Embedding results pertaining to strong approximation of fourier series. II

Установлены три теоремы, являушиеся в определенном смысле обратными к теоремам влозения, докаэанным в нащеи предыдушеи работе, но первые две теоремы относятся толяко к непрерывным функциям, коеффициенты Фуряе которых являутся КВУ-последователянос тями, см. (13).     

Embedding results pertaining to strong approximation of fourier series. III

Докаэано влозение гдеβ≥р(r+1) или 0<β<р(r+1) и последователяностя $\left\{ {n^{\beta - pr} \omega \left( {\frac{1}{n}} \right)^p } \right\}$ почти монотонно воэрастает.     

Improved approximation results on standard quartic polynomial optimization

In this paper, we consider the problem of approximately solving standard quartic polynomial optimization (SQPO). Using its reformulation as a copositive tensor programming, we show how to approximate the optimal solution of SQPO by using a series of polyhedral cones to approximate the cone of copositive tensors. The established quality of approximation is sharper than the ones studied in the literature. As an interesting extension, we also propose some approximation bounds on multi-homogenous polynomial optimization problems.     

Improved results on the robustness of stochastic approximation algorithms

This paper is a continuation of the research carried out in [1]–[2], where the robustness analysis for stochastic approximation algorithms is given for two cases: 1. The regression function and the Liapunov function are not zero at the sought-forx ⁰; 2. lim sup _n a131_{n i=1} ⁿ _i+1 is not zero, where { _i } are the measurement errors and {a _n} are the weighting coefficients in the algorithm. Allowing these deviations from zero to occur simultaneously but to remain small, this paper shows that the estimation error is still small even for a class of measurement errors more general than that considered in [2].This project is supported by the National Natural Science Foundation of China.     

Truncated sparse approximation property and truncated q-norm minimization

This paper considers approximately sparse signal and low-rank matrix's recovery via truncated norm minimization min_x ||x_T||_q and min_X||X_T||_(S_q) from noisy measurements. We first introduce truncated sparse approximation property, a more general robust null space property,and establish the stable recovery of signals and matrices under the truncated sparse approximation property. We also explore the relationship between the restricted isometry property and truncated sparse approximation property. And we also prove that if a measurement matrix A or linear map A satisfies truncated sparse approximation property of order k, then the first inequality in restricted isometry property of order k and of order 2 k can hold for certain different constants δ_k and δ_(2k), respectively. Last, we show that if δ_(s(~k+|T~c|)) ((s-1)/s)~(1/2) for some s≥ 4/3, then measurement matrix A and linear map A satisfy truncated sparse approximation property of order k. It should be pointed out that when T~c= Ф, our conclusion implies that sparse approximation property of order k is weaker than restricted isometry property of order sk.     

Making sparse matrices sparser: Computational results

Many optimization algorithms involve repeated processing of a fixed set of linear constraints. If we pre-process the constraint matrixA to be sparser, then algebraic operations onA will become faster. We consider the problem of making a given matrix as sparse as possible, theSparsity Problem (SP). In a companion paper with S. Frank Chang, we developed some theoretical algorithms for SP under a non-degeneracy assumption (McCormick and Chang, 1988). Here we investigate what must be done to make those algorithms applicable in practice. We report encouraging computational results in making linear programming constraint matrices sparser. We also find that the Simplex Algorithm can solve the reduced LPs faster. Comparisons are made to a heuristic algorithm for SP of Adler et al. (1989).This work was partially supported by NSF Grants ECS-84-04350 and CDR-84-21402, and by ONR Contract N0014-87-K0214.     

Fixed income linked life insurance policies with minimum guarantees: Pricing models and numerical results

We propose a pricing model for life insurance policies in which the benefits are linked to the performance of a portfolio of interest rate sensitive assets (reference fund), and a minimum guarantee provision is present. The model is cast in the celebrated term structure framework developed by Cox, Ingersoll and Ross (1985). As for the behaviour of the investment component, we analyse two polar cases. In the first one the payments due on the reference fund when the contract is still “alive” are not reinvested, while in the second case we propose a reinvestment policy. We show how to obtain a closed form solution for the single premium in the no-reinvestment case, and how to implement a simulation approach to calculate numerically the single premium in the reinvestment case. We illustrate our analysis with numerical results that help in understanding the comparative static properties of the models proposed.     

Approximating smooth and sparse functions by deep neural networks: Optimal approximation rates and saturation

Constructing neural networks for function approximation is a classical and longstanding topic in approximation theory. In this paper, we aim at constructing deep neural networks with three hidden layers using a sigmoidal activation function to approximate smooth and sparse functions. Specifically, we prove that the constructed deep nets with controllable magnitude of free parameters can reach the optimal approximation rate in approximating both smooth and sparse functions. In particular, we prove that neural networks with three hidden layers can avoid the phenomenon of saturation, i.e., the phenomenon that for some neural network architectures, the approximation rate stops improving for functions of very high smoothness.     

SOME CONVERSE RESULTS ON ONESIDED APPROXIMATION ： JUSTIFICATIONS

The present paper deals with best onesided approximation rate in Lp spaces ^~En（f）Lp of f∈C2π.Although it is clear that the estimate ^~E(f)Lp≤C‖f‖Lp cannot be correct for all f∈L^p2π in case p＜∞, the question whether ~En(f)Lp≤Cω(f,n^-l)Lp or ^~En(f)Lp≤CEn(f)Lp holds for f∈ C2π remains totally untouched.Therefore it forms a basic problem to justify onesided approximation. The present paper will provide an answer to settle down the basis.     

Improved bounds for the traveling umpire problem: A stronger formulation and a relax-and-fix heuristic

Given a double round-robin tournament, the traveling umpire problem (TUP) consists of determining which games will be handled by each one of several umpire crews during the tournament. The objective is to minimize the total distance traveled by the umpires, while respecting constraints that include visiting every team at home, and not seeing a team or venue too often. We strengthen a known integer programming formulation for the TUP and use it to implement a relax-and-fix heuristic that improves the quality of 24 out of 25 best-known feasible solutions to instances in the TUP benchmark. We also improve all best-known lower bounds for those instances and, for the first time, provide lower bounds for instances with more than 16 teams.     

Improved estimates of statistical regularization parameters in fourier differentiation and smoothing

Summary We investigate the statistical methods of cross-validation (CV) and maximum-likelihood (ML) for estimating optimal regularization parameters in the numerical differentiation and smoothing of non-exact data. Various criteria for optimality are examined, and the (asymptotic) notions of strong optimality, weak optimality and suboptimality are introduced relative to these criteria. By restricting attention to certainN-dimensional Hilbert spaces of smooth and stochastic functions, whereN is the number of data, we give regularity conditions on the data under which _CV, the regularization parameter predicted by CV, is strongly optimal with respect to the predictive mean-square signal error. We show that _ML is at best weakly optimal with respect to this criterion but is strongly optimal with respect to the innovation variance of the data. For numerical differentiation, _CV and _ML are both shown to be suboptimal with respect to the predictive mean-square derivative error.     

On the purely algebraic data‐sparse approximation of the inverse and the triangular factors of sparse matrices

The approximation of the inverse and the factors of the LU decomposition of general sparse matrices by hierarchical matrices is investigated. In this first approach, we present and motivate a new matrix partitioning algorithm which is based on the matrix graph by proving logarithmic‐linear complexity of the approximant in the case of bounded condition numbers. In contrast to the usual partitioning, the new algorithm allows to treat general grids if the origin of the sparse matrix is the finite element discretization of differential operators. Numerical examples indicate that the restriction to bounded condition numbers has only technical reasons. Copyright © 2010 John Wiley & Sons, Ltd.