We consider the problem of approximating a subset $M$ of a Hilbert space $X$ by a low-dimensional manifold $M_n$, using samples from $M$. We propose a nonlinear approximation method where $M_n $ is defined as the range of a smooth nonlinear decoder $D$ defined on $\mathbb{R}^n$ with values in a possibly high-dimensional linear space $X_N$, and a linear encoder $E$ which associates to an element from $ M$ its coefficients $E(u)$ on a basis of a $n$-dimensional subspace $X_n \subset X_N$, where $X_n$ and $X_N$ are optimal or near to optimal linear spaces, depending on the selected error measure. The linearity of the encoder allows to easily obtain the parameters $E(u)$ associated with a given element $u$ in $M$. The proposed decoder is a polynomial map from $\mathbb{R}^n$ to $X_N$ which is obtained by a tree-structured composition of polynomial maps, estimated sequentially from samples in $M$. Rigorous error and stability analyses are provided, as well as an adaptive strategy for constructing a decoder that guarantees an approximation of the set $M$ with controlled mean-squared or wort-case errors, and a controlled stability (Lipschitz continuity) of the encoder and decoder pair. We demonstrate the performance of our method through numerical experiments.

翻译：我们考虑利用来自集合$M$的样本，在希尔伯特空间$X$中通过低维流形$M_n$逼近其子集$M$的问题。我们提出一种非线性逼近方法，其中$M_n$被定义为光滑非线性解码器$D$的值域，该解码器定义在$\mathbb{R}^n$上，取值于可能的高维线性空间$X_N$；同时引入线性编码器$E$，它将$M$中元素$u$映射到$n$维子空间$X_n \subset X_N$基上的系数$E(u)$。此处$X_n$与$X_N$是根据所选误差度量达到最优或接近最优的线性空间。编码器的线性特性使得我们能便捷地获取给定元素$u \in M$对应的参数$E(u)$。所提出的解码器是从$\mathbb{R}^n$到$X_N$的多项式映射，通过树状结构的多项式映射复合构成，并依据$M$中的样本进行序贯估计。我们提供了严格的误差与稳定性分析，以及构建解码器的自适应策略，该策略能保证以受控的均方误差或最坏情况误差逼近集合$M$，同时确保编码器-解码器对的稳定性（Lipschitz连续性）。我们通过数值实验验证了所提方法的性能。