OASIcs.SOSA.2018.15.pdf
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Simple Analyses of the Sparse Johnson-Lindenstrauss Transform
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1st Symposium on Simplicity in Algorithms (SOSA 2018)
Series: Open Access Series in Informatics (OASIcs) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2018-01-05
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Michael B. Cohen, T.S. Jayram, and Jelani Nelson. Simple Analyses of the Sparse Johnson-Lindenstrauss Transform. In 1st Symposium on Simplicity in Algorithms (SOSA 2018). Open Access Series in Informatics (OASIcs), Volume 61, pp. 15:1-15:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/OASIcs.SOSA.2018.15
https://doi.org/10.4230/OASIcs.SOSA.2018.15
Abstract
For every n-point subset X of Euclidean space and target distortion 1+eps for 0<eps<1, the Sparse Johnson Lindenstrauss Transform (SJLT) of (Kane, Nelson, J. ACM 2014) provides a linear dimensionality-reducing map f:X-->l_2^m where f(x) = Ax for A a matrix with m rows where (1) m = O((log n)/eps^2), and (2) each column of A is sparse, having only O(eps m) non-zero entries. Though the constructions given for such A in (Kane, Nelson, J. ACM 2014) are simple, the analyses are not, employing intricate combinatorial arguments. We here give two simple alternative proofs of their main result, involving no delicate combinatorics. One of these proofs has already been tested pedagogically, requiring slightly under forty minutes by the third author at a casual pace to cover all details in a blackboard course lecture.
Keywords
- dimensionality reduction
- Johnson-Lindenstrauss
- Sparse Johnson-Lindenstrauss Transform
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