Space Of Square Integrable Functions Research Articles

Best Choices for Regularization Parameters in Learning Theory: On the Bias—Variance Problem

Abstract. No abstract.

On efficient estimation of linear functionals of a bivariate distribution with known marginals

In this paper we construct efficient estimators for linear functionals of a bivariate distribution with known marginals. Previously, Bickel et al. (Ann. Statist. 19 (1991) 1316) constructed such estimators using the modified minimum chi-square principle. Our estimators utilize the least-squares principle and orthonormal bases for the Hilbert spaces of square integrable functions under the known marginal distributions and are easy to compute. Simulations indicate that in the moderate sample sizes considered our estimator compares favorably with the one proposed by Bickel et al.

Shift-invariant, DWT-based "projection" method for estimation of ultrasound pulse power spectrum.

An approach to computing estimates of the ultrasound pulse spectrum from echo-ultrasound RF sequences, measured from biological tissues, is proposed. It is computed by a "projection" algorithm based on the Discrete Wavelet Transform (DWT) using averaging over a range of linear shifts. It is shown that the robust, shift invariant estimate of the ultrasound pulse power spectrum can be obtained by the projection of RF line log spectrum on an appropriately chosen subspace of L2(R) (i.e., the space of square-integrable functions) that is spanned by a redundant collection of compactly supported, scaling functions. This redundant set is formed from the traditional (in Wavelet analysis) orthogonal set of scaling functions and also by all its linear (discrete) shifts. A proof is given that the estimate, so obtained, could be viewed as the average of the orthogonal projections of the RF line log spectrum, computed for all significant linear shifts of the RF line log spectrum in frequency domain. It implies that the estimate is shift-invariant. A computationally efficient scheme is presented for calculating the estimate. Proof is given that the averaged, shift-invariant estimate can be obtained simply by a convolution with a kernel, which can be viewed as the discretized auto-correlation function of the scaling function, appropriate to the particular subspace being considered. It implies that the computational burden is at most O(n log2 n), where n is the problem size, making the estimate quite suitable for real-time processing. Because of the property of the wavelet transform to suppress polynomials of orders lower than the number of the vanishing moments of the wavelet used, the presented approach can be considered as a local polynomial fitting. This locality plays a crucial role in the performance of the algorithm, improving the robustness of the estimation. Moreover, it is shown that the "averaging" nature of the proposed estimation allows using (relatively) poorly regular wavelets (i.e., short filters), without affecting the estimation quality. The latter is of importance whenever the number of calculations is crucial.

A Monte Carlo EM approach for partially observable diffusion processes: theory and applications to neural networks.

We present a Monte Carlo approach for training partially observable diffusion processes. We apply the approach to diffusion networks, a stochastic version of continuous recurrent neural networks. The approach is aimed at learning probability distributions of continuous paths, not just expected values. Interestingly, the relevant activation statistics used by the learning rule presented here are inner products in the Hilbert space of square integrable functions. These inner products can be computed using Hebbian operations and do not require backpropagation of error signals. Moreover, standard kernel methods could potentially be applied to compute such inner products. We propose that the main reason that recurrent neural networks have not worked well in engineering applications (e.g., speech recognition) is that they implicitly rely on a very simplistic likelihood model. The diffusion network approach proposed here is much richer and may open new avenues for applications of recurrent neural networks. We present some analysis and simulations to support this view. Very encouraging results were obtained on a visual speech recognition task in which neural networks outperformed hidden Markov models.

A Modified and a Finite Index Weber Transforms

This paper introduces, by way of constructing, specific finite and infinite integral transforms with Bessel functions J_\nu and Y_\nuä in their kernels. The infinite transform and its reciprocal look deceptively similar to the known Weber transform and its reciprocal, respectively, but fundamentally differ from them. The new transform enjoys an operational property that makes it useful for applications to some problems in differential equations with non-constant coefficients. The paper gives a characterization of the image of some spaces of square integrable functions with respect to some measure under the infinite and finite transforms.

A generalization of the Itô formula

The classical Itô formula is generalized to some anticipating processes. The processes we consider are in a Sobolev space which is a subset of the space of square integrable functions over a white noise space. The proof of the result uses white noise techniques.

Simplified quantum oscillator solution

To find the eigenvalues and eigenvectors of the quantum oscillator Hamiltonian, it is enough to construct eigenvectors with the raising operator and see that these eigenvectors are complete. The subspace they span must be the entire space of square-integrable functions because it is invariant for the raising and lowering operators and therefore invariant for the position and momentum operators. This shows that the Hamiltonian has no continuous spectrum.

The Bargmann representation for the quantum mechanics on a sphere

The Bargmann representation is constructed corresponding to the coherent states for a particle on a sphere introduced in Kowalski and Rembieliński, J. Phys. A: 33, 6035 (2000). The connection is discussed between the introduced formalism and the standard approach based on the Hilbert space of square integrable functions on a sphere S2.

Fast numerical solution of the linearized Molodensky problem

When standard boundary element methods (BEM) are used in order to solve the linearized vector Molodensky problem we are confronted with two problems: (1) the absence of O(|x|−2) terms in the decay condition is not taken into account, since the single-layer ansatz, which is commonly used as representation of the disturbing potential, is of the order O(|x|−1) as x→∞. This implies that the standard theory of Galerkin BEM is not applicable since the injectivity of the integral operator fails; (2) the N×N stiffness matrix is dense, with N typically of the order 105. Without fast algorithms, which provide suitable approximations to the stiffness matrix by a sparse one with O(N(logN) s ), s≥0, non-zero elements, high-resolution global gravity field recovery is not feasible. Solutions to both problems are proposed. (1) A proper variational formulation taking the decay condition into account is based on some closed subspace of co-dimension 3 of the space of square integrable functions on the boundary surface. Instead of imposing the constraints directly on the boundary element trial space, they are incorporated into a variational formulation by penalization with a Lagrange multiplier. The conforming discretization yields an augmented linear system of equations of dimension N+3×N+3. The penalty term guarantees the well-posedness of the problem, and gives precise information about the incompatibility of the data. (2) Since the upper left submatrix of dimension N×N of the augmented system is the stiffness matrix of the standard BEM, the approach allows all techniques to be used to generate sparse approximations to the stiffness matrix, such as wavelets, fast multipole methods, panel clustering etc., without any modification. A combination of panel clustering and fast multipole method is used in order to solve the augmented linear system of equations in O(N) operations. The method is based on an approximation of the kernel function of the integral operator by a degenerate kernel in the far field, which is provided by a multipole expansion of the kernel function. Numerical experiments show that the fast algorithm is superior to the standard BEM algorithm in terms of CPU time by about three orders of magnitude for N=65 538 unknowns. Similar holds for the storage requirements. About 30 iterations are necessary in order to solve the linear system of equations using the generalized minimum residual method (GMRES). The number of iterations is almost independent of the number of unknowns, which indicates good conditioning of the system matrix.

Approximation inL 2 Sobolev spaces on the 2-sphere by quasi-interpolation

In this article we consider a simple method of radial quasi-interpolation by polynomials on the unit sphere in ℝ3, and present rates of covergence for this method in Sobolev spaces of square integrable functions. We write the discrete Fourier series as a quasi-interpolant and hence obtain convergence rates, in the aforementioned Sobolev spaces, for the discrete Fourier projection. We also discuss some typical practical examples used in the context of spherical wavelets.

Plemelj Projection Operators over Domain Manifolds

Plemelj projection operators are introduced for spaces of square integrable functions defined over the boundaries of a class of compact real n-dimensional manifolds lying in C^n. These manifolds posses many properties similar to domains in R^n, and are consequently called domain manifolds. The key ingredients used here are techniques from both real and complex Clifford analysis. Analogues of the Kerzman-Stein kernel and Szego projection operators ar introduced, and their conformal covariance is described.

IRREDUCIBLE UNITARY REPRESENTATIONS OF SUp,q II: THE CONTINUOUS SERIES

A class of irreducible unitary representations belonging to the continuous series of SUp,q is explicitly determined in a space composed of the Kronecker product of three spaces of square integrable functions. The continuous series corresponds to a Cartan subgroup whose vector part has maximal dimension. These representations are distinguished by a parameter r = 1, 2, …, p for p ≤ q in SUp,q. For r = 0, one obtains the representations in the discrete series as in [5], and all representations in the continuous series, for r ≠ 0, are obtained explicitly.

Plemelj Projection Operators over Domain Manifolds

Plemelj projection operators are introduced for spaces of square integrable functions defined over the boundaries of a class of compact real n-dimensional manifolds lying in C^n. These manifolds posses many properties similar to domains in R^n, and are consequently called domain manifolds. The key ingredients used here are techniques from both real and complex Clifford analysis. Analogues of the Kerzman-Stein kernel and Szego projection operators ar introduced, and their conformal covariance is described.

Generalization of a Theorem of Boas to a Class of Integral Transforms

R. Boas obtained a characterization of the class of all square integrable functions vanishing on some interval of the form [−σ, σ] in terms of the behavior of its Fourier transform. An analogue of Boas’ theorem is obtained. A characterization of the image of the space of square integrable functions vanishing on some interval under various integral transformations is derived. The class of integral transformations considered is related to singular Sturm-Liouville boundary-value problems on a half line.

The concept of a noncommutative Riemann surface

We consider the compactification M(atrix) theory on a Riemann surface Σ of genus g>1. A natural generalization of the case of the torus leads to construct a projective unitary representation of π 1( Σ), realized on the Hilbert space of square integrable functions on the upper half-plane. A uniquely determined gauge connection, which in turn defines a gauged sl 2( R) algebra, provides the central extension. This has a geometric interpretation as the gauge length of a geodesic triangle, and corresponds to a 2-cocycle of the 2nd Hochschild cohomology group of the Fuchsian group uniformizing Σ. Our construction can be seen as a suitable double-scaling limit N→∞, k→−∞ of a U( N) representation of π 1( Σ), where k is the degree of the associated holomorphic vector bundle, which can be seen as the higher-genus analog of 't Hooft's clock and shift matrices of QCD. We compare the above mentioned uniqueness of the connection with the one considered in the differential-geometric approach to the Narasimhan–Seshadri theorem provided by Donaldson. We then use our infinite dimensional representation to construct a C ★-algebra which can be interpreted as a noncommutative Riemann surface Σ θ . Finally, we comment on the extension to higher genus of the concept of Morita equivalence.

Modulated Hermite series expansions and the time-bandwidth product

The harmonically modulated Hermite series constitute an orthonormal basis in the Hilbert space of square-integrable functions. This basis comprises three free parameters, namely a translation, a modulation, and a scale factor. In practical situations, we are interested in series expansions that are as compact as possible. We can use the free parameters as the means to obtain a compact series expansion for a given function. We choose as the compactness criterion the first-order moment of the energy distribution in the transform domain. It is shown that, in that case, the optimum compaction parameters can be given in a simple analytic form depending on signal measurements only. Furthermore, these parameters have a clear physical interpretation, and the minimum of the compactness criterion is directly related to the time-bandwidth product.

Quantum anti-de Sitter space and sphere at roots of unity

An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined which is covariant under U_q(so(2,D-1)), for q a root of unity. The star-structure is studied in detail. The scalar fields have an intrinsic high-energy cutoff, and arise most naturally as fields on orbifolds AdS_q^D \times S^D/G if D is odd, and AdS_q^D \times S_{\chi}^{2D-1}/G if D is even. Here G is a finite abelian group, and S_{\chi} is a certain ``chiral sector'' of the classical sphere. Hilbert spaces of square integrable functions are discussed. Analogous results are found for the q-deformed sphere S_q^D.

Spectral Analysis of Fokker--Planck and Related Operators Arising From Linear Stochastic Differential Equations

We study spectral properties of certain families of linear second-order differential operators arising from linear stochastic differential equations. We construct a basis in the Hilbert space of square-integrable functions using modified Hermite polynomials, and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed. In particular, we completely describe the spectrum of the Fokker--Planck operator on an appropriate invariant subspace of rapidly decaying functions. The eigenvalues of the Fokker--Planck operator provide information about the speed of convergence of the corresponding probability distribution to steady state, which is important for stochastic estimation and control applications. We show that the operator families under consideration can be realized as solutions of differential equations in the double bracket form on an operator Lie algebra, which leads to a simple expression for the flow of their eigenfunctions.

The central limit theorem for random perturbations of rotations

We prove a functional central limit theorem for stationary random sequences given by the transformations on the two-dimensional torus. This result is based on a functional central limit theorem for ergodic stationary martingale differences with values in a separable Hilbert space of square integrable functions.

The Black-Scholes pricing formula in the quantum context.

A natural explanation for extreme irregularities in the evolution of prices in financial markets is provided by quantum effects. The lack of simultaneous observability of relevant variables and the interference of attempted observation with the values of these variables represent such effects. These characteristics have been noted by traders and economists and appear intrinsic to market dynamics. This explanation is explored here in terms of a corresponding generalization of the Wiener process and its role in the Black-Scholes-Merton theory. The differentiability of the Wiener process as a sesquilinear form on a dense domain in the Hilbert space of square-integrable functions over Wiener space is shown and is extended to the quantum context. This provides a basis for a corresponding generalization of the Ito theory of stochastic integration. An extension of the Black-Scholes option pricing formula to the quantum context is deduced.