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165
A tutorial on support vector machines for pattern recognition
 Data Mining and Knowledge Discovery
, 1998
"... The tutorial starts with an overview of the concepts of VC dimension and structural risk minimization. We then describe linear Support Vector Machines (SVMs) for separable and nonseparable data, working through a nontrivial example in detail. We describe a mechanical analogy, and discuss when SV ..."
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Cited by 3324 (12 self)
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The tutorial starts with an overview of the concepts of VC dimension and structural risk minimization. We then describe linear Support Vector Machines (SVMs) for separable and nonseparable data, working through a nontrivial example in detail. We describe a mechanical analogy, and discuss when SVM solutions are unique and when they are global. We describe how support vector training can be practically implemented, and discuss in detail the kernel mapping technique which is used to construct SVM solutions which are nonlinear in the data. We show how Support Vector machines can have very large (even infinite) VC dimension by computing the VC dimension for homogeneous polynomial and Gaussian radial basis function kernels. While very high VC dimension would normally bode ill for generalization performance, and while at present there exists no theory which shows that good generalization performance is guaranteed for SVMs, there are several arguments which support the observed high accuracy of SVMs, which we review. Results of some experiments which were inspired by these arguments are also presented. We give numerous examples and proofs of most of the key theorems. There is new material, and I hope that the reader will find that even old material is cast in a fresh light.
Subalgebras of Calgebras. III. Multivariable operator theory
 Acta Math
, 1998
"... Abstract. A dcontraction is a dtuple (T1,..., Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · ·+ Tdξd‖2 ≤ ‖ξ1‖2 + ‖ξ2‖2 + · · ·+ ‖ξd‖2 for all ξ1, ξ2,..., ξd ∈ H. These are the higher dimensional counterparts of contractions. We show that man ..."
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Cited by 99 (4 self)
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Abstract. A dcontraction is a dtuple (T1,..., Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · ·+ Tdξd‖2 ≤ ‖ξ1‖2 + ‖ξ2‖2 + · · ·+ ‖ξd‖2 for all ξ1, ξ2,..., ξd ∈ H. These are the higher dimensional counterparts of contractions. We show that many of the operatortheoretic aspects of function theory in the unit disk generalize to the unit ball Bd in complex dspace, including von Neumann’s inequality and the model theory of contractions. These results depend on properties of the dshift, a distinguished dcontraction which acts on a new H2 space associated with Bd, and which is the higher dimensional counterpart of the unilateral shift. H 2 and the dshift are highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the dshift relative to its generated C∗algebra we find that there is more uniqueness in dimension d ≥ 2 than there is in dimension one.
A Unifying Construction of Orthonormal Bases for System Identification
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 1994
"... In this paper we develop a general and very simple construction for complete orthonormal bases for system identification. This construction provides a unifying formulation of all known orthonormal bases since the common FIR and recently popular Laguerre and Kautz model structures are restrictive spe ..."
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Cited by 79 (20 self)
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In this paper we develop a general and very simple construction for complete orthonormal bases for system identification. This construction provides a unifying formulation of all known orthonormal bases since the common FIR and recently popular Laguerre and Kautz model structures are restrictive special cases of our construction as is another construction method based on balanced realisations of all pass functions. However, in contrast to these special cases, the basis vectors in our unifying construction can have nearly arbitrary magnitude frequency response according to the prior information the user wishes to inject into the problem. We also provide results characterising the completeness properties of our bases.
Subalgebras of C ∗ algebras III: Multivariable operator theory
 Acta Math
, 1998
"... Abstract. A dcontraction is a dtuple (T1,..., Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · · + Tdξd ‖ 2 ≤ ‖ξ1 ‖ 2 + ‖ξ2 ‖ 2 + · · · + ‖ξd ‖ 2 for all ξ1, ξ2,..., ξd ∈ H. These are the higher dimensional counterparts of contractions. We sho ..."
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Cited by 77 (7 self)
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Abstract. A dcontraction is a dtuple (T1,..., Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · · + Tdξd ‖ 2 ≤ ‖ξ1 ‖ 2 + ‖ξ2 ‖ 2 + · · · + ‖ξd ‖ 2 for all ξ1, ξ2,..., ξd ∈ H. These are the higher dimensional counterparts of contractions. We show that many of the operatortheoretic aspects of function theory in the unit disk generalize to the unit ball Bd in complex dspace, including von Neumann’s inequality and the model theory of contractions. These results depend on properties of the dshift, a distinguished dcontraction which acts on a new H 2 space associated with Bd, and which is the higher dimensional counterpart of the unilateral shift. H 2 and the dshift are highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the dshift relative to its generated C ∗algebra we find that there is more uniqueness in dimension d ≥ 2 than there is in dimension one.
Determinantal probability measures
, 2002
"... Abstract. Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We initiate a detailed study of the discrete analogue, the most prominent example of which has been the uniform spanning tree measure. Our main results concern relationship ..."
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Cited by 36 (4 self)
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Abstract. Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We initiate a detailed study of the discrete analogue, the most prominent example of which has been the uniform spanning tree measure. Our main results concern relationships with matroids, stochastic domination, negative association, completeness for infinite matroids, tail triviality, and a method for extension of results from orthogonal projections to positive contractions. We also present several new avenues for further investigation, involving Hilbert spaces, combinatorics, homology,
Higherrank numerical ranges and compression problems
 Linear Algebra Appl
"... Abstract. We consider higherrank versions of the standard numerical range for matrices. A central motivation for this investigation comes from quantum error correction. We develop the basic structure theory for the higherrank numerical ranges, and give a complete description in the Hermitian case. ..."
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Cited by 32 (3 self)
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Abstract. We consider higherrank versions of the standard numerical range for matrices. A central motivation for this investigation comes from quantum error correction. We develop the basic structure theory for the higherrank numerical ranges, and give a complete description in the Hermitian case. We also consider associated projection compression problems. 1.
Quantum Exponential Function
 Rev. Math. Phys
, 2000
"... A special function playing an essential role in the construction of quantum \ax + b"group is introduced and investigated. The function is denoted by F~(r; %), where ~ is a constant such that the deformation parameter q2 = e−i~. The rst variable r runs over nonzero real numbers; the range of t ..."
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Cited by 31 (1 self)
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A special function playing an essential role in the construction of quantum \ax + b"group is introduced and investigated. The function is denoted by F~(r; %), where ~ is a constant such that the deformation parameter q2 = e−i~. The rst variable r runs over nonzero real numbers; the range of the second one depends on the sign of r: % = 0 for r> 0 and % = 1 for r < 0. After the holomorphic continuation the function satises the functional equation F~(ei~r; %) = (1 + ei~=2r)F~(r;−%): The name \exponential function " is justied by the formula: F~(R; )F~(S; ) = F~([R + S]; e); where R; S are selfadjoint operators satisfying certain commutation relations and [R+S] is a selfadjoint extension of the sum R+ S determined by operators and appearing in the formula. This formula will be used in a forthcoming paper to construct a unitary operator W satisfying the pentagonal equation of Baaj and Skandalis. Quantum exponential functions appear in the following setting. One considers pairs of closed operators (R;S) acting on a Hilbert space H satisfying certain carefully
Fitzpatrick functions and continuous linear monotone operators
, 2006
"... The notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex, lower semicontinuous, and proper function and any (not necessarily symmetric) continuous linear positive operator. It was recently discovered that most fundamental resul ..."
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Cited by 31 (16 self)
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The notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex, lower semicontinuous, and proper function and any (not necessarily symmetric) continuous linear positive operator. It was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing Fitzpatrick functions. In this paper, we study Fitzpatrick functions of continuous linear monotone operators defined on a Hilbert space. A novel characterization of skew operators is presented. A result by Brézis and Haraux is reproved using the Fitzpatrick function. We investigate the Fitzpatrick function of the sum of two operators, and we show that a known upper bound is actually exact in finitedimensional and more general settings. Cyclic monotonicity properties are also analyzed, and closed forms of the Fitzpatrick functions of all orders are provided for all rotators in the Euclidean plane.
A CartanHadamard Theorem for BanachFinsler Manifolds
 Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), Geom. Dedicata 95
, 2002
"... . In this paper we study Banach{Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive dierential in every point. In this context we generalize the classical theorem of Cartan{ Hadamard, saying t ..."
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Cited by 30 (1 self)
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. In this paper we study Banach{Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive dierential in every point. In this context we generalize the classical theorem of Cartan{ Hadamard, saying that the exponential function is a covering map. We apply this to symmetric spaces and thus obtain criteria for Banach{Lie groups with an involution to have a polar decomposition. Typical examples of symmetric Finsler manifolds with seminegative curvature are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure which in general is not Riemannian. Introduction Let M = G=K be a nitedimensional noncompact Riemannian symmetric space, where K is the group of xed points of an involution on G . Then G has a polar decomposition in the sense that the decomposition g = k + p of its Lie algebra into the eigenspaces of the involution d(1) leads to a di...
Higherrank numerical ranges of unitary and normal matrices, preprint
, 608
"... Abstract. We verify a conjecture on the structure of higherrank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higherrank numerical ranges for a generic unitary matrix are given by complex polygons determined by ..."
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Cited by 26 (3 self)
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Abstract. We verify a conjecture on the structure of higherrank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higherrank numerical ranges for a generic unitary matrix are given by complex polygons determined by the spectral structure of the matrix. We discuss applications of the results to quantum error correction, specifically to the problem of identification and construction of codes for binary unitary noise models. 1.