18.S096: Johnson-Lindenstrauss Lemma and Gordon’s Theorem
نویسنده
چکیده
Suppose one has n points, X = {x1, . . . , xn}, in Rd (with d large). If d > n, since the points have to lie in a subspace of dimension n it is clear that one can consider the projection f : Rd → Rn of the points to that subspace without distorting the geometry of X. In particular, for every xi and xj , ‖f(xi)− f(xj)‖ = ‖xi − xj‖, meaning that f is an isometry in X. Suppose now we allow a bit of distortion, and look for f : Rd → Rk that is an −isometry, meaning that (1− )‖xi − xj‖ ≤ ‖f(xi)− f(xj)‖ ≤ (1 + )‖xi − xj‖. (1)
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