Abstract not found

We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble is related to a two-dimensional directed growth model, and the Charlier ensemble is related to the lengths of weakly increasing subsequences in random words. The Krawtchouk ensemble occurs in connection with zig-zag paths in random domino tilings of the Aztec diamond, and also in a certain simplified directed first-passage percolation model. We use the Charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of Tracy and Widom. As a limit of the Meixner ensemble or the Charlier ensemble we obtain the Plancherel measure on partitions, and using this we prove a conjecture of Baik, Deift and Johansson that under the Plancherel measure, the distribution of the lengths of the first k rows in the partition, appropriately scaled, converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the Gaussian Unitary Ensemble. In this problem a certain discrete kernel, which we call the discrete Bessel kernel, plays an important role.

1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a representation π ∈ G ∧ the weight (dim π) 2 /|G|. For the symmetric group S(n), the set S(n) ∧ is the set of partitions λ of the number

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.

A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.

We apply the exponential weight algorithm, introduced and Littlestone and Warmuth [17] and by Vovk [24] to the problem of predicting a binary sequence almost as well as the best biased coin. We first show that for the case of the logarithmic loss, the derived algorithm is equivalent to the Bayes algorithm with Jeffrey's prior, that was studied by Xie and Barron under probabilistic assumptions [26]. We derive a uniform bound on the regret which holds for any sequence. We also show that if the empirical distribution of the sequence is bounded away from 0 and from 1, then, as the length of the sequence increases to infinity, the difference between this bound and a corresponding bound on the average case regret of the same algorithm (which is asymptotically optimal in that case) is only 1=2. We show that this gap of 1=2 is necessary by calculating the regret of the min-max optimal algorithm for this problem and showing that the asymptotic upper bound is tight. We also study the application...

We consider error estimates for the interpolation by a special class of compactly supported radial basis functions. These functions consist of a univariate polynomial within their support and are of minimal degree depending on space dimension and smoothness. Their associated "native" Hilbert spaces are shown to be norm-equivalent to Sobolev spaces. Thus we can derive approximation orders for functions from Sobolev spaces which are comparable to those of thin-plate-spline interpolation. Finally, we investigate the numerical stability of the interpolation process.

We construct classes of nonstationary wavelets generated by what we call spherical basis functions (SBFs), which comprise a subclass of Schoenberg 's positive definite functions on the m-sphere. The wavelets are intrinsically defined on the m-sphere, and are independent of the choice of coordinate system. In addition, they may be orthogonalized easily, if desired. We will discuss decomposition, reconstruction, and localization for these wavelets. In the special case of the 2-sphere, we derive an uncertainty principle that expresses the trade-off between localization and the presence of high harmonics---or high frequencies---in expansions in spherical harmonics. We discuss the application of this principle to the wavelets that we construct. I. Introduction Geophyiscal or meteorological data collected over the surface of the earth via satellites or ground stations will invariably come from scattered sites. Synthesizing and analyzing such data is the motivation for the work that is pr...

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this paper, log 2 stands for the logarithm to the base 2. ffl Theorem 2.1, which is formulated for random fractals of limsup type in [0; 1], has an obvious generalization to random `fractals of limsup type' in [0; 1] d . In this setup we can take '(r) to be any gauge function which is regularly varying of index ff 2 (0; d) as r # 0, and replace 10 A. Dembo, Y. Peres, J. Rosen and O. Zeitouni (2.1) and (2.2) by Var(Mn (I)) i(n)IE(Mn (I)) = i(n)p n 2 d(n\Gammam) and 2 \Gammadn i(n)=('(2 \Gamman )p n ) ! 0 respectively. The proof of such a generalization is basically identical to the proof of Theorem 2.1. To establish Theorem 2.1 we need two lemmas. The first one is a version of the well-known connection between energy and Hausdorff measure. For the reader's convenience, we include the brief proof. Lemma 2.2. Fix an increasing gauge function ' such that '(0) = 0. Suppose that B is a Borel set in [0; 1], and is a probability measure on B. If the dyadic energy E' () := 1 X m=1 X J2Dm (J) 2 '(2 \Gammam ) of is finite, then H ' (B) ? 0. (In fact H ' (B) = 1, but that is unimportant for our purpose). See [14] for the connection of E' () to more traditional expressions for energy. Proof: Let \Psi(x) := 1 X m=1 X J2Dm (J) '(2 \Gammam ) 1 J (x) : Since R B \Psi(x) d = E' (), taking C = 2E' (), the set BC := fx 2 B fi fi fi \Psi(x) Cg has (BC ) 1=2. The restriction C of to BC satisfies C (J) C'(2 \Gammam ) for every J 2 Dm for all m. Since any interval I ae [0; 1] can be covered by three shorter dyadic intervals, it follows that C (I) 3C'(jI j) for any interval I . Hence, if A is any countable collection of intervals with BC ` [A I , then 1 2 (BC ) X A C (I) 3C X A '(jI j) which implies that 1=(6C) H ' (BC ). Alternati...