2 research outputs found
Nearly-optimal bounds for sparse recovery in generic norms, with applications to -median sketching
We initiate the study of trade-offs between sparsity and the number of
measurements in sparse recovery schemes for generic norms. Specifically, for a
norm , sparsity parameter , approximation factor , and
probability of failure , we ask: what is the minimal value of so that
there is a distribution over matrices with the property that
for any , given , we can recover a -sparse approximation to in
the given norm with probability at least ? We give a partial answer to
this problem, by showing that for norms that admit efficient linear sketches,
the optimal number of measurements is closely related to the doubling
dimension of the metric induced by the norm on the set of all
-sparse vectors. By applying our result to specific norms, we cast known
measurement bounds in our general framework (for the norms, ) as well as provide new, measurement-efficient schemes (for the
Earth-Mover Distance norm). The latter result directly implies more succinct
linear sketches for the well-studied planar -median clustering problem.
Finally, our lower bound for the doubling dimension of the EMD norm enables us
to address the open question of [Frahling-Sohler, STOC'05] about the space
complexity of clustering problems in the dynamic streaming model.Comment: 29 page
Metric dimension reduction: A snapshot of the Ribe program
The purpose of this article is to survey some of the context, achievements,
challenges and mysteries of the field of metric dimension reduction, including
new perspectives on major older results as well as recent advances.Comment: proceedings of ICM 201