2 research outputs found
Space lower bounds for linear prediction in the streaming model
We show that fundamental learning tasks, such as finding an approximate
linear separator or linear regression, require memory at least \emph{quadratic}
in the dimension, in a natural streaming setting. This implies that such
problems cannot be solved (at least in this setting) by scalable
memory-efficient streaming algorithms. Our results build on a memory lower
bound for a simple linear-algebraic problem -- finding orthogonal vectors --
and utilize the estimates on the packing of the Grassmannian, the manifold of
all linear subspaces of fixed dimension.Comment: Added a minor correction in referencing the prior wor
Exponentially Improved Dimensionality Reduction for : Subspace Embeddings and Independence Testing
Despite many applications, dimensionality reduction in the -norm is
much less understood than in the Euclidean norm. We give two new oblivious
dimensionality reduction techniques for the -norm which improve
exponentially over prior ones:
1. We design a distribution over random matrices , where , such that given any
matrix , with probability at least ,
simultaneously for all , . Note
that is linear, does not depend on , and maps into .
Our distribution provides an exponential improvement on the previous best known
map of Wang and Woodruff (SODA, 2019), which required ,
even for constant and . Our bound is optimal, up to a
polynomial factor in the exponent, given a known lower bound for
constant and .
2. We design a distribution over matrices ,
where , such that given any
-mode tensor , one can estimate the
entrywise -norm from . Moreover, and so given vectors , one can compute
in time , which is much faster
than the time required to form . Our linear map gives a streaming algorithm for independence testing using
space , improving the previous
doubly exponential space bound of
Braverman and Ostrovsky (STOC, 2010).Comment: To appear in COLT 2021; v2: minor fixes for camera ready versio