We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves two parameters q and $q^{\prime }$. The first one controls the mean number of kept vertices in the downsampling, while the second one is a tuning parameter between space localization and frequency localization. We provide an explicit reconstruction formula, bounds on the reconstruction operator norm and on the error in the intertwining relation, and a Jackson-like inequality. These bounds lead to recommend a way to choose the parameters q and $q^{\prime }$. We illustrate the method by numerical experiments.

我们提出了一种新方法，用于对在有限连接加权图的顶点上定义的函数执行多尺度分析。我们的方法依靠随机生成的森林对顶点集进行下采样，并依靠马尔可夫交织关系的近似解来提供子图结构和导致该函数集小波基础的滤波器组。我们的构造涉及两个参数q和$q^{\prime }$。第一个控制下采样中保留顶点的平均数量，而第二个是空间定位和频率定位之间的调整参数。我们提供了一个显式的重构公式，它限制了重构算子范数和交织关系中的误差，并给出了杰克逊式不等式。这些界限导致推荐一种选择参数q和的方法$q^{\prime }$。我们通过数值实验说明了该方法。