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- W4299168148 abstract "Let $Phiinmathbb{R}^{mtimes n}$ be a sparse Johnson-Lindenstrauss transform [KN14] with $s$ non-zeroes per column. For a subset $T$ of the unit sphere, $varepsilonin(0,1/2)$ given, we study settings for $m,s$ required to ensure $$ mathop{mathbb{E}}_Phi sup_{xin T} left||Phi x|_2^2 - 1 right| < varepsilon , $$ i.e. so that $Phi$ preserves the norm of every $xin T$ simultaneously and multiplicatively up to $1+varepsilon$. We introduce a new complexity parameter, which depends on the geometry of $T$, and show that it suffices to choose $s$ and $m$ such that this parameter is small. Our result is a sparse analog of Gordon's theorem, which was concerned with a dense $Phi$ having i.i.d. Gaussian entries. We qualitatively unify several results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and Fourier-based restricted isometries. Our work also implies new results in using the sparse Johnson-Lindenstrauss transform in numerical linear algebra, classical and model-based compressed sensing, manifold learning, and constrained least squares problems such as the Lasso." @default.
- W4299168148 created "2022-10-02" @default.
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- W4299168148 date "2013-11-11" @default.
- W4299168148 modified "2023-10-16" @default.
- W4299168148 title "Toward a unified theory of sparse dimensionality reduction in Euclidean space" @default.
- W4299168148 doi "https://doi.org/10.48550/arxiv.1311.2542" @default.
- W4299168148 hasPublicationYear "2013" @default.
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