A classic problem is the estimation of a set of parameters from measurements collected by only a few sensors. The number of sensors is often limited by physical or economical constraints and their placement is of fundamental importance to obtain accurate estimates. Unfortunately, the selection of the optimal sensor locations is intrinsically combinatorial and the available approximation algorithms are not guaranteed to generate good solutions in all cases of interest. We propose FrameSense, a greedy algorithm for the selection of optimal sensor locations. The core cost function of the algorithm is the frame potential, a scalar property of matrices that measures the orthogonality of its rows. Notably, FrameSense is the first algorithm that is near-optimal in terms of mean square error, meaning that its solution is always guaranteed to be close to the optimal one. Moreover, we show with an extensive set of numerical experiments that FrameSense achieves state-of-the-art performance while having the lowest computational cost, when compared to other greedy methods.

The following contains instructions to reproduce the results of the paper
*Near-Optimal Sensor Placement for Linear Inverse Problems*.

First download the code (and uncompress the files if necessary). Then, open up *MATLAB* and navigate to the folder where the code is located. Then, for figure *X* run

```
>> figX_plot
```

Note that the data for the figures is located in the `/data`

folder. If you want to run the algorithms from scratch, simply execute

```
>> figX_algo.m
```

To get the results for different panels in *Figure 2* (*Figure 4*), you can specify the `matrix_type`

parameter within the `fig2_plot.m`

(`fig4_plot.m`

). The options are:

`matrix_type=0`

: to reproduce the figure for*random normalized matrix*`matrix_type=1`

: to reproduce the figure for*random matrix*`matrix_type=2`

: to reproduce the figure for*DCT randomly selected matrix*`matrix_type=3`

: to reproduce the figure for*random orthogonalized matrix*`matrix_type=4`

: to reproduce the figure for*Bernoulli 0.5*

- Microsoft Windows 8, MATLAB R2015b
- Max Book Pro 2012, Matlab R2015b

Juri Ranieri, Amina Chebira and Martin Vetterli Laboratory for Audiovisual Communications (LCAV) at EPFL.

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