Local reconstruction of low‐rank matrices and subspaces

We study the problem of reconstructing a low‐rank matrix, where the input is an n × m matrix M over a field and the goal is to reconstruct a (near‐optimal) matrix that is low‐rank and close to M under some distance function Δ. Furthermore, the reconstruction must be local, i.e., provides access to any desired entry of by reading only a few entries of the input M (ideally, independent of the matrix dimensions n and m). Our formulation of this problem is inspired by the local reconstruction framework of Saks and Seshadhri (SICOMP, 2010). Our main result is a local reconstruction algorithm for the case where Δ is the normalized Hamming distance (between matrices). Given M that is ‐close to a matrix of rank (together with d and ), this algorithm computes with high probability a rank‐d matrix that is ‐close to M. This is a local algorithm that proceeds in two phases. The preprocessing phase reads only random entries of M, and stores a small data structure. The query phase deterministically outputs a desired entry by reading only the data structure and 2d additional entries of M. We also consider local reconstruction in an easier setting, where the algorithm can read an entire matrix column in a single operation. When Δ is the normalized Hamming distance between vectors, we derive an algorithm that runs in polynomial time by applying our main result for matrix reconstruction. For comparison, when Δ is the truncated Euclidean distance and , we analyze sampling algorithms by using statistical learning tools. A preliminary version of this paper appears appears in ECCC, see: http://eccc.hpi-web.de/report/2015/128/ © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 607–630, 2017     

弱Orlicz鞅空间的弱原子分解及其应用

对3类由凹函数生成的弱Orlicz鞅空间建立了相应的弱原子分解.作为应用,首先给出了这些弱Orlicz鞅空间上次线性算子有界的一个充分条件,并在此基础上证明了一些弱型鞅不等式,然后证明了关于这些弱Orlicz鞅空间的Marcinkiewicz型插值定理.     

次线性期望下的一般中心极限定理

受Peng-中心极限定理的启发,本文主要应用G-正态分布的概念,放宽Peng-中心极限定理的条件,在次线性期望下得到形式更为一般的中心极限定理.首先,将均值条件E[X_n]=ε[X_n]=0放宽为|E[X_n]|+|ε[X_n]|=O(1/n);其次,应用随机变量截断的方法,放宽随机变量的2阶矩与2+δ阶矩条件;最后,将该定理的Peng-独立性条件进行放宽,得到卷积独立随机变量的中心极限定理.     

弱Orlicz鞅空间的原子分解

本文对于几种类型的弱Orlicz 鞅空间建立了强型和弱型的原子分解定理, 证明了这些空间上的次线性算子的有界性以及这些空间彼此的连续嵌入关系. 弱Orlicz 空间是一类拟Banach 空间, 有关结论扩展了现有的关于Orlicz 空间和弱型Lorentz 空间的相关结论.     

一类分数次积分算子在Herz型Hardy空间上的有界性

通过放宽尺寸条件,得到了一类具有分数次积分性质的次线性算子从Herz型Hardy空间到(弱)Herz型Hardy空间有界性的判定条件,以及端点处的弱型估计.     

How big are the increments of G-Brownian motion?

In this paper,we investigate the problem:How big are the increments of G-Brownian motion.We obtain the Csrg and R′ev′esz’s type theorem for the increments of G-Brownian motion.As applications of this result,we get the law of iterated logarithm and the Erds and R′enyi law of large numbers for G-Brownian motion.Furthermore,it turns out that our theorems are natural extensions of the classical results obtained by Csrg and R′ev′esz(1979).     

Existence and multiplicity results for infinitely many solutions for Kirchhoff‐type problems in

In this paper, we deal with existence and multiplicity of solutions of Kirchhoff‐type problems Under more relaxed assumptions on f, we establish some existence criteria to guarantee that the preceding problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical point theory. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Existence of Positive Solution for n OrderSingular Boundary Value Problems

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｅｘｉｓｔｅｎｃｅｆｏｒｔｈｅｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｈａｖｅｂｅｅｎｗｉｄｅｌｙｓｔｕｄｉｅｄｒｅｃｅｎｔｌｙ．Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｗｉｌｌｄｉｓｃｕｓｓｔｈｅＢＶＰｏｆｔｈｅｆｏｌｌｏｗｉｎｇｃｏｎｄｉｔｉｏｎｓ：ｕ（ｎ）＋ａ（ｔ）ｆ（ｕ）＝０，０＜ｔ＜１，ｕ（ｋ）（０）＝０，０ｋｎ－２（１．１）ｕ（１）＝０，　ｗｈｅｒｅ（ｈ１）ａ：（０，１）→（０，∞）ｉｓａｃｏｎｔｉｎｕｏｕｓｆｕｎｃｔｉｏｎ．（ｈ２）ｆ：［０，∞）→［…