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 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 function on the generalization performance of support vector machines.

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 Corson-Rikert, 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

We establish learning rates up to the order of n −1 for support vector machines with hinge loss (L1-SVMs) 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

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.

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 Hilbert-Schmidt or trace class, respectively. Our method consists in a factorization argument which is based on calculating the semigroup difference via the Feynman-Kac formula. We also derive trace class estimates for differences of resolvent powers, provided the underlying semigroup has finite local dimension.

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 Lp-sections. 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

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 non-limiting 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...

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

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Abstract. 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 θ-Hilbertian spaces for some θ> 0 (see Corollary 6.7), where θ-Hilbertian 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 θ-Hilbertian 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. pseudo-measures) on