24,913 research outputs found
Open problem: Tightness of maximum likelihood semidefinite relaxations
We have observed an interesting, yet unexplained, phenomenon: Semidefinite
programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend
to be tight in recovery problems with noisy data, even when MLE cannot exactly
recover the ground truth. Several results establish tightness of SDP based
relaxations in the regime where exact recovery from MLE is possible. However,
to the best of our knowledge, their tightness is not understood beyond this
regime. As an illustrative example, we focus on the generalized Procrustes
problem
The Gravity Dual of the Ising Model
We evaluate the partition function of three dimensional theories of gravity
in the quantum regime, where the AdS radius is Planck scale and the central
charge is of order one. The contribution from the AdS vacuum sector can - with
certain assumptions - be computed and equals the vacuum character of a minimal
model CFT. The torus partition function is given by a sum over geometries which
is finite and computable. For generic values of Newton's constant G and the AdS
radius L the result has no Hilbert space interpretation, but in certain cases
it agrees with the partition function of a known CFT. For example, the
partition function of pure Einstein gravity with G=3L equals that of the Ising
model, providing evidence that these theories are dual. We also present
somewhat weaker evidence that the 3-state and tricritical Potts models are dual
to pure higher spin theories of gravity based on SL(3) and E_6, respectively.Comment: 42 page
Blind Compressed Sensing Over a Structured Union of Subspaces
This paper addresses the problem of simultaneous signal recovery and
dictionary learning based on compressive measurements. Multiple signals are
analyzed jointly, with multiple sensing matrices, under the assumption that the
unknown signals come from a union of a small number of disjoint subspaces. This
problem is important, for instance, in image inpainting applications, in which
the multiple signals are constituted by (incomplete) image patches taken from
the overall image. This work extends standard dictionary learning and
block-sparse dictionary optimization, by considering compressive measurements,
e.g., incomplete data). Previous work on blind compressed sensing is also
generalized by using multiple sensing matrices and relaxing some of the
restrictions on the learned dictionary. Drawing on results developed in the
context of matrix completion, it is proven that both the dictionary and signals
can be recovered with high probability from compressed measurements. The
solution is unique up to block permutations and invertible linear
transformations of the dictionary atoms. The recovery is contingent on the
number of measurements per signal and the number of signals being sufficiently
large; bounds are derived for these quantities. In addition, this paper
presents a computationally practical algorithm that performs dictionary
learning and signal recovery, and establishes conditions for its convergence to
a local optimum. Experimental results for image inpainting demonstrate the
capabilities of the method
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