Journal of Complexity最新文献

A plethora of applications from diverse disciplines reduce to solving generalized equations involving Banach space valued operators. These equations are solved mostly iteratively, when a sequence is generated approximating a solution provided that certain conditions are valid on the starting point and the operators appearing on the method. Secant-type methods are developed whose specializations reduce to well known methods such as Newton, modified Newton, Secant, Kurchatov and Steffensen to mention a few. Unified local as well as semi-local analysis of these methods is presented using the celebrated contraction mapping principle in combination with the Aubin property of a set valued operator, and generalized continuity assumption on the operators on these methods. Numerical applications complement the theory.

The paper concerns problems of the recovery of operators from noisy information in weighted $L_q$-spaces with homogeneous weights. A number of general theorems are proved and applied to finding exact constants in multidimensional Carlson type inequalities with several weights and problems of the recovery of differential operators from a noisy Fourier transform. In particular, optimal methods are obtained for the recovery of powers of generalized Laplace operators from a noisy Fourier transform in the $L_p$-metric.

As an extension of the standard paradigm in statistical learning theory, we introduce the concept of r-learnability, $0<r\le 1$, which is a notion very closely related to that of nonexact oracle inequalities (see Lecue and Mendelson (2012) [7]). The r-learnability concept can enable so-called fast learning rates (along with corresponding sample complexity-type bounds) to be established at the cost of multiplying the approximation error term by an extra $(1+r)$-factor in the learning error estimate. We establish a new, general r-learning bound (nonexact oracle inequality) yielding fast learning rates in probability (up to at most a logarithmic factor) for proper learning in the general setting of an agnostic model, essentially only assuming a uniformly bounded squared loss function and a hypothesis class of finite VC-dimension (that is, finite pseudo-dimension).

A new algorithm is presented for computing the resultant of two generic bivariate polynomials over an arbitrary field. For $p,q$ in $K[x,y]$ of degree d in x and n in y, the resultant with respect to y is computed using $O\left(n^{1.458}d\right)$ arithmetic operations if $d=O\left(n^{1/3}\right)$. For $d=1$, the complexity estimate is therefore reconciled with the estimates of Neiger et al. 2021 for the related problems of modular composition and characteristic polynomial in a univariate quotient algebra. The 3/2 barrier in the exponent of n is crossed for the first time for the resultant. The problem is related to that of computing determinants of structured polynomial matrices. We identify new advanced aspects of structure for a polynomial Sylvester matrix. This enables to compute the determinant by mixing the baby steps/giant steps approach of Kaltofen and Villard 2005, until then restricted to the case $d=1$ for characteristic polynomials, and the high-order lifting strategy of Storjohann 2003 usually reserved for dense polynomial matrices.

We investigate the approximation of weighted integrals over $R^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to n integration nodes for functions from these spaces. In the one-dimensional case $(d=1)$, we obtain the right convergence rate of optimal quadratures. For $d\ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $R^d$.

Mordechay B. Levin (1999) has constructed a number λ which is normal in base 2, and such that the sequence ${\left(\left\{2^n\lambda \right\}\right)}_{n=0,1,2,\dots }$ has very small discrepancy $N\cdot D_N=O\left({(\log N)}^2\right)$. This construction technique was generalized by Becher and Carton (2019), who generated normal numbers via nested perfect necklaces, for which the same upper discrepancy estimate holds. In this paper we derive an upper discrepancy bound for so-called semi-perfect nested necklaces and show that for Levin's normal number in arbitrary prime base p this upper bound for the discrepancy is best possible. This result generalizes a previous result by the authors (2022) in base 2.

Our result for Levin's normal number in any prime base might support the guess that $O\left({(\log N)}^2\right)$ is the best order in N that can be achieved by a normal number, while generalizing the class of known normal numbers by introducing semi-perfect necklaces on the other hand might help for the search of normal numbers that satisfy smaller discrepancy bounds.