Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018. / Cataloged from PDF version of thesis. / Includes bibliographical references (pages 151-167). / Motivated by various applications such as seriation, network alignment and ranking from pairwise comparisons, we study the problem of estimating a structured matrix with rows and columns shuffled by latent permutations, given noisy and incomplete observations of its entries. This problem is at the intersection of shape constrained estimation which has a long history in statistics, and latent permutation learning which has driven a recent surge of interest in the machine learning community. Shape constraints on matrices, such as monotonicity and smoothness, are generally more robust than parametric assumptions, and often allow for adaptive and efficient estimation in high dimensions. On the other hand, latent permutations underlie many graph matching and assignment problems that are computationally intractable in the worst-case and not yet well-understood in the average-case. Therefore, it is of significant interest to both develop statistical approaches and design efficient algorithms for problems where shape constraints meet latent permutations. In this work, we consider three specific models: the statistical seriation model, the noisy sorting model and the strong stochastic transitivity model. First, statistical seriation consists in permuting the rows of a noisy matrix in such a way that all its columns are approximately monotone, or more generally, unimodal. We study both global and adaptive rates of estimation for this model, and introduce an efficient algorithm for the monotone case. Next, we move on to ranking from pairwise comparisons, and consider the noisy sorting model. We establish the minimax rates of estimation for noisy sorting, and propose a near-linear time multistage algorithm that achieves a near-optimal rate. Finally, we study the strong stochastic transitivity model that significantly generalizes the noisy sorting model for estimation from pairwise comparisons. Our efficient algorithm achieves the rate (n- 3 /4 ), narrowing a gap between the statistically optimal rate Õ(n-1 ) and the state-of-the-art computationally efficient rate [Theta] (n- 1/ 2 ). In addition, we consider the scenario where a fixed subset of pairwise comparisons is given. A dichotomy exists between the worst-case design, where consistent estimation is often impossible, and an average-case design, where we show that the optimal rate of estimation depends on the degree sequence of the comparison topology. / by Cheng Mao. / Ph. D.
| Identifer | oai:union.ndltd.org:MIT/oai:dspace.mit.edu:1721.1/117863 |
| Date | January 2018 |
| Creators | Mao, Cheng, Ph. D. Massachusetts Institute of Technology |
| Contributors | Philippe Rigollet., Massachusetts Institute of Technology. Department of Mathematics., Massachusetts Institute of Technology. Department of Mathematics. |
| Publisher | Massachusetts Institute of Technology |
| Source Sets | M.I.T. Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 167 pages, application/pdf |
| Rights | MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission., http://dspace.mit.edu/handle/1721.1/7582 |
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