标题： 神经状态空间模型的可观测性条件与其特征值及其单位根 显示英文标题

标题： Observability conditions for neural state-space models with eigenvalues and their roots of unity

摘要： 我们通过常微分方程和控制理论的视角研究神经状态空间模型和Mamba架构背景下可观测性的概念。 我们开发了策略以强制实现可观测性，这些策略针对学习环境进行了定制，特别是在初始时刻可学习的隐藏状态以及在整个连续时间内和高维情况下的可学习性。 我们还强调我们的方法侧重于特征值、单位根或两者兼有。 当强制实现可观测性时，我们的方法实现了计算效率，有时是在大规模情况下。 我们根据经典控制理论制定了基于机器学习的可观测性条件，并讨论了其计算复杂度。 我们的非平凡结果是五方面的。 我们通过使用可学习矩阵的排列在神经应用中讨论了可观测性，无需高精度。 我们提出了两个基于傅里叶变换的结果，这些结果能够以高概率实现可观测性，直到学习中的随机性。 这些结果是通过傅里叶空间中表示及其特征结构、非线性映射和可观测性矩阵之间的相互作用来实现的。 我们提出了一个关于Mamba的结果，类似于Hautus型条件，但采用了范德蒙德矩阵而非特征向量的论证方法。 我们的最终结果是Mamba系统的共享参数构造，在高指数运算中具有计算效率。 我们开发了一种与此耦合的训练算法，证明在某些正交条件下满足Robbins-Monro条件，而更传统的训练过程则无法满足具有高Lipschitz常数的收缩条件。 显示英文摘要

摘要： We operate through the lens of ordinary differential equations and control theory to study the concept of observability in the context of neural state-space models and the Mamba architecture. We develop strategies to enforce observability, which are tailored to a learning context, specifically where the hidden states are learnable at initial time, in conjunction to over its continuum, and high-dimensional. We also highlight our methods emphasize eigenvalues, roots of unity, or both. Our methods effectuate computational efficiency when enforcing observability, sometimes at great scale. We formulate observability conditions in machine learning based on classical control theory and discuss their computational complexity. Our nontrivial results are fivefold. We discuss observability through the use of permutations in neural applications with learnable matrices without high precision. We present two results built upon the Fourier transform that effect observability with high probability up to the randomness in the learning. These results are worked with the interplay of representations in Fourier space and their eigenstructure, nonlinear mappings, and the observability matrix. We present a result for Mamba that is similar to a Hautus-type condition, but instead employs an argument using a Vandermonde matrix instead of eigenvectors. Our final result is a shared-parameter construction of the Mamba system, which is computationally efficient in high exponentiation. We develop a training algorithm with this coupling, showing it satisfies a Robbins-Monro condition under certain orthogonality, while a more classical training procedure fails to satisfy a contraction with high Lipschitz constant.