  Low rank approximation has been extensively studied in the past. It is most
suitable to reproduce rectangular like structures in the data. In this work we
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. Exemplary, a seismic wave can be recorded at $N$ sensors
with different time of arrival $\lambda$; Or a car moves through a video
changing its position in every frame. We present theoretical results as well as
an efficient algorithm to calculate a shifted rank-1 approximation in $O(NM
\log M)$. The benefit of the proposed method is demonstrated in numerical
experiments. A comparison to other sparse approximation methods is given.
Finally, we illustrate the utility of the extracted parameters for direct
information extraction in several applications including video processing or
non-destructive testing.
