arXiv subjects

Laya Ghodrati

Publications and source records attributed to Laya Ghodrati.

4 recordsLinked to original sources

arXiv2024

Transportation of Measure Regression in Higher Dimensions

We present an optimal transport framework for performing regression when both the covariate and the response are probability distributions on a compact Euclidean subset $Ω\subset\mathbb{R}^d$, where $d>1$. Extending beyond compactly supported distributions, this method also applies when both the predictor and responses are Gaussian distributions on $\mathbb{R}^d$. Our approach generalizes an existing transportation-based regression model to higher dimensions. This model postulates that the conditional Fréchet mean of the response distribution is linked to the covariate distribution via an optimal transport map. We establish an upper bound for the rate of convergence of a plug-in estimator. We propose an iterative algorithm for computing the estimator, which is based on DC (Difference of Convex Functions) Programming. In the Gaussian case, the estimator achieves a parametric rate of convergence, and the computation of the estimator simplifies to a finite-dimensional optimization over positive definite matrices, allowing for an efficient solution. The performance of the estimator is demonstrated in a simulation study.

math.ST↗

arXiv2023

On Distributional Autoregression and Iterated Transportation

We consider the problem of defining and fitting models of autoregressive time series of probability distributions on a compact interval of $\mathbb{R}$. An order-$1$ autoregressive model in this context is to be understood as a Markov chain, where one specifies a certain structure (regression) for the one-step conditional Fréchet mean with respect to a natural probability metric. We construct and explore different models based on iterated random function systems of optimal transport maps. While the properties and interpretation of these models depend on how they relate to the iterated transport system, they can all be analyzed theoretically in a unified way. We present such a theoretical analysis, including convergence rates, and illustrate our methodology using real and simulated data. Our approach generalises or extends certain existing models of transportation-based regression and autoregression, and in doing so also provides some additional insights on existing models.

stat.ME↗

arXiv2022

Minimax Rate for Optimal Transport Regression Between Distributions

Distribution-on-distribution regression considers the problem of formulating and estimating a regression relationship where both covariate and response are probability distributions. The optimal transport distributional regression model postulates that the conditional Fréchet mean of the response distribution is linked to the covariate distribution via an optimal transport map. We establish the minimax rate of estimation of such a regression function, by deriving a lower-bound that matches the convergence rate attained by the Fréchet least squares estimator.

math.ST↗

arXiv2021

Distribution-on-Distribution Regression via Optimal Transport Maps

We present a framework for performing regression when both covariate and response are probability distributions on a compact interval $Ω\subset\mathbb{R}$. Our regression model is based on the theory of optimal transportation and links the conditional Fréchet mean of the response distribution to the covariate distribution via an optimal transport map. We define a Fréchet-least-squares estimator of this regression map, and establish its consistency and rate of convergence to the true map, under both full and partial observation of the regression pairs. Computation of the estimator is shown to reduce to an isotonic regression problem, and thus our regression model can be implemented with ease. We illustrate our methodology using real and simulated data.

stat.ME↗