Learning non-linear systems from noisy, limited, and/or dependent data is an important task across various scientific fields including statistics, engineering, computer science, mathematics, and many more. In general, this learning task is ill-posed; however, additional information about the data’s structure or on the behavior of the unknown function can make the task well-posed. In this work, we study the problem of learning nonlinear functions from corrupted and weakly dependent data. The learning problem is recast as a sparse robust linear regression problem where we incorporate both the unknown coefficients and the corruptions in a basis pursuit framework. The main contribution of our paper is to provide a reconstruction guarantee for the associated ${\ell }_1$-optimization problem where the sampling matrix is formed from weakly dependent data. Specifically, we prove that the sampling matrix satisfies the null space property and the stable null space property, provided that the data is compact and satisfies a suitable concentration inequality. We show that our recovery results are applicable to various types of weakly dependent data such as exponentially strongly $\alpha$-mixing data, geometrically $\mathcal{C}$-mixing data, and uniformly ergodic Markov chain. Our theoretical results are verified via several numerical simulations.

从嘈杂的，有限的和/或依赖的数据中学习非线性系统是横跨包括统计，工程，计算机科学，数学等许多科学领域的一项重要任务。通常，这种学习任务是不适当的；但是，有关数据结构或未知功能行为的其他信息可以使任务处于适当状态。在这项工作中，我们研究了从损坏的和弱相关的数据中学习非线性函数的问题。学习问题被重塑为稀疏的鲁棒线性回归问题，在该问题中，我们将未知系数和分解都纳入了基本追求框架中。我们论文的主要贡献是为相关${\ell }_{\mathrm{1个}}$-优化问题，其中采样矩阵由弱相关数据形成。具体而言，我们证明采样矩阵满足零空间性质和稳定的零空间性质，前提是数据紧凑且满足适当的浓度不等式。我们表明，我们的恢复结果适用于各种类型的弱相关数据，例如指数强$\alpha$几何混合数据 $\mathcal{C}$-混合数据，并遍历遍历马尔可夫链。我们的理论结果通过几个数值模拟得到了验证。