1. High-Dimensional Matched Subspace Detection When Data are Missing
Reading Group
High-Dimensional Matched Subspace Detection When Data are Missing
(Univ Wisconsin-Madison: Laura Balzano, Benjamin Recht, and Robert Nowak, Feb 2010, 1 self-citation)
Presenter: Zhe Chen
ECE / CMR
Tennessee Technological University
April 8, 2011

2. Outline Introduction Main Contribution of This Paper
Numerical Experiments
Conclusion
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3. The Problem (1/2)
This paper considers the problem of deciding whether a highly incomplete signal lies within a given subspace.
Let denote a signal and let , where w is a noise of known distribution. We are given a subspace and we wish to decide if or not, based on x.
Consider a variation where it is prohibitive or impossible to measure v completely. Assume that only a small subset
of the elements of v are observed (with or without noise), and based on these observations we want to test whether
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4. The Problem (2/2)
For example, a large networked system. It may be impossible to obtain every measurement from every point in the network.
(Cognitive Radio Network, Smart Grid, Wireless Tomography…)
This paper focuses on an estimator of the energy of v in S based on only observing the elements
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5. Energy Estimation from Incomplete Data (1/2)
There are two natural estimators of based on , where is the vector of dimension comprised of the elements , and denotes the projection operator onto S.
The first is simply to form the vector with elements if
and zero if This ‘zero-filled’ vector yields the simple estimator Unfortunately, the estimator is fundamentally flawed. Even if , the zero-filled vector does not necessarily lie in S.
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6. Energy Estimation from Incomplete Data (2/2)
A better estimator: Where ,
It follows immediately that if , then
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7. Main Theorem
Coherence of a subspace S:
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8. Lemmas for the Main Theorem
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9. Numerical Experiments
Incoherent subspace (spanned by the columns of the Fourier basis). r = 10, n = 100
Coherent subspace. r = 10, n = 120
Each of the simulations had a fixed vector and a particular sample size m but different sample set , and we show the mean, minimum and maximum projection error of those 100 simulations.
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10. Conclusion
It is possible to detect whether a highly incomplete vector has energy outside a subspace.
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11. Tennessee Technological University
Reading Group
Online Identification and Tracking of Subspaces from Highly Incomplete Information
(Univ of Wisconsin-Madison: Laura Balzano, Robert Nowak and Benjamin Recht, Jun 2009, 0 citation)
Presenter: Zhe Chen
ECE / CMR
Tennessee Technological University
April 8, 2011

12. Outline Introduction Main Contribution of This Paper
Numerical Experiments
Discussion
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13. The Problem (1/2)
This work presents an efficient online algorithm for tracking subspaces from highly incomplete observations, named GROUSE (Grassmanian Rank-One Update Subspace Estimation).
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14. The Problem (2/2)
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15. Stochastic Gradient Decent on the Grassmannian
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16. Numerical Experiments (1/2)
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17. Numerical Experiments (2/2)
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18. Discussion
Though we can guarantee that there is a basin of attraction around the global minimum, it is not yet clear how to characterize when GROUSE will end up in this basin.
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19. Thank you!
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