Bounds on margin distributions in learning problems

Let be a probability space and let P_n be the empirical measure based on i.i.d. sample (X₁,…,X_n) from P. Let be a class of measurable real valued functions on For define F_f(t):=P{ft} and F_n,f(t):=P_n{ft}. Given γ(0,1], define _n,γ(δ):=1/(n^1−γ/2δ^γ). We show that if the L₂(P_n)-entropy of the class grows as ^−α for some α(0,2), then, for all and all δ(0,Δ_n), Δ_n=O(n^1/2),

and

where and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define

Then for all

uniformly in and with probability 1 (for the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory.     

Precise orders of strong unicity constants for a class of rational functions

Let denote a certain class of rational functions. For each f ε , consider the polynomial of degree at most n that best approximates f in the uniform norm. The corresponding strong unicity constant is denoted by M_n(f). Then there exist positive constants α and β, not depending on n, such that an M_n(f) βn, N = 1,2,….     

Nonparametric Approach for Non-Gaussian Vector Stationary Processes

Suppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrixf(λ). In this paper we consider the testing problemH: ∫^π_−π K{f(λ)} dλ=cagainstA: ∫^π_−π K{f(λ)} dλ≠c, whereK{·} is an appropriate function andcis a given constant. For this problem we propose a testT_nbased on ∫^π_−π K{f(λ)} dλ=c, wheref(λ) is a nonparametric spectral estimator off(λ), and we define an efficacy ofT_nunder a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectraf₄^Zofz(t). If it does not depend onf₄^Z, we say thatT_nis non-Gaussian robust. We will give sufficient conditions forT_nto be non-Gaussian robust. Since our test setting is very wide we can apply the result to many problems in time series. We discuss interrelation analysis of the components of {z(t)} and eigenvalue analysis off(λ). The essential point of our approach is that we do not assume the parametric form off(λ). Also some numerical studies are given and they confirm the theoretical results.     

A sufficient condition for pairing problem of generators in symmetrizable Kac–Moody algebras

In a symmetrizable Kac–Moody algebra g(A), let α=∑_i=1ⁿk_iα_i be an imaginary root satisfying k_i>0 and α,α_i<0 for i=1,2,…,n. In this paper, it is proved that for any x_αg_α{0}, satisfying [x_α,f_n]≠0 and [x_α,f_i]=0 for i=1,2,…,n−1, there exists a vector y such that the subalgebra generated by x_α and y contains g′(A), the derived subalgebra of g(A).     

Sharp inequalities for convolution-type operators

Let μ be a probability measure on [− a, a], a > 0, and let x₀ε[− a, a], f ε Cⁿ([−2a, 2a]), n 0 even. Using moment methods we derive best upper bounds to ¦∫_−a^a ([f(x₀ + y) + f(x₀ − y)]/2) μ(dy) − f(x₀)¦, leading to sharp inequalities that are attainable and involve the second modulus of continuity of f⁽ⁿ⁾ or an upper bound of it.     

Omega result for the mean square of the Riemann zeta function

A recent method of Soundararajan enables one to obtain improved Ω-result for finite series of the form ∑_nf(n) cos (2πλ_nx+β) where 0≤λ₁≤λ₂≤. . . and β are real numbers and the coefficients f(n) are all non-negative. In this paper, Soundararajan’s method is adapted to obtain improved Ω-result for E(t), the remainder term in the mean-square formula for the Riemann zeta-function on the critical line. The Atkinson series for E(t) is of the above type, but with an oscillating factor (−1)ⁿ attached to each of its terms.     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.     

Fractional Modified Dyadic Integral and Derivative on ℝ<Subscript>+</Subscript>

For functions in the Lebesgue space L(ℝ₊), a modified strong dyadic integral J _α and a modified strong dyadic derivative D ^(α) of fractional order α > 0 are introduced. For a given function f ∈ L(ℝ₊), criteria for the existence of these integrals and derivatives are obtained. A countable set of eigenfunctions for the operators J _α and D ^(α) is indicated. The formulas D ^(α)(J ^α(f)) = f and J _α(D ^(α)(f)) = f are proved for each α > 0 under the condition that . We prove that the linear operator is unbounded, where is the natural domain of J _α. A similar statement for the operator is proved. A modified dyadic derivative d ^(α)(f)(x) and a modified dyadic integral j _α(f)(x) are also defined for a function f ∈ L(ℝ₊) and a given point x ∈ ℝ₊. The formulas d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) are shown to be valid at each dyadic Lebesgue point x ∈ ℝ₊ of f.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 64–70, 2005Original Russian Text Copyright © by B. I. GolubovSupported by the Russian Foundation for Basic Research (grant no. 05-01-00206).     

On the convergence of averaging Hermite interpolators

We investigate two sequences of polynomial operators, H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x), of degrees 2n − 2 and 2n − 3, respectively, defined by interpolatory conditions similar to those of the classical Hermite-Féjer interpolators H_{2n − 1}(f, x). If H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x) are based on the zeros of the jacobi polynomials P_n^(α,β)(x), their convergence behaviour is similar to that of H_{2n − 1}(f;, x). If they are based on the zeros of (1 − x²)T_{n − 2}(x), their convergence behaviour is better, in some sense, than that of H_{2n − 1}(f, x).     

Polynomial approximation using projections whose kernels contain the Chebyshev polynomials

Let P be any projection of f ε C^{n + 1}[−1, 1] onto _n such that P(T_{n + 1}) = 0. It is shown that, for a wide class of such projections, f − Pf may be expressed in terms of f^{n + 1} in the same manner as E_n(f), the error of minimax approximation. Furthermore, a general necessary condition is obtained for this phenomenon.     

On almost sure convergence of Cesaro averages of subsequences of vector-valued functions

A remarkable theorem proved by Komlòs [4] states that if {f_n} is a bounded sequence in L¹(R), then there exists a subsequence {f_nk} and f L¹(R) such that f_nk (as well as any further subsequence) converges Cesaro to f almost everywhere. A similar theorem due to Révész [6] states that if {f_n} is a bounded sequence in L²(R), then there is a subsequence {f_nk} and f L²(R) such that Σ_k=1^∞ a_k(f_nk − f) converges a.e. whenever Σ_k=1^∞ | a_k |² < ∞. In this paper, we generalize these two theorems to functions with values in a Hilbert space (Theorems 3.1 and 3.3).     

Pseudoprime values of the Fibonacci sequence, polynomials and the Euler function

We show that if α > 1 is any fixed integer, then for a sufficiently large x > 1, the nth Fibonacci number F_n is a base α pseudopfime only for at most (4 + o(1))π(x) of posifive integers n x. The same result holds for Mersenne numbers 2ⁿ — 1 and for one more general class of Lucas sequences. A slight modification of our method also leads to similar results for polynomial sequences f(n), where f ∊ [X]. Finally, we use a different technique to get a much sharper upper bound on the counting fimction of the positive integers n such that φ(n) is a base α pseudoprime, where φ is the Euler function.     

On the Uniform Approximation of a Class of Analytic Functions by Bruwier Series

For a class of analytic functions f(z) defined by Laplace–Stieltjes integrals the uniform convergence on compact subsets of the complex plane of the Bruwier series (B-series) ∑^∞_n=0 λ_n(f) , λ_n(f)=f⁽ⁿ⁾(nc)+cf⁽ⁿ⁺¹⁾(nc), generated by f(z) and the uniform approximation of the generating function f(z) by its B-series in cones |arg z|< is shown.     

On Lp-Generalization of a Theorem of Adamyan, Arov, and Kreın

The paper deals with problems relating to the theory of Hankel operators. Let G be a bounded simple connected domain with the boundary Γ consisting of a closed analytic Jordan curve. Denote by _n,p(G), 1p<∞, the class of all meromorphic functions on G that can be represented in the form h=β/α, where β belongs to the Smirnov class E_p(G), α is a polynomial degree at most n, α0. We obtain estimates of s-numbers of the Hankel operator A_f constructed from fL_p(Γ), 1p<∞, in terms of the best approximation Δ_n,p of f in the space L_p(Γ) by functions belonging to the class _n,p(G).     

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

On an infinite series of Abel occurring in the theory of interpolation

The purpose of this paper is to show that for a certain class of functions f which are analytic in the complex plane possibly minus (−∞, −1], the Abel series f(0) + Σ_{n = 1}^∞ f⁽ⁿ⁾(nβ) z(z − nβ)^{n − 1}/n! is convergent for all β>0. Its sum is an entire function of exponential type and can be evaluated in terms of f. Furthermore, it is shown that the Abel series of f for small β>0 approximates f uniformly in half-planes of the form Re(z) − 1 + δ, δ>0. At the end of the paper some special cases are discussed.     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

Convexity preserving and predicting by Bernstein polynomials

It is known that the Bernstein polynomials of a function f defined on [0, 1 ] preserve its convexity properties, i.e., if f⁽ⁿ⁾ 0 then for m n, (B_mf)⁽ⁿ⁾ 0. Moreover, if f is n-convex then (B_mf)⁽ⁿ⁾ 0. While the converse is not true, we show that if f is bounded on (a, b) and if for every subinterval [α, β] (a, b) the nth derivative of the mth Bernstein polynomial of f on [α, β] is nonnegative then f is n-convex.     

Error bounds for least squares approximation by polynomials

Let f ε Cⁿ⁺¹[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials q_k associated with a distribution dα on [−1, 1]. It is shown that if q_n+1/q_n max(q_n+1(1)/q_n(1), −q_n+1(−1)/q_n(−1)), then f − H[f] f^{n + 1} · q_n+1/q_{n + 1}^{(n + 1)}, where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)^α (1 + t)^β dt, α, β > −1, the condition on q_n+1/q_n is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.     

On the nonexistence of entire solutions of certain type of nonlinear differential equations

By utilizing Nevanlinna's value distribution theory of meromorphic functions, it is shown that the following type of nonlinear differential equations:

fⁿ(z)+P_n−3(f)=p₁e^α₁z+p₂e^α₂z