Série géométrique | EPFL Graph Search

thumb|Preuve sans mots de l'égalité1/2 + 1/4 + 1/8 + 1/16 + ⋯ = 1 thumb|Illustration de l'égalité 1/4 + 1/16 + 1/64 + 1/256 + ⋯ = 1/3 :chacun des carrés violets mesure 1/4 de la surface du grand carré le plus proche (1/2× = 1/4, 1/4×1/4 = 1/16, etc.). Par ailleurs, la somme des aires des carrés violets est égale à un tiers de la superficie du grand carré. En mathématiques, la série géométrique est l'un des exemples de série numérique les plus simples. C'est la série associée à une suite géométrique, c'est-à-dire la suite des sommes partielles des termes de cette suite. Intuitivement, une série géométrique est une série avec un ratio constant des termes successifs. Par exemple, la série :\frac12+\frac14+\frac18+\frac1{16}+\cdots est géométrique, parce que chaque terme est le produit du précédent par 1/2. Elle admet, dans les algèbres de Banach, une généralisation qui permet d'étudier les variations de l'inverse d'un élément. Définition dans le cor

Resolution-robust Large Mask In painting with Fourier Convolutions

Anastasia Remizova

Modern image inpainting systems, despite the significant progress, often struggle with large missing areas, complex geometric structures, and high-resolution images. We find that one of the main reasons for that is the lack of an effective receptive field in both the inpainting network and the loss function. To alleviate this issue, we propose a new method called large mask inpainting (LaMa). LaMa is based on i) a new inpainting network architecture that uses fast Fourier convolutions (FFCs), which have the imagewide receptive field; ii) a high receptive field perceptual loss; iii) large training masks, which unlocks the potential of the first two components. Our inpainting network improves the state-of-the-art across a range of datasets and achieves excellent performance even in challenging scenarios, e.g. completion of periodic structures. Our model generalizes surprisingly well to resolutions that are higher than those seen at train time, and achieves this at lower parameter&time costs than the competitive baselines. The code is available at https://github.com/saic-mdal/lama.

IEEE COMPUTER SOC2022

Frequency domain theory for functional time series: Variance decomposition and an invariance principle

Neda Mohammadi Jouzdani

This paper is concerned with frequency domain theory for functional time series, which are temporally dependent sequences of functions in a Hilbert space. We consider a variance decomposition, which is more suitable for such a data structure than the variance decomposition based on the Karhunen-Loeve expansion. The decomposition we study uses eigenvalues of spectral density operators, which are functional analogs of the spectral density of a stationary scalar time series. We propose estimators of the variance components and derive convergence rates for their mean square error as well as their asymptotic normality. The latter is derived from a frequency domain invariance principle for the estimators of the spectral density operators. This principle is established for a broad class of linear time series models. It is a main contribution of the paper.

INT STATISTICAL INST2020

Procrustes Metrics and Optimal Transport for Covariance Operators

Valentina Masarotto

Covariance operators play a fundamental role in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on the space of such operators, as well as with their intrinsic infinite-dimensionality. We will show that we can identify the space of infinite-dimensional covariance operators equipped with the Procrustes size-and-shape metric from shape theory, with that of centred Gaussian processes, equipped with the Wasserstein metric of optimal transportation. We then describe key geometrical and topological aspects of the space of covariance operators endowed with the Procrustes metric. Through the notion of multicoupling of Gaussian measures, we establish existence, uniqueness and stability for the Fréchet mean of covariance operators with respect to the Procrustes metric. Furthermore, we will provide generative models that are canonical for such metric. We then turn to the problem of comparing several samples of stochastic processes with respect to their second-order structure, and we subsequently describe the main modes of variation in this second order structure. These two tasks are carried out via an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of covariance operators respectively. In order to perform ANOVA, we introduce a novel approach based on optimal (multi)transport and identify each covariance with an optimal transport map. These maps are then contrasted with the identity with respect to a norm-induced distance. The resulting test statistic, calibrated by permutation, outperforms the state-of-the-art in the functional case. If the null hypothesis postulating equality of the operators is rejected, thanks to a geometric interpretation of the transport maps we can construct a PCA on the tangent space with the aim of understanding the sample variability. Finally, we provide a further example of use of the optimal transport framework, by applying it to the problem of clustering of operators. Two different clustering algorithms are presented, one of which is innovative. The transportation ANOVA, PCA and clustering are validated both on simulated scenarios and real dataset.

EPFL2019

Procrustes Metrics and Optimal Transport for Covariance Operators

Valentina Masarotto

EPFL2019