Optimized Quantization in Distributed Graph Signal Processing

Abstract

Distributed graph signal processing methods require that the graph nodes communicate by exchanging messages. These messages have a finite precision in a realistic network, which may necessitate to implement quantization. Quantization, in turn, generates errors in the distributed processing tasks, compared to perfect settings. This paper proposes a novel method to minimize the quantization error without compromising the communication costs by bounding the exchanged messages along with allocating a limited bit budget through the network in an optimized way. In particular, the quantization adapts to the network topology and message importance in the iterative distributed processing algorithm. Our results show that the proposed method is efficient in minimizing the quantization error and that it outperforms baseline algorithms when the bit budget is limited.

Quantization Analysis and Robust Design for Distributed Graph Filters

“…(37) Under the RES graph model and given the zero-mean quantization noise, it can be easily shown from (33) and ( 34) that E[w q t ] = E[w t ]; i.e., in expectation both the quantized and unquantized ARMA filters give the same output. However, the quantization impacts on the second order moment of the filter output error ε yt in (36). We analyze next the MSE of the latter, which by simple algebra, can be split as:…”

“…However, in consensus, the goal is to exchange quantized data to reach a consensus with respect to some global quantity, while with graph filters the goal is to exchange quantized data to perform any graph filtering task. The importance of quantization from the graph signal processing perspective has been recently recognized in [35][36][37]. In particular, [35] -the most related to our workdiscusses the impact of fixed-stepsize quantization on FIR graph filters.…”

“…The work in [37] approximates the graph spectral dictionaries as polynomials of the graph Laplacian operator and learns polynomial dictionaries that are robust to signal quantization. Finally, [36] develops an adaptive quantization scheme for FIR graph filters that minimizes the quantization errors by bounding the exchanged messages and optimizing the bit allocation. While being relevant contributions on the quantization aspects of graph filtering, the limitation of these works is that they focus solely on FIR graph filters and fixedstepsize quantization.…”

Task-Based Graph Signal Compression

“…1) Greedy Bit Assignment: The proposed greedy algorithm starts by setting the minimal bit allocation for all the quantizers, i.e., M (0) i = 1 for each i ∈ P. Then, it increments the number of levels for the quantizer which contributes most to the objective (17). While this procedure does not impose the constraint M i ≥ M i+1 for each i ∈ {1, 2, ..., P − 1}, it is implicitly maintained due to the descending order of {λ Γ,i }.…”

“…While this procedure does not impose the constraint M i ≥ M i+1 for each i ∈ {1, 2, ..., P − 1}, it is implicitly maintained due to the descending order of {λ Γ,i }. Specifically, the gradient of the objective (17)…”