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Quantization plays an important role as an interface between analog and digital environments. Since quantization is a many to few mapping, it is a non-linear irreversible process. This made, in addition of the quantization noise signal dependency, the traditional methods of system identification no longer applicable. In this work, we propose a method for parsimonious system identification when only quantized measurements of the output are observable. More precisely, we develop an algorithm that aims at identifying a low order system that is compatible with a priori information on the system and the collected quantized output information. Moreover, the proposed approach can be used even if only fragmented information on the quantized output is available. The proposed algorithm relies on an ADMM approach to p quasi-norm optimization. Numerical results highlight the performance of the proposed approach when compared to the 1 minimization in terms of the sparsity of the induced solution.

Sparsity finds applications in areas as diverse as statistics, machine learning, and signal processing.Computations over sparse structures are less complex compared to their dense counterparts, and their storage consumes less space. This paper proposes a heuristic method for retrieving sparse approximate solutions of optimization problems via minimizing the ℓ p quasi-norm, where 0 < p < 1. An iterative two-block ADMM algorithm for minimizing the ℓ p quasi-norm subject to convex constraints is proposed. For p = s/q < 1, s, q ∈ Z + , the proposed algorithm requires solving for the roots of a scalar degree 2q polynomial as opposed to applying a soft thresholding operator in the case of ℓ 1 . The merit of that algorithm relies on its ability to solve the ℓ p quasi-norm minimization subject to any convex set of constraints. However, it suffers from low speed, due to a convex projection step in each iteration, and the lack of mathematical convergence guarantee. We then aim to vanquish these shortcomings by relaxing the assumption on the constraints set to be the set formed due to convex and differentiable, with Lipschitz continuous gradient, functions, i.e. specifically, polytope sets. Using a proximal gradient step, we mitigate the convex projection step and hence enhance the algorithm speed while proving its convergence. We then present various applications where the proposed algorithm excels, namely, matrix rank minimization, sparse signal reconstruction from noisy measurements, sparse binary classification, and system identification. The results demonstrate the significant gains obtained by the proposed algorithm compared to those via ℓ 1 minimization.

Quantization is the process of mapping an input signal from an infinite continuous set to a countable set with a finite number of elements. It is a non-linear irreversible process, which makes the traditional methods of system identification no longer applicable. In this work, we propose a method for parsimonious linear time invariant system identification when only quantized observations, discerned from noisy data, are available. More formally, given a priori information on the system, represented by a compact set containing the poles of the system, and quantized realizations, our algorithm aims at identifying the least order system that is compatible with the available information. The proposed approach takes also into account that the available data can be subject to fragmentation. Our proposed algorithm relies on an ADMM approach to solve a p , (0 < p < 1), quasi-norm objective problem. Numerical results highlight the performance of the proposed approach when compared to the 1 minimization in terms of the sparsity of the induced solution.

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