We investigate the unsupervised learning of non-invertible observation functions in nonlinear state-space models. Assuming abundant data of the observation process along with the distribution of the state process, we introduce a nonparametric generalized moment method to estimate the observation function via constrained regression. The major challenge comes from the non-invertibility of the observation function and the lack of data pairs between the state and observation. We address the fundamental issue of identifiability from quadratic loss functionals and show that the function space of identifiability is the closure of a RKHS that is intrinsic to the state process. Numerical results show that the first two moments and temporal correlations, along with upper and lower bounds, can identify functions ranging from piecewise polynomials to smooth functions, leading to convergent estimators. The limitations of this method, such as non-identifiability due to symmetry and stationarity, are also discussed.
翻译:我们调查了非线性国家空间模型中非垂直观测功能的未经监督的学习情况。 假设观测过程的大量数据以及国家进程的分布,我们采用非对称通用瞬时法,通过限制回归来估计观测功能。 重大挑战在于观察功能的不可忽略以及国家和观察之间缺乏数据配对。 我们解决了四级损失功能的可识别性这一根本问题,并表明可识别性的功能空间是国家进程所固有的RKHS的关闭。 数字结果显示,前两个时刻和时间相关性,连同上下界,可以辨别从片度多面到平滑功能的功能,导致一致的估算。 也讨论了这一方法的局限性,如因对称性和稳定性造成的不可识别性。