24,112 research outputs found
Randomized Riemannian Preconditioning for Orthogonality Constrained Problems
Optimization problems with (generalized) orthogonality constraints are
prevalent across science and engineering. For example, in computational science
they arise in the symmetric (generalized) eigenvalue problem, in nonlinear
eigenvalue problems, and in electronic structures computations, to name a few
problems. In statistics and machine learning, they arise, for example, in
canonical correlation analysis and in linear discriminant analysis. In this
article, we consider using randomized preconditioning in the context of
optimization problems with generalized orthogonality constraints. Our proposed
algorithms are based on Riemannian optimization on the generalized Stiefel
manifold equipped with a non-standard preconditioned geometry, which
necessitates development of the geometric components necessary for developing
algorithms based on this approach. Furthermore, we perform asymptotic
convergence analysis of the preconditioned algorithms which help to
characterize the quality of a given preconditioner using second-order
information. Finally, for the problems of canonical correlation analysis and
linear discriminant analysis, we develop randomized preconditioners along with
corresponding bounds on the relevant condition number
Robust canonical correlations: a comparative study.
Several approaches for robust canonical correlation analysis will be presented and discussed. A first method is based on the definition of canonical correlation analysis as looking for linear combinations of two sets of variables having maximal (robust) correlation. A second method is based on alternating robust regressions. These methods are discussed in detail and compared with the more traditional approach to robust canonical correlation via covariance matrix estimates. A simulation study compares the performance of the different estimators under several kinds of sampling schemes. Robustness is studied as well by breakdown plots.Alternating regression; Canonical correlations; Correlation measures; Projection-pursuit; Robust covariance estimation; Robust regression; Robustness;
Automatic vehicle tracking and recognition from aerial image sequences
This paper addresses the problem of automated vehicle tracking and
recognition from aerial image sequences. Motivated by its successes in the
existing literature focus on the use of linear appearance subspaces to describe
multi-view object appearance and highlight the challenges involved in their
application as a part of a practical system. A working solution which includes
steps for data extraction and normalization is described. In experiments on
real-world data the proposed methodology achieved promising results with a high
correct recognition rate and few, meaningful errors (type II errors whereby
genuinely similar targets are sometimes being confused with one another).
Directions for future research and possible improvements of the proposed method
are discussed
Onsager's missing steps retraced
Onsager's paper on phase transition and phase coexistence in anisotropic
colloidal systems is a landmark in the theory of lyotropic liquid crystals.
However, an uncompromising scrutiny of Onsager's original derivation reveals
that it would be rigorously valid only for ludicrous values of the system's
number density (of the order of the reciprocal of the number of particles)
Based on Penrose's tree identity and an appropriate variant of the mean-field
approach for purely repulsive, hard-core interactions, our theory shows that
Onsager's theory is indeed valid for a reasonable range of densities
Emergent gauge dynamics of highly frustrated magnets
Condensed matter exhibits a wide variety of exotic emergent phenomena such as
the fractional quantum Hall effect and the low temperature cooperative behavior
of highly frustrated magnets. I consider the classical Hamiltonian dynamics of
spins of the latter phenomena using a method introduced by Dirac in the 1950s
by assuming they are constrained to their lowest energy configurations as a
simplifying measure. Focusing on the kagome antiferromagnet as an example, I
find it is a gauge system with topological dynamics and non-locally connected
edge states for certain open boundary conditions similar to doubled
Chern-Simons electrodynamics expected of a spin liquid. These dynamics
are also similar to electrons in the fractional quantum Hall effect. The
classical theory presented here is a first step towards a controlled
semi-classical description of the spin liquid phases of many pyrochlore and
kagome antiferromagnets and towards a description of the low energy classical
dynamics of the corresponding unconstrained Heisenberg models.Comment: Updated with some appendices moved to the main body of the paper and
some additional improvements. 21 pages, 5 figure
Virtues and Flaws of the Pauli Potential
Quantum simulations of complex fermionic systems suffer from a variety of
challenging problems. In an effort to circumvent these challenges, simpler
``semi-classical'' approaches have been used to mimic fermionic correlations
through a fictitious ``Pauli potential''. In this contribution we examine two
issues. First, we address some of the inherent difficulties in a widely used
version of the Pauli potential. Second, we refine such a potential in a manner
consistent with the most basic properties of a cold Fermi gas, such as its
momentum distribution and its two-body correlation function.Comment: 16 pages, 6 figure
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