Yoav Zemel
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View article: Transportation-based functional ANOVA and PCA for covariance operators
Transportation-based functional ANOVA and PCA for covariance operators Open
We consider the problem of comparing several samples of stochastic processes with respect to their second-order structure, and describing the main modes of variation in this second order structure, if present. These tasks can be seen as an…
View article: Non-injectivity of Bures--Wasserstein barycentres in infinite dimensions
Non-injectivity of Bures--Wasserstein barycentres in infinite dimensions Open
We construct a counterexample to the injectivity conjecture of Masarotto et al (2018). Namely, we construct a class of examples of injective covariance operators on an infinite-dimensional separable Hilbert space for which the Bures--Wasse…
View article: Transportation-Based Functional ANOVA and PCA for Covariance Operators
Transportation-Based Functional ANOVA and PCA for Covariance Operators Open
We consider the problem of comparing several samples of stochastic processes with respect to their second-order structure, and describing the main modes of variation in this second order structure, if present. These tasks can be seen as an…
View article: Randomized Wasserstein Barycenter Computation: Resampling with Statistical Guarantees
Randomized Wasserstein Barycenter Computation: Resampling with Statistical Guarantees Open
We propose a hybrid resampling method to approximate finitely supported Wasserstein barycenters on large-scale datasets, which can be combined with any exact solver. Nonasymptotic bounds on the expected error of the objective value as well…
View article: Bayesian semiparametric modelling of phase-varying point processes
Bayesian semiparametric modelling of phase-varying point processes Open
We propose a Bayesian semiparametric approach for registration of multiple point processes. Our approach entails modelling the mean measures of the phase-varying point processes with a Bernstein–Dirichlet prior, which induces a prior on th…
View article: Randomised Wasserstein Barycenter Computation: Resampling with Statistical Guarantees
Randomised Wasserstein Barycenter Computation: Resampling with Statistical Guarantees Open
We propose a hybrid resampling method to approximate finitely supported Wasserstein barycenters on large-scale datasets, which can be combined with any exact solver. Nonasymptotic bounds on the expected error of the objective value as well…
View article: Limit Laws for Empirical Optimal Solutions in Stochastic Linear Programs
Limit Laws for Empirical Optimal Solutions in Stochastic Linear Programs Open
We consider a general linear program in standard form whose right-hand side constraint vector is subject to random perturbations. This defines a stochastic linear program for which, under general conditions, we characterize the fluctuation…
View article: Optimal Transport
Optimal Transport Open
In this preliminary chapter, we introduce the problem of optimal transport, which is the main concept behind Wasserstein spaces. General references on this topic are the books by Rachev and Rüschendorf [107], Villani [124, 125], Ambrosio e…
View article: Fréchet Means in the Wasserstein Space $$\mathcal W_2$$
Fréchet Means in the Wasserstein Space $$\mathcal W_2$$ Open
If H is a Hilbert space (or a closed convex subspace thereof) and x 1, …, x N ∈ H, then the empirical mean $$\overline x_N=N^{-1}\sum x_i$$ is the unique element of H that minimises the sum of squared distances from the x i's.
View article: An Invitation to Statistics in Wasserstein Space
An Invitation to Statistics in Wasserstein Space Open
This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspect…
View article: Construction of Fréchet Means and Multicouplings
Construction of Fréchet Means and Multicouplings Open
When given measures μ 1, …, μ N are supported on the real line, computing their Fréchet mean $$\bar \mu $$ is straightforward (Sect. 3.1.4 ). This is in contrast to the multivariate case, where, apart from the important yet special case of…
View article: The Wasserstein Space
The Wasserstein Space Open
The Kantorovich problem described in the previous chapter gives rise to a metric structure, the Wasserstein distance, in the space of probability measures $$P(\mathcal X)$$ on a space $$\mathcal X$$ . The resulting metric space, a subspace…
View article: Phase Variation and Fréchet Means
Phase Variation and Fréchet Means Open
Why is it relevant to construct the Fréchet mean of a collection of measures with respect to the Wasserstein metric? A simple answer is that this kind of average will often express a more natural notion of "typical" realisation of a random…
View article: Fréchet means and Procrustes analysis in Wasserstein space
Fréchet means and Procrustes analysis in Wasserstein space Open
We consider two statistical problems at the intersection of functional and non-Euclidean data analysis: the determination of a Fréchet mean in the Wasserstein space of multivariate distributions; and the optimal registration of deformed ra…
View article: Bayesian semiparametric modelling of phase-varying point processes
Bayesian semiparametric modelling of phase-varying point processes Open
We propose a Bayesian semiparametric approach for registration of multiple point processes. Our approach entails modelling the mean measures of the phase-varying point processes with a Bernstein-Dirichlet prior, which induces a prior on th…
View article: Statistical Aspects of Wasserstein Distances
Statistical Aspects of Wasserstein Distances Open
Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in or…
View article: Optimal Transport: Fast Probabilistic Approximation with Exact Solvers
Optimal Transport: Fast Probabilistic Approximation with Exact Solvers Open
We propose a simple subsampling scheme for fast randomized approximate computation of optimal transport distances. This scheme operates on a random subset of the full data and can use any exact algorithm as a black-box back-end, including …
View article: Fréchet means in Wasserstein space : theory and algorithms
Fréchet means in Wasserstein space : theory and algorithms Open
This work studies the problem of statistical inference for Fréchet means in the Wasserstein space of measures on Euclidean spaces, $\mathcal W_2 ( \mathbb R^d )$. This question arises naturally from the problem of separating amplitude and …
View article: A unified measure of linear and nonlinear selection on quantitative traits
A unified measure of linear and nonlinear selection on quantitative traits Open
Summary Lande and Arnold's approach to quantifying natural selection has become a standard tool in evolutionary biology due to its simplicity and generality. It treats linear and nonlinear selection in two separate frameworks, generating c…
View article: Amplitude and phase variation of point processes
Amplitude and phase variation of point processes Open
We develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping, known as the problem of separation of amplitude and phase variation. The amplitude variation of a real random f…