Hutch++: Optimal Stochastic Trace Estimation.

We study the problem of estimating the trace of a matrix A that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch++, which computes a (1 ± ε) approximation to tr( A ) for any positive semidefinite (PSD) A using just O(1/ε) matrix-vector products. This improves on the ubiquitous Hutchinson's estimator, which requires O(1/ε 2) matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinson's estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinson's method in experiments. While our theory requires A to be positive semidefinite, empirical gains extend to applications involving non-PSD matrices, such as triangle estimation in networks.

[Is there a benefit to continue pap smear screening for cervical cancer after 65 years of age? A retrospective study on 53,644 women]. / Existe-t-il un intérêt à poursuivre les frottis de dépistage du cancer du col utérin après 65 ans ? Étude rétrospective sur 53 644 femmes.

BACKGROUND: One third of cervical cancers are discovered after age 65. French guidelines allow women to cease having cervical pap smears at age 65, providing they have had a minimum of two consecutive negative smears. AIM: To study the frequency of pathological smears in the group of women aged over 65 with a normal and sufficient cytological follow-up, according to the guidelines. To analyze the smear results and the former follow-up of this population to determine whether there is an interest to continue the smear screening after age 65. METHODS: This is a retrospective multicenter study based on collecting data from three pathology laboratories of the region Rhone-Alpes. We study a population of women 65 and older with smears that have been made between 2004 and 2008. RESULTS: Precancerous lesions and cervical cancer can be discovered after age 65 despite an adequate former follow-up. The rate of these women for having a pathological smear is 14,2‰, including 1,2‰ with cytological abnormalities in favour of cancer. CONCLUSION: These results show that there is interest to continue the smear tests after age 65 at least, while considering medical and financial issues of a reasonable screening strategy.