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236
Training Support Vector Machines: an Application to Face Detection
, 1997
"... We investigate the application of Support Vector Machines (SVMs) in computer vision. SVM is a learning technique developed by V. Vapnik and his team (AT&T Bell Labs.) that can be seen as a new method for training polynomial, neural network, or Radial Basis Functions classifiers. The decision surface ..."
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Cited by 561 (1 self)
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We investigate the application of Support Vector Machines (SVMs) in computer vision. SVM is a learning technique developed by V. Vapnik and his team (AT&T Bell Labs.) that can be seen as a new method for training polynomial, neural network, or Radial Basis Functions classifiers. The decision surfaces are found by solving a linearly constrained quadratic programming problem. This optimization problem is challenging because the quadratic form is completely dense and the memory requirements grow with the square of the number of data points. We present a decomposition algorithm that guarantees global optimality, and can be used to train SVM's over very large data sets. The main idea behind the decomposition is the iterative solution of subproblems and the evaluation of optimality conditions which are used both to generate improved iterative values, and also establish the stopping criteria for the algorithm. We present experimental results of our implementation of SVM, and demonstrate the ...
Pedestrian Detection Using Wavelet Templates
 in Computer Vision and Pattern Recognition
, 1997
"... This paper presents a trainable object detection architecture that is applied to detecting people in static images of cluttered scenes. This problem poses several challenges. People are highly nonrigid objects with a high degree of variability in size, shape, color, and texture. Unlike previous app ..."
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Cited by 225 (23 self)
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This paper presents a trainable object detection architecture that is applied to detecting people in static images of cluttered scenes. This problem poses several challenges. People are highly nonrigid objects with a high degree of variability in size, shape, color, and texture. Unlike previous approaches, this system learns from examples and does not rely on any a priori (handcrafted) models or on motion. The detection technique is based on the novel idea of the wavelet template that defines the shape of an object in terms of a subset of the wavelet coefficients of the image. It is invariant to changes in color and texture and can be used to robustly define a rich and complex class of objects such as people. We show how the invariant properties and computational efficiency of the wavelet template make it an effective tool for object detection. 1 Introduction The problem of object detection has seen a high degree of interest over the years. The fundamental problem is how to characte...
On the distribution of the largest eigenvalue in principal components analysis
 Ann. Statist
, 2001
"... Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribu ..."
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Cited by 197 (2 self)
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Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts. 1. Introduction. The
Communication over fading channels with delay constraints
 IEEE Transactions on Information Theory
, 2002
"... We consider a user communicating over a fading channel with perfect channel state information. Data is assumed to arrive from some higher layer application and is stored in a buffer until it is transmitted. We study adapting the user's transmission rate and power based on the channel state informati ..."
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Cited by 170 (7 self)
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We consider a user communicating over a fading channel with perfect channel state information. Data is assumed to arrive from some higher layer application and is stored in a buffer until it is transmitted. We study adapting the user's transmission rate and power based on the channel state information as well as the buffer occupancy; the objectives are to regulate both the longterm average transmission power and the average buffer delay incurred by the traffic. Two models for this situation are discussed; one corresponding to fixedlength/variablerate codewords and one corresponding to variablelength codewords. The tradeoff between the average delay and the average transmission power required for reliable communication is analyzed. A dynamic programming formulation is given to find all Pareto optimal power/delay operating points. We then quantify the behavior of this tradeoff in the regime of asymptotically large delay. In this regime we characterize simple buffer control policies which exhibit optimal characteristics. Connections to the delaylimited capacity and the expected capacity of fading channels are also discussed.
Regularized estimation of large covariance matrices
 Ann. Statist
, 2008
"... This paper considers estimating a covariance matrix of p variables from n observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance. We show that these estimates are consistent in the operator norm as long as (log p)/n → ..."
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Cited by 89 (13 self)
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This paper considers estimating a covariance matrix of p variables from n observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance. We show that these estimates are consistent in the operator norm as long as (log p)/n → 0, and obtain explicit rates. The results are uniform over some fairly natural wellconditioned families of covariance matrices. We also introduce an analogue of the Gaussian white noise model and show that if the population covariance is embeddable in that model and wellconditioned, then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix. The results can be extended to smooth versions of banding and to nonGaussian distributions with sufficiently short tails. A resampling approach is proposed for choosing the banding parameter in practice. This approach is illustrated numerically on both simulated and real data. 1. Introduction. Estimation
On asymptotic stability of solitary waves for nonlinear Schrödinger equations,Ann
 Inst. Henri Poincaré, Anal. Non Linéaire
"... The longtime asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to a solitary wave, the solution converges to a sum of another s ..."
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Cited by 80 (2 self)
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The longtime asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schrödinger equation. The proofs use the strategy of BuslaevPerelman [3, 4]: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection and method of majorants. 1
GALERKIN FINITE ELEMENT APPROXIMATIONS OF STOCHASTIC ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
, 2004
"... We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the ..."
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Cited by 77 (5 self)
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We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method generates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The second method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h or pversion, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.
Generalized Model Sets and Dynamical Systems
 CRM Monograph Series
, 1999
"... It is shown that the dynamical systems approach to the diffraction properties of model sets can be generalized to regular model sets in arbitrary sigmacompact Abelian groups with arbitrary locally compact Abelian groups as internal spaces. It is then shown that these regular model sets possess pure ..."
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Cited by 64 (0 self)
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It is shown that the dynamical systems approach to the diffraction properties of model sets can be generalized to regular model sets in arbitrary sigmacompact Abelian groups with arbitrary locally compact Abelian groups as internal spaces. It is then shown that these regular model sets possess pure point diffraction spectra.
Computable bounds for geometric convergence rates of Markov chains
, 1994
"... Recent results for geometrically ergodic Markov chains show that there exist constants R ! 1; ae ! 1 such that sup jfjV j Z P n (x; dy)f(y) \Gamma Z ß(dy)f(y)j RV (x)ae n where ß is the invariant probability measure and V is any solution of the drift inequalities Z P (x; dy)V (y) V (x) ..."
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Cited by 51 (7 self)
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Recent results for geometrically ergodic Markov chains show that there exist constants R ! 1; ae ! 1 such that sup jfjV j Z P n (x; dy)f(y) \Gamma Z ß(dy)f(y)j RV (x)ae n where ß is the invariant probability measure and V is any solution of the drift inequalities Z P (x; dy)V (y) V (x) + b1l C (x) which are known to guarantee geometric convergence for ! 1; b ! 1 and a suitable small set C. In this paper we identify for the first time computable bounds on R and ae in terms of ; b and the minorizing constants which guarantee the smallness of C. In the simplest case where C is an atom ff with P (ff; ff) ffi we can choose any ae ? # where [1 \Gamma #] \Gamma1 = 1 (1 \Gamma ) 2 h 1 \Gamma + b + b 2 + i ff (b(1 \Gamma ) + b 2 ) i and i ff i 34 \Gamma 8ffi 2 ffi 3 ji b 1 \Gamma j 2 ; and we can then choose R ae=[ae \Gamma #]. The bounds for general small sets C are similar but more complex. We apply these to simple queueing models and Markov chain Mo...
On the Distribution of the Largest Principal Component
 ANN. STATIST
, 2000
"... Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degr ..."
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Cited by 48 (0 self)
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Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n=p = 1: When centered by p = ( p n 1+ p p) 2 and scaled by p = ( p n 1+ p p)(1= p n 1+1= p p) 1=3 the distribution of x (1) approaches the TracyWidom law of order 1, which is dened in terms of the Painleve II dierential equation, and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to ...