arXiv research

Through the direct study of the analysis estimator we derive oracle inequalities with fast and slow rates by adapting the arguments involving projections by Dalalyan, Hebiri and Lederer (2017). We then extend the theory to the square root analysis estimator. Finally, we focus on (square root) total variation regularize…

Unified framework for fair regression in aware and unaware settings.

We establish adaptive results for trend filtering: least squares estimation with a penalty on the total variation of $(k-1)^{\rm th}$ order differences. Our approach is based on combining a general oracle inequality for the $\ell_1$-penalized least squares estimator with "interpolating vectors" to upper-bound the "effe…

We generalize to tree graphs obtained by connecting path graphs an oracle result obtained for the Fused Lasso over the path graph. Moreover we show that it is possible to substitute in the oracle inequality the minimum of the distances between jumps by their harmonic mean. In doing so we prove a lower bound on the comp…

The use of machine-learning in neuroimaging offers new perspectives in early diagnosis and prognosis of brain diseases. Although such multivariate methods can capture complex relationships in the data, traditional approaches provide irregular (l2 penalty) or scattered (l1 penalty) predictive pattern with a very limited…

We introduce an iterative optimization scheme for convex objectives consisting of a linear loss and a non-separable penalty, based on the expectation-consistent approximation and the vector approximate message-passing (VAMP) algorithm. Specifically, the penalties we approach are convex on a linear transformation of the…

We present a convex approach to probabilistic segmentation and modeling of time series data. Our approach builds upon recent advances in multivariate total variation regularization, and seeks to learn a separate set of parameters for the distribution over the observations at each time point, but with an additional pena…

The total variation (TV) penalty, as many other analysis-sparsity problems, does not lead to separable factors or a proximal operatorwith a closed-form expression, such as soft thresholding for the $\ell\_1$ penalty. As a result, in a variational formulation of an inverse problem or statisticallearning estimation, it l…

New method detects changepoints in PDEs using optimized neural networks.

Identification of regions of interest (ROI) associated with certain disease has a great impact on public health. Imposing sparsity of pixel values and extracting active regions simultaneously greatly complicate the image analysis. We address these challenges by introducing a novel region-selection penalty in the framew…

In this paper, we provide a Banach-space formulation of supervised learning with generalized total-variation (gTV) regularization. We identify the class of kernel functions that are admissible in this framework. Then, we propose a variation of supervised learning in a continuous-domain hybrid search space with gTV regu…

New scalable algorithm for non-negative linear regression with entropy-regularized OT loss.

We consider estimating a piecewise-constant image, or a gradient-sparse signal on a general graph, from noisy linear measurements. We propose and study an iterative algorithm to minimize a penalized least-squares objective, with a penalty given by the "l_0-norm" of the signal's discrete graph gradient. The method proce…

Although the Lasso has been extensively studied, the relationship between its prediction performance and the correlations of the covariates is not fully understood. In this paper, we give new insights into this relationship in the context of multiple linear regression. We show, in particular, that the incorporation of …

New privacy mechanism for user-level discrete distributions with reduced penalty.

Paper addresses Byzantine attacks in decentralized optimization over networks.

Principal component analysis (PCA) is an exploratory tool widely used in data analysis to uncover dominant patterns of variability within a population. Despite its ability to represent a data set in a low-dimensional space, the interpretability of PCA remains limited. However, in neuroimaging, it is essential to uncove…

Ideas from the image processing literature have recently motivated a new set of clustering algorithms that rely on the concept of total variation. While these algorithms perform well for bi-partitioning tasks, their recursive extensions yield unimpressive results for multiclass clustering tasks. This paper presents a g…

Optimal pre-processing reduces disparate impact by minimizing total variation distance.

Predictive models can be used on high-dimensional brain images for diagnosis of a clinical condition. Spatial regularization through structured sparsity offers new perspectives in this context and reduces the risk of overfitting the model while providing interpretable neuroimaging signatures by forcing the solution to …

Study extends DRO with IPMs, linking robustness to regularization and GANs.

The paper studies curves in Riemannian manifolds using total variation flow.

We introduce the binacox, a prognostic method to deal with the problem of detecting multiple cut-points per features in a multivariate setting where a large number of continuous features are available. The method is based on the Cox model and combines one-hot encoding with the binarsity penalty, which uses total-variat…

The total variation distance is a core statistical distance between probability measures that satisfies the metric axioms, with value always falling in $[0,1]$. This distance plays a fundamental role in machine learning and signal processing: It is a member of the broader class of $f$-divergences, and it is related to …

We consider the problem of estimating a function defined over $n$ locations on a $d$-dimensional grid (having all side lengths equal to $n^{1/d}$). When the function is constrained to have discrete total variation bounded by $C_n$, we derive the minimax optimal (squared) $\ell_2$ estimation error rate, parametrized by …

SaR-SVM-STV improves hyperspectral image classification with shape-adaptive reconstruction and denoising.

Motivated by distributed machine learning settings such as Federated Learning, we consider the problem of fitting a statistical model across a distributed collection of heterogeneous data sets whose similarity structure is encoded by a graph topology. Precisely, we analyse the case where each node is associated with fi…

Estimates parameters of interconnected linear systems using total variation penalization.

Sharp inequality between TV and Hellinger distances for Gaussian mixtures.

Paper proposes a method to estimate total variation distance for synthetic data fidelity.

We consider the problem of minimizing the sum of submodular set functions assuming minimization oracles of each summand function. Most existing approaches reformulate the problem as the convex minimization of the sum of the corresponding Lovász extensions and the squared Euclidean norm, leading to algorithms requiring …

Proximal algorithms applied to current deformation into cycles.

Example shows learnable distributions not privately learnable.

Graphs with non-negative Ollivier-Ricci curvature cannot be expanders.

We study the theoretical properties of image denoising via total variation penalized least-squares. We define the total vatiation in terms of the two-dimensional total discrete derivative of the image and show that it gives rise to denoised images that are piecewise constant on rectangular sets. We prove that, if the t…

The paper shows diffusion models can converge faster to a target distribution with low-dimensional structure.

A main task in data analysis is to organize data points into coherent groups or clusters. The stochastic block model is a probabilistic model for the cluster structure. This model prescribes different probabilities for the presence of edges within a cluster and between different clusters. We assume that the cluster ass…

The split Bregman (SB) method [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323-43] is a fast splitting-based algorithm that solves image reconstruction problems with general l1, e.g., total-variation (TV) and compressed sensing (CS), regularizations by introducing a single variable split to decouple …

The nonnegative matrix factorization (NMF) is widely used in signal and image processing, including bio-informatics, blind source separation and hyperspectral image analysis in remote sensing. A great challenge arises when dealing with a nonlinear formulation of the NMF. Within the framework of kernel machines, the mod…

Regularization methods are often employed in deep learning neural networks (DNNs) to prevent overfitting. For penalty based DNN regularization methods, convex penalties are typically considered because of their optimization guarantees. Recent theoretical work have shown that nonconvex penalties that satisfy certain reg…

Error estimates found between SGD with momentum and Langevin diffusion.

New method relaxes TV distance for two-sample testing without distributional assumptions.

We focus on the maximum regularization parameter for anisotropic total-variation denoising. It corresponds to the minimum value of the regularization parameter above which the solution remains constant. While this value is well know for the Lasso, such a critical value has not been investigated in details for the total…

Paper tackles Byzantine attacks in distributed learning with a new ADMM method.

Hypergraphs allow one to encode higher-order relationships in data and are thus a very flexible modeling tool. Current learning methods are either based on approximations of the hypergraphs via graphs or on tensor methods which are only applicable under special conditions. In this paper, we present a new learning frame…

New bounds on neural network convergence using information theory.

Algorithm learns affine transformations robustly from corrupted samples.

Efficiently estimates binary product distributions with privacy.