Distributed estimation of an unknown signal is a common task in sensor networks. The scenario usually envisioned consists of several nodes, each making an observation correlated with the signal of interest. The acquired data is then wirelessly transmitted to a central reconstruction point that aims at estimating the desired signal within a prescribed accuracy. Motivated by the obvious processing limitations inherent to such distributed infrastructures, we seek to find efficient compression schemes that account for limited available power and communication bandwidth. In this paper, we propose a transform-based approach to this problem where each sensor provides the central reconstruction point with a low-dimensional approximation of its local observation by means of a suitable linear transform. Under the mean-squared error criterion, we derive the optimal solution to apply at one sensor assuming all else being fixed. This naturally leads to an iterative algorithm whose optimality properties are exemplified using a simple though illustrative correlation model. The stationarity issue is also investigated. Under restrictive assumptions, we then provide an asymptotic distortion analysis, as the size of the observed vectors becomes large. Our derivation relies on a variation of the Toeplitz distribution theorem which allows to provide a reverse "water-filling" perspective to the problem of optimal dimensionality reduction. We illustrate, with a first-order Gauss-Markov model, how our findings allow to compute analytical closed-form distortion formulas that provide an accurate estimation of the reconstruction error obtained in the finite dimensional regime.

First, open **MATLAB** and navigate to the folder containing the code. Then, run the following command to reproduce *Figure X*

```
>> run genfig_X
```

Note that some figures are

- FR: fast reproducibility (taking less than 5 minutes)
- SR: slow reproducibility (taking more than 5 minutes)

The difficulty level for each figure is listed below

`genfig_2a`

: Reproduces Figure 2(a) (FR)`genfig_2b`

: Reproduces Figure 2(b) (FR)`genfig_3`

: Reproduces Figure 3 (SR)`genfig_5a`

: Reproduces Figure 5(a) (FR)`genfig_5b`

: Reproduces Figure 5(b) (FR)`genfig_5c`

: Reproduces Figure 5(c) (FR)`genfig_6a`

: Reproduces Figure 6(a) (FR)`genfig_6b`

: Reproduces Figure 6(b) (FR)`genfig_6c`

: Reproduces Figure 6(c) (FR)`genfig_7a`

: Reproduces Figure 7(a) (FR)`genfig_7b`

: Reproduces Figure 7(b) (FR)`genfig_7c`

: Reproduces Figure 7(c) (FR)`genfig_8`

: Reproduces Figure 8 (SR)

Also, note that you can run *Algorithm 1* in the paper using the MATLAB file `algo1.m`

(if is fast to reproduce).

- Microsoft Windows 8, with MATLAV R2014b

Copyright (c) 2008, Olivier Roy and Martin Vetterli Ecole Polytechnique Federale de Lausanne (EPFL),

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This software is distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose. See the GNU General Public License for more details (enclosed in the file GPL).

Olivier Roy and Martin Vetterli Laboratory for Audiovisual Communications (LCAV) at EPFL.

Comment | Rating | Date |

4 | 2015-07-07 |

2008

Easy to reproduce figures

★★★★★

★★★★★

Well-commented code

★★★★☆

★★★★☆

Olivier Roy; Martin Vetterli, "Dimensionality Reduction for Distributed Estimation in the Infinite Dimensional Regime", IEEE Transactions on Information Theory, Vol. 54, No. 2, pp. 1655-1669