Random forests are a statistical learning method widely used in many areas of scientific research because of its ability to learn complex relationships between input and output variables and also its capacity to handle high-dimensional data. However, current random forest approaches are not flexible enough to handle heterogeneous data such as curves, images and shapes. In this paper, we introduce Fréchet trees and Fréchet random forests, which allow to handle data for which input and output variables take values in general metric spaces. To this end, a new way of splitting the nodes of trees is introduced and the prediction procedures of trees and forests are generalized. Then, random forests out-of-bag error and variable importance score are naturally adapted. A consistency theorem for Fréchet regressogram predictor using data-driven partitions is given and applied to Fréchet purely uniformly random trees. The method is studied through several simulation scenarios on heterogeneous data combining longitudinal, image and scalar data. Finally, one real data set about air quality is used to illustrate the use of the proposed method in practice.

We consider the problem of estimating a low-rank signal matrix from noisy measurements under the assumption that the distribution of the data matrix belongs to an exponential family. In this setting, we derive generalized Stein's unbiased risk estimation (SURE) formulas that hold for any spectral estimators which shrink or threshold the singular values of the data matrix. This leads to new data-driven spectral estimators, whose optimality is discussed using tools from random matrix theory and through numerical experiments. Under the spiked population model and in the asymptotic setting where the dimensions of the data matrix are let going to infinity, some theoretical properties of our approach are compared to recent results on asymptotically optimal shrinking rules for Gaussian noise. It also leads to new procedures for singular values shrinkage in finite-dimensional matrix denoising for Gamma- distributed and Poisson-distributed measurements.

Compressed sensing is a theory which guarantees the exact recovery of sparse signals from a small number of linear projections. The sampling schemes suggested by current compressed sensing theories are often of little practical relevance, since they cannot be implemented on real acquisition systems. In this paper, we study a new random sampling approach that consists of projecting the signal over blocks of sensing vectors. A typical example is the case of blocks made of horizontal lines in the 2-D Fourier plane. We provide the theoretical results on the number of blocks that are sufficient for exact sparse signal reconstruction. This number depends on two properties named intra- and inter-support block coherence. We then show that our bounds coincide with the best so far results in a series of examples, including Gaussian measurements or isolated measurements. We also show that the result is sharp when used with specific blocks in time-frequency bases, in the sense that the minimum required amount of blocks to reconstruct sparse signals cannot be improved up to a multiplicative logarithmic factor. The proposed results provide a good insight on the possibilities and limits of block compressed sensing in imaging devices, such as magnetic resonance imaging, radio-interferometry, or ultra-sound imaging.

Reducing acquisition time is of fundamental importance in various imaging modalities. Variable density sampling (VDS) provides an appealing framework for addressing this issue. It was justified recently from a theoretical point of view in the compressed sensing (CS) literature. Unfortunately, the sampling schemes suggested by current CS theories may not be relevant since they do not take the acquisition constraints (for example, continuity of the acquisition trajectory in magnetic resonance imaging (MRI)) into account. In this paper, we propose a numerical method to perform variable density sampling with block constraints. Our main contribution is a new way to draw the blocks in order to mimic CS strategies based on isolated measurements. The basic idea is to minimize a tailored dissimilarity measure between a probability distribution defined on the set of isolated measurements and a probability distribution defined on a set of blocks of measurements. This problem turns out to be convex and solvable in high dimension. Our second contribution is to define an efficient minimization algorithm based on Nesterov's accelerated gradient descent in metric spaces. We carefully study the choice of the metrics and of the prox-function. We show that the optimal choice may depend on the type of blocks under consideration. Finally, we show that we can obtain better MRI reconstruction results using our sampling schemes than with standard strategies such as equiangularly distributed radial lines.

We describe a method for analyzing the principal modes of geometric variability of images. For this purpose, we propose a general framework based on the use of deformation operators for modeling the geometric variability of images around a reference mean pattern. In this setting, we describe a simple algorithm for estimating the geometric variability of a set of images. Some numerical experiments on real data are proposed for highlighting the benefits of this approach. The consistency of this procedure is also analyzed in statistical deformable models.

This paper considers the problem of estimating a mean pattern in the setting of Grenander's pattern theory. Shape variability in a dataset of curves or images is modeled by the random action of elements in a compact Lie group on an infinite dimensional space. In the case of observations contaminated by an additive Gaussian white noise, it is shown that estimating a reference template in the setting of Grenanders pattern theory falls into the category of deconvolution problems over Lie groups. To obtain this result, we build an estimator of a mean pattern by using Fourier deconvolution and harmonic analysis on compact Lie groups. In an asymptotic setting where the number of observed curves or images tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Sobolev balls. This rate depends on the smoothness of the density of the random Lie group elements representing shape variability in the data, which makes a connection between estimating a mean pattern and standard deconvolution problems in nonparametric statistics.

In this paper we focus on extended Euclidean registration of a set of noisy images. We provide an appropriate statistical model for this kind of registration problem, and a new criterion based on Fourier-type transforms is proposed to estimate the translation, rotation, and scaling parameters to align a set of images. This criterion is a two-step procedure which does not require the use of a reference template onto which all the images are aligned. Our approach is based on M-estimation, and we prove the consistency of the resulting estimators. A small-scale simulation study and real examples are used to illustrate the numerical performances of our procedure.