arXiv subjects

Clarice Poon

Publications and source records attributed to Clarice Poon.

At least 19 recordsLinked to original sources

arXiv2026

Sample complexity of unbalanced entropic OT

Optimal transport (OT) has become a central language for comparing probability measures, but exact balanced OT is often both too rigid for data with missing, created, or destroyed mass and subject to unfavorable high-dimensional sample complexity. Entropic regularization and unbalanced relaxations address these limitations in complementary ways. Entropy smooths the geometry, improves statistical behavior, and enables fast Sinkhorn-type algorithms, while unbalanced marginal penalties replace hard conservation constraints by divergence terms adapted to noisy empirical data. This paper studies the sample complexity of entropic unbalanced OT at the level of the optimal coupling, rather than only the scalar transport value. We develop a translation-invariant dual formulation, prove compactness and strong convexity properties for the intrinsic dual variables, and convert these geometric estimates into high-probability finite-sample bounds for empirical couplings. The results clarify why regularization is a practical necessity in machine learning applications: it softens the curse of dimensionality, reduces the number of samples needed for stable transport estimation, and keeps the resulting estimators compatible with scalable Sinkhorn-type solvers.

math.ST↗

arXiv2026

On the global convergence of gradient flow for wide shallow models beyond homogeneous nonlinearities

A surprising phenomenon in the training of neural networks is the ability of gradient descent to find global minimizers of the training loss despite its non-convexity. Following earlier work, we investigate this behavior for wide shallow models. Existing global convergence results primarily concern models with positively one-homogeneous nonlinearities, such as ReLU activations, and models with scalar output weights and bounded nonlinearities, such as sigmoid activations. We study a broader class of models, including multi-head attention layers and two-layer networks with bounded or asymptotically positively one-homogeneous activations and vector output weights. Building upon [Chizat and Bach, 2018], we prove that, in the limit of many hidden neurons or attention heads, non-global minimizers of the training loss are unstable under mean-field gradient flow dynamics by constructing "escape regions" in the parameter space. Our global convergence statements are conditional in the following sense: if the mean-field gradient flow converges in W2, then its limit must be a global minimizer. We revisit the bounded nonlinearity, scalar-output setting of [CB18], giving an escape region construction adapted to unbounded nonlinear parameter domains. We also propose new constructions for nonlinearities with at most linear growth under a non-degeneracy assumption and for asymptotically positively one-homogeneous nonlinearities. Finally, we show the well-posedness and stability estimates for the mean-field training dynamics under sub-Gaussian initializations.

math.OC↗

arXiv2026

Curvature of optimal transport with respect to the cost and applications to inverse optimal transport

We study the inverse optimal transport problem of recovering the ground cost from an optimal transport plan. In discrete settings, this problem reduces to inverse linear programming and is intrinsically ill-posed, exhibiting non-identifiability and flat directions. We show that in the continuous setting, the regularity of the marginals fundamentally alters the structure of the inverse problem. Assuming smooth positive densities for the source and target measures, we characterize the second variation of the optimal transport functional with respect to the ground cost in H\"older spaces. In particular, we show that it is non-degenerate modulo the natural transport invariances, yielding a strict curvature property that is absent in discrete transport. As a consequence, we obtain local identifiability and stability results for inverse optimal transport. For the structured family of bilinear costs (i.e. Mahalanobis parametrizations), the ground cost can be uniquely recovered up to the intrinsic invariances from a single optimal coupling under a natural spanning condition. We further show that this identifiability property is generic under arbitrarily small perturbations of the marginals, while settings where the optimal transport map is affine (for instance Gaussian or elliptical marginals) remain degenerate. Finally, we establish precise bounds on the bias and statistical variance of inverse optimal transport under entropic regularization. These results reveal a structural parallel between forward and inverse optimal transport: regularity of the marginals ensures smooth optimal maps in the forward problem, while non-degeneracy of the induced transport plan yields curvature and local invertibility in the inverse problem.

math.OC↗

arXiv2026

Adaptive Dimension Reduction for Overlapping Group Sparsity

Typical dimension reduction techniques for nonoverlapping sparse optimization involve screening or sieving strategies based on a dual certificate derived from the first-order optimality condition, approximating the gradients or exploiting certain inherent low-dimensional structure of the sparse solution. In comparison, dimension reduction rules for overlapping group sparsity are generally less developed because the subgradient structure is more complex, making the link between sparsity pattern and the dual variable indirect due to the non-separability. In this work, we propose new dual certificates for overlapping group sparsity and a novel adaptive scheme for identifying the support of the overlapping group LASSO. We demonstrate how this scheme can be integrated into and significantly accelerate existing algorithms, including Primal-Dual splitting method, alternating direction method of multipliers and a recently developed variable projection scheme based on over-parameterization. We provide convergence analysis of the method and verify its practical effectiveness through experiments on standard datasets.

math.OC↗

arXiv2025

On the non-convexity issue in the radial Calder\'on problem

A classical approach to the Calder\'on problem is to estimate the unknown conductivity by solving a nonlinear least-squares problem. It leads to a nonconvex optimization problem which is generally believed to be riddled with bad local minimums. We revisit this issue in the case of piecewise constant radial conductivities and prove that, contrary to previous claims, there are no spurious critical points in the case of two scalar unknowns with no measurement noise. We also provide a partial proof of this result in the general setting which holds under a numerically verifiable assumption. Finally, we investigate whether a recently proposed approach based on convexification yields better reconstructions. For the first time, we propose a way to implement it in practice and show that it is consistently outperformed by some least squares solvers, which are also faster and require less measurements.

math.NA↗

arXiv2025

Learning from samples: inverse problems over measures

We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle. Such problems arise in learning costs, energies, and dynamics from distributional data, but the associated forward solution map is typically nonlinear and implicit. We show that its optimality gap nevertheless yields convex empirical objectives for finite-dimensional potential classes, and we introduce sharpened Fenchel--Young losses that add a data-dependent discrepancy inside the forward problem. This keeps the estimator calibrated while improving the local geometry of the loss. Our main stability theorem separates the inverse error analysis into measurement error, forward perturbation, and empirical curvature. We instantiate this principle for inverse entropic unbalanced optimal transport and for inverse Jordan--Kinderlehrer--Otto (JKO) learning from independent snapshot samples, obtaining high-probability parameter recovery bounds. JKO schemes discretize Wasserstein gradient flows through a sequence of variational problems over measures, making them a natural language for population dynamics observed through snapshots. In this JKO case, the sharpened objective reduces to an unbalanced transport problem, which also clarifies the connection between variational gap losses and quadratic iJKO\(^\star\) surrogates. Numerical experiments illustrate the conditioning effect of sharpening and its benefits for sparse inverse-gradient-flow recovery.

cs.LG↗

arXiv2024

Hadamard Langevin dynamics for sampling the l1-prior

Priors with non-smooth log-densities, such as the l1-prior, are widely used in Bayesian inverse problems for their sparsity-inducing properties. Existing Langevin-based sampling methods typically rely on proximal mappings or smooth approximations, which alter the target distribution. We propose an alternative approach based on a Hadamard product parameterization of the l1-norm, leading to a smooth but nonconvex and non-globally Lipschitz potential whose marginal law exactly recovers the desired posterior. The resulting Hadamard Langevin dynamics (HLD) defines a diffusion process that is analytically distinct from proximal or mirror-type Langevin schemes. Our main contribution is a rigorous well-posedness theory for both the continuous and discrete HLD. We establish existence and uniqueness of strong solutions, geometric ergodicity of the continuous dynamics, and convergence of the discretized scheme as the step size tends to zero. These results provide the first theoretical foundation for sampling from nonconvex, nonsmooth posteriors through overparameterized Langevin dynamics.

math.NA↗

arXiv2023

Super-resolved Lasso

Super-resolution of pointwise sources is of utmost importance in various areas of imaging sciences. Specific instances of this problem arise in single molecule fluorescence, spike sorting in neuroscience, astrophysical imaging, radar imaging, and nuclear resonance imaging. In all these applications, the Lasso method (also known as Basis Pursuit or \( \ell^1 \)-regularization) is the de facto baseline method for recovering sparse vectors from low-resolution measurements. This approach requires discretization of the domain, which leads to quantization artifacts and consequently, an overestimation of the number of sources. While grid-less methods, such as Prony-type methods or non-convex optimization over the source position, can mitigate this, the Lasso remains a strong baseline due to its versatility and simplicity. In this work, we introduce a simple extension of the Lasso, termed ``super-resolved Lasso" (SR-Lasso). Inspired by the Continuous Basis Pursuit (C-BP) method, our approach introduces an extra parameter to account for the shift of the sources between grid locations. Our method is more comprehensive than C-BP, accommodating both arbitrary real-valued or complex-valued sources. Furthermore, it can be solved similarly to the Lasso as it boils down to solving a group-Lasso problem. A notable advantage of SR-Lasso is its theoretical properties, akin to grid-less methods. Given a separation condition on the sources and a restriction on the shift magnitude outside the grid, SR-Lasso precisely estimates the correct number of sources.

math.OC↗

arXiv2023

Sparsistency for Inverse Optimal Transport

Optimal Transport is a useful metric to compare probability distributions and to compute a pairing given a ground cost. Its entropic regularization variant (eOT) is crucial to have fast algorithms and reflect fuzzy/noisy matchings. This work focuses on Inverse Optimal Transport (iOT), the problem of inferring the ground cost from samples drawn from a coupling that solves an eOT problem. It is a relevant problem that can be used to infer unobserved/missing links, and to obtain meaningful information about the structure of the ground cost yielding the pairing. On one side, iOT benefits from convexity, but on the other side, being ill-posed, it requires regularization to handle the sampling noise. This work presents an in-depth theoretical study of the l1 regularization to model for instance Euclidean costs with sparse interactions between features. Specifically, we derive a sufficient condition for the robust recovery of the sparsity of the ground cost that can be seen as a far reaching generalization of the Lasso's celebrated Irrepresentability Condition. To provide additional insight into this condition, we work out in detail the Gaussian case. We show that as the entropic penalty varies, the iOT problem interpolates between a graphical Lasso and a classical Lasso, thereby establishing a connection between iOT and graph estimation, an important problem in ML.

math.ST↗

arXiv2023

Compressed online Sinkhorn

The use of optimal transport (OT) distances, and in particular entropic-regularised OT distances, is an increasingly popular evaluation metric in many areas of machine learning and data science. Their use has largely been driven by the availability of efficient algorithms such as the Sinkhorn algorithm. One of the drawbacks of the Sinkhorn algorithm for large-scale data processing is that it is a two-phase method, where one first draws a large stream of data from the probability distributions, before applying the Sinkhorn algorithm to the discrete probability measures. More recently, there have been several works developing stochastic versions of Sinkhorn that directly handle continuous streams of data. In this work, we revisit the recently introduced online Sinkhorn algorithm of [Mensch and Peyr\'e, 2020]. Our contributions are twofold: We improve the convergence analysis for the online Sinkhorn algorithm, the new rate that we obtain is faster than the previous rate under certain parameter choices. We also present numerical results to verify the sharpness of our result. Secondly, we propose the compressed online Sinkhorn algorithm which combines measure compression techniques with the online Sinkhorn algorithm. We provide numerical experiments to show practical numerical gains, as well as theoretical guarantees on the efficiency of our approach.

cs.LG↗

arXiv2022

Smooth over-parameterized solvers for non-smooth structured optimization

Non-smooth optimization is a core ingredient of many imaging or machine learning pipelines. Non-smoothness encodes structural constraints on the solutions, such as sparsity, group sparsity, low-rank and sharp edges. It is also the basis for the definition of robust loss functions and scale-free functionals such as square-root Lasso. Standard approaches to deal with non-smoothness leverage either proximal splitting or coordinate descent. These approaches are effective but usually require parameter tuning, preconditioning or some sort of support pruning. In this work, we advocate and study a different route, which operates a non-convex but smooth over-parametrization of the underlying non-smooth optimization problems. This generalizes quadratic variational forms that are at the heart of the popular Iterative Reweighted Least Squares (IRLS). Our main theoretical contribution connects gradient descent on this reformulation to a mirror descent flow with a varying Hessian metric. This analysis is crucial to derive convergence bounds that are dimension-free. This explains the efficiency of the method when using small grid sizes in imaging. Our main algorithmic contribution is to apply the Variable Projection (VarPro) method which defines a new formulation by explicitly minimizing over part of the variables. This leads to a better conditioning of the minimized functional and improves the convergence of simple but very efficient gradient-based methods, for instance quasi-Newton solvers. We exemplify the use of this new solver for the resolution of regularized regression problems for inverse problems and supervised learning, including total variation prior and non-convex regularizers.

math.OC↗

arXiv2021

Smooth Bilevel Programming for Sparse Regularization

Iteratively reweighted least square (IRLS) is a popular approach to solve sparsity-enforcing regression problems in machine learning. State of the art approaches are more efficient but typically rely on specific coordinate pruning schemes. In this work, we show how a surprisingly simple reparametrization of IRLS, coupled with a bilevel resolution (instead of an alternating scheme) is able to achieve top performances on a wide range of sparsity (such as Lasso, group Lasso and trace norm regularizations), regularization strength (including hard constraints), and design matrices (ranging from correlated designs to differential operators). Similarly to IRLS, our method only involves linear systems resolutions, but in sharp contrast, corresponds to the minimization of a smooth function. Despite being non-convex, we show that there is no spurious minima and that saddle points are "ridable", so that there always exists a descent direction. We thus advocate for the use of a BFGS quasi-Newton solver, which makes our approach simple, robust and efficient. We perform a numerical benchmark of the convergence speed of our algorithm against state of the art solvers for Lasso, group Lasso, trace norm and linearly constrained problems. These results highlight the versatility of our approach, removing the need to use different solvers depending on the specificity of the ML problem under study.

stat.ML↗

arXiv2021

Screening for Sparse Online Learning

Sparsity promoting regularizers are widely used to impose low-complexity structure (e.g. l1-norm for sparsity) to the regression coefficients of supervised learning. In the realm of deterministic optimization, the sequence generated by iterative algorithms (such as proximal gradient descent) exhibit "finite activity identification", namely, they can identify the low-complexity structure in a finite number of iterations. However, most online algorithms (such as proximal stochastic gradient descent) do not have the property owing to the vanishing step-size and non-vanishing variance. In this paper, by combining with a screening rule, we show how to eliminate useless features of the iterates generated by online algorithms, and thereby enforce finite activity identification. One consequence is that when combined with any convergent online algorithm, sparsity properties imposed by the regularizer can be exploited for computational gains. Numerically, significant acceleration can be obtained.

cs.LG↗

arXiv2020

An off-the-grid approach to multi-compartment magnetic resonance fingerprinting

We propose a novel numerical approach to separate multiple tissue compartments in image voxels and to estimate quantitatively their nuclear magnetic resonance (NMR) properties and mixture fractions, given magnetic resonance fingerprinting (MRF) measurements. The number of tissues, their types or quantitative properties are not a-priori known, but the image is assumed to be composed of sparse compartments with linearly mixed Bloch magnetisation responses within voxels. Fine-grid discretisation of the multi-dimensional NMR properties creates large and highly coherent MRF dictionaries that can challenge scalability and precision of the numerical methods for (discrete) sparse approximation. To overcome these issues, we propose an off-the-grid approach equipped with an extended notion of the sparse group lasso regularisation for sparse approximation using continuous (non-discretised) Bloch response models. Further, the nonlinear and non-analytical Bloch responses are approximated by a neural network, enabling efficient back-propagation of the gradients through the proposed algorithm. Tested on simulated and in-vivo healthy brain MRF data, we demonstrate effectiveness of the proposed scheme compared to the baseline multicompartment MRF methods.

cs.CV↗

arXiv2020

Geometry of First-Order Methods and Adaptive Acceleration

First-order operator splitting methods are ubiquitous among many fields through science and engineering, such as inverse problems, signal/image processing, statistics, data science and machine learning, to name a few. In this paper, we study a geometric property of first-order methods when applying to solve non-smooth optimization problems. With the tool of "partial smoothness", we design a framework to analyze the trajectory of the fixed-point sequence generated by first-order methods and show that locally, the fixed-point sequence settles onto a regular trajectory such as a straight line or a spiral. Based on this finding, we discuss the limitation of current widely used "inertial acceleration" technique, and propose a trajectory following adaptive acceleration algorithm. Global convergence is established for the proposed acceleration scheme based on the perturbation of fixed-point iteration. Locally, we first build connections between the acceleration scheme and the well-studied "vector extrapolation technique" in the field of numerical analysis, and then discuss local acceleration guarantees of the proposed acceleration scheme. Moreover, our result provides a geometric interpretation of these vector extrapolation techniques. Numerical experiments on various first-order methods are provided to demonstrate the advantage of the proposed adaptive acceleration scheme.

math.OC↗

arXiv2019

Degrees of freedom for off-the-grid sparse estimation

A central question in modern machine learning and imaging sciences is to quantify the number of effective parameters of vastly over-parameterized models. The degrees of freedom is a mathematically convenient way to define this number of parameters. Its computation and properties are well understood when dealing with discretized linear models, possibly regularized using sparsity. In this paper, we argue that this way of thinking is plagued when dealing with models having very large parameter spaces. In this case it makes more sense to consider "off-the-grid" approaches, using a continuous parameter space. This type of approach is the one favoured when training multi-layer perceptrons, and is also becoming popular to solve super-resolution problems in imaging. Training these off-the-grid models with a sparsity inducing prior can be achieved by solving a convex optimization problem over the space of measures, which is often called the Beurling Lasso (Blasso), and is the continuous counterpart of the celebrated Lasso parameter selection method. In previous works, the degrees of freedom for the Lasso was shown to coincide with the size of the smallest solution support. Our main contribution is a proof of a continuous counterpart to this result for the Blasso. Our findings suggest that discretized methods actually vastly over-estimate the number of intrinsic continuous degrees of freedom. Our second contribution is a detailed study of the case of sampling Fourier coefficients in 1D, which corresponds to a super-resolution problem. We show that our formula for the degrees of freedom is valid outside of a set of measure zero of observations, which in turn justifies its use to compute an unbiased estimator of the prediction risk using the Stein Unbiased Risk Estimator (SURE).

stat.ML↗

arXiv2019

Trajectory of Alternating Direction Method of Multipliers and Adaptive Acceleration

The alternating direction method of multipliers (ADMM) is one of the most widely used first-order optimisation methods in the literature owing to its simplicity, flexibility and efficiency. Over the years, numerous efforts are made to improve the performance of the method, such as the inertial technique. By studying the geometric properties of ADMM, we discuss the limitations of current inertial accelerated ADMM and then present and analyze an adaptive acceleration scheme for the method. Numerical experiments on problems arising from image processing, statistics and machine learning demonstrate the advantages of the proposed acceleration approach.

math.OC↗

arXiv2019

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.

cs.CV↗