The wave functions of a quantum isotropic harmonic oscillator in N-space modified by barriers at the coordinate hyperplanes can be expressed in terms of certain generalized spherical harmonics. These are associated with a product-type weight function on the sphere. Their analysis is carried out by means of differential-difference operators. The symmetries of this system involve the Weyl group o...

Consider a N × n random matrix Yn = (Y n ij ) where the entries are given by Y n ij = σ(i/N,j/n) √ n Xn ij , the X n ij being centered i.i.d. and σ : [0, 1] 2 → [0,∞) being a continuous function called a variance profile. Consider now a deterministic N ×n matrix Λn = ( Λij ) whose off-diagonal entries are zero. Denote by Σn the non-centered matrix Yn+Λn and by N ∧n = min(N,n). Then under the as...

iii Contents Abstract v 1. Introduction 1 2. Optimal aggregation 5 3. From micro to macro 9 4. From macro to semi-and non-parametric 11 5. Statistical learning 17 6. Conclusion 23 Appendix. The dual quadratic program 23 References 27 v Abstract Support Vector (SV)-regression, a common tool in statistical learning, estimates functions as linear combinations of given (nonlinear) kernel functions....

It is well known that dimensionality reduction based on multivariate analysis methods and their kernelized extensions can be formulated as generalized eigenvalue problems of scatter matrices, Gram matrices or their augmented matrices. This paper provides a generic and theoretical framework of multivariate analysis introducing a new expression for scatter matrices and Gram matrices, called Gener...

We present a novel family of language model (LM) estimation techniques named Sparse Non-negative Matrix (SNM) estimation. A first set of experiments empirically evaluating it on the One Billion Word Benchmark [Chelba et al., 2013] shows that SNM n-gram LMs perform almost as well as the well-established Kneser-Ney (KN) models. When using skip-gram features the models are able to match the state-...

In this paper, we present a new compactly supported kernel for SVM based image recognition. This kernel which we called Geometric Compactly Supported (GCS) can be viewed as a generalization of spherical kernels to higher dimensions. The construction of the GCS kernel is based on a geometric approach using the intersection volume of two n-dimensional balls. The compactness property of the GCS ke...

We consider the actuator placement problem for linear systems. Specifically, we aim to identify an actuator which requires the least amount of control energy to drive the system from an arbitrary initial condition to the origin in the worst case. Said otherwise, we investigate the minimax problem of minimizing the control energy over the worst possible initial conditions. Recall that the least ...

The open problem of determining the exact value of the n-th linear polarization constant cn of R n has received considerable attention over the past few years. This paper makes a contribution to the subject by providing a new lower bound on the value of sup‖y‖=1 | 〈x1,y〉 · · · 〈xn,y〉 |, where x1, . . . ,xn are unit vectors in R. The new estimate is given in terms of the eigenvalues of the Gram ...

In this paper we introduce an improved variant of the LLL algorithm. Using the Gram matrix to avoid expensive correction steps necessary in the Schnorr-Euchner algorithm and introducing the use of buffered transformations allows us to obtain a major improvement in reduction time. Unlike previous work, we are able to achieve the improvement while obtaining a strong reduction result and maintaini...

In linear system theory, the Hankel singular values are often computed in a state space setting using the product of Gramian matrices. They are known, however, to be intrinsically dependent only on the input-output map and not on any choice of state space coordinates. In the nonlinear case, there are well defined notions of singular value functions and a Hankel operator, but the connections bet...