Consistency of coefficient‐based spectral clustering with l1‐regularizer

In recent years there has been an increasing interest in learning with the coefficient‐based space spanned by a kernel function, since it provides great flexibility for the learning process and is adopted easily to other algorithms. We investigate the spectral clustering algorithms by learning with the $l^1$ ‐regularizer scheme in a coefficient‐based hypothesis space. The main difficulty to study spectral clustering in our setting is that the hypothesis space not only depends on a coefficient‐based space, but also depends on some constrained conditions. We technically overcome this difficultly by a local polynomial reproduction formula and a construction method. The consistency of spectral clustering algorithms given consideration to sparsity is stated in terms of properties of the data space, the underlying measure, the kernel as well as the regularity of a target function.