Distributed Karhunen-Loeve Transform With Nested Subspaces
A network in which sensors observe a common Gaussian source is analyzed. Using a fixed linear transform, each sensor compresses its high-dimensional observation into a low-dimensional representation. The latter is provided to a central decoder that reconstructs the source according to a mean squared error (MSE) distortion metric. The Distributed Karhunen-Loeve Transform (d-KLT) has been shown to provide a (locally) optimal linear solution for compression at each sensor. While the d-KLT achieves the lowest distortion linear reconstruction known, it does not maintain a nested subspace structure. In the case of ideal links to the decoder, this paper presents transforms that maintain nested subspaces, allowing the decoder to approximate a delay-limited source in an online fashion according to a desired sensor schedule. A distortion envelope for one distributed transform with nested subspace properties (d-nested-KLT) is provided. In the case of i.i.d. noise to the decoder, under assumptions of power allocation over subspaces, it is also possible to achieve nested subspaces utilizing correlations between sensors' observations. Results are applicable for data access over networks, and online information processing in sensor networks.