Search arXivSearch

arXiv · math/0701092

A Conversation with Ulf Grenander

Abstract

Ulf Grenander was born in Vastervik, Sweden, on July 23, 1923. He started his undergraduate education at Uppsala University, and earned his B.A. degree in 1946, the Fil. Lic. degree in 1948 and the Fil. Dr. degree in 1950, all from the University of Stockholm. His Ph.D. thesis advisor was Harald Cramér. Professor Grenander is well known for pathbreaking research in a number of areas including pattern theory, computer vision, inference in stochastic processes, probabilities on algebraic structures and actuarial mathematics. He has published more than one dozen influential books, of which Statistical Analysis of Stationary Time Series (1957, coauthored with M. Rosenblatt), Probabilities on Algebraic Structures (1963; also in Russian) and Abstract Inference (1981b) are regarded as classics. His three-volume lecture notes, namely, Pattern Synthesis (vol. I, 1976), Pattern Analysis (vol. II, 1978) and Regular Structures (vol. III, 1981a; also in Russian) created and nurtured a brand new area of research. During 1951--1966, Professor Grenander's career path took him to the University of Chicago (1951--1952), the University of California--Berkeley (1952--1953), the University of Stockholm (1953--1957), Brown University (1957--1958) and the Institute for Insurance Mathematics and Mathematical Statistics (1958--1966) as its Professor and Director. From 1966 until his retirement he was L. Herbert Ballou University Professor at Brown University. Professor Grenander also held the position of Scientific Director (1971--1973) of the Swedish Institute of Applied Mathematics. He has earned many honors and awards, including Arhennius Fellow (1948), Fellow of the Institute of Mathematical Statistics (1953), Prize of the Nordic Actuaries (1961), Arnberger Prize of the Royal Swedish Academy of Science (1962), Member of the Royal Swedish Academy of Science (1965), Guggenheim Fellowship (1979) and Honorary Fellow of the Royal Statistical Society, London (1989). He has delivered numerous prestigious lectures, including the Rietz Lecture (1985), the Wald Lectures (1995) and the Mahalanobis Lecture (2004). Professor Grenander received an Honorary D.Sc. degree (1993) from the University of Chicago and is a Fellow of both the American Academy of Arts and Sciences (1995) and the National Academy of Sciences, U.S.A. (1998). Professor Grenander's career, life, passion and hobbies can all be summarized by one simple word: Mathematics.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nitis Mukhopadhyay. 2007-01-03. A Conversation with Ulf Grenander. https://doi.org/10.1214/088342305000000313

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Sample complexity and weak limits of nonsmooth multimarginal Schrödinger system with application to optimal transport barycenter

Multimarginal optimal transport (MOT) has emerged as a useful framework for many applied problems. However, compared to the well-studied classical two-marginal optimal transport theory, analysis of MOT is far more challenging and remains much less developed. In this paper, we study the statistical estimation and inference problems for the entropic MOT (EMOT), whose optimal solution is characterized by the multimarginal Schrödinger system. Assuming only boundedness of the cost function, we derive sharp sample complexity for estimating several key quantities pertaining to EMOT (cost functional and Schrödinger coupling) from point clouds that are randomly sampled from the input marginal distributions. Moreover, with substantially weaker smoothness assumption on the cost function than the existing literature, we derive distributional limits and bootstrap validity of various key EMOT objects. As an application, we propose the multimarginal Schrödinger barycenter as a new and natural way to regularize the exact Wasserstein barycenter and demonstrate its statistical optimality.

math.ST

Nonparametric spectral density estimation using interactive mechanisms under local differential privacy

We study the problem of estimating the spectral density of a centered stationary Gaussian time series under local differential privacy constraints. Specifically, we propose new interactive privacy mechanisms for three tasks: recovering a single covariance coefficient, recovering the spectral density at a fixed frequency, and global recovery. Our approach achieves faster rates through a two-stage process: we first apply the Laplace mechanism to the truncated value, and then use the resulting privatized sample to learn about the dependence mechanism in the time series. For spectral densities belonging to Hölder and Sobolev smoothness classes, we demonstrate that our algorithms improve upon the non-interactive mechanism of Kroll (2024) for small privacy parameter $α$, since the pointwise rates depend on $nα^2$ instead of $nα^4$. Moreover, we show that the rate $(nα^4)^{-1}$ is optimal for estimating a covariance coefficient with non-interactive mechanisms. However, the $L_2$ rate of our interactive estimator is slower than the pointwise rate. We show how to use these procedures to provide a bona fide locally differentially private estimator of the entire covariance matrix. A simulation study validates our findings.

math.ST

Estimating eigenvectors and eigenspaces of covariance matrices: Optimal Bounds and Conditions for Consistency

Let $X = [ ξ_1, \,\, ξ_2,...\,\, ,ξ_d]^\top$ be a zero-mean random vector of large dimension $d$ ($d \rightarrow \infty$) with (hidden) covariance matrix $M = (m_{ij})_{1 \leq i, j \leq d},$ where $m_{ij} = m_{ji} = \textbf{Cov}(ξ_i, ξ_j).$ Let $X_1, X_2, \dots, X_n$ be $n$ iid samples of $X$. Consider the sample covariance matrix $$\textstyle \tilde{M} := \frac{1}{n} \sum_{i=1}^{n} X_i X_i^\top.$$ In practice, one frequently uses the eigenvectors and eigenspaces of $\tilde M$ as estimators for those of $M$. A central task is to provide an error analysis for these estimators. In this paper, we provide an optimal error analysis, obtaining upper and lower bounds of matching order of magnitude, for a wide range of parameters $d$ and $n$, under mild assumptions on $M$. As corollaries, we obtain new necessary and sufficient conditions for the consistency of the estimators. In these conditions, we only require the number of samples $n$ to depend linearly on the effective rank of $M$, which can be much smaller than the dimension $d$.

math.ST