Application of fast spherical Fourier transform to density estimation

This paper is on density estimation on the 2-sphere, S², using the orthogonal series estimator corresponding to spherical harmonics. In the standard approach of truncating the Fourier series of the empirical density, the Fourier transform is replaced with a version of the discrete fast spherical Fourier transform, as developed by Driscoll and Healy. The fast transform only applies to quantitative data on a regular grid. We will apply a kernel operator to the empirical density, to produce a function whose values at the vertices of such a grid will be the basis for the density estimation. The proposed estimation procedure also contains a deconvolution step, in order to reduce the bias introduced by the initial kernel operator. The main issue is to find necessary conditions on the involved discretization and the bandwidth of the kernel operator, to preserve the rate of convergence that can be achieved by the usual computationally intensive Fourier transform. Density estimation is considered in L²(S²) and more generally in Sobolev spaces H_v(S²), any v?0, with the regularity assumption that the probability density to be estimated belongs to H_s(S²) for some s>v. The proposed technique to estimate the Fourier transform of an unknown density keeps computing cost down to order O(n), where n denotes the sample size.     

The spectrum of Jacobi matrices in terms of its associated weight function

Using the so-called Lanczos procedure of orthogonalization a method is developed to calculate the elements of a N-dimensional Jacobi matrix and/or the coefficients of the three-term recurrence relation of a system of orthogonal polynomials {P_m(x), m = 0, 1, 2, ?, N} in terms of the moments μ_r^′(1) of its associated weight function. The eigenvalue density ?^(N)(x) and its asymptotical limit, i.e. when N tends to infinite, are also calculated in terms of μ_r^′(1). The method is used to determine the functions ?^(N)(x) and ?(x) for some known weight functions, like the normal distribution, the uniform distribution, the semicircular distribution and the gamma or Pearson type III distribution. As a byproduct the asymptotical density of zeros of Chebyshev, Legendre and generalized Laguerre polynomials are found.     

Orthonormal expansions of invariant densities for expanding maps

We give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding C^ω Markov maps. This is a classical problem, with one of the standard approaches being Ulam's method [Problems in Modern Mathematics, Interscience, New York, 1960] of phase space discretisation.Our method hinges instead on the expansion of the density function with respect to an L² orthonormal basis, and the computation of the expansion coefficients in terms of the periodic orbits of the expanding map. The efficiency of the method, and its extension to C^k expanding maps, are also discussed.     

On the Determinant of a Uniformly Distributed Complex Matrix

We derive the joint density for the singular values of a random complex matrix A uniformly distributed on ||A||_F = 1 This joint density allows us to obtain the conditional expectation of det(A^HA) = |det A|² given the smallest singular value. This result has been used by Shub and Smale in their analysis of the complexity of Bezout′s theorem.     

The complexity of detecting fixed-density clusters

We study the complexity of finding a subgraph of a certain size and a certain density, where density is measured by the average degree. Let γ:N→Q₊ be any density function, i.e., γ is computable in polynomial time and satisfies γ(k)?k-1 for all k∈N. Then γ-CLUSTER is the problem of deciding, given an undirected graph G and a natural number k, whether there is a subgraph of G on k vertices that has average degree at least γ(k). For γ(k)=k-1, this problem is the same as the well-known CLIQUE problem, and thus NP-complete. In contrast to this, the problem is known to be solvable in polynomial time for γ(k)=2. We ask for the possible functions γ such that γ-CLUSTER remains NP-complete or becomes solvable in polynomial time. We show a rather sharp boundary: γ CLUSTER is NP-complete if γ=2+Ω(1/k^1-ε) for some ε>0 and has a polynomial-time algorithm for γ=2+O(1/k). The algorithm also shows that for γ=2+O(1/k^1-o(1)), γ-CLUSTER is solvable in subexponential time 2^no(1).     

A borderline Gaussian random Fourier series for the sample convergence in variation

We give an example of a Gaussian random Fourier series, of which the normalized remainders have their sojourn times converging in variation, namely the convergence in the space L¹(R) of the related density distributions, to the Gaussian density. The convergence in the space L^∞(R) is also proved. In our approach, we use local times of Gaussian random Fourier series.     

Membership of distributions of polynomials in the Nikolskii–Besov class

The main result of this paper asserts that the distribution density of any non-constant polynomial f(ξ₁,ξ₂,...) of degree d in independent standard Gaussian random variables ξ₁ (possibly, in infinitely many variables) always belongs to the Nikol’skii–Besov space B^1/d (R¹) of fractional order 1/d (see the definition below) depending only on the degree of the polynomial. A natural analog of this assertion is obtained for the density of the joint distribution of k polynomials of degree d, also with a fractional order that is independent of the number of variables, but depends only on the degree d and the number of polynomials. We also give a new simple sufficient condition for a measure on R^k to possess a density in the Nikol’skii–Besov class B^α(R)^k. This result is applied for obtaining an upper bound on the total variation distance between two probability measures on R^k via the Kantorovich distance between them and a certain Nikol’skii–Besov norm of their difference. Applications are given to estimates of distributions of polynomials in Gaussian random variables.     

Precision of neural timing: The small ? limit

We explore the precision of neural timing in a model neural system with n identical input neurons whose firing time in response to stimulation is chosen from a density f. These input neurons stimulate a target cell which fires when it receives m hits within ? msec. We prove that the density of the firing time of the target cell converges as ?→0 to the input density f raised to the mth and normalized. We give conditions for convergence of the density in L¹, pointwise, and uniformly as well as conditions for the convergence of the standard deviations.     

Polynomial indexing of integer lattice-points II. Nonexistence results for higher-degree polynomials

Denoting the nonnegative (resp. signed) integers by N (resp. Z) and the real numbers by R, let S ? R² and f: R² → R. This current portion of our work specializes the basic concepts of Part I: Here f is a storing (resp. packing) function on S when f|(Z² ∩ S) is an injection into (resp. bijection onto) N; and the densityS ÷ f is $\text{lim}{}_{\mathrm{r\to \infty }}\text{(}\text{1}\text{n}\text{) \# {Z}{}^2\text{∩ S ∩ f}{}^{\mathrm{?1}}\text{([?n, +n])}}$. Take a rational sector of the plane R² to mean a closed sector with rational boundary rays, and consider the plane itself as a rational sector. Define the function f to be sectorially increasing on a set S if this f is eventually increasing on some rational sectors S_i constituting finite family which covers S. This paper obtains the following conclusions: A nonquadratic polynomial f(x, y) on a rational sector S with nonvoid interior, provided either f is sectorially increasing or deg(f) ≤ 4, cannot be a storing function with unit density on the sector S. Our proofs of these conclusions involve a strong criterion for zero density. These results support a nonexistence conjecture in Part I of our work, and extend an earlier remark of Pólya and Szegö. Indeed these results, under any one of our auxiliary assumptions, exclude nonquadratic storing functions with unit density on the arrays Z², Z × N, N².     

Reduction of variance for Gaussian densities via restriction to convex sets

Let X be a random vector with values in $\text{R}$ⁿ and a Gaussian density f. Let Y be a random vector whose density can be factored as k · f, where k is a logarithmically concave function on $\text{R}$ⁿ. We prove that the covariance matrix of X dominates the covariance matrix of Y by a positive semidefinite matrix. When k is the indicator function of a compact convex set A of positive measure the difference is positive definite. If A and X are both symmetric Var(a · X) is bounded above by an expression which is always strictly less than Var(a · X) for every a ∈ $\text{R}$ⁿ. Finally some counterexamples are given to show that these results cannot be extended to the general case where f is any logarithmically concave density.     

Polynomial indexing of integer lattice-points I. General concepts and quadratic polynomials

Denoting the nonnegative integers by N and the signed integers by Z, we let S be a subset of Z^m for m = 1, 2,… and f be a mapping from S into N. We call f a storing function on S if it is injective into N, and a packing function on S if it is bijective onto N. Motivation for these concepts includes extendible storage schemes for multidimensional arrays, pairing functions from recursive function theory, and, historically earliest, diagonal enumeration of Cartesian products. Indeed, Cantor's 1878 denumerability proof for the product N² exhibits the equivalent packing functions $\text{f}{}_{\mathrm{<mtext>Cantor</mtext>}}\text{(x, y) = {}\text{either}\text{x}\text{or}\text{y} +}\text{(x + y)(x + y + 1)}\text{2}$ on the domain N², and a 1923 Fueter-Pólya result, in our terminology, shows f_Cantor the only quadratic packing function on N². This paper extends the preceding result. For any real-valued function f on S we define a density $\text{S ÷ f =}\text{lim}{}_{\mathrm{n\to \infty }}\text{(}\text{1}\text{n}\text{)\#{S ? f}{}^{\mathrm{?1}}\text{([?n, +n])}}$, and for any packing function f on S we observe the fact S ÷ f = 1. Using properties of this density, and invoking Davenport's lemma from geometric number theory, we find all polynomial storing functions with unit density on N, and exclude any polynomials with these properties on Z, then find all quadratic storing functions with unit density on N², and exclude any quadratics with these properties on Z × N, Z². The admissible quadratics on N² are all nonnegative translates of f_Cantor. An immediate sequel to this paper excludes some higher-degree polynomials on subsets of Z².     

Ranges of operators and derivatives

We show a unified method of proving the existence of C¹-Fréchet smooth and Lipschitz mappings which are surjective or whose range of the derivative contains the whole dual unit ball. As an application, under Martin's Maximum axiom, we obtain a complete result for those spaces with density character ω₁.     

On positivity preservation for free quantum fields

Let A be a real Bose or Fermi one-particle operator with ∥ A ∥ ? I. Using Kaplansky's density theorem, a simple proof is given of the fact that Γ(A), the operator in Fock space induced by A, is positivity preserving in the relevant L²-space.     

Toeplitz operators on Herglotz wave functions

The space of Herglotz wave functions in R² consists of all the solutions of the Helmholtz equation that can be represented as the Fourier transform in R² of a measure supported in the circle and with density in L²(S¹). This space has a structure of a Hilbert space with reproducing kernel. The purpose of this article is to study Toeplitz operators with nonnegative radial symbols, defined on this space. We study the symbols defining bounded and compact Toeplitz operators as well as the Toeplitz operators belonging to the Schatten classes sp.     

On the largest principal angle between random subspaces

Formulas are derived for the probability density function and the probability distribution function of the largest canonical angle between two p-dimensional subspaces of Rⁿ chosen from the uniform distribution on the Grassmann manifold (which is the unique distribution invariant by orthogonal transformations of Rⁿ). The formulas involve the gamma function and the hypergeometric function of a matrix argument.     

Concentration of measures supported on the cube

We prove a log-Sobolev inequality for a certain class of log-concave measures in high dimension. These are the probability measures supported on the unit cube [0, 1] ⁿ ? ? ⁿ whose density takes the form exp(?ψ), where the function ψ is assumed to be convex (but not strictly convex) with bounded pure second derivatives. Our argument relies on a transportation-cost inequality á la Talagrand.     

Bernoulli convolutions associated with certain non-Pisot numbers

The Bernoulli convolution ν_λ measure is shown to be absolutely continuous with L² density for almost all , and singular if λ⁻¹ is a Pisot number. It is an open question whether the Pisot type Bernoulli convolutions are the only singular ones. In this paper, we construct a family of non-Pisot type Bernoulli convolutions ν_λ such that their density functions, if they exist, are not L². We also construct other Bernoulli convolutions whose density functions, if they exist, behave rather badly.     

Hölder Continuous Solutions to Complex Hessian Equations

We prove the Hölder continuity of the solution to complex Hessian equation with the right hand side in L ^p , \(p>\frac {n}{m}\) , 1 < m < n, in a m-strongly pseudoconvex domain in ? ⁿ under some additional conditions on the density near the boundary and on the boundary data.     

ON possible Turán densities

The Turán density π(F) of a family F of k-graphs is the limit as n → ∞ of the maximum edge density of an F-free k-graph on n vertices. Let Π _∞ ^(k) consist of all possible Turán densities and let Π _fin ^(k) ? Π _∞ ^(k) be the set of Turán densities of finite k-graph families. Here we prove that Π _fin ^(k) contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that Π _fin ^(k) contains an irrational number for each k ≥ 3. Also, we show that Π _∞ ^(k) has cardinality of the continuum. In particular, Π _∞ ^(k) ≠ Π _fin ^(k) .     

A bias corrected nonparametric regression estimator

In this article, we propose a new method of bias reduction in nonparametric regression estimation. The proposed new estimator has asymptotic bias order h⁴, where h is a smoothing parameter, in contrast to the usual bias order h² for the local linear regression. In addition, the proposed estimator has the same order of the asymptotic variance as the local linear regression. Our proposed method is closely related to the bias reduction method for kernel density estimation proposed by Chung and Lindsay (2011). However, our method is not a direct extension of their density estimate, but a totally new one based on the bias cancelation result of their proof.