arXiv subjects

Claudio Mayrink Verdun

Publications and source records attributed to Claudio Mayrink Verdun.

25 records · Page 2Linked to original sources

arXiv2024

Non-Asymptotic Uncertainty Quantification in High-Dimensional Learning

Uncertainty quantification (UQ) is a crucial but challenging task in many high-dimensional regression or learning problems to increase the confidence of a given predictor. We develop a new data-driven approach for UQ in regression that applies both to classical regression approaches such as the LASSO as well as to neural networks. One of the most notable UQ techniques is the debiased LASSO, which modifies the LASSO to allow for the construction of asymptotic confidence intervals by decomposing the estimation error into a Gaussian and an asymptotically vanishing bias component. However, in real-world problems with finite-dimensional data, the bias term is often too significant to be neglected, resulting in overly narrow confidence intervals. Our work rigorously addresses this issue and derives a data-driven adjustment that corrects the confidence intervals for a large class of predictors by estimating the means and variances of the bias terms from training data, exploiting high-dimensional concentration phenomena. This gives rise to non-asymptotic confidence intervals, which can help avoid overestimating uncertainty in critical applications such as MRI diagnosis. Importantly, our analysis extends beyond sparse regression to data-driven predictors like neural networks, enhancing the reliability of model-based deep learning. Our findings bridge the gap between established theory and the practical applicability of such debiased methods.

cs.LG↗

arXiv2024

High-Dimensional Confidence Regions in Sparse MRI

One of the most promising solutions for uncertainty quantification in high-dimensional statistics is the debiased LASSO that relies on unconstrained $\ell_1$-minimization. The initial works focused on real Gaussian designs as a toy model for this problem. However, in medical imaging applications, such as compressive sensing for MRI, the measurement system is represented by a (subsampled) complex Fourier matrix. The purpose of this work is to extend the method to the MRI case in order to construct confidence intervals for each pixel of an MR image. We show that a sufficient amount of data is $n \gtrsim \max\{ s_0\log^2 s_0\log p, s_0 \log^2 p \}$.

eess.SP↗

arXiv2023

Uncertainty quantification for learned ISTA

Model-based deep learning solutions to inverse problems have attracted increasing attention in recent years as they bridge state-of-the-art numerical performance with interpretability. In addition, the incorporated prior domain knowledge can make the training more efficient as the smaller number of parameters allows the training step to be executed with smaller datasets. Algorithm unrolling schemes stand out among these model-based learning techniques. Despite their rapid advancement and their close connection to traditional high-dimensional statistical methods, they lack certainty estimates and a theory for uncertainty quantification is still elusive. This work provides a step towards closing this gap proposing a rigorous way to obtain confidence intervals for the LISTA estimator.

stat.ML↗

arXiv2023

Uncertainty quantification for sparse Fourier recovery

One of the most prominent methods for uncertainty quantification in high-dimen-sional statistics is the desparsified LASSO that relies on unconstrained $\ell_1$-minimization. The majority of initial works focused on real (sub-)Gaussian designs. However, in many applications, such as magnetic resonance imaging (MRI), the measurement process possesses a certain structure due to the nature of the problem. The measurement operator in MRI can be described by a subsampled Fourier matrix. The purpose of this work is to extend the uncertainty quantification process using the desparsified LASSO to design matrices originating from a bounded orthonormal system, which naturally generalizes the subsampled Fourier case and also allows for the treatment of the case where the sparsity basis is not the standard basis. In particular we construct honest confidence intervals for every pixel of an MR image that is sparse in the standard basis provided the number of measurements satisfies $n \gtrsim\max\{ s\log^2 s\log p, s \log^2 p \}$ or that is sparse with respect to the Haar Wavelet basis provided a slightly larger number of measurements.

cs.IT↗

arXiv2021

Iteratively Reweighted Least Squares for Basis Pursuit with Global Linear Convergence Rate

The recovery of sparse data is at the core of many applications in machine learning and signal processing. While such problems can be tackled using $\ell_1$-regularization as in the LASSO estimator and in the Basis Pursuit approach, specialized algorithms are typically required to solve the corresponding high-dimensional non-smooth optimization for large instances. Iteratively Reweighted Least Squares (IRLS) is a widely used algorithm for this purpose due its excellent numerical performance. However, while existing theory is able to guarantee convergence of this algorithm to the minimizer, it does not provide a global convergence rate. In this paper, we prove that a variant of IRLS converges with a global linear rate to a sparse solution, i.e., with a linear error decrease occurring immediately from any initialization, if the measurements fulfill the usual null space property assumption. We support our theory by numerical experiments showing that our linear rate captures the correct dimension dependence. We anticipate that our theoretical findings will lead to new insights for many other use cases of the IRLS algorithm, such as in low-rank matrix recovery.

math.OC↗

arXiv2021

A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples

We propose an iterative algorithm for low-rank matrix completion that can be interpreted as an iteratively reweighted least squares (IRLS) algorithm, a saddle-escaping smoothing Newton method or a variable metric proximal gradient method applied to a non-convex rank surrogate. It combines the favorable data-efficiency of previous IRLS approaches with an improved scalability by several orders of magnitude. We establish the first local convergence guarantee from a minimal number of samples for that class of algorithms, showing that the method attains a local quadratic convergence rate. Furthermore, we show that the linear systems to be solved are well-conditioned even for very ill-conditioned ground truth matrices. We provide extensive experiments, indicating that unlike many state-of-the-art approaches, our method is able to complete very ill-conditioned matrices with a condition number of up to $10^{10}$ from few samples, while being competitive in its scalability.

math.OC↗

arXiv2018

Denoising and Completion of Structured Low-Rank Matrices via Iteratively Reweighted Least Squares

We propose a new Iteratively Reweighted Least Squares (IRLS) algorithm for the problem of completing or denoising low-rank matrices that are structured, e.g., that possess a Hankel, Toeplitz or block-Hankel/Toeplitz structure. The algorithm optimizes an objective based on a non-convex surrogate of the rank by solving a sequence of quadratic problems. Our strategy combines computational efficiency, as it operates on a lower dimensional generator space of the structured matrices, with high statistical accuracy which can be observed in experiments on hard estimation and completion tasks. Our experiments show that the proposed algorithm StrucHMIRLS exhibits an empirical recovery probability close to 1 from fewer samples than the state-of-the-art in a Hankel matrix completion task arising from the problem of spectral super-resolution of badly separated frequencies. Furthermore, we explain how the proposed algorithm for structured low-rank recovery can be used as preprocessing step for improved robustness in frequency or line spectrum estimation problems.

math.OC↗