10,940 research outputs found
Image registration with sparse approximations in parametric dictionaries
We examine in this paper the problem of image registration from the new
perspective where images are given by sparse approximations in parametric
dictionaries of geometric functions. We propose a registration algorithm that
looks for an estimate of the global transformation between sparse images by
examining the set of relative geometrical transformations between the
respective features. We propose a theoretical analysis of our registration
algorithm and we derive performance guarantees based on two novel important
properties of redundant dictionaries, namely the robust linear independence and
the transformation inconsistency. We propose several illustrations and insights
about the importance of these dictionary properties and show that common
properties such as coherence or restricted isometry property fail to provide
sufficient information in registration problems. We finally show with
illustrative experiments on simple visual objects and handwritten digits images
that our algorithm outperforms baseline competitor methods in terms of
transformation-invariant distance computation and classification
Sparse recovery on Euclidean Jordan algebras
This paper is concerned with the problem of sparse recovery on Euclidean Jordan algebra (SREJA), which includes the sparse signal recovery problem and the low-rank symmetric matrix recovery problem as special cases. We introduce the notions of restricted isometry property (RIP), null space property (NSP), and s-goodness for linear transformations in s-SREJA, all of which provide sufficient conditions for s-sparse recovery via the nuclear norm minimization on Euclidean Jordan algebra. Moreover, we show that both the s-goodness and the NSP are necessary and sufficient conditions for exact s-sparse recovery via the nuclear norm minimization on Euclidean Jordan algebra. Applying these characteristic properties, we establish the exact and stable recovery results for solving SREJA problems via nuclear norm minimization
Regularity Properties for Sparse Regression
Statistical and machine learning theory has developed several conditions
ensuring that popular estimators such as the Lasso or the Dantzig selector
perform well in high-dimensional sparse regression, including the restricted
eigenvalue, compatibility, and sensitivity properties. However, some
of the central aspects of these conditions are not well understood. For
instance, it is unknown if these conditions can be checked efficiently on any
given data set. This is problematic, because they are at the core of the theory
of sparse regression.
Here we provide a rigorous proof that these conditions are NP-hard to check.
This shows that the conditions are computationally infeasible to verify, and
raises some questions about their practical applications.
However, by taking an average-case perspective instead of the worst-case view
of NP-hardness, we show that a particular condition, sensitivity, has
certain desirable properties. This condition is weaker and more general than
the others. We show that it holds with high probability in models where the
parent population is well behaved, and that it is robust to certain data
processing steps. These results are desirable, as they provide guidance about
when the condition, and more generally the theory of sparse regression, may be
relevant in the analysis of high-dimensional correlated observational data.Comment: Manuscript shortened and more motivation added. To appear in
Communications in Mathematics and Statistic
Rotating Rindler-AdS Space
If the Hamiltonian of a quantum field theory is taken to be a timelike
isometry, the vacuum state remains empty for all time. We search for such
stationary vacua in anti-de Sitter space. By considering conjugacy classes of
the Lorentz group, we find interesting one-parameter families of stationary
vacua in three-dimensional anti-de Sitter space. In particular, there exists a
family of rotating Rindler vacua, labeled by the rotation parameter beta, which
are related to the usual Rindler vacuum by non-trivial Bogolubov
transformations. Rotating Rindler-AdS space possesses not only an
observer-dependent event horizon but even an observer-dependent ergosphere. We
also find rotating vacua in global AdS provided a certain region of spacetime
is excluded.Comment: 14 pages, LaTe
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