Valiant Metric Embeddings , Dimension Reduction
نویسنده
چکیده
In the previous lecture notes, we saw that any metric (X, d) with |X| = n can be embedded into R 2 n) under any the `1 metric (actually, the same embedding works for any `p metic), with distortion O(log n). Here, we describe an extremely useful approach for reducing the dimensionality of a Euclidean (`2) metric, while incurring very little distortion. Such dimension reduction is useful for a number of reasons: on the practical side, many geometric algorithms have runtimes that scale poorly with the dimension of the space in which they operate. From a theoretical perspective these dimension-reduction procedures have been used numerous times as components within other algorithms (e.g. Locality Sensitive Hashing).
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