arXiv research

Unified analysis simplifies Johnson-Lindenstrauss lemma for data reduction.

This paper surveys various methods for dimensionality reduction and nearest neighbor search.

A new method for private query release using Johnson-Lindenstrauss projection.

We develop a probabilistic framework for sequential random projection.

The Johnson-Lindenstrauss Lemma allows for the projection of $n$ points in $p-$dimensional Euclidean space onto a $k-$dimensional Euclidean space, with $k \ge \frac{24\ln \emph{n}}{3ε^2-2ε^3}$, so that the pairwise distances are preserved within a factor of $1\pmε$. Here, working directly with the distributions of the …

The paper examines how well node similarities are preserved by random projections in graph embeddings.

Let $Φ\in\mathbb{R}^{m\times n}$ be a sparse Johnson-Lindenstrauss transform [KN14] with $s$ non-zeroes per column. For a subset $T$ of the unit sphere, $\varepsilon\in(0,1/2)$ given, we study settings for $m,s$ required to ensure $\mathop{\mathbb{E}}_Φ\sup_{x\in T} \left|\|Φx\|_2^2 - 1 \right| < \varepsilon ,$ i.e…

Unified bounds for sketched bilinear forms in machine learning and statistics.

Derandomization reveals structure in neural networks, reducing sample complexity.

Aggregates predictions from multiple regression models using random projections and kernel methods.

We develop a novel methodology based on the marriage between the Bhattacharyya distance, a measure of similarity across distributions of random variables, and the Johnson-Lindenstrauss Lemma, a technique for dimension reduction. The resulting technique is a simple yet powerful tool that allows comparisons between data-…

We introduce sparse random projection, an important dimension-reduction tool from machine learning, for the estimation of discrete-choice models with high-dimensional choice sets. Initially, high-dimensional data are compressed into a lower-dimensional Euclidean space using random projections. Subsequently, estimation …

Let $\varepsilon\in(0,1)$ and $X\subset\mathbb R^d$ be arbitrary with $|X|$ having size $n>1$. The Johnson-Lindenstrauss lemma states there exists $f:X\rightarrow\mathbb R^m$ with $m = O(\varepsilon^{-2}\log n)$ such that $\forall x\in X\ \forall y\in X, \|x-y\|_2 \le \|f(x)-f(y)\|_2 \le (1+\varepsilon)\|x-y\|_2 .$…

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 recover and, in several cases, improve results for sets of sparse and structured spars…

Optimization can learn Johnson-Lindenstrauss embeddings without randomization.

In this paper we show that the computational complexity of the Iterative Thresholding and K-residual-Means (ITKrM) algorithm for dictionary learning can be significantly reduced by using dimensionality-reduction techniques based on the Johnson-Lindenstrauss lemma. The dimensionality reduction is efficiently carried out…

The main contribution of the paper is to show that Gaussian sketching of a kernel-Gram matrix $\boldsymbol K$ yields an operator whose counterpart in an RKHS $\mathcal H$, is a \emph{random projection} operator---in the spirit of Johnson-Lindenstrauss (J-L) lemma. To be precise, given a random matrix $Z$ with i.i.d. Ga…

Several important families of computational and statistical results in machine learning and randomized algorithms rely on uniform bounds on quadratic forms of random vectors or matrices. Such results include the Johnson-Lindenstrauss (J-L) Lemma, the Restricted Isometry Property (RIP), randomized sketching algorithms, …

Novel privatization framework for high-dimensional variable selection with differential privacy.

Random projections help in representing sparse graphs efficiently.

Principal Components Regression (PCR) is a traditional tool for dimension reduction in linear regression that has been both criticized and defended. One concern about PCR is that obtaining the leading principal components tends to be computationally demanding for large data sets. While random projections do not possess…

From a fresh data science perspective, this thesis discusses the prediction of coronary artery disease based on genetic variations at the DNA base pair level, called Single-Nucleotide Polymorphisms (SNPs), collected from the Ontario Heart Genomics Study (OHGS). First, the thesis explains two commonly used supervised le…

The paper studies how norms of random vectors are preserved by random projections.

Tensorized Rademacher projections outperform Gaussian projections in reducing tensor dimensions.

Explains the Schwarz lemma in lecture notes.

HD-BWDM improves clustering validation in high-dimensional data.

We consider the problem of efficient randomized dimensionality reduction with norm-preservation guarantees. Specifically we prove data-dependent Johnson-Lindenstrauss-type geometry preservation guarantees for Ho's random subspace method: When data satisfy a mild regularity condition -- the extent of which can be estima…

Author provides an alternate proof of the free ribbon lemma.

The paper extends Schwarz's lemma to RC-positivity and complex manifolds.

Survey on strong closing lemmas in Hamiltonian dynamics.

Unified Schwarz lemma in Kähler and Hermitian geometry.

Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.

Formulates Index III lemma and Rauch III theorem with applications.

New Schwarz Lemma for Bergman metrics in bounded domains.

Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rel…

Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.

The paper improves Zakalyukin's lemma for frontals and applies it to surface singularities.

Meridian lemma extended to fully alternating links in thickened surfaces.

Proves a quantitative closing lemma for negatively curved manifolds.

Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.

Extends Margulis Lemma to RCD(K,N) spaces.

New method for computing terminal embeddings in sublinear time.

This work improves tensor decomposition methods, especially for large datasets.

For a symplectic manifold $(M,ω)$, not necessarily hard Lefschetz, we prove a version of the Merkulov $dδ$--lemma. We also study the $dδ$--lemma and related cohomologies for compact symplectic solvmanifolds.

The paper characterizes when the $\partial \overline{\partial}$-lemma holds for twistor spaces.

Positive representations on surfaces have positive cross-ratios and satisfy a collar lemma.

Proves a generalized Whitehead cut vertex lemma for tree groups.

Proves a general ∂∂̄-lemma and applies it to a Fujino conjecture.