In this paper, we develop a local rank correlation measure which quantifies the performance of dimension reduction methods. The local rank correlation is easily interpretable, and robust against the extreme skewness of nearest neighbor distributions in high dimensions. Some benchmark datasets are studied. We find that the local rank correlation closely corresponds to our visual interpretation o...

We consider the problem of approximating a given matrix by a low-rank matrix so as to minimize the entry-wise `p-approximation error, for any p ≥ 1; the case p = 2 is the classical SVD problem. We obtain the first provably good approximation algorithms for this version of lowrank approximation that work for every value of p ≥ 1, including p =∞. Our algorithms are simple, easy to implement, work...

Though algebraic geometry over C is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the n × n × n tensors of rank n over C, which has as a dense subset the orbit of a single tensor under a natural group action. We construct polynomial invariants under this group action whose non-v...

Let n > 1 be an integer and p be a prime number. Denote by Cpn the class of non-thin association p-schemes of degree pn. A sharp upper and lower bounds on the rank of schemes in Cpn with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound...

Tensor decomposition has important applications in various disciplines, but it remains an extremely challenging task even to this date. A slightly more manageable endeavor has been to find a low rank approximation in place of the decomposition. Even for this less stringent undertaking, it is an established fact that tensors beyond matrices can fail to have best low rank approximations, with the...

In this paper, we consider the so-called structured low rank approximation (SLRA) problem as a problem of optimization on the set of either matrices or vectors. Briefly, SLRA is defined as follows. Given an initial matrix with a certain structure (for example, Hankel), the aim is to find a matrix of specified lower rank that approximates this initial matrix, whilst maintaining the initial struc...

BACKGROUND To evaluate the efficacy of postoperative radiotherapy (RT) on local failure-free survival (LFFS), distant metastasis-free survival (DMFS) and overall survival (OS) in patients with localized primary soft tissue sarcoma (STS) and to identify prognostic factors. METHODS AND MATERIALS Between January 2000 and July 2010, 220 consecutive patients with localized primary STS, who receive...