unknown can be appropriately represented in a basis with a limited number of significant components, while most coefficients are close to zero. This occurrence is frequently observed in real-world scenarios, such as with piecewise smooth signals. In this study, we propose a probabilistic sparsity prior formulated as a mixture of degenerate Gaussians, capable of modeling sparsity with respect to a generic basis. Under this premise, we design a neural network that can be interpreted as the Bayes estimator for linear inverse problems. Additionally, we put forth both a supervised and an unsupervised training strategy to estimate the parameters of this network. To evaluate the effectiveness of our approach, we conduct a numerical comparison with commonly employed sparsity-promoting regularization techniques, namely LASSO, group LASSO, iterative hard thresholding, and sparse coding/dictionary learning. Notably, our reconstructions consistently exhibit lower mean square error values across all $1$D datasets utilized for the comparisons, even in cases where the datasets significantly deviate from a Gaussian mixture model.

stat.ML↗

arXiv2023

Manifold Learning by Mixture Models of VAEs for Inverse Problems

Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.

cs.LG↗