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The many proofs of an identity on the norm of oblique projections
 Numer. Algorithms
"... Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P 2 = P, which is neither null nor the identity, it holds that ‖P ‖ = ‖I − P ‖. This useful equality, while not widelyknown, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler o ..."
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Cited by 23 (1 self)
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Given an oblique projector P on a Hilbert space, i.e., an operator satisfying P 2 = P, which is neither null nor the identity, it holds that ‖P ‖ = ‖I − P ‖. This useful equality, while not widelyknown, has been proven repeatedly in the literature. Many published proofs are reviewed, and simpler ones are presented.
Damage identification using inverse methods
 Special Issue of the Royal Society Philosophical Transactions on Structural Health Monitoring and Damage Prognosis
, 2007
"... Abstract This chapter gives an overview of the use of inverse methods in damage detection and location, using measured vibration data. Inverse problems require the use of a model and the identification of uncertain parameters of this model. Damage is often local in nature and although the effect of ..."
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Cited by 16 (2 self)
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Abstract This chapter gives an overview of the use of inverse methods in damage detection and location, using measured vibration data. Inverse problems require the use of a model and the identification of uncertain parameters of this model. Damage is often local in nature and although the effect of the loss of stiffness may require only a small number of parameters, the lack of knowledge of the location means that a large number of candidate parameters must be included. This leads to potential illconditioning problems, and this topic is reviewed in this chapter. This chapter then goes on to discuss a number of problems that exist with the inverse approach to structural health monitoring, including modelling errors, environmental effects, damage localisation, regularisation, models of damage and sensor validation. 1 Introduction to Inverse Methods Inverse methods combine an initial model of the structure and measured data to improve the model or test an hypothesis. In practice the model is based on finite element analysis and the measurements are acceleration and force data, often in the form of a modal database, although frequency response function (FRF) data may also be used.
Exact subspace segmentation and outlier detection by lowrank representation
 Journal of Machine Learning Research
, 2011
"... In this work, we address the following matrix recovery problem: suppose we are given a set of data points containing two parts, one part consists of samples drawn from a union of multiple subspaces and the other part consists of outliers. We do not know which data points are outliers, or how many ou ..."
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Cited by 13 (5 self)
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In this work, we address the following matrix recovery problem: suppose we are given a set of data points containing two parts, one part consists of samples drawn from a union of multiple subspaces and the other part consists of outliers. We do not know which data points are outliers, or how many outliers there are. The rank and number of the subspaces are unknown either. Can we detect the outliers and segment the samples into their right subspaces, efficiently and exactly? We utilize a socalled LowRank Representation (LRR) method to solve this problem, and prove that under mild technical conditions, any solution to LRR exactly recovers the row space of the samples and detect the outliers as well. Since the subspace membership is provably determined by the row space, this further implies that LRR can perform exact subspace segmentation and outlier detection, in an efficient way. 1
Majorization for changes in angles between subspaces, Ritz values, and graph Laplacian spectra
 SIAM J. MATRIX ANAL. APPL
, 2006
"... Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ≺w. We explain these ideas and apply them in several areas, angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisin ..."
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Cited by 12 (5 self)
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Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ≺w. We explain these ideas and apply them in several areas, angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let Θ(X, Y) be the vector of principal angles in nondecreasing order between subspaces X and Y of a finite dimensional space H with a scalar product. We consider the change in principal angles between subspaces X and Z, where we let X be perturbed to give Y. We measure the change using weak majorization. We prove that
Improved Image Set Classification via Joint Sparse Approximated Nearest Subspaces
"... Existing multimodel approaches for image set classification extract local models by clustering each image set individually only once, with fixed clusters used for matching with other image sets. However, this may result in the two closest clusters to represent different characteristics of an object ..."
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Cited by 11 (2 self)
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Existing multimodel approaches for image set classification extract local models by clustering each image set individually only once, with fixed clusters used for matching with other image sets. However, this may result in the two closest clusters to represent different characteristics of an object, due to different undesirable environmental conditions (such as variations in illumination and pose). To address this problem, we propose to constrain the clustering of each query image set by forcing the clusters to have resemblance to the clusters in the gallery image sets. We first define a Frobenius norm distance between subspaces over Grassmann manifolds based on reconstruction error. We then extract local linear subspaces from a gallery image set via sparse representation. For each local linear subspace, we adaptively construct the corresponding closest subspace from the samples of a probe image set by joint sparse representation. We show that by minimising the sparse representation reconstruction error, we approach the nearest point on a Grassmann manifold. Experiments on Honda, ETH80 and CambridgeGesture datasets show that the proposed method consistently outperforms several other recent techniques, such as Affine Hull based Image
Jordan’s principal angles in complex vector spaces
, 2006
"... We analyse the possible recursive definitions of principal angles and vectors in complex vector spaces and give a new projector based definition. This enables us to derive important properties of the principal vectors and to generalize a result of Björck and Golub (Math. Comput. 1973; 27(123):579–59 ..."
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Cited by 9 (0 self)
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We analyse the possible recursive definitions of principal angles and vectors in complex vector spaces and give a new projector based definition. This enables us to derive important properties of the principal vectors and to generalize a result of Björck and Golub (Math. Comput. 1973; 27(123):579–594), which is the basis of today’s computational procedures in real vector spaces. We discuss other angle definitions
RAYLEIGHRITZ MAJORIZATION ERROR BOUNDS WITH APPLICATIONS TO FEM AND SUBSPACE ITERATIONS
, 2008
"... The RayleighRitz method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is Ainvariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspace ..."
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Cited by 8 (4 self)
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The RayleighRitz method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is Ainvariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is Ainvariant, the absolute changes in the Ritz values of A with respect to X compared to the Ritz values with respect to Y represent the absolute eigenvalue approximation error. A recent paper [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 548559] by M. Argentati et al. bounds the error in terms of the principal angles between X and Y using weak majorization, e.g., a sharp bound is proved if X corresponds to a contiguous set of extreme eigenvalues of A. In this paper, we extend this sharp bound to dimX ≤ dimY and to the general case of an arbitrary Ainvariant subspace X, which was a conjecture in this previous paper. We present our RayleighRitz majorization error bound in the context of the finite element method (FEM), and show how it can improve known FEM eigenvalue error bounds. We derive a new majorizationtype convergence rate bound of subspace iterations and combine it with the previous result to obtain a similar bound for the block Lanczos method.
A Formula for Angles between Subspaces of Inner Product Spaces. Contributions to Algebra and Geometry
"... Abstract. We present an explicit formula for angles between two subspaces of inner product spaces. Our formula serves as a correction for, as well as an extension of, the formula proposed by Risteski and Trenčevski [13]. As a consequence of our formula, a generalized CauchySchwarz inequality is ob ..."
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Abstract. We present an explicit formula for angles between two subspaces of inner product spaces. Our formula serves as a correction for, as well as an extension of, the formula proposed by Risteski and Trenčevski [13]. As a consequence of our formula, a generalized CauchySchwarz inequality is obtained.
Two key estimation techniques for the brokenarrows watermarking scheme
 IN PROC. OF 11TH ACM MULTIMEDIA AND SECURITY WORKSHOP
, 2009
"... This paper presents two different key estimation attacks targeted for the image watermarking system proposed for the BOWS2 contest. Ten thousands images are used in order to estimate the secret key and remove the watermark while minimizing the distortion. Two different techniques are proposed. The ..."
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Cited by 6 (1 self)
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This paper presents two different key estimation attacks targeted for the image watermarking system proposed for the BOWS2 contest. Ten thousands images are used in order to estimate the secret key and remove the watermark while minimizing the distortion. Two different techniques are proposed. The first one combines a regressionbased denoising process to filter out the component of the original images and a clustering algorithm to compute the different components of the key. The second attack is based on an inline subspace estimation algorithm, which estimates the subspace associated with the secret key without computing eigen decomposition. The key components are then estimated using Independent Component Analysis and a strategy designed to leave efficiently the detection region is presented. On six test images, the two attacks are able to remove the mark with very small distortions (between 41.8 dB and 49 dB).
Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix
 SIAM J. Matrix Anal. Appl
"... Abstract. The RayleighRitz method is widely used for eigenvalue approximation. Given a matrix X with columns that form an orthonormal basis for a subspace X, and a Hermitian matrix A, the eigenvalues of X H AX are called Ritz values of A with respect to X. If the subspace X is Ainvariant then the ..."
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Abstract. The RayleighRitz method is widely used for eigenvalue approximation. Given a matrix X with columns that form an orthonormal basis for a subspace X, and a Hermitian matrix A, the eigenvalues of X H AX are called Ritz values of A with respect to X. If the subspace X is Ainvariant then the Ritz values are some of the eigenvalues of A. If the Ainvariant subspace X is perturbed to give rise to another subspace Y, then the vector of absolute values of changes in Ritz values of A represents the absolute eigenvalue approximation error using Y. We bound the error in terms of principal angles between X and Y. We capitalize on ideas from a recent paper [DOI: 10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute values of differences between Ritz values for subspaces X and Y was weakly (sub)majorized by a constant times the sine of the vector of principal angles between X and Y, the constant being the spread of the spectrum of A. In that result no assumption was made on either subspace being Ainvariant. It was conjectured there that if one of the trial subspaces is Ainvariant then an analogous weak majorization bound should be much stronger as it should only involve terms of the order of sine squared. Here we confirm this conjecture. Specifically we prove that the absolute eigenvalue error is weakly majorized by a constant times the sine squared of the vector of principal angles between the subspaces X and Y, where the constant is proportional to the spread of the spectrum of A. For many practical cases we show that the proportionality factor is simply one, and that this bound is sharp. For the general case we can only prove the result with a slightly larger constant, which we believe is artificial. Key words. Hermitian matrices, angles between subspaces, majorization, Lidskii’s eigenvalue theorem, perturbation bounds, Ritz values, RayleighRitz method, invariant subspace.