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Researcher profile

Matti Lassas

Matti Lassas contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.APmath.DGMachine Learningmath.OCeess.IVmath-phmath.MPmath.PRmath.SPmath.STStatistics Theorymath.FAmath.NAMethodologyNumerical Analysis

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Published work

21 published item(s)

preprint2026arXiv

Denoising data using convex relaxations

We study the problem of denoising observations \(Y_i=X_i+Z_i\), where the latent variables \(X_i\) are sampled from a low-dimensional manifold in \(\mathbb{R}^n\) and the noise variables \(Z_i\) are isotropic Gaussian. We propose a convex-relaxation estimator that first reduces dimension by principal component analysis and then projects the observations onto the convex hull of the projected latent manifold. We construct a statistical oracle that estimates its supporting hyperplanes from empirical Gaussian tail probabilities of the noisy sample. Under a lower-mass condition on the latent distribution, we prove finite-sample guarantees for the oracle and derive error bounds for the resulting denoiser. The analysis combines risk bounds for least-squares projection under convex constraints with entropy bounds for convex hulls. We also verify the assumptions of the framework for a Cryo-Electron Microscopy observation model by establishing suitable covering number and Lipschitz estimates for the associated group action and imaging operators.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Function graph transformers universally approximate operators between function spaces

We study the approximation of nonlinear operators between function spaces by transformers. Our approach is to lift functions to measures supported on their graphs and leverage a recently introduced measure-theoretic view of transformers. A function $h$ is represented by its graph measure $γ_h$, with finite tokens $\{(x_j,h(x_j))\}_{j=1}^N$ being its empirical approximations. We show that this framework elegantly models discretization refinement via convergence of measures and provides a natural setting for operator learning. Within this framework, we introduce function graph transformers, a graph-preserving subclass of measure-theoretic transformers that maps graph measures to graph measures, which is to say that outputs remain single-valued functions. Crucially, this additional structure does not reduce generality: we prove that the resulting graph-preserving maps can be approximated by finite compositions of standard softmax self-attention layers and pointwise MLPs, yielding universal approximation results for broad classes of nonlinear operators. Unlike existing theoretical approaches to operator learning with transformers, the measure-theoretic framework also accommodates regularized negative-order Sobolev inputs for which discretization invariance is particularly challenging, as well as query points on different output domains. Overall, function graph transformers provide a continuum viewpoint and mathematical toolkit for transformer-based operator learning, clarifying the roles of positional encodings, graph structure, regularization, and ensuring consistency across discretizations.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Lens rigidity in 2D: The reconstruction of a Riemann surface from its geodesic lengths

We address the question of whether a Riemannian manifold-with-boundary (M,g) in dimension two is uniquely determined from knowledge of the distances between points on its boundary. An affirmative answer is called boundary rigidity for (M,g); it is closely related to lens rigidity. The latter question originates in the problem of reconstructing the speed of sound in an unknown medium from measurements of the travel time of sound waves that are sent in and ultimately return to the boundary. We prove essentially optimal results on these rigidity questions: Our first result answers proves rigidity locally, near a convex portion of the boundary. Our second result proves rigidity globally, for manifolds with convex boundary, in the absence of trapping (closed geodesics), thus confirming a conjecture of Uhlmann. Our final result proves the optimal reconstruction for convex boundaries even in the presence of trapping, showing rigidity up to outermost trapped geodesics. Our results thus extend the classical work of Pestov and Uhlmann on rigidity of simple 2-manifolds, as well as the many prior results on injectivity of the X-ray transform, which address linearized versions of the rigidity problem. \par Our method is to treat the (non-linear) rigidity problem directly, where we simultaneously re-cast the lens data as generalized Riemannian circles, and obtain rigidity for these ``pseudo-circles'', by studying a system of equations that we show these objects must satisfy. The rigidity we obtain ultimately is proven via {novel} estimates that are reminiscent of energy-type estimates for hyperbolic equations.

0 citations0 reviewsNo code linkedOpen access
preprint2024arXiv

On the lack of external response of a nonlinear medium in the second-harmonic generation process

This paper concerns the scattering problem for a nonlinear medium of compact support, $D$, with second-harmonic generation. Such a medium, when probed with monochromatic light beams at frequency $ω$, generates additional waves at frequency $2ω$. The response of the medium is governed by a system of two coupled semilinear partial differential equations for the electric fields at frequency $ω$ and $2ω$. We investigate whether there are situations in which the generated $2ω$ wave is localized inside $D$, that is, the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated in $D$ with non-standard boundary conditions. The analysis presented here sets up a mathematical framework needed to investigate a multitude of questions related to nonlinear scattering with second-harmonic generation.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Inverse problems for discrete heat equations and random walks for a class of graphs

We study the inverse problem of determining a finite weighted graph $(X,E)$ from the source-to-solution map on a vertex subset $B\subset X$ for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on $B$. In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under a novel geometric condition (called the Two-Points Condition), the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on $B$, or from the observation on $B$ of one realization of the random walk.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

An inverse problem for a semi-linear wave equation: a numerical study

We consider an inverse problem of recovering a potential associated to a semi-linear wave equation with a quadratic nonlinearity in $1 + 1$ dimensions. We develop a numerical scheme to determine the potential from a noisy Dirichlet-to-Neumann map on the lateral boundary. The scheme is based on the recent higher order linearization method [20]. We also present an approach to numerically estimating two-dimensional derivatives of noisy data via Tikhonov regularization. The methods are tested using synthetic noisy measurements of the Dirichlet-to-Neumann map. Various examples of reconstructions of the potential functions are given.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Deep learning architectures for nonlinear operator functions and nonlinear inverse problems

We develop a theoretical analysis for special neural network architectures, termed operator recurrent neural networks, for approximating nonlinear functions whose inputs are linear operators. Such functions commonly arise in solution algorithms for inverse boundary value problems. Traditional neural networks treat input data as vectors, and thus they do not effectively capture the multiplicative structure associated with the linear operators that correspond to the data in such inverse problems. We therefore introduce a new family that resembles a standard neural network architecture, but where the input data acts multiplicatively on vectors. Motivated by compact operators appearing in boundary control and the analysis of inverse boundary value problems for the wave equation, we promote structure and sparsity in selected weight matrices in the network. After describing this architecture, we study its representation properties as well as its approximation properties. We furthermore show that an explicit regularization can be introduced that can be derived from the mathematical analysis of the mentioned inverse problems, and which leads to certain guarantees on the generalization properties. We observe that the sparsity of the weight matrices improves the generalization estimates. Lastly, we discuss how operator recurrent networks can be viewed as a deep learning analogue to deterministic algorithms such as boundary control for reconstructing the unknown wavespeed in the acoustic wave equation from boundary measurements.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Fitting a manifold of large reach to noisy data

Let ${\mathcal M}\subset {\mathbb R}^n$ be a $C^2$-smooth compact submanifold of dimension $d$. Assume that the volume of ${\mathcal M}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${\mathcal M}$ is greater than $τ$. Moreover, let $μ$ be a probability measure on ${\mathcal M}$ whose density on ${\mathcal M}$ is a strictly positive Lipschitz-smooth function. Let $x_j\in {\mathcal M}$, $j=1,2,\dots,N$ be $N$ independent random samples from distribution $μ$. Also, let $ξ_j$, $j=1,2,\dots, N$ be independent random samples from a Gaussian random variable in ${\mathbb R}^n$ having covariance $σ^2I$, where $σ$ is less than a certain specified function of $d, V$ and $τ$. We assume that we are given the data points $y_j=x_j+ξ_j,$ $j=1,2,\dots,N$, modelling random points of ${\mathcal M}$ with measurement noise. We develop an algorithm which produces from these data, with high probability, a $d$ dimensional submanifold ${\mathcal M}_o\subset {\mathbb R}^n$ whose Hausdorff distance to ${\mathcal M}$ is less than $Cdσ^2/τ$ and whose reach is greater than $cτ/d^6$ with universal constants $C,c > 0$. The number $N$ of random samples required depends almost linearly on $n$, polynomially on $σ^{-1}$ and exponentially on $d$.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Inverse problems for locally perturbed lattices -- Discrete Hamiltonian and quantum graph

We consider the inverse scattering problems for two types of Schrödinger operators on locally perturbed periodic lattices. For the discrete Hamiltonian, the knowledge of the S-matrix for all energies determines the graph structure and the coefficients of the Hamiltonian. For locally perturbed equilateral metric graphs, the knowledge of the S-matrix for all energies determines the graph structure.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Learning a microlocal prior for limited-angle tomography

Limited-angle tomography is a highly ill-posed linear inverse problem. It arises in many applications, such as digital breast tomosynthesis. Reconstructions from limited-angle data typically suffer from severe stretching of features along the central direction of projections, leading to poor separation between slices perpendicular to the central direction. A new method is introduced, based on machine learning and geometry, producing an estimate for interfaces between regions of different X-ray attenuation. The estimate can be presented on top of the reconstruction, indicating more reliably the true form and extent of features. The method uses directional edge detection, implemented using complex wavelets and enhanced with morphological operations. By using machine learning, the visible part of the wavefront set is first extracted and then extended to the full domain, filling in the parts of the wavefront set that would otherwise be hidden due to the lack of measurement directions.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Universal Joint Approximation of Manifolds and Densities by Simple Injective Flows

We study approximation of probability measures supported on $n$-dimensional manifolds embedded in $\mathbb{R}^m$ by injective flows -- neural networks composed of invertible flows and injective layers. We show that in general, injective flows between $\mathbb{R}^n$ and $\mathbb{R}^m$ universally approximate measures supported on images of extendable embeddings, which are a subset of standard embeddings: when the embedding dimension m is small, topological obstructions may preclude certain manifolds as admissible targets. When the embedding dimension is sufficiently large, $m \ge 3n+1$, we use an argument from algebraic topology known as the clean trick to prove that the topological obstructions vanish and injective flows universally approximate any differentiable embedding. Along the way we show that the studied injective flows admit efficient projections on the range, and that their optimality can be established "in reverse," resolving a conjecture made in Brehmer and Cranmer 2020.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

A foliated and reversible Finsler manifold is determined by its broken scattering relation

The broken scattering relation consists of the total lengths of broken geodesics that start from the boundary, change direction once inside the manifold, and propagate to the boundary. We show that if two reversible Finsler manifolds satisfying a convex foliation condition have the same broken scattering relation, then they are isometric. This implies that some anisotropic material parameters of the Earth can be in principle reconstructed from single scattering measurements at the surface.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Random tree Besov priors -- Towards fractal imaging

We propose alternatives to Bayesian a priori distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well defined infinite-dimensional random variables, and can be approximated by finite-dimensional random variables. We introduce a new wavelet-based model, where the non zero coefficient are chosen in a systematic way so that prior draws have certain fractal behaviour. We show that realisations of this new prior take values in some Besov spaces and have singularities only on a small set $τ$ that has a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator, arising from the the new prior, in denoising problem.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Construction of artificial point sources for a linear wave equation in unknown medium

We study the wave equation on a bounded domain of $\mathbb R^m$ and on a compact Riemannian manifold $M$ with boundary. We assume that the coefficients of the wave equation are unknown but that we are given the hyperbolic Neumann-to-Dirichlet map $Λ$ that corresponds to the physical measurements on the boundary. Using the knowledge of $Λ$ we construct a sequence of Neumann boundary values so that at a time $T$ the corresponding waves converge to zero while the time derivative of the waves converge to a delta distribution. Such waves are called an artificial point source. The convergence of the wave takes place in the function spaces naturally related to the energy of the wave. We apply the results for inverse problems and demonstrate the focusing of the waves numerically in the 1-dimensional case.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Deep neural networks for inverse problems with pseudodifferential operators: an application to limited-angle tomography

We propose a novel convolutional neural network (CNN), called $Ψ$DONet, designed for learning pseudodifferential operators ($Ψ$DOs) in the context of linear inverse problems. Our starting point is the Iterative Soft Thresholding Algorithm (ISTA), a well-known algorithm to solve sparsity-promoting minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling and convolution, which characterize our $Ψ$DONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited angle X-ray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of $Ψ$DONet on simulated data from limited-angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are $Ψ$DOs or Fourier integral operators.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Inverse scattering on non-compact manifolds with general metric

The problems we address in this paper are the spectral theory and the inverse problems associated with Laplacians on non-compact Riemannian manifolds and more general manifolds admitting conic singularities. In particular, we study the inverse scattering problem where one observes the asymptotic behavior of the solutions of the Helmholtz equation on the manifold. These observations are analogous to Heisenberg's scattering matrix in quantum mechanics. We then show that the knowledge of the scattering matrix determines the topology and the metric of the manifold. In the paper we develop a unified approach to consider scattering problems on manifolds that can have very different type of infinities, such as regular hyperbolic ends, cusps, and cylindrical ends related to models encountered in the study of wave guides. We allow the manifold to have also conic singularities. Due to this, the studied class of manifolds include orbifolds. Such non-smooth structures arise in the study of the stability of inverse problems and of the geometrical collapse.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Uniqueness and stability of an inverse problem for a semi-linear wave equation

We consider the recovery of a potential associated with a semi-linear wave equation on $\mathbb{R}^{n+1}$, $n\geq 1$. We show a Hölder stability estimate for the recovery of an unknown potential $a$ of the wave equation $\square u +a u^m=0$ from its Dirichlet-to-Neumann map. We show that an unknown potential $a(x,t)$, supported in $Ω\times[t_1,t_2]$, of the wave equation $\square u +a u^m=0$ can be recovered in a Hölder stable way from the map $u|_{\partial Ω\times [0,T]}\mapsto \langleψ,\partial_νu|_{\partial Ω\times [0,T]}\rangle_{L^2(\partial Ω\times [0,T])}$. This data is equivalent to the inner product of the Dirichlet-to-Neumann map with a measurement function $ψ$. We also prove similar stability result for the recovery of $a$ when there is noise added to the boundary data. The method we use is constructive and it is based on the higher order linearization. As a consequence, we also get a uniqueness result. We also give a detailed presentation of the forward problem for the equation $\square u +a u^m=0$.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Reconstruction and stability in Gel'fand's inverse interior spectral problem

Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $Δ_g$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We then construct a stable approximation to the manifold $(M,g)$. Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from $M$ when the above data are given with a small error. We give an explicit $\log\log$-type stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gel'fand's inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric Boundary Control method.

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