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674
The Gram matrix is
"... x x ⊤ dP(x) Estimate G is equivalent to estimate N(θ) = 〈x, θ 〉 2 dP(x) since N(θ) = θ ⊤ Gθ P is unknown X1,..., Xn ∈ R d ∼ P i.i.d. sample Goal: Estimate N(θ) for every θ ∈ R d from the sampleGeneral Setting Let P ∈ M 1 +(R d). ..."
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x x ⊤ dP(x) Estimate G is equivalent to estimate N(θ) = 〈x, θ 〉 2 dP(x) since N(θ) = θ ⊤ Gθ P is unknown X1,..., Xn ∈ R d ∼ P i.i.d. sample Goal: Estimate N(θ) for every θ ∈ R d from the sampleGeneral Setting Let P ∈ M 1 +(R d).
GRAM MATRIX IN C ∗-MODULES
, 905
"... Abstract. Let (X, 〈·, ·〉) be a semi-inner product module over a C∗-algebra A. For arbitrary n ∈ N and x1,...,xn ∈ X we study the so-called n × n Gram matrix [〈xi, xj〉] with entries in A, construct a non-decreasing sequence of positive matrices in Mn(A) which is majorized by [〈xi, xj〉] and apply it t ..."
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Abstract. Let (X, 〈·, ·〉) be a semi-inner product module over a C∗-algebra A. For arbitrary n ∈ N and x1,...,xn ∈ X we study the so-called n × n Gram matrix [〈xi, xj〉] with entries in A, construct a non-decreasing sequence of positive matrices in Mn(A) which is majorized by [〈xi, xj〉] and apply
Concentration properties of the eigenvalues of the Gram matrix
, 2009
"... We consider the concentration of the eigenvalues of the Gram matrix for a sample of iid vectors distributed in the unit ball of a Hilbert space. The square-root term in the deviation bound is shown to scale with the largest eigenvalue, the remaining term decaying as n1. This result is the consequenc ..."
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We consider the concentration of the eigenvalues of the Gram matrix for a sample of iid vectors distributed in the unit ball of a Hilbert space. The square-root term in the deviation bound is shown to scale with the largest eigenvalue, the remaining term decaying as n1. This result
On the Nyström Method for Approximating a Gram Matrix for Improved Kernel-Based Learning
- JOURNAL OF MACHINE LEARNING RESEARCH
, 2005
"... A problem for many kernel-based methods is that the amount of computation required to find the solution scales as O(n³), where n is the number of training examples. We develop and analyze an algorithm to compute an easily-interpretable low-rank approximation to an nn Gram matrix G such that compu ..."
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Cited by 188 (11 self)
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A problem for many kernel-based methods is that the amount of computation required to find the solution scales as O(n³), where n is the number of training examples. We develop and analyze an algorithm to compute an easily-interpretable low-rank approximation to an nn Gram matrix G
On the eigenspectrum of the gram matrix and its relationship to the operator eigenspectrum
- ALT 2002, LNAI 2533
, 2002
"... In this paper we analyze the relationships between the eigenvalues of the m × m Gram matrix K for a kernel k(·, ·) corresponding to a sample x1,...,xm drawn from a density p(x) and the eigenvalues of the corresponding continuous eigenproblem. We bound the differences between the two spectra and pr ..."
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Cited by 17 (7 self)
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In this paper we analyze the relationships between the eigenvalues of the m × m Gram matrix K for a kernel k(·, ·) corresponding to a sample x1,...,xm drawn from a density p(x) and the eigenvalues of the corresponding continuous eigenproblem. We bound the differences between the two spectra
On the Eigenspectrum of the Gram Matrix and the Generalisation Error of Kernel PCA
- IEEE Transactions on Information Theory
, 2005
"... Abstract — In this paper the relationships between the eigenvalues of the m × m Gram matrix K for a kernel κ(·, ·) corresponding to a sample x1,..., xm drawn from a density p(x) and the eigenvalues of the corresponding continuous eigenproblem is analysed. The differences between the two spectra are ..."
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Cited by 13 (1 self)
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Abstract — In this paper the relationships between the eigenvalues of the m × m Gram matrix K for a kernel κ(·, ·) corresponding to a sample x1,..., xm drawn from a density p(x) and the eigenvalues of the corresponding continuous eigenproblem is analysed. The differences between the two spectra
The Gram matrix of a PT-symmetric quantum system ∗)
, 2003
"... The eigenstates of a diagonalizable PT-symmetric Hamiltonian satisfy unconventional completeness and orthonormality relations. These relations reflect the properties of a pair of bi-orthonormal bases associated with non-hermitean diagonalizable operators. In a similar vein, such a dual pair of bases ..."
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of bases is shown to possess, in the presence of PT symmetry, a Gram matrix of a particular structure: its inverse is obtained by simply swapping the signs of some its matrix elements. PACS: 03.67.–w Key words: Gram Matrix, PT symmetry, bi-orthonormal basis The spectrum of a non-hermitean Hamiltonian H ̂
Kernels for Measures Defined on the Gram Matrix of their Support
, 2009
"... We present in this work a new family of kernels to compare positive measures on arbitrary spaces X endowed with a positive kernel κ, which translates naturally into kernels between histograms or clouds of points. We first cover the case where X is Euclidian, and focus on kernels which take into acco ..."
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on functions which are invariant to the addition of a null eigenvalue to the spectrum of the variance matrix, we can define kernels between atomic measures on arbitrary spaces X endowed with a kernel κ by using directly the eigenvalues of the centered Gram matrix of the joined support of the compared measures
SPECTRAL ANALYSIS OF THE GRAM MATRIX OF MIXTURE MODELS
"... Abstract. This text is devoted to the asymptotic study of some spectral properties of the Gram matrix WTW built upon a collection w1,..., wn ∈ Rp of random vectors (the columns of W), as both the number n of observations and the dimension p of the observations tend to infinity and are of similar ord ..."
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Cited by 1 (1 self)
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Abstract. This text is devoted to the asymptotic study of some spectral properties of the Gram matrix WTW built upon a collection w1,..., wn ∈ Rp of random vectors (the columns of W), as both the number n of observations and the dimension p of the observations tend to infinity and are of similar
Results 1 - 10
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