5,638 research outputs found
Sparse Recovery with Very Sparse Compressed Counting
Compressed sensing (sparse signal recovery) often encounters nonnegative data
(e.g., images). Recently we developed the methodology of using (dense)
Compressed Counting for recovering nonnegative K-sparse signals. In this paper,
we adopt very sparse Compressed Counting for nonnegative signal recovery. Our
design matrix is sampled from a maximally-skewed p-stable distribution (0<p<1),
and we sparsify the design matrix so that on average (1-g)-fraction of the
entries become zero. The idea is related to very sparse stable random
projections (Li et al 2006 and Li 2007), the prior work for estimating summary
statistics of the data.
In our theoretical analysis, we show that, when p->0, it suffices to use M=
K/(1-exp(-gK) log N measurements, so that all coordinates can be recovered in
one scan of the coordinates. If g = 1 (i.e., dense design), then M = K log N.
If g= 1/K or 2/K (i.e., very sparse design), then M = 1.58K log N or M = 1.16K
log N. This means the design matrix can be indeed very sparse at only a minor
inflation of the sample complexity.
Interestingly, as p->1, the required number of measurements is essentially M
= 2.7K log N, provided g= 1/K. It turns out that this result is a general
worst-case bound
Ricci Curvature on Alexandrov spaces and Rigidity Theorems
In this paper, we introduce a new notion for lower bounds of Ricci curvature
on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's
maximal diameter theorem to Alexandrov spaces under this Ricci curvature
condition.Comment: final versio
Lipschitz continuity of harmonic maps between Alexandrov spaces
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every
energy minimizing harmonic map from an Alexandrov space with curvature bounded
from below to an Alexandrov space with non-positive curvature is locally
H\"older continuous. In [39], F. H. Lin proposed a challenge problem: Can the
H\"older continuity be improved to Lipschitz continuity? J. Jost also asked a
similar problem about Lipschitz regularity of harmonic maps between singular
spaces (see Page 38 in [28]). The main theorem of this paper gives a complete
resolution to it.Comment: We remove the assumption in the previous version that the domain
space has nonnegative generalized Ricci curvature. This solves Lin's
conjecture completely. To appear in Invent. Mat
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