Hadamard Matrices and Their Applications
A. S. Hedayat,W. D. Wallis +1 more
TLDR
Hadamard matrices have been widely studied in the literature and many of their applications can be found in this paper, e.g., incomplete block designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III (SRSIII), optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects.Abstract:
An $n \times n$ matrix $H$ with all its entries $+1$ and $-1$ is Hadamard if $HH' = nI$. It is well known that $n$ must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every $n$ which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the last three decades. These matrices can be transformed to produce incomplete block designs, $t$-designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III designs, optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects. In this paper we survey the existence of Hadamard matrices and many of their applications.read more
Citations
More filters
Posted Content
Hadamard Response: Estimating Distributions Privately, Efficiently, and with Little Communication
TL;DR: Hadamard Response (HR) as mentioned in this paper is a local privatization scheme that requires no shared randomness and is symmetric with respect to the users, and has order optimal sample complexity for all ε = O(n + 2 ) bits per user.
Journal ArticleDOI
Some characterizations of affinely full-dimensional factorial designs
Satoshi Aoki,Akimichi Takemura +1 more
TL;DR: In this article, a new class of two-level non-regular fractional factorial designs is defined, called affinely full-dimensional factorial design, and the properties of this class are investigated from the viewpoint of D -optimality.
Posted Content
Statistical and Algorithmic Perspectives on Randomized Sketching for Ordinary Least-Squares -- ICML
TL;DR: In this article, the authors consider the statistical and algorithmic aspects of solving large-scale least-squares (LS) problems using randomized sketching algorithms, and provide a rigorous comparison of both perspectives leading to insights on how they differ.
Posted Content
SPRIGHT: A Fast and Robust Framework for Sparse Walsh-Hadamard Transform
TL;DR: It is shown that there is no extra price that needs to be paid for being robust to noise other than a constant factor, and the same sample complexity and computational complexity can be maintained as those of the noiseless case using the SParse Robust Iterative Graph-based Hadamard Transform (SPRIGHT) algorithm.
Posted Content
Stochastic Newton and Quasi-Newton Methods for Large Linear Least-squares Problems.
TL;DR: Stochastic Newton iterates and stochastic quasi-Newton iterates are described to efficiently solve large linear least-squares problems where the very large data sets present a significant computational burden (e.g., the size may exceed computer memory or data are collected in real-time).