551 research outputs found
New bounds on the Lieb-Thirring constants
Improved estimates on the constants , for ,
in the inequalities for the eigenvalue moments of Schr\"{o}dinger
operators are established
Rayleigh-Ritz majorization error bounds of the mixed type
The absolute change in the Rayleigh quotient (RQ) for a Hermitian matrix with
respect to vectors is bounded in terms of the norms of the residual vectors and
the angle between vectors in [\doi{10.1137/120884468}]. We substitute
multidimensional subspaces for the vectors and derive new bounds of absolute
changes of eigenvalues of the matrix RQ in terms of singular values of residual
matrices and principal angles between subspaces, using majorization. We show
how our results relate to bounds for eigenvalues after discarding off-diagonal
blocks or additive perturbations.Comment: 20 pages, 1 figure. Accepted to SIAM Journal on Matrix Analysis and
Application
The MM Alternative to EM
The EM algorithm is a special case of a more general algorithm called the MM
algorithm. Specific MM algorithms often have nothing to do with missing data.
The first M step of an MM algorithm creates a surrogate function that is
optimized in the second M step. In minimization, MM stands for
majorize--minimize; in maximization, it stands for minorize--maximize. This
two-step process always drives the objective function in the right direction.
Construction of MM algorithms relies on recognizing and manipulating
inequalities rather than calculating conditional expectations. This survey
walks the reader through the construction of several specific MM algorithms.
The potential of the MM algorithm in solving high-dimensional optimization and
estimation problems is its most attractive feature. Our applications to random
graph models, discriminant analysis and image restoration showcase this
ability.Comment: Published in at http://dx.doi.org/10.1214/08-STS264 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Simple Iterative Algorithm for Parsimonious Binary Kernel Fisher Discrimination
By applying recent results in optimization theory variously known as optimization transfer or majorize/minimize algorithms, an algorithm for binary, kernel, Fisher discriminant analysis is introduced that makes use of a non-smooth penalty on the coefficients to provide a parsimonious solution. The problem is converted into a smooth optimization that can be solved iteratively with no greater overhead than iteratively re-weighted least-squares. The result is simple, easily programmed and is shown to perform, in terms of both accuracy and parsimony, as well as or better than a number of leading machine learning algorithms on two well-studied and substantial benchmarks
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