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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

The Paulsen problem is a basic problem in operator theory that was resolved in a recent tour-de-force work of Kwok, Lau, Lee and Ramachandran. In particular, they showed that every epsilon-nearly equal norm Parseval frame in d dimensions is within squared distance O(epsilon d^{13/2}) of an equal norm Parseval frame. We give a dramatically simpler proof based on the notion of radial isotropic position, and along the way show an improved bound of O(epsilon d^2).

Linus Hamilton and Ankur Moitra. The Paulsen Problem Made Simple. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 41:1-41:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hamilton_et_al:LIPIcs.ITCS.2019.41, author = {Hamilton, Linus and Moitra, Ankur}, title = {{The Paulsen Problem Made Simple}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {41:1--41:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.41}, URN = {urn:nbn:de:0030-drops-101347}, doi = {10.4230/LIPIcs.ITCS.2019.41}, annote = {Keywords: radial isotropic position, operator scaling, Paulsen problem} }

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Invited Talk

**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

In every corner of machine learning and statistics, there is a need for estimators that work not just in an idealized model but even when their assumptions are violated. Unfortunately in high-dimensions, being provably robust and efficiently computable are often at odds with each other. In this talk, we give the first efficient algorithm for estimating the parameters of a high-dimensional Gaussian which is able to tolerate a constant fraction of corruptions that is independent of the dimension. Prior to our work, all known estimators either needed time exponential in the dimension to compute, or could tolerate only an inverse polynomial fraction of corruptions. Not only does our algorithm bridge the gap between robustness and algorithms, it turns out to be highly practical in a variety of settings.

Ankur Moitra. Robustness Meets Algorithms (Invited Talk). In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{moitra:LIPIcs.SWAT.2018.3, author = {Moitra, Ankur}, title = {{Robustness Meets Algorithms}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {3:1--3:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.3}, URN = {urn:nbn:de:0030-drops-88294}, doi = {10.4230/LIPIcs.SWAT.2018.3}, annote = {Keywords: robust estimators, machine learning algorithms} }

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Invited Talk

**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Here we study some of the statistical and algorithmic problems that arise in recommendation systems. We will be interested in what happens when we move beyond the matrix setting, to work with higher order objects — namely, tensors. To what extent does inference over more complex objects yield better predictions, but at the expense of the running time? We will explore the computational vs. statistical tradeoffs for some basic problems about recovering approximately low rank tensors from few observations, and will show that our algorithms are nearly optimal among all polynomial time algorithms, under natural complexity-theoretic assumptions.
This is based on joint work with Boaz Barak.

Ankur Moitra. Beyond Matrix Completion (Invited Talk). In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, p. 8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{moitra:LIPIcs.FSTTCS.2015.8, author = {Moitra, Ankur}, title = {{Beyond Matrix Completion}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {8--8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.8}, URN = {urn:nbn:de:0030-drops-56650}, doi = {10.4230/LIPIcs.FSTTCS.2015.8}, annote = {Keywords: matrix completion, recommendation systems, tensor prediction} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

We show that for any odd k and any instance I of the max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a 1/2 + Omega(1/sqrt(D)) fraction of I's constraints, where D is a bound on the number of constraints that each variable occurs in.
This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a quantum algorithm to find an assignment satisfying a 1/2 Omega(D^{-3/4}) fraction of the equations.
For arbitrary constraint satisfaction problems, we give a similar result for "triangle-free" instances; i.e., an efficient algorithm that finds an assignment satisfying at least a mu + Omega(1/sqrt(degree)) fraction of constraints, where mu is the fraction that would be satisfied by a uniformly random assignment.

Boaz Barak, Ankur Moitra, Ryan O’Donnell, Prasad Raghavendra, Oded Regev, David Steurer, Luca Trevisan, Aravindan Vijayaraghavan, David Witmer, and John Wright. Beating the Random Assignment on Constraint Satisfaction Problems of Bounded Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 110-123, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{barak_et_al:LIPIcs.APPROX-RANDOM.2015.110, author = {Barak, Boaz and Moitra, Ankur and O’Donnell, Ryan and Raghavendra, Prasad and Regev, Oded and Steurer, David and Trevisan, Luca and Vijayaraghavan, Aravindan and Witmer, David and Wright, John}, title = {{Beating the Random Assignment on Constraint Satisfaction Problems of Bounded Degree}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {110--123}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.110}, URN = {urn:nbn:de:0030-drops-52981}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.110}, annote = {Keywords: constraint satisfaction problems, bounded degree, advantage over random} }

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