859 research outputs found
Proximal boosting and its acceleration
Gradient boosting is a prediction method that iteratively combines weak
learners to produce a complex and accurate model. From an optimization point of
view, the learning procedure of gradient boosting mimics a gradient descent on
a functional variable. This paper proposes to build upon the proximal point
algorithm when the empirical risk to minimize is not differentiable to
introduce a novel boosting approach, called proximal boosting. Besides being
motivated by non-differentiable optimization, the proposed algorithm benefits
from Nesterov's acceleration in the same way as gradient boosting [Biau et al.,
2018]. This leads to a variant, called accelerated proximal boosting.
Advantages of leveraging proximal methods for boosting are illustrated by
numerical experiments on simulated and real-world data. In particular, we
exhibit a favorable comparison over gradient boosting regarding convergence
rate and prediction accuracy
Robust Lasso-Zero for sparse corruption and model selection with missing covariates
We propose Robust Lasso-Zero, an extension of the Lasso-Zero methodology
[Descloux and Sardy, 2018], 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
On the asymptotic rate of convergence of Stochastic Newton algorithms and their Weighted Averaged versions
The majority of machine learning methods can be regarded as the minimization
of an unavailable risk function. To optimize the latter, given samples provided
in a streaming fashion, we define a general stochastic Newton algorithm and its
weighted average version. In several use cases, both implementations will be
shown not to require the inversion of a Hessian estimate at each iteration, but
a direct update of the estimate of the inverse Hessian instead will be favored.
This generalizes a trick introduced in [2] for the specific case of logistic
regression, by directly updating the estimate of the inverse Hessian. Under
mild assumptions such as local strong convexity at the optimum, we establish
almost sure convergences and rates of convergence of the algorithms, as well as
central limit theorems for the constructed parameter estimates. The unified
framework considered in this paper covers the case of linear, logistic or
softmax regressions to name a few. Numerical experiments on simulated data give
the empirical evidence of the pertinence of the proposed methods, which
outperform popular competitors particularly in case of bad initializa-tions.Comment: Computational Optimization and Applications, 202
Sampling by blocks of measurements in compressed sensing
Various acquisition devices impose sampling blocks of measurements. A typical example is parallel magnetic resonance imaging (MRI) where several radio-frequency coils simultaneously acquire a set of Fourier modulated coefficients. We study a new random sampling approach that consists in selecting a set of blocks that are predefined by the application of interest. We provide theoretical results on the number of blocks that are required for exact sparse signal reconstruction. We finish by illustrating these results on various examples, and discuss their connection to the literature on CS
Convergence and error analysis of PINNs
Physics-informed neural networks (PINNs) are a promising approach that
combines the power of neural networks with the interpretability of physical
modeling. PINNs have shown good practical performance in solving partial
differential equations (PDEs) and in hybrid modeling scenarios, where physical
models enhance data-driven approaches. However, it is essential to establish
their theoretical properties in order to fully understand their capabilities
and limitations. In this study, we highlight that classical training of PINNs
can suffer from systematic overfitting. This problem can be addressed by adding
a ridge regularization to the empirical risk, which ensures that the resulting
estimator is risk-consistent for both linear and nonlinear PDE systems.
However, the strong convergence of PINNs to a solution satisfying the physical
constraints requires a more involved analysis using tools from functional
analysis and calculus of variations. In particular, for linear PDE systems, an
implementable Sobolev-type regularization allows to reconstruct a solution that
not only achieves statistical accuracy but also maintains consistency with the
underlying physics
Block-constrained compressed sensing
Dans cette thèse, nous visons à combiner les théories d'échantillonnage compressé (CS) avec une structure d'acquisition par blocs de mesures. D'une part, nous obtenons des résultats théoriques de CS avec contraintes d'acquisition par blocs, pour la reconstruction de tout vecteur s-parcimonieux et pour la reconstruction d'un vecteur x de support S fixé. Nous montrons que l'acquisition structurée peut donner de bons résultats de reconstruction théoriques, à condition que le signal à reconstruire présente une structure de parcimonie, adaptée aux contraintes d'échantillonnage. D'autre part, nous proposons des méthodes numériques pour générer des schémas d'échantillonnage efficaces reposant sur des blocs de mesures. Ces méthodes s'appuient sur des techniques de projection de mesure de probabilité.This PhD. thesis is dedicated to combine compressed sensing with block structured acquisition. In the first part of this work, theoretical CS results are derived with blocks acquisition constraints, for the recovery of any s-sparse signal and for the recovery of a vector with a given support S.We show that structured acquisition can be successfully used in a CS framework, provided that the signal to reconstruct presents an additional structure in its sparsity, adapted to the sampling constraints.In the second part of this work, we propose numerical methods to generate efficient block sampling schemes. This approach relies on the measure projection on admissible measures
HYR2PICS: Hybrid Regularized Reconstruction for combined Parallel Imaging and Compressive Sensing in MRI
International audienceBoth parallel Magnetic Resonance Imaging~(pMRI) and Compressed Sensing (CS) are emerging techniques to accelerate conventional MRI by reducing the number of acquired data in the -space. So far, first attempts to combine sensitivity encoding (SENSE) imaging in pMRI with CS have been proposed in the context of Cartesian trajectories. Here, we extend these approaches to non-Cartesian trajectories by jointly formulating the CS and SENSE recovery in a hybrid Fourier/wavelet framework and optimizing a convex but nonsmooth criterion. On anatomical MRI data, we show that HYRPICS outperforms wavelet-based regularized SENSE reconstruction. Our results are also in agreement with the Transform Point Spread Function (TPSF) criterion that measures the degree of incoherence of -space undersampling schemes
Missing Data Imputation using Optimal Transport
Missing data is a crucial issue when applying machine learning algorithms to
real-world datasets. Starting from the simple assumption that two batches
extracted randomly from the same dataset should share the same distribution, we
leverage optimal transport distances to quantify that criterion and turn it
into a loss function to impute missing data values. We propose practical
methods to minimize these losses using end-to-end learning, that can exploit or
not parametric assumptions on the underlying distributions of values. We
evaluate our methods on datasets from the UCI repository, in MCAR, MAR and MNAR
settings. These experiments show that OT-based methods match or out-perform
state-of-the-art imputation methods, even for high percentages of missing
values
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