##### Department of Mathematics,

University of California San Diego

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### Math 278B - Seminar on Mathematics of Information, Data, and Signals

## Florian Bossmann

#### Harbin Institute of Technology

## Shifted rank-$1$ approximation for seismic data

##### Abstract:

Low rank approximation has been extensively studied in the past. It is most suitable to reproduce rectangular like structures in the data. In this talk I introduce a generalization using ``shifted'' rank-1 matrices to approximate $A\in\mathbb{C}^{M\times N}$. These matrices are of the form $S_{\lambda}(uv^*)$ where $u\in\mathbb{C}^M$, $v\in\mathbb{C}^N$ and $\lambda\in\mathbb{Z}^N$. The operator $S_{\lambda}$ circularly shifts the $k$-th column of $uv^*$ by $\lambda_k$. These kind of shifts naturally appear in applications, where an object $u$ is observed in $N$ measurements at different positions indicated by the shift $\lambda$. The vector $v$ gives the observation intensity. This model holds e.g., for seismic waves that are recorded at $N$ sensors at different times $\lambda$. The main difficulty of the above stated problem lies in finding a suitable shift vector $\lambda$. Once the shift is known, a simple singular value decomposition can be applied to reconstruct $u$ and $v$. We propose a greedy method to reconstruct $\lambda$ and validate our approach in numerical examples. Reference: F. Boss mann, J. Ma, Enhanced image approximation using shifted rank-1 reconstruction, Inverse Problems and Imaging, accepted 2019, https://arxiv.org/abs/1810.01681

Caroline Moosmueller

### January 9, 2020

### 10:00 AM

### AP&M 2402

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