On the Logarithmic Factor in Smoothing Inequalities for the Lévy and Lévy–Prokhorov Distances

1987 ◽  
Vol 31 (4) ◽  
pp. 691-693 ◽  
Author(s):  
A. Yu. Zaitsev
Keyword(s):  
Logarithmic Factor

scholarly journals Decentralized and parallel primal and dual accelerated methods for stochastic convex programming problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Darina Dvinskikh ◽  
Alexander Gasnikov
Keyword(s):  
Convex Optimization ◽  
Inverse Problems ◽  
Convex Programming ◽  
Data Science ◽  
Optimization Problems ◽  
Stochastic Gradient ◽  
Logarithmic Factor ◽  
Accelerated Methods ◽  
Convex Optimization Problems ◽  
Primal And Dual

Abstract We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles, the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique, we show that the proposed methods with stochastic oracle can be additionally parallelized at each node. The considered algorithms can be applied to many data science problems and inverse problems.

On the rate of Poisson process approximation to a Bernoulli process

1997 ◽  
Vol 34 (4) ◽  
pp. 898-907 ◽  
Author(s):  
Aihua Xia
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Discrete Time ◽  
Wasserstein Distance ◽  
Stein’S Method ◽  
Upper Bounds ◽  
Stein's Method ◽  
Bernoulli Process ◽  
Logarithmic Factor ◽  
Process Approximation

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.

scholarly journals Dispersing Hash Functions

2000 ◽  
Vol 7 (36) ◽  
Author(s):  
Rasmus Pagh
Keyword(s):  
Lower Bounds ◽  
Random Function ◽  
Hash Functions ◽  
Expected Value ◽  
Independent Interest ◽  
Upper And Lower Bounds ◽  
Related Issue ◽  
Logarithmic Factor

A new hashing primitive is introduced: dispersing hash functions. A family<br />of hash functions F is dispersing if, for any set S of a certain size and random<br />h in F, the expected value of |S|−|h[S]| is not much larger than the expectancy<br />if h had been chosen at random from the set of all functions.<br />We give tight, up to a logarithmic factor, upper and lower bounds on the<br />size of dispersing families. Such families previously studied, for example <br />universal families, are significantly larger than the smallest dispersing families,<br />making them less suitable for derandomization. We present several applications<br /> of dispersing families to derandomization (fast element distinctness, set<br />inclusion, and static dictionary initialization). Also, a tight relationship <br />between dispersing families and extractors, which may be of independent interest,<br />is exhibited.<br />We also investigate the related issue of program size for hash functions<br />which are nearly perfect. In particular, we exhibit a dramatic increase in<br />program size for hash functions more dispersing than a random function.

scholarly journals Optimal Time-Space Trade-Offs for Sorting

1998 ◽  
Vol 5 (10) ◽  
Author(s):  
Jakob Pagter ◽  
Theis Rauhe
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Fundamental Problem ◽  
Full Range ◽  
Optimal Time ◽  
Sorting Algorithm ◽  
Logarithmic Factor ◽  
Time Space ◽  
Trade Offs ◽  
Space Product

We study the fundamental problem of sorting in a sequential model of computation and in particular consider the time-space trade-off (product of time and space) for this problem.<br />Beame has shown a lower bound of  Omega(n^2) for this product leaving a gap of a logarithmic factor up to the previously best known upper bound of O(n^2 log n) due to Frederickson. Since then, no progress has been made towards tightening this gap.<br />The main contribution of this paper is a comparison based sorting algorithm which closes this gap by meeting the lower bound of Beame. The time-space product O(n^2) upper bound holds for the full range of space bounds between log n and n/log n. Hence in this range our algorithm is optimal for comparison based models as well as for the very powerful general models considered by Beame.

scholarly journals Multilevel weighted least squares polynomial approximation

2020 ◽  
Vol 54 (2) ◽  
pp. 649-677 ◽  
Author(s):  
Abdul-Lateef Haji-Ali ◽  
Fabio Nobile ◽  
Raúl Tempone ◽  
Sören Wolfers
Keyword(s):  
Least Squares ◽  
Polynomial Approximation ◽  
Weighted Least Squares ◽  
A Priori ◽  
Computational Cost ◽  
Discretization Error ◽  
Logarithmic Factor ◽  
Complexity Bounds ◽  
Single Level ◽  
Random Samples

Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.

scholarly journals Constrained Ramsey Numbers

2009 ◽  
Vol 18 (1-2) ◽  
pp. 247-258 ◽  
Author(s):  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Ramsey Theorem ◽  
Logarithmic Factor ◽  
Subgraph Isomorphic ◽  
Rate Of Growth ◽  
Special Cases ◽  
Edge Colouring

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang and Ling showed that f(S, T) ≤ O(st2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f(S, Pt) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.

scholarly journals THE FINITE PART OF DIVERGENT INTEGRALS WITH LOGARITHMIC FACTORS

2011 ◽  
Vol 16 (4) ◽  
pp. 537-557 ◽  
Author(s):  
Kaido Latt
Keyword(s):  
Singular Function ◽  
Finite Part ◽  
Logarithmic Factor ◽  
Change Of Variables

The concepts of the finite part (f.p.) and analytic finite part (a.f.p.) of divergent integrals are defined in the situation where the singular function in the integral has a logarithmic factor. The change of variables in f.p.- and a.f.p-integrals is examined.

scholarly journals Note on Long Paths in Eulerian Digraphs

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Charlotte Knierim ◽  
Maxime Larcher ◽  
Anders Martinsson
Keyword(s):  
Recent Result ◽  
Short Note ◽  
Average Degree ◽  
Logarithmic Factor ◽  
Long Paths

Long paths and cycles in Eulerian digraphs have received a lot of attention recently. In this short note, we show how to use methods from [Knierim, Larcher, Martinsson, Noever, JCTB 148:125--148] to find paths of length $d/(\log d+1)$ in Eulerian digraphs with average degree $d$, improving  the recent result of $\Omega(d^{1/2+1/40})$. Our result is optimal up to at most a logarithmic factor.  

scholarly journals On the Rate of Convergence of Difference Schemes for the Poisson Equation with Dynamic Interface Conditions

2003 ◽  
Vol 3 (1) ◽  
pp. 177-188 ◽  
Author(s):  
Boško Jovanovič ◽  
Lubin G. Vulkov
Keyword(s):  
Rate Of Convergence ◽  
Difference Schemes ◽  
Interface Condition ◽  
Problem Solution ◽  
Two Dimensional ◽  
Differential Problem ◽  
Logarithmic Factor ◽  
The Poisson Equation ◽  
Dynamic Interface ◽  
Sobolev Norms

AbstractThe convergence of difference schemes for the two–dimensional weakly parabolic equation (elliptic equation with a dynamic interface condition) is studied. Estimates for the rate of convergence “almost” (except for the logarithmic factor) compatible with the smoothness of the differential problem solution in special discrete Sobolev norms are obtained.

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