A generalized diffusion frame for parsimonious representation of functions on data defined manifolds
A generalized diffusion frame for parsimonious representation of functions on data defined manifolds
复制标题
数据定义流形上函数简约表示的广义扩散框架
DOI：
10.1016/j.neunet.2010.12.007
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
H. Mhaskar
中科院分区：
文献类型：
--
作者：
H. Mhaskar
半监督学习中的一种标准技术是将高维数据视为嵌入在高维环境空间中的低维流形的子集，并使用数据在扩散映射的特征空间上的投影。本文的动机是最近的工作Coifman和Maggioni扩散小波来完成这样的预测近似使用迭代的热核。在更一般的情况下，我们考虑一个拟度量测度空间X（代替流形），和一个定义在X上的可积函数类上的非常一般的算子T（代替扩散映射）。我们用T的迭代的线性组合来表示X上的函数。我们的构造避免了需要计算本征值和本征函数的运营商。此外，函数f的局部光滑性由f的表示中的项的局部范数行为来表征。这一性质类似于经典小波表示。尽管算子T利用了整个空间上的目标函数的值，但这种能力导致自动的“特征检测”，从而导致目标函数的简约表示。在X是光滑紧致流形（无边界）的情况下，我们的理论允许T是与热算子交换的任何算子，但其特征值必须满足某些条件。特别地，T可以被选择为热算子本身，或者对应于合适的伪微分算子的绿色算子。
One of the now standard techniques in semi-supervised learning is to think of a high dimensional data as a subset of a low dimensional manifold embedded in a high dimensional ambient space, and to use projections of the data on eigenspaces of a diffusion map. This paper is motivated by a recent work of Coifman and Maggioni on diffusion wavelets to accomplish such projections approximately using iterates of the heat kernel. In greater generality, we consider a quasi-metric measure space X (in place of the manifold), and a very general operator T defined on the class of integrable functions on X (in place of the diffusion map). We develop a representation of functions on X in terms of linear combinations of iterates of T. Our construction obviates the need to compute the eigenvalues and eigenfunctions of the operator. In addition, the local smoothness of a function f is characterized by the local norm behavior of the terms in our representation of f. This property is similar to that of the classical wavelet representations. Although the operator T utilizes the values of the target function on the entire space, this ability results in automatic “feature detection”, leading to a parsimonious representation of the target function. In the case when X is a smooth compact manifold (without boundary), our theory allows T to be any operator that commutes with the heat operator, subject to certain conditions on its eigenvalues. In particular, T can be chosen to be the heat operator itself, or a Green’s operator corresponding to a suitable pseudo-differential operator.