Sparse System Identification

• Sparse system identification is the process of estimating a dynamic system's model by assuming only a few nonzero coefficients, ensuring parsimony and robustness.
• It leverages techniques such as compressed sensing, convex relaxations, and sparse regression (e.g., SINDy) to accurately recover system parameters even in the presence of noise.
• Applications span wireless communications, biomedicine, and physics, with ongoing research addressing real-time, hybrid, and uncertainty quantification challenges.

Sparse system identification is the process of estimating a dynamical system’s structure and parameters under the explicit assumption that only a small subset of the possible system coefficients are nonzero. This structural prior serves to regularize ill-posed estimation problems, promote interpretability, accelerate convergence, and provide robustness in high-dimensional, potentially contaminated, or subsampled settings common to modern applications in signal processing, control, communications, biomedicine, and physics-informed data analysis.

1. Problem Formulation and Sparse Priors

Sparse system identification is posed as the recovery of system parameters or equations under the assumption of sparsity in the underlying model. In discrete-time LTI settings, the canonical input–output model is

$y = Hx + e + w$

where $H$ is typically Toeplitz-structured, $x$ is the parameter vector to be identified, $e$ is a sparse outlier (gross error) vector, and $w$ is random observation noise (Xu et al., 2012). In continuous-time, the system is modeled as

$\dot{x} = f(x,u)$

with $f$ parameterized sparsely in a user- or physics-informed function dictionary (Du et al., 13 Feb 2025, Kaptanoglu et al., 2023).

Sparsity assumptions enter either directly—as cardinality constraints, e.g., via $\ell_0$ penalties—or through convex relaxations such as $\ell_1$-norm or block/group regularization, or in Bayesian frameworks by imposing sparsity-inducing priors (e.g., Laplace or group-automatic relevance determination) (Zhou et al., 2021). This promotes global models that are both parsimonious and robust to overfitting.

2. Algorithmic Methodologies

Algorithmic approaches to sparse system identification vary by system type, data setting, and application requirements:

• Compressed Sensing for Adaptive Filters: Combining random or structured measurement matrices (e.g., via filtering and decimation) with sparse recovery solvers exploits the denoising and underdetermined estimation capabilities of compressed sensing techniques (Hosseini et al., 2012). The process targets a compressed representation of the system, applies adaptive filtering in the reduced domain, and recovers the full parameter vector through $\ell_1$-regularized or Bayesian estimators.
• Convex Relaxation and Block/Group Regularization: High-dimensional LTI system estimation employs block- or group-sparse regularization solving

$$\widehat{\Theta} = \arg\min_{\Theta} \tfrac{1}{2d}\|Y - X\Theta\|_F^2 + \lambda \|\Theta\|_{\text{block}}$$

where $\|\cdot\|_{\text{block}}$ is typically a sum over block or group norms (e.g., $\ell_{1}/\ell_{\infty}$) that reflects known structural sparsity (Fattahi et al., 2018).

• Sparse Regression in Nonlinear Dynamics: In nonlinear system identification, the SINDy (Sparse Identification of Nonlinear Dynamics) framework formulates

$\dot{x} = \Theta(x) \Xi$

with $\Theta(x)$ a dictionary of candidate functions (derived analytically for physical consistency in PC-SINDy (Du et al., 13 Feb 2025) or polynomials/trigonometric functions for generic applications (Kaptanoglu et al., 2023)). Sequential thresholded least squares, Lasso, mixed-integer optimization, and other sparse regression solvers are applied for coefficient recovery. Recent advances leverage weak formulations and ensembling to boost robustness to noise and sampling deficiencies (Kaptanoglu et al., 2023).

• Sparsity-Aware Adaptive Filters: Standard LMS (Least Mean Square) algorithms can be augmented with sparsity-promoting terms—such as $\ell_1$, $\ell_0$, $p$-norm, or partial-norm penalties—to form variants like ZA-LMS, $l_0$-LMS, p-LLMs, or adaptive convex combinations of multiple filters (Jin et al., 2013, Gu et al., 2013, Feng et al., 2015, Feng et al., 2015, Feng et al., 2015, Gui et al., 2013, Gogineni, 2015). Gradient comparators, partial updating, adaptive penalty parameter selection, and dynamic windowing are common techniques to balance convergence speed and steady-state misadjustment.
• Regularization-Based Approaches under Feedback: For stochastic feedback systems, identification with adaptive weighted $\ell_1$ penalties on the LS estimate ensures set convergence (support recovery) and parameter convergence (consistency), even when current inputs depend on past outputs and exogenous noise (Zhao et al., 2019).
• Bayesian Deep Learning and Low-Rank Approximations: Sparse Bayesian neural networks—with group sparsity-inducing priors, recursive Hessian computation, and Laplace approximation—achieve both robustness and model selection, supporting structured inference in high-dimensional and nonlinear dynamical processes (Zhou et al., 2021). Low-rank matrix approximations and iterative optimization (e.g., Levenberg–Marquardt) with backward elimination yield sparse models suitable for systems with irregular sampling or partial observability (Haring et al., 2022, Vides, 2021).

3. Theoretical Guarantees and Performance Bounds

Theoretical analysis addresses sample complexity, recovery guarantees, and error bounds:

• Null Space and Restricted Isometry Properties: For Toeplitz/convolutional measurement operators in compressed sensing or robust estimation, a null space property ensures that support recovery via $\ell_1$ minimization is achievable for outlier fractions below a calculable threshold (Xu et al., 2012).
• Sample Complexity in High-Dimensional Systems: Block-regularized estimators achieve small per-element error and exact support recovery with a number of sample trajectories that scales polynomially with the block size and per-block sparsity but only logarithmically with the system dimension—significantly less than the sample size required for standard least squares (Fattahi et al., 2018, Fattahi et al., 2019).
• Finite-Time Support and Value Recovery: Under mutual incoherence and stability-type conditions, Lasso-like estimators guarantee, with high probability, both exact sparsity pattern recovery and explicit $\ell_\infty$ error bounds, as soon as data length exceeds a computable (polylogarithmic in system size) threshold (Fattahi et al., 2019).
• Consistency under Noise and Outliers: Algorithms based on adaptive weighted $\ell_1$ penalties and iterative LS estimation achieve almost sure convergence of null coefficients to zero and nonzero coefficient estimates to true values—even under feedback and in presence of nonvanishing noise (Zhao et al., 2019).

4. Practical Applications and Robustness

Sparse system identification methodologies have demonstrated utility in:

• Wireless Communications and Channel Estimation: Identification of sparse impulse responses (as in multipath channels) yields fast convergence and reduced error under low SNR or partial pilot data (Hosseini et al., 2012, Feng et al., 2015).
• Echo Cancellation and Adaptive Filtering: Sparse adaptive and proportionate filters empirically outperform standard LMS in acoustic echo reduction and change tracking in telecommunication and audio systems (Jin et al., 2013, Gogineni, 2015).
• Power Systems and Microgrids: Physically consistent SINDy-based methods extract interpretable nonlinear dynamical models from noisy PMU data, enabling robust prediction and stability analysis even under large, untrained disturbances and incomplete knowledge of DER configurations (Du et al., 13 Feb 2025).
• Systems Biology, Biomedicine, and Physics: Sparse identification approaches with analytically constructed dictionaries support the extraction of governing equations from limited, noisy, and unevenly sampled data (Yue et al., 2016, Vides, 2021).

Robustness to outliers, colored input, and time-varying structure is achieved through strategies such as: denoising by compressed sensing recovery, adaptive filter windowing and convex combinations, and ensembling over noise-perturbed datasets.

5. Benchmarking, Evaluation, and Comparisons

Systematic benchmarking leveraging standardized chaotic dynamical system databases (e.g., dysts (Kaptanoglu et al., 2023)) has established the relative efficacy of sparse regression algorithms (STLSQ, Lasso, SR3, MIOSR, weak SINDy):

Algorithm	Strengths	Limitations
STLSQ	Speed, robustness, accuracy on clean data	Greedy errors under noise
MIOSR	Exact sparse recovery, best-in-class accuracy	Potentially higher runtime
Lasso, SR3	Simplicity, convexity	Sensitivity to parameter tuning
Weak SINDy	Superior noise robustness, improved coefficient recovery	None significant

Performance is largely independent of underlying system chaos, scale separation, or nonlinearity, highlighting the generality of the sparse identification paradigm when candidate libraries are sufficiently expressive.

6. Extensions, Limitations, and Future Directions

Current research expands sparse system identification toward:

• Nonlinear, Hybrid, and Time-Varying Systems: Sparse identification methods are being extended to handle systems with varying or switching dynamics, hybrid discrete–continuous models, or models with complex nonlinearities, through block, group, and adaptive regularization (Fattahi et al., 2018).
• Online and Real-Time Implementation: Efficient updates, parallelizable optimization routines, and recursive methods are enabling real-time operation for large-scale systems and streaming data (Haring et al., 2022, Vides, 2021).
• Integration of Physical Priors and Consistency: Data-driven identification frameworks explicitly encode domain-specific physics in candidate libraries, as in PC-SINDy, improving interpretability and control utility (Du et al., 13 Feb 2025).
• Uncertainty Quantification and Bayesian Model Averaging: Sparse Bayesian deep learning approaches quantify both epistemic and aleatoric uncertainty, vital for safety-critical applications (Zhou et al., 2021).
• Open Problems: Nonconvexity of certain objective functions, parameter selection for regularization weights and thresholds, and rigorous consistency analysis in non-i.i.d., partial observation, or feedback settings remain areas of active investigation.

Sparse system identification thus provides a principled framework for constructing reliable, interpretable, and computationally efficient models across disciplines characterized by high-dimensional, structured, and noisy data.