# Seminar

# Mathematical Theory of Model-Based Inference from Incomplete Observations

작성자Author

관리자

작성일Date

2017-11-09 14:36

조회Views

695

분야Field | 수학과 전임교원 임용후보자공개 세미나 | ||
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날짜Date | 2017-11-08 | 시간Time | 5:00 ~ 6:30 |

장소Place | Math. Bldg. 404 | 초청자Host | 수학과 |

연사Speaker | 이기륭 | 소속Affiliation | Georgia Institute of Technology/Research Scientist II |

TOPIC | Mathematical Theory of Model-Based Inference from Incomplete Observations | ||

소개 및 안내사항Content | Title : Mathematical Theory of Model-Based Inference from Incomplete ObservationsAbstract : In the era of big data, there are many situations where one has to infer from incomplete observations. The imperfection of available data arises in various forms. In this talk, we focus on the following two scenarios. First, we consider compressed learning of high dimensional data. When the original data in high dimension follow a simple geometric model, statistical learning on compressed features in low dimension provides comparable generalization bound. The restricted isometry property (RIP) preserves the geometry of the underlying model and has been an integral tool for the mathematical theory of compressed learning.We propose generalized notions of sparsity and provide a unified framework for the RIP of structured random measurements given by isotropic group actions. Our results extend the RIP for partial Fourier measurements to a much broader context and identify a sufficient number of group structured measurements for the RIP on generalized sparsity models. We illustrate the main results on an infinite dimensional example, where the sparsity represented by a condition that approximates the total variation. Second, we consider the scenario where the desired information as a time series is accessed as indirect observations through a time-invariant system with uncertainty. The measurements in this case is given in the form of the convolution with an unknown kernel and the estimation is cast as blind deconvolution. Particularly, we study the mathematical theory of multichannel blind deconvolution where we observe the output of multiple channels that are all excited with the same unknown input source. From these observations, we wish to estimate the source and the impulse responses of each of the channels simultaneously. We show that this problem is well-posed if the channel impulse responses follow a simple geometric model. Under these models, we show how the channel estimates can be found by solving corresponding non-convex optimization problems. We analyze methods for solving these non-convex programs, and provide performance guarantees for each. |

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