A lower - bound on ` 2 dimensionality reduction
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چکیده
A lower-bound on `2 dimensionality reduction The main focus of this lecture is a lower bound on the dimension when doing dimensionality reduction with -distortion in `1 and `2. In particular, it will be shown that there exist graphs requiring Ω(log n) dimensions to embed with any fixed desired distortion in Euclidean space. The main technical result is: Theorem 1 (Alon [1]). Let v1, . . . , vn+1 ∈ Rd and 1/ √ n ≤ < 1/3 be given, such that 1 ≤ ‖vi − vj‖ ≤ 1 + for all i 6= j ∈ [n + 1]. Then the subspace spanned by v1, . . . , vn+1 has dimension d = Ω (
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تاریخ انتشار 2006