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Face Recognition Based on Texture Discrimination by Using Geodesic Distance Approximations Between Multivariate Normal Distributions
Full text linkGeodesic distances are a natural dissimilarity measure between
probability distributions of a fixed type, and are used to discriminate texture in
several image-based measurements. Besides, since there is no known
closed-form solution for the geodesic distance between general
multivariate normal distributions, we propose two efficient
approximations to discriminate textures in the context of face
recognition. Unlike the typical appearance-based approach that uses
low-resolution grayscale face images, we propose a novel generative
approach for face recognition based on texture discrimination. In
the proposed approach, sparse facial features are extracted from
high-resolution color face images using predefined landmark
topologies, in which landmarks are in discriminative locations of
face images. By adopting a common landmark topology, the
dissimilarity between distinct face images can be scored in terms of
the dissimilarities between the texture in their corresponding
landmark vicinities. The proposed multivariate normal distributions
represent the color intensities around each landmark location. The
classification of new face samples occurs by determining the face
image sample in the training set which minimizes the dissimilarity
score. The proposed face recognition method was compared to methods
representative of the state-of-the-art using color and grayscale
face images, and presented higher recognition rates. Moreover, the
proposed measures to discriminate textures tend to be efficient in
face recognition and in general texture discrimination (e.g.,
texture recognition of material images), as our experiments sugges
Fusion in 2-cores of Maximal Parabolic Subgroups of the Baby Monster
No full textIn this paper the fusion of the 2-cores of the four maximal parabolic subgroups of the Baby
Monster, namely those of shape 29+16Sp8(2), 23+32(L3(2) � Sym(5)), 22+10+20(Sym(3) � M22 : 2)
and 21+22Co2, are determined
Information distance estimation between mixtures of multivariate Gaussians
Full text linkThere are efficient software programs for extracting from large data sets and image
sequences certain mixtures of probability distributions, such as multivariate Gaussians, to represent
the important features and their mutual correlations needed for accurate document retrieval from
databases. This note describes a method to use information geometric methods for distance measures
between distributions in mixtures of arbitrary multivariate Gaussians. There is no general analytic
solution for the information geodesic distance between two k-variate Gaussians, but for many purposes
the absolute information distance may not be essential and comparative values suffice for proximity
testing and document retrieval. Also, for two mixtures of different multivariate Gaussians we must
resort to approximations to incorporate the weightings. In practice, the relation between a reasonable
approximation and a true geodesic distance is likely to be monotonic, which is adequate for many
applications. Here we consider some choices for the incorporation of weightings in distance estimation
and provide illustrative results from simulations of differently weighted mixtures of multivariate
Gaussians
Optimality of the Paterson-Stockmeyer method for evaluating matrix polynomials and rational matrix functions
Full text linkMany state-of-the-art algorithms reduce the computation of transcendental
matrix functions to the evaluation of polynomial or rational approximants at a
matrix argument. This task can be accomplished efficiently by recurring to the
Paterson-Stockmeyer method, an evaluation scheme originally developed for
matrix polynomials that extends quite naturally to rational functions. An
important feature of these techniques is that the number of matrix
multiplications required to evaluate an approximant of order n grows slower
than n itself, with the result that different approximants yield the same
asymptotic computational cost.
We analyze the number of matrix multiplications required by the
Paterson-Stockmeyer method and by two widely used generalizations, one for
evaluating diagonal Padé approximants of generic functions and one
specifically tailored to those of the exponential. In all three cases, we
identify the approximants of maximum order for any given computational cost
An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential
Full text linkThe most popular algorithms for computing the matrix exponential are those based on the scaling and squaring technique. For optimal efficiency these are usually tuned to a particular precision of floating-point arithmetic. We design a new scaling and squaring algorithm that takes the unit roundoff of the arithmetic as input and chooses the algorithmic parameters in order to keep the forward error in the underlying Padé approximation below the unit roundoff. To do so, we derive an explicit expression for all the coefficients in an error expansion for Padé approximants to the exponential and use it to obtain a new bound for the truncation error. We also derive a new technique for selecting the internal parameters used by the algorithm, which at each step decides whether to scale or to increase the degree of the approximant. The algorithm can employ diagonal Padé approximants or Taylor approximants and can be used with a Schur decomposition or in transformation-free form. Our numerical experiments show that the new algorithm performs in a forward stable way for a wide range of precisions and that the most accurate of our implementations, the Taylor-based transformation-free variant, is superior to existing alternatives
Computing the Wave-Kernel Matrix Functions
No full textWe derive an algorithm for computing the wave-kernel functions and for an arbitrary square matrix , where . The algorithm is based on Pad\'{e} approximation and the use of double angle formulas. We show that the backward error of any approximation to can be explicitly expressed in terms of a hypergeometric function. To bound the backward error we derive and exploit a new bound for that is sharper than one previously obtained by Al-Mohy and Higham (\textit{SIAM J. Matrix Anal.\ Appl.}, 31(3):970--989, 2009). The amount of scaling and the degree of the Pade approximant are chosen to minimize the computational cost subject to achieving backward stability for in exact arithmetic. Numerical experiments show that the algorithm behaves in a forward stable manner in floating-point arithmetic and is superior in this respect to the general purpose Schur--Parlett algorithm applied to these functions
Optimal iterative solvers for linear nonsymmetric systems and nonlinear systems with PDE origins: Balanced black-box stopping tests
Full text linkThis paper discusses the design of efficient algorithms for solving linear nonsymmetric systems and nonlinear systems associated with FEM approximation of elliptic PDEs. The novel feature of the designed linear solvers like GMRES, BICGSTAB(l), TFQMR, and nonlinear solvers like Newton and Picard, is the incorporation of error control in the ‘natural norm’ in combination with an effective a posteriori estimator for the PDE approximation error. This leads to robust and optimal black-box stopping criteria: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error
An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential
No full textThe most popular algorithms for computing the matrix exponential are those based on the scaling and squaring technique. For optimal efficiency these are usually tuned to a particular precision of floating-point arithmetic. We design a new scaling and squaring algorithm that takes the unit roundoff of the arithmetic as input and chooses the algorithmic parameters in order to keep the forward error in the underlying Padé approximation below the unit roundoff. To do so, we derive an explicit expression for all the coefficients in an error expansion for Padé approximants to the exponential and use it to obtain a new bound for the truncation error. We also derive a new technique for selecting the internal parameters used by the algorithm, which at each step decides whether to scale or to increase the degree of the approximant. The algorithm can employ diagonal Padé approximants or Taylor approximants and can be used with a Schur decomposition or in transformation-free form. Our numerical experiments show that the new algorithm performs in a forward stable way for a wide range of precisions and that the most accurate of our implementations, the Taylor-based transformation-free variant, is superior to existing alternatives
Non-backtracking PageRank
No full textThe PageRank algorithm, which has been
``bringing order to the web" for more
than twenty years, computes the steady state of
a classical random walk plus teleporting.
Here we consider a variation of
PageRank that uses a non-backtracking random walk.
To do this, we first
reformulate PageRank in terms of the associated line graph.
A non-backtracking analog then emerges naturally.
Comparing the resulting steady states, we find that,
even for undirected graphs,
non-backtracking
generally leads to a different ranking of the nodes.
We then focus
on computational issues, deriving
an explicit representation of the new algorithm
that can exploit structure and sparsity
in the underlying network.
Finally, we assess effectiveness and
efficiency of this
approach on some real-world networks
