Related papers: Guarantees for the Kronecker Fast Johnson-Lindenst…
The seminal result of Johnson and Lindenstrauss on random embeddings has been intensively studied in applied and theoretical computer science. Despite that vast body of literature, we still lack of complete understanding of statistical…
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…
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…
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…
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 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…
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…
We study the effect of Johnson-Lindenstrauss transforms in various projective clustering problems, generalizing recent results which only applied to center-based clustering [MMR19]. We ask the general question: for a Euclidean optimization…
The paper re-analyzes a version of the celebrated Johnson-Lindenstrauss Lemma, in which matrices are subjected to constraints that naturally emerge from neuroscience applications: a) sparsity and b) sign-consistency. This particular variant…
Recent work of [Dasgupta-Kumar-Sarlos, STOC 2010] gave a sparse Johnson-Lindenstrauss transform and left as a main open question whether their construction could be efficiently derandomized. We answer their question affirmatively by giving…
A refinement of so-called fast Johnson-Lindenstrauss transform, due to Ailon and Chazelle (2006), and Matou\v{s}ek (2008), is proposed. While it preserves the time efficiency and simplicity of implementation of the original construction, it…
We derive a Kronecker product approximation for the micromagnetic long range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this operator for structured tensors (Canonical format, Tucker format,…
This paper presents a new method for estimating high dimensional covariance matrices. The method, permuted rank-penalized least-squares (PRLS), is based on a Kronecker product series expansion of the true covariance matrix. Assuming an…
Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such…
The celebrated Johnson-Lindenstrauss lemma states that for all $\varepsilon \in (0,1)$ and finite sets $X \subseteq \mathbb{R}^N$ with $n>1$ elements, there exists a matrix $\Phi \in \mathbb{R}^{m \times N}$ with…
We present a new paradigm for speeding up randomized computations of several frequently used functions in machine learning. In particular, our paradigm can be applied for improving computations of kernels based on random embeddings. Above…
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 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 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 examine a class of embeddings based on structured random matrices with orthogonal rows which can be applied in many machine learning applications including dimensionality reduction and kernel approximation. For both the…