Results 1  10
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50
Generalization Performance of Regularization Networks and Support . . .
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2001
"... We derive new bounds for the generalization error of kernel machines, such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers. The proofs make use of a viewpoint that is apparently novel in the field of statistical learning theory. The hy ..."
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Cited by 73 (20 self)
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We derive new bounds for the generalization error of kernel machines, such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers. The proofs make use of a viewpoint that is apparently novel in the field of statistical learning theory. The hypothesis class is described in terms of a linear operator mapping from a possibly infinitedimensional unit ball in feature space into a finitedimensional space. The covering numbers of the class are then determined via the entropy numbers of the operator. These numbers, which characterize the degree of compactness of the operator, can be bounded in terms of the eigenvalues of an integral operator induced by the kernel function used by the machine. As a consequence, we are able to theoretically explain the effect of the choice of kernel function on the generalization performance of support vector machines.
Fast rates for support vector machines using gaussian kernels
 Ann. Statist
, 2004
"... We establish learning rates up to the order of n −1 for support vector machines with hinge loss (L1SVMs) and nontrivial distributions. For the stochastic analysis of these algorithms we use recently developed concepts such as Tsybakov’s noise assumption and local Rademacher averages. Furthermore we ..."
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Cited by 51 (7 self)
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We establish learning rates up to the order of n −1 for support vector machines with hinge loss (L1SVMs) and nontrivial distributions. For the stochastic analysis of these algorithms we use recently developed concepts such as Tsybakov’s noise assumption and local Rademacher averages. Furthermore we introduce a new geometric noise condition for distributions that is used to bound the approximation error of Gaussian kernels in terms of their widths. 1
Analytic Methods for Simulated Light Transport
, 1995
"... this dissertation was conducted. Special thanks to Erin Shaw, Steve Westin, Pete Shirley, and John Hughes for carefully reading various portions of this work and offering detailed comments. Many thanks to my coauthors Julie Dorsey, Dave Kirk, Kevin Novins, David Salesin, Francois Sillion, Brian Sini ..."
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Cited by 34 (9 self)
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this dissertation was conducted. Special thanks to Erin Shaw, Steve Westin, Pete Shirley, and John Hughes for carefully reading various portions of this work and offering detailed comments. Many thanks to my coauthors Julie Dorsey, Dave Kirk, Kevin Novins, David Salesin, Francois Sillion, Brian Sinits, Ken Torrance, and Steve Westin, from whom I have learned so much over the years, and to Pete Shirley, Dani Lischinski, Bill Gropp, and Jim Ferwerda for enumerable stimulating vi vii discussions. Thanks also to Ben Trumbore and Albert Dicruttalo for modeling and software support, to Dan Kartch for all the help with document preparation, to Jonathan CorsonRikert, Ellen French, Linda Stephens, and Judy Smith for admin istrative support, and to Hurf Sheldon for many years of cheerful and professional systems support. From my days at Apollo Computer, I'd like to thank A1 Lopez, Fabio Pettinati, Ken Severson, and Terry Lindgren for all their encouragement. Many fellow students and assorted friends have also helped and inspired me along the way, including Lenny Pitt, Mukesh Prasad, Michael Monks, Ken Musgrave, Andrew Glassner, Mimi and Noel Mateo, and Susan Vonderheide
Entropy Numbers, Operators and Support Vector Kernels
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1998
"... We derive new bounds for the generalization error of feature space machines, such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers. The proofs are based on a viewpoint that is apparently novel in the field of statistical learning theory ..."
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Cited by 11 (3 self)
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We derive new bounds for the generalization error of feature space machines, such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers. The proofs are based on a viewpoint that is apparently novel in the field of statistical learning theory. The hypothesis class is described in terms of a linear operator mapping from a possibly infinite dimensional unit ball in feature space into a finite dimensional space. The covering numbers of the class are then determined via the entropy numbers of the operator. These numbers, which characterize the degree of compactness of the operator, can be bounded in terms of the eigenvalues of an integral operator induced by the kernel function used by the machine. As a consequence we are able to theoretically explain the effect of the choice of kernel functions on the generalization performance of support vector machines.
Hyperplane conjecture for quotient spaces of Lp
 Forum Math
, 1994
"... We give a positive solution for the hyperplane conjecture of quotient spaces F of Lp, where 1 < p ≤ ∞. vol(BF) n−1 n ≤ c0 p ′ sup H hyperplane vol(BF ∩ H). This result is extended to Banach lattices which does not contain ℓn 1 ’s uniformly. Our main tools are tensor products and minimal volume ratio ..."
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Cited by 8 (1 self)
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We give a positive solution for the hyperplane conjecture of quotient spaces F of Lp, where 1 < p ≤ ∞. vol(BF) n−1 n ≤ c0 p ′ sup H hyperplane vol(BF ∩ H). This result is extended to Banach lattices which does not contain ℓn 1 ’s uniformly. Our main tools are tensor products and minimal volume ratio with respect to Lpsections. An open problem in the theory of convex sets is the so called Hyperplane problem: Does there exist a universal constant c> 0 such that for all n ∈ IN and all convex, symmetric bodies K ⊂ IR n one has K  n−1
Scattering at Obstacles of Finite Capacity
, 1996
"... For very general generators of diffusion semigroups we show that the essential and absolutely continuous spectra do not change when one adds an extra Dirichlet boundary condition on a "small" set. This is done by proving that the corresponding semigroup differences are HilbertSchmidt or trace class ..."
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Cited by 7 (7 self)
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For very general generators of diffusion semigroups we show that the essential and absolutely continuous spectra do not change when one adds an extra Dirichlet boundary condition on a "small" set. This is done by proving that the corresponding semigroup differences are HilbertSchmidt or trace class, respectively. Our method consists in a factorization argument which is based on calculating the semigroup difference via the FeynmanKac formula. We also derive trace class estimates for differences of resolvent powers, provided the underlying semigroup has finite local dimension.
Complex interpolation between Hilbert, Banach and operator spaces
, 2008
"... Motivated by a question of Vincent Lafforgue, we study the Banach spaces X satisfying the following property: there is a function ε → ∆X(ε) tending to zero with ε> 0 such that every operator T: L2 → L2 with ‖T ‖ ≤ ε that is simultaneously contractive (i.e. of norm ≤ 1) on L1 and on L ∞ must be of ..."
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Cited by 7 (0 self)
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Motivated by a question of Vincent Lafforgue, we study the Banach spaces X satisfying the following property: there is a function ε → ∆X(ε) tending to zero with ε> 0 such that every operator T: L2 → L2 with ‖T ‖ ≤ ε that is simultaneously contractive (i.e. of norm ≤ 1) on L1 and on L ∞ must be of norm ≤ ∆X(ε) on L2(X). We show that ∆X(ε) ∈ O(ε α) for some α> 0 iff X is isomorphic to a quotient of a subspace of an ultraproduct of θHilbertian spaces for some θ> 0 (see Corollary 6.7), where θHilbertian is meant in a slightly more general sense than in our previous paper [43]. Let Br(L2(µ)) be the space of all regular operators on L2(µ). We are able to describe the complex interpolation space (Br(L2(µ)),B(L2(µ))) θ. We show that T: L2(µ) → L2(µ) belongs to this space iff T ⊗ idX is bounded on L2(X) for any θHilbertian space X. More generally, we are able to describe the spaces (B(ℓp0),B(ℓp1))θ or (B(Lp0),B(Lp1))θ for any pair 1 ≤ p0,p1 ≤ ∞ and 0 < θ < 1. In the same vein, given a locally compact Abelian group G, let M(G) (resp. PM(G)) be the space of complex measures (resp. pseudomeasures) on
Some Limiting Embeddings in Weighted Function Spaces and Related Entropy Numbers
, 1997
"... The paper deals with weighted function spaces of type B s p;q (R n ; w(x)) and F s p;q (R n ; w(x)), where w(x) is a weight function of at most polynomial growth. Of special interest are weight functions of type w(x) = (1 + jxj 2 ) ff=2 (log(2 + jxj)) with ff 0 and 2 R. Our main resu ..."
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Cited by 6 (3 self)
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The paper deals with weighted function spaces of type B s p;q (R n ; w(x)) and F s p;q (R n ; w(x)), where w(x) is a weight function of at most polynomial growth. Of special interest are weight functions of type w(x) = (1 + jxj 2 ) ff=2 (log(2 + jxj)) with ff 0 and 2 R. Our main result deals with estimates for the entropy numbers of compact embeddings between spaces of this type; more precisely, we may extend and tighten some of our previous results in [12]. AMS Subject Classification: 46E 35 Key Words: weighted function spaces, compact embeddings, entropy numbers Introduction 1 Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Weighted embeddings  the nonlimiting case 3 2 Limiting embeddings, entropy numbers 7 2.1 Estimates from above, an approach via duality arguments . . . . . . . . . . . . . . 8 2.2 Estimates from above, an approach via approximation numbers . . . . . . . . . . . 15 2.3 Estimates...
Bennett–Carl inequalities for symmetric Banach sequence spaces and unitary ideals, submitted
, 1998
"... We prove an abstract interpolation theorem which interpolates the (r, 2)summing and (s, 2)mixing norm of a fixed operator in the image and range space. Combined with interpolation formulas for spaces of operators we obtain as an application the original Bennett–Carl inequalities for identities act ..."
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Cited by 5 (4 self)
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We prove an abstract interpolation theorem which interpolates the (r, 2)summing and (s, 2)mixing norm of a fixed operator in the image and range space. Combined with interpolation formulas for spaces of operators we obtain as an application the original Bennett–Carl inequalities for identities acting between Minkowski spaces ℓu as well as their analogues for Schatten classes Su. Furthermore, our techniques motivate a study of Bennett–Carl inequalities within a more general setting of symmetric Banach sequence spaces and unitary ideals. In [Ben73] and [Car74] Bennett and Carl independently proved the following inequalities: For 1 ≤ u ≤ 2 and 1 ≤ u ≤ v ≤ ∞ the identity operator id: ℓu ֒ → ℓv is absolutely (r,2)summing, i. e. there is a constant c> 0 such that for each set of finitely many x1,...,xn ∈ ℓu ( n ∑ ‖xk ‖ r ℓv k=1) 1/r ≤ c · sup
Amenability of Banach algebras of compact operators
 Isrl. J. Math
, 1994
"... Abstract. In this paper we study conditions on a Banach space X that ensure that the Banach algebra K(X) of compact operators is amenable. We give a symmetrized approximation property of X which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including a ..."
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Cited by 5 (2 self)
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Abstract. In this paper we study conditions on a Banach space X that ensure that the Banach algebra K(X) of compact operators is amenable. We give a symmetrized approximation property of X which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties of X are implied by the amenability of K(X). Amenability is a cohomological property of Banach algebras which was introduced in [J]. The definition is given below. It may be thought of as