A Sparser Johnson-Lindenstrauss Transform
نویسندگان
چکیده
We give a Johnson-Lindenstrauss transform with column sparsity s = Θ(ε−1 log(1/δ)) into optimal dimension k = O(ε−2 log(1/δ)) to achieve distortion 1±ε with success probability 1−δ. This is the first distribution to provide an asymptotic improvement over the Θ(k) sparsity bound for all values of ε, δ. Previous work of [Dasgupta-Kumar-Sarlós, STOC 2010] gave a distribution with s = Õ(ε−1 log(1/δ))1, with tighter analyses later in [Kane-Nelson, CoRR abs/1006.3585] and [Braverman-Ostrovsky-Rabani, CoRR abs/1011.2590] showing that their construction achieves s = Õ(ε−1 log(1/δ)). As in the previous work, our scheme only requires limited independence hash functions. In fact, potentially one of our hash functions could be made deterministic given an explicit construction of a sufficiently good error-correcting code. Our linear dependence on log(1/δ) in the sparsity allows us to plug our construction into algorithms of [Clarkson-Woodruff, STOC 2009] to achieve the fastest known streaming algorithms for numerical linear algebra problems such as approximate linear regression and best rank-k approximation. Their reductions to the Johnson-Lindenstrauss lemma require exponentially small δ, and thus a superlinear dependence on log(1/δ) in s leads to significantly slower algorithms.
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عنوان ژورنال:
- CoRR
دوره abs/1012.1577 شماره
صفحات -
تاریخ انتشار 2010