arXiv subjects

Sylvain Sardy

Publications and source records attributed to Sylvain Sardy.

At least 19 recordsLinked to original sources

arXiv2026

The Pivotal Information Criterion

The Bayesian and Akaike information criteria aim at finding a good balance between under- and over-fitting. They are extensively used every day by practitioners. Yet we contend they suffer from at least two afflictions: their penalty parameter $\lambda=\log n$ and $\lambda=2$ are too small, leading to many false discoveries, and their inherent (best subset) discrete optimization is infeasible in high dimension. We alleviate these issues with the pivotal information criterion: PIC is defined as a continuous optimization problem, and the PIC penalty parameter $\lambda$ is selected at the detection boundary (under pure noise). PIC's choice of $\lambda$ is the quantile of a statistic that we prove to be (asymptotically) pivotal, provided the loss function is appropriately transformed. As a result, simulations show a phase transition in the probability of exact support recovery with PIC, a phenomenon studied with no noise in compressed sensing. Applied on real data, for similar predictive performances, PIC selects the least complex model among state-of-the-art learners.

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arXiv2025

A pivotal transform for the high-dimensional location-scale model

We study the high-dimensional linear model with noise distribution known up to a scale parameter. With an $\ell_1$-penalty on the regression coefficients, we show that a transformation of the log-likelihood allows for a choice of the tuning parameter not depending on the scale parameter. This transformation is a generalization of the square root Lasso for quadratic loss. The tuning parameter can asymptotically be taken at the detection edge. We establish an oracle inequality, variable selection and asymptotic efficiency of the estimator of the scale parameter and the intercept. The examples include Subbotin distributions and the Gumbel distribution.

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arXiv2025

Survival of the fittest Cox model: Pivotal variable selection for time-to-event data

We revisit Cox's proportional hazards model to improve variable selection in survival analysis. A square-root transformation of the partial likelihood renders the selection of the regularization parameter pivotal, free of the unknown baseline hazard and censoring mechanism. The resulting criterion borrows from information criteria such as BIC and from penalized regression methods such as the lasso, taking the best of both. On simulated and real data, our method substantially improves upon state-of-the-art approaches used daily in support recovery.

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arXiv2024

Validation-Free Sparse Learning: A Phase Transition Approach to Feature Selection

The growing environmental footprint of artificial intelligence (AI), especially in terms of storage and computation, calls for more frugal and interpretable models. Sparse models (e.g., linear, neural networks) offer a promising solution by selecting only the most relevant features, reducing complexity, preventing over-fitting and enabling interpretation-marking a step towards truly intelligent AI. The concept of a right amount of sparsity (without too many false positive or too few true positive) is subjective. So we propose a new paradigm previously only observed and mathematically studied for compressed sensing (noiseless linear models): obtaining a phase transition in the probability of retrieving the relevant features. We show in practice how to obtain this phase transition for a class of sparse learners. Our approach is flexible and applicable to complex models ranging from linear to shallow and deep artificial neural networks while supporting various loss functions and sparsity-promoting penalties. It does not rely on cross-validation or on a validation set to select its single regularization parameter. For real-world data, it provides a good balance between predictive accuracy and feature sparsity. A Python package is available at https://github.com/VcMaxouuu/HarderLASSO containing all the simulations and ready-to-use models.

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arXiv2024

The $\infty$-S test via regression quantile affine LASSO

A novel test in the linear $\ell_1$ (LAD) and quantile regressions is proposed, based on the scores provided by the dual variables (signs) arising in the calculation of the (so-called) affine-lasso estimate--a Rao-type, Lagrange multiplier test using the thresholding, towards the null hypothesis of the test, function of the latter estimate.

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arXiv2024

Tomographic reconstruction of a disease transmission landscape via GPS recorded random paths

Identifying areas in a landscape where individuals have a higher likelihood of disease infection is key to managing diseases. Unlike conventional methods relying on ecological assumptions, we perform a novel epidemiological tomography for the estimation of landscape propensity to disease infection, using GPS animal tracks in a manner analogous to tomographic techniques in positron emission tomography (PET). Treating tracking data as random Radon transforms, we analyze Cervid movements in a game preserve, paired with antibody levels for epizootic hemorrhagic disease virus (EHDV) -- a vector-borne disease transmitted by biting midges. After discretizing the field and building the regression matrix of the time spent by each deer (row) at each point of the lattice (column), we model the binary response (infected or not) as a binomial linear inverse problem where spatial coherence is enforced with a total variation regularization. The smoothness of the reconstructed propensity map is selected by the quantile universal threshold. To address limitations of small sample sizes and evaluate significance of our estimates, we quantify uncertainty using a bootstrap-based data augmentation procedure. Our method outperforms alternative ones when using simulated and real data. This tomographic framework is novel, with no established statistical methods tailored for such data.

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arXiv2022

Sparse additive models in high dimensions with wavelets

In multivariate regression, when covariates are numerous, it is often reasonable to assume that only a small number of them has predictive information. In some medical applications for instance, it is believed that only a few genes out of thousands are responsible for cancers. In that case, the aim is not only to propose a good fit, but also to select the relevant covariates (genes). We propose to perform model selection with additive models in high dimensions (sample size and number of covariates). Our approach is computationally efficient thanks to fast wavelet transforms, it does not rely on cross validation, and it solves a convex optimization problem for a prescribed penalty parameter, called the quantile universal threshold. We also propose a second rule based on Stein unbiased risk estimation geared towards prediction. We use Monte Carlo simulations and real data to compare various methods based on false discovery rate (FDR), true positive rate (TPR) and mean squared error. Our approach is the only one to handle high dimensions, and has the best FDR--TPR trade-off.

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arXiv2022

Robust Lasso-Zero for sparse corruption and model selection with missing covariates

We propose Robust Lasso-Zero, an extension of the Lasso-Zero methodology, initially introduced for sparse linear models, to the sparse corruptions problem. We give theoretical guarantees on the sign recovery of the parameters for a slightly simplified version of the estimator, called Thresholded Justice Pursuit. The use of Robust Lasso-Zero is showcased for variable selection with missing values in the covariates. In addition to not requiring the specification of a model for the covariates, nor estimating their covariance matrix or the noise variance, the method has the great advantage of handling missing not-at random values without specifying a parametric model. Numerical experiments and a medical application underline the relevance of Robust Lasso-Zero in such a context with few available competitors. The method is easy to use and implemented in the R library lass0.

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arXiv2022

A phase transition for finding needles in nonlinear haystacks with LASSO artificial neural networks

To fit sparse linear associations, a LASSO sparsity inducing penalty with a single hyperparameter provably allows to recover the important features (needles) with high probability in certain regimes even if the sample size is smaller than the dimension of the input vector (haystack). More recently learners known as artificial neural networks (ANN) have shown great successes in many machine learning tasks, in particular fitting nonlinear associations. Small learning rate, stochastic gradient descent algorithm and large training set help to cope with the explosion in the number of parameters present in deep neural networks. Yet few ANN learners have been developed and studied to find needles in nonlinear haystacks. Driven by a single hyperparameter, our ANN learner, like for sparse linear associations, exhibits a phase transition in the probability of retrieving the needles, which we do not observe with other ANN learners. To select our penalty parameter, we generalize the universal threshold of Donoho and Johnstone (1994) which is a better rule than the conservative (too many false detections) and expensive cross-validation. In the spirit of simulated annealing, we propose a warm-start sparsity inducing algorithm to solve the high-dimensional, non-convex and non-differentiable optimization problem. We perform precise Monte Carlo simulations to show the effectiveness of our approach.

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arXiv2020

What needles do sparse neural networks find in nonlinear haystacks

Using a sparsity inducing penalty in artificial neural networks (ANNs) avoids over-fitting, especially in situations where noise is high and the training set is small in comparison to the number of features. For linear models, such an approach provably also recovers the important features with high probability in regimes for a well-chosen penalty parameter. The typical way of setting the penalty parameter is by splitting the data set and performing the cross-validation, which is (1) computationally expensive and (2) not desirable when the data set is already small to be further split (for example, whole-genome sequence data). In this study, we establish the theoretical foundation to select the penalty parameter without cross-validation based on bounding with a high probability the infinite norm of the gradient of the loss function at zero under the zero-feature assumption. Our approach is a generalization of the universal threshold of Donoho and Johnstone (1994) to nonlinear ANN learning. We perform a set of comprehensive Monte Carlo simulations on a simple model, and the numerical results show the effectiveness of the proposed approach.

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arXiv2019

Model selection with lasso-zero: adding straw to the haystack to better find needles

The high-dimensional linear model $y = X β^0 + ε$ is considered and the focus is put on the problem of recovering the support $S^0$ of the sparse vector $β^0.$ We introduce Lasso-Zero, a new $\ell_1$-based estimator whose novelty resides in an "overfit, then threshold" paradigm and the use of noise dictionaries concatenated to $X$ for overfitting the response. To select the threshold, we employ the quantile universal threshold based on a pivotal statistic that requires neither knowledge nor preliminary estimation of the noise level. Numerical simulations show that Lasso-Zero performs well in terms of support recovery and provides an excellent trade-off between high true positive rate and low false discovery rate compared to competitors. Our methodology is supported by theoretical results showing that when no noise dictionary is used, Lasso-Zero recovers the signs of $β^0$ under weaker conditions on $X$ and $S^0$ than the Lasso and achieves sign consistency for correlated Gaussian designs. The use of noise dictionary improves the procedure for low signals.

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arXiv2018

Low-rank model with covariates for count data analysis

Count data are collected in many scientific and engineering tasks including image processing, single-cell RNA sequencing and ecological studies. Such data sets often contain missing values, for example because some ecological sites cannot be reached in a certain year. In addition, in many instances, side information is also available, for example covariates about ecological sites or species. Low-rank methods are popular to denoise and impute count data, and benefit from a substantial theoretical background. Extensions accounting for covariates have been proposed, but to the best of our knowledge their theoretical and empirical properties have not been thoroughly studied, and few softwares are available for practitioners. We propose a complete methodology called LORI (Low-Rank Interaction), including a Poisson model, an algorithm, and automatic selection of the regularization parameter, to analyze count tables with covariates. We also derive an upper bound on the estimation error. We provide a simulation study with synthetic data, revealing empirically that LORI improves on state of the art methods in terms of estimation and imputation of the missing values. We illustrate how the method can be interpreted through visual displays with the analysis of a well-know plant abundance data set, and show that the LORI outputs are consistent with known results. Finally we demonstrate the relevance of the methodology by analyzing a water-birds abundance table from the French national agency for wildlife and hunting management (ONCFS). The method is available in the R package lori on the Comprehensive Archive Network (CRAN).

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arXiv2018

denoiseR: A Package for Low Rank Matrix Estimation

We introduce denoiseR, an R package that provides a unified implementation of several state-of-the-art proposals for regularized low rank matrix estimation, along with automatic selection of the regularization parameters. We also extend these methods to allow for missing values. The regularization schemes discussed in this paper are built around singular-value shrinkage and bootstrap-based stability arguments. We illustrate how to use out package by applying it to several real and simulated datasets, and highlight strengths and weaknesses of the different implemented methods.

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arXiv2018

Thresholding tests

We derive a new class of statistical tests for generalized linear models based on thresholding point estimators. These tests can be employed whether the model includes more parameters than observations or not. For linear models, our tests rely on pivotal statistics derived from model selection techniques. Affine lasso, a new extension of lasso, allows to unveil new tests and to develop in the same framework parametric and nonparametric tests. Our tests for generalized linear models are based on new asymptotically pivotal statistics. A composite thresholding test attempts to achieve uniformly most power under both sparse and dense alternatives with success. In a simulation, we compare the level and power of these tests under sparse and dense alternative hypotheses. The thresholding tests have a better control of the nominal level and higher power than existing tests.

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arXiv2017

Quantile universal threshold for model selection

Efficient recovery of a low-dimensional structure from high-dimensional data has been pursued in various settings including wavelet denoising, generalized linear models and low-rank matrix estimation. By thresholding some parameters to zero, estimators such as lasso, elastic net and subset selection allow to perform not only parameter estimation but also variable selection, leading to sparsity. Yet one crucial step challenges all these estimators: the choice of the threshold parameter~$λ$. If too large, important features are missing; if too small, incorrect features are included. Within a unified framework, we propose a new selection of $λ$ at the detection edge under the null model. To that aim, we introduce the concept of a zero-thresholding function and a null-thresholding statistic, that we explicitly derive for a large class of estimators. The new approach has the great advantage of transforming the selection of $λ$ from an unknown scale to a probabilistic scale with the simple selection of a probability level. Numerical results show the effectiveness of our approach in terms of model selection and prediction.

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arXiv2017

Nonparametric estimation of galaxy cluster's emissivity and point source detection in astrophysics with two lasso penalties

Astrophysicists are interested in recovering the 3D gas emissivity of a galaxy cluster from a 2D image taken by a telescope. A blurring phenomenon and presence of point sources make this inverse problem even harder to solve. The current state-of-the-art technique is two step: first identify the location of potential point sources, then mask these locations and deproject the data. We instead model the data as a Poisson generalized linear model (involving blurring, Abel and wavelets operators) regularized by two lasso penalties to induce sparse wavelet representation and sparse point sources. The amount of sparsity is controlled by two quantile universal thresholds. As a result, our method outperforms the existing one.

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arXiv2016

Threshold Selection for Total Variation Denoising

Total variation (TV) denoising is a nonparametric smoothing method that has good properties for preserving sharp edges and contours in objects with spatial structures like natural images. The estimate is sparse in the sense that TV reconstruction leads to a piecewise constant function with a small number of jumps. A threshold parameter controls the number of jumps and the quality of the estimation. In practice, this threshold is often selected by minimizing a goodness-of-fit criterion like cross-validation, which can be costly as it requires solving the high-dimensional and non-differentiable TV optimization problem many times. We propose instead a two step adaptive procedure via a connection to large deviation of stochastic processes. We also give conditions under which TV denoising achieves exact segmentation. We then apply our procedure to denoise a collection of 1D and 2D test signals verifying the effectiveness of our approach in practice.

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arXiv2014

Quantile universal threshold: model selection at the detection edge for high-dimensional linear regression

To estimate a sparse linear model from data with Gaussian noise, consilience from lasso and compressed sensing literatures is that thresholding estimators like lasso and the Dantzig selector have the ability in some situations to identify with high probability part of the significant covariates asymptotically, and are numerically tractable thanks to convexity. Yet, the selection of a threshold parameter $λ$ remains crucial in practice. To that aim we propose Quantile Universal Thresholding, a selection of $λ$ at the detection edge. We show with extensive simulations and real data that an excellent compromise between high true positive rate and low false discovery rate is achieved, leading also to good predictive risk.

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