# 2013-14: LDHD: SAMSI-CRM Workshop on Geometric Aspects of High-dimensional Inference: March 31-April 2, 2014

## Workshop Information

The workshop was held at SAMSI in Research Triangle Park, NC, and was a joint venture between SAMSI and Centre de Recerca Matemàtica (CRM) in Barcelona, Spain.

The development of methods of statistical inference for high-dimensional data has become a focal point of research in statistics and machine learning in the recent years. One of the crucial problems is to understand how to estimate efficiently a very high-dimensional object under certain "low complexity" constraints that make the estimation possible. In specific settings, "low complexity" could mean, for instance, sparsity of a vector in a high-dimensional space or low-rank properties of a large matrix. The goal is to develop methods that would be adaptive to the underlying "low complexity" structure. Such problems are extremely important both in statistics -- for instance, in high-dimensional regression or in covariance matrix estimation -- and in a variety of applications, including compressed sensing, collaborative filtering, and quantum state tomography.

Theoretical analysis of methods of high-dimensional inference often relies on deep understanding of the underlying geometry of high-dimensional spaces that leads to highly nontrivial problems of geometric nature. Similar problems have occurred and have been studied in such areas of mathematics as high-dimensional probability, random matrix theory, asymptotic geometric analysis, convex geometry, and additive combinatorics. Some of the tools developed in these areas proved to be extremely useful in high-dimensional statistics; these tools include empirical processes methods, concentration inequalities, and various techniques from the theory of random matrices. There is also great potential for applications of other tools developed in recent years, such as generic chaining bounds for stochastic processes or an emerging theory of log-concave distributions in high-dimensional spaces.

The goal of this workshop was to bring together researchers actively working on the development of high-dimensional inference in statistics, machine learning, compressed sensing, and other related areas with mathematicians who made major contributions to high-dimensional probability and asymptotic geometric analysis in recent years. This provided a great opportunity for fruitful discussions of cutting edge problems in high-dimensional statistics and major advances in our understanding of geometry of high-dimensional spaces.

Topics covered by the workshop may included:

- low rank matrix estimation;
- sparse recovery and compressed sensing;
- covariance estimation for high-dimensional data;
- model selection and oracle inequalities in high-dimensional statistics;
- statistical inference for log-concave distributions in high dimensions;
- non-asymptotic theory of random matrices;
- concentration inequalities, generic chaining, and empirical processes methods in high-dimensional statistics;
- hypotheses testing for high-dimensional objects.

If you need further information please send an email to ldhd@samsi.info .

Schedule

Participant List

Speaker Titles and Abstracts

Poster Titles

### Schedule

**Monday, March 31, 2014 at SAMSI, Room 150**

8:30-8:55 a.m. | Registration |

8:55-9:00 | Welcome |

9:00-10:30 | Roman Vershynin, University of MichiganHigh-Dimensional Estimation: Geometric and Probabilistic Insights |

10:30-11:00 | Break |

11:00-11:45 | Sourav Chatterjee, Stanford UniversityLeast Squares under Convex Constraint |

11:45-1:00 | Lunch (at SAMSI) |

1:00-1:45 | Witold Bednorz, University of WarsawBernoulli Theorem and Empirical Processes |

1:45-2:00 | Break |

2:00-2:45 | Christian Houdre, Georgia Institute of TechnologyOn the Limiting Law of the Length of the Longest Common and Increasing Subsequence |

2:45-3:00 | Break |

3:00-3:45 | Guillaume Lecue, University Marne-le-ValleeCompressed Sensing Under Weak Moment Conditions |

3:45-4:00 | Break |

4:00-5:00 | Stas Minsker, Duke UniversityGeometric Median: Applications to Robust and Scalable Statistical Estimation |

5:00-6:30 | Poster Session and Reception SAMSI will provide poster presentation boards and tape. The board dimensions are 4 ft. wide by 3 ft. high. They are tri-fold with each side being 1 ft. wide and the center 2 ft. wide. Please make sure your poster fits the board. The boards can accommodate up to 16 pages of paper measuring 8.5 inches by 11 inches. |

**Tuesday, April 1, 2014 at SAMSI, Room 150**

9:00-10:30 a.m. | Gabor Lugosi, Pompeu Fabra UniversityDetection of Correlations and High-Dimensional Random Geometric Graphs |

10:30-11:00 | Break |

11:00-11:45 | Andrew Nobel, University of North CarolinaLarge Average Submatrices of a Gaussian Random Matrix: Landscapes and Local Optima |

11:45-1:00 | Lunch (at SAMSI) |

1:00-1:45 | Zongming Ma, University of PennsylvaniaEstimating High-dimensional Matrices: Convex Geometry and Computational Barriers |

1:45-2:00 | Break |

2:00-2:45 | Shuheng Zhou, University of MichiganGEMINI: Graph Estimation with Matrix Variate Normal Instances |

2:45-3:00 | Break |

3:00-3:45 | Harrison Zhou, Yale UniversitySparse Canonical Correlation Analysis |

3:45-4:00 | Break |

4:00-4:45 | Alexander Rakhlin, University of PennsylvaniaUniform Martingale LLN, Sequential Complexities, and Applications to Online Learning |

**Wednesday, April 2, 2014 at SAMSI, Room 150**

9:00-9:45 a.m. | Justin Romberg, Georgia Institute of TechnologyCompressed Subspace Matching on the Continuum |

9:45-10:00 | Break |

10:00-10:45 | Rob Nowak, University of WisconsinLogistic Regression with Structured Sparsity |

10:45-11:00 | Break |

11:00-11:45 | Johannes Lederer, Cornell UniversityDon't Fall for Tuning Parameters |

11:45-1:00 | Lunch (at SAMSI) |