99 research outputs found
Sparser Johnson-Lindenstrauss Transforms
We give two different and simple constructions for dimensionality reduction
in via linear mappings that are sparse: only an
-fraction of entries in each column of our embedding matrices
are non-zero to achieve distortion with high probability, while
still achieving the asymptotically optimal number of rows. These are the first
constructions to provide subconstant sparsity for all values of parameters,
improving upon previous works of Achlioptas (JCSS 2003) and Dasgupta, Kumar,
and Sarl\'{o}s (STOC 2010). Such distributions can be used to speed up
applications where dimensionality reduction is used.Comment: v6: journal version, minor changes, added Remark 23; v5: modified
abstract, fixed typos, added open problem section; v4: simplified section 4
by giving 1 analysis that covers both constructions; v3: proof of Theorem 25
in v2 was written incorrectly, now fixed; v2: Added another construction
achieving same upper bound, and added proof of near-tight lower bound for DKS
schem
Optimality of the Johnson-Lindenstrauss Lemma
For any integers and , we show the existence of a set of vectors such that any embedding satisfying
must have This lower bound matches the upper bound given by the Johnson-Lindenstrauss
lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of
of interest, since there is always an isometric embedding into
dimension (either the identity map, or projection onto
).
Previously such a lower bound was only known to hold against linear maps ,
and not for such a wide range of parameters [LN16]. The
best previously known lower bound for general was [Wel74, Lev83, Alo03], which
is suboptimal for any .Comment: v2: simplified proof, also added reference to Lev8
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Sketching and streaming algorithms for processing massive data
The rate at which electronic information is generated in the world is exploding. In this article we explore techniques known as sketching and streaming for processing massive data both quickly and memory-efficiently.Engineering and Applied Science
Bounded Independence Fools Degree-2 Threshold Functions
Let x be a random vector coming from any k-wise independent distribution over
{-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is
determined up to an additive epsilon for k = poly(1/epsilon). This answers an
open question of Diakonikolas et al. (FOCS 2009). Using standard constructions
of k-wise independent distributions, we obtain a broad class of explicit
generators that epsilon-fool the class of degree-2 threshold functions with
seed length log(n)*poly(1/epsilon).
Our approach is quite robust: it easily extends to yield that the
intersection of any constant number of degree-2 threshold functions is
epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if
the entries of x are k-wise independent standard normals, implying for example
that bounded independence derandomizes the Goemans-Williamson hyperplane
rounding scheme.
To achieve our results, we introduce a technique we dub multivariate
FT-mollification, a generalization of the univariate form introduced by Kane et
al. (SODA 2010) in the context of streaming algorithms. Along the way we prove
a generalized hypercontractive inequality for quadratic forms which takes the
operator norm of the associated matrix into account. These techniques may be of
independent interest.Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was
wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem
8.1 to m^4, not m^
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