Related papers: The Fast Johnson-Lindenstrauss Transform is Even F…
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…
The Johnson-Lindenstrauss (JL) lemma is a fundamental result in dimensionality reduction, ensuring that any finite set $X \subseteq \mathbb{R}^d$ can be embedded into a lower-dimensional space $\mathbb{R}^k$ while approximately preserving…
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 $f(X)$…
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…
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…
The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $\mathbb{R}^d$ into $m=O(\varepsilon^{-2} \lg n)$ dimensions while preserving all pairwise…
In this work, we analyze dimension reduction algorithms based on the Kac walk and discrete variants. (1) For $n$ points in $\mathbb{R}^{d}$, we design an optimal Johnson-Lindenstrauss (JL) transform based on the Kac walk which can be…
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…
The Johnson-Lindenstrauss (JL) Lemma introduced the concept of dimension reduction via a random linear map, which has become a fundamental technique in many computational settings. For a set of $n$ points in $\mathbb{R}^d$ and any fixed…
The Johnson-Lindenstrauss (JL) lemma is a cornerstone of dimensionality reduction in Euclidean space, but its applicability to non-Euclidean data has remained limited. This paper extends the JL lemma beyond Euclidean geometry to handle…
The \emph{Sparse Johnson-Lindenstrauss Transform} of Kane and Nelson (SODA 2012) provides a linear dimensionality-reducing map $A \in \mathbb{R}^{m \times u}$ in $\ell_2$ that preserves distances up to distortion of $1 + \varepsilon$ with…
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 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…
We show an analog to the Fast Johnson-Lindenstrauss Transform for Nearest Neighbor Preserving Embeddings in $\ell_2$. These are sparse, randomized embeddings that preserve the (approximate) nearest neighbors. The dimensionality of the…
Dimension reduction is a key algorithmic tool with many applications including nearest-neighbor search, compressed sensing and linear algebra in the streaming model. In this work we obtain a {\em sparse} version of the fundamental tool in…
For any $n>1$ and $0<\varepsilon<1/2$, we show the existence of an $n^{O(1)}$-point subset $X$ of $\mathbb{R}^n$ such that any linear map from $(X,\ell_2)$ to $\ell_2^m$ with distortion at most $1+\varepsilon$ must have $m = \Omega(\min\{n,…
The Johnson-Lindenstrauss Lemma allows for the projection of $n$ points in $p-$dimensional Euclidean space onto a $k-$dimensional Euclidean space, with $k \ge \frac{24\ln \emph{n}}{3\epsilon^2-2\epsilon^3}$, so that the pairwise distances…
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 Kronecker product is an important matrix operation with a wide range of applications in supporting fast linear transforms, including signal processing, graph theory, quantum computing and deep learning. In this work, we introduce a…
The problems of random projections and sparse reconstruction have much in common and individually received much attention. Surprisingly, until now they progressed in parallel and remained mostly separate. Here, we employ new tools from…