相关论文: A Sparse Johnson-Lindenstrauss Transform using Fas…
For a family of interpolation norms $\| \cdot \|_{1,2,s}$ on $\mathbb{R}^n$, we provide a distribution over random matrices $\Phi_s \in \mathbb{R}^{m \times n}$ parametrized by sparsity level $s$ such that for a fixed set $X$ of $K$ points…
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…
We give near-tight lower bounds for the sparsity required in several dimensionality reducing linear maps. First, consider the JL lemma which states that for any set of n vectors in R there is a matrix A in R^{m x d} with m = O(eps^{-2}log…
The Johnson-Lindenstrauss property ({\sf JLP}) of random matrices has immense application in computer science ranging from compressed sensing, learning theory, numerical linear algebra, to privacy. This paper explores the properties and…
Feature hashing and other random projection schemes are commonly used to reduce the dimensionality of feature vectors. The goal is to efficiently project a high-dimensional feature vector living in $\mathbb{R}^n$ into a much…
For a set $X$ of $N$ points in $\mathbb{R}^D$, the Johnson-Lindenstrauss lemma provides random linear maps that approximately preserve all pairwise distances in $X$ -- up to multiplicative error $(1\pm \epsilon)$ with high probability --…
The famous Johnson-Lindenstrauss lemma states that for any set of n vectors, there is a linear transformation into a space of dimension O(log n) that approximately preserves all their lengths. In fact, a Haar random unitary transformation…
Allen-Zhu, Gelashvili, Micali, and Shavit construct a sparse, sign-consistent Johnson-Lindenstrauss distribution, and prove that this distribution yields an essentially optimal dimension for the correct choice of sparsity. However, their…
We present a simplified and unified analysis of the Johnson-Lindenstrauss (JL) lemma, a cornerstone of dimensionality reduction for managing high-dimensional data. Our approach simplifies understanding and unifies various constructions…
The Johnson-Lindenstrauss transform is a fundamental method for dimension reduction in Euclidean spaces, that can map any dataset of $n$ points into dimension $O(\log n)$ with low distortion of their distances. This dimension bound is tight…
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…
A classical result of Johnson and Lindenstrauss states that a set of $n$ high dimensional data points can be projected down to $O(\log n/\epsilon^2)$ dimensions such that the square of their pairwise distances is preserved up to a small…
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…
We present a novel algorithm, named the 2D-FFAST, to compute a sparse 2D-Discrete Fourier Transform (2D-DFT) featuring both low sample complexity and low computational complexity. The proposed algorithm is based on mixed concepts from…
A new iterative low complexity algorithm has been presented for computing the Walsh-Hadamard transform (WHT) of an $N$ dimensional signal with a $K$-sparse WHT, where $N$ is a power of two and $K = O(N^\alpha)$, scales sub-linearly in $N$…
In this paper we make a novel use of the Johnson-Lindenstrauss Lemma. The Lemma has an existential form saying that there exists a JL transformation $f$ of the data points into lower dimensional space such that all of them fall into…
Locality-sensitive hashing (LSH) is an effective randomized technique widely used in many machine learning tasks. The cost of hashing is proportional to data dimensions, and thus often the performance bottleneck when dimensionality is high…
Johnson and Lindenstrauss (Contemporary Mathematics, 1984) showed that for $n > m$, a scaled random projection $\mathbf{A}$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ is an approximate isometry on any set $S$ of size at most exponential in $m$.…
We present a theory for Euclidean dimensionality reduction with subgaussian matrices which unifies several restricted isometry property and Johnson-Lindenstrauss type results obtained earlier for specific data sets. In particular, we…
Can we sense our location in an unfamiliar environment by taking a sublinear-size sample of our surroundings? Can we efficiently encrypt a message that only someone physically close to us can decrypt? To solve this kind of problems, we…