{"id":606756,"date":"2019-09-02T14:40:39","date_gmt":"2019-09-02T21:40:39","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=606756"},"modified":"2020-08-28T17:00:05","modified_gmt":"2020-08-29T00:00:05","slug":"completion-of-high-rank-ultrametric-matrices-using-selective-entries","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/completion-of-high-rank-ultrametric-matrices-using-selective-entries\/","title":{"rendered":"Completion of high-rank ultrametric matrices using selective entries"},"content":{"rendered":"
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Ultrametric matrices are hierarchically structured matrices that arise naturally in many scenarios, e.g. delay covariance of packets sent from a source to a set of clients in a computer network, interactions between multi-scale communities in a social network, and genome sequence alignment scores in phylogenetic tree reconstruction problems. In this work, we show that it is possible to complete \\(\\)n\\times n[\\latex] ultrametric matrices using only \\(\\)n log^2(n)[\\latex] entries. Since ultrametric matrices are high- rank matrices, our results extend recent work on completion of \\(\\) n \\times n[\\latex] low-rank matrices that requires \\(\\)n log(n)[\\latex] randomly sampled entries. In the ultrametric setting, a random sampling of entries does not suffice, and we require selective sampling of entries using feedback obtained from entries observed at a previous stage.<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Ultrametric matrices are hierarchically structured matrices that arise naturally in many scenarios, e.g. delay covariance of packets sent from a source to a set of clients in a computer network, interactions between multi-scale communities in a social network, and genome sequence alignment scores in phylogenetic tree reconstruction problems. In this work, we show that it 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