This paper gives a comprehensive treatment of the convergence rates of penalized spline estimators for simultaneously estimating several leading principal component functions, when the functional data is sparsely observed. The penalized spline estimators are defined as the solution of a penalized empirical risk minimization problem, where the loss function belongs to a general class of loss functions motivated by the matrix Bregman divergence, and the penalty term is the integrated squared derivative. The theory reveals that the asymptotic behavior of penalized spline estimators depends on the interesting interplay between several factors, i.e., the smoothness of the unknown functions, the spline degree, the spline knot number, the penalty order, and the penalty parameter. The theory also classifies the asymptotic behavior into seven scenarios and characterizes whether and how the minimax optimal rates of convergence are achievable in each scenario.

翻译：本文全面研究了当函数数据稀疏观测时，同时估计多个主要主成分函数的惩罚样条估计量的收敛速率。惩罚样条估计量定义为惩罚经验风险最小化问题的解，其中损失函数属于由矩阵布雷格曼散度启发的一般损失函数类，而惩罚项是积分平方导数。理论表明，惩罚样条估计量的渐近行为取决于多个因素之间的有趣相互作用，即未知函数的平滑性、样条次数、样条节点数、惩罚阶数及惩罚参数。该理论还将渐近行为分为七种情形，并刻画了每种情形下是否以及如何实现极小化最优收敛速率。