Related papers: New bounds for circulant Johnson-Lindenstrauss emb…
We prove a variant of a Johnson-Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimise the randomness used, is easy to implement and provides good running times. The price to be paid is the higher…
The Johnson-Lindenstrauss (JL) lemma allows subsets of a high-dimensional space to be embedded into a lower-dimensional space while approximately preserving all pairwise Euclidean distances. This important result has inspired an extensive…
Consider an m by N matrix Phi with the Restricted Isometry Property of order k and level delta, that is, the norm of any k-sparse vector in R^N is preserved to within a multiplicative factor of 1 +- delta under application of Phi. We show…
We continue our study of the Johnson-Lindenstrauss lemma and its connection to circulant matrices started in \cite{HV}. We reduce the bound on $k$ from $k=O(\epsilon^{-2}\log^3n)$ proven there to $k=O(\epsilon^{-2}\log^2n)$. Our technique…
This work constructs Jonson-Lindenstrauss embeddings with best accuracy, as measured by variance, mean-squared error and exponential concentration of the length distortion. Lower bounds for any data and embedding dimensions are determined,…
The Johnson-Lindenstrauss lemma is a fundamental result in probability with several applications in the design and analysis of algorithms in high dimensional geometry. Most known constructions of linear embeddings that satisfy the…
The Johnson-Lindenstrauss lemma is one of the corner stone results in dimensionality reduction. It says that given $N$, for any set of $N$ vectors $X \subset \mathbb{R}^n$, there exists a mapping $f : X \to \mathbb{R}^m$ such that $f(X)$…
In this paper new general modewise Johnson-Lindenstrauss (JL) subspace embeddings are proposed that are both considerably faster to generate and easier to store than traditional JL embeddings when working with extremely large vectors and/or…
In the recent paper [Jin, Kolda & Ward, arXiv:1909.04801], it is proved that the Kronecker fast Johnson-Lindenstrauss transform (KFJLT) is, in fact, a Johnson-Lindenstrauss transform, which had previously only been conjectured. In this…
Let $\mathcal{M}$ be a smooth $d$-dimensional submanifold of $\mathbb{R}^N$ with boundary that's equipped with the Euclidean (chordal) metric, and choose $m \leq N$. In this paper we consider the probability that a random matrix $A \in…
Embeddings play a pivotal role across various disciplines, offering compact representations of complex data structures. Randomized methods like Johnson-Lindenstrauss (JL) provide state-of-the-art and essentially unimprovable theoretical…
The Johnson-Lindenstrauss (JL) theorem states that a set of points in high-dimensional space can be embedded into a lower-dimensional space while approximately preserving pairwise distances with high probability Johnson and Lindenstrauss…
This paper investigates theoretical properties of subsampling and hashing as tools for approximate Euclidean norm-preserving embeddings for vectors with (unknown) additive Gaussian noises. Such embeddings are sometimes called…
The Johnson--Lindenstrauss (JL) lemma is a powerful tool for dimensionality reduction in modern algorithm design. The lemma states that any set of high-dimensional points in a Euclidean space can be flattened to lower dimensions while…
The Johnson-Lindenstrauss transform allows one to embed a dataset of $n$ points in $\mathbb{R}^d$ into $\mathbb{R}^m,$ while preserving the pairwise distance between any pair of points up to a factor $(1 \pm \varepsilon)$, provided that $m…
Uniform bounds on sketched inner products of vectors or matrices underpin several important computational and statistical results in machine learning and randomized algorithms, including the Johnson-Lindenstrauss (J-L) lemma, the Restricted…
In this paper, we study a fast approximation method for {\it large-scale high-dimensional} sparse least-squares regression problem by exploiting the Johnson-Lindenstrauss (JL) transforms, which embed a set of high-dimensional vectors into a…
It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance.…
The seminal Fast Johnson-Lindenstrauss (Fast JL) transform by Ailon and Chazelle (SICOMP'09) embeds a set of $n$ points in $d$-dimensional Euclidean space into optimal $k=O(\varepsilon^{-2} \ln n)$ dimensions, while preserving all pairwise…
Johnson-Lindenstrauss embeddings are widely used to reduce the dimension and thus the processing time of data. To reduce the total complexity, also fast algorithms for applying these embeddings are necessary. To date, such fast algorithms…