In Yoshua Belgio's 2003 technical report http://www.iro.umontreal.ca/~lisa/pointeurs/TR1232.pdf, and subsequent 2004 paper http://www.iro.umontreal.ca/~lisa/pointeurs/bengio_eigenfunctions_nc_2004.pdf, he showed that both spectral clustering and kernel PCA are special cases of learning eigenfunctions of the kernel integral operator: $K[f](x):=\int_{\mathbb{R}^p}k(x,y)f(y)dP(y)$ where $P$ is a probability measure on $\mathbb{R}^p$. In the second link (the paper), he also mentions and proves certain results on the convergence of the eigenspectra of random matrices $[k(X_i,X_j)]_{n \times n}$ to the eigenspectrum of the kernel integral operator $K$. Note that, here $X_1,...X_n$ are i.i.d. random variables from an unknown distribution.

In the discussion and open questions sections of both the report and the paper, he talks about using a smoother distribution than the empirical distribution to approximate the eigenspectrum of the kernel integral operator $K$ defined above. He also poses the question of possible applicability of this kind of smoother approximation. See pp. 28 of the paper (second link), point 1.

I'd like to know if there's some recent developments or advancements, research wprks regarding using a smoother approximation to kernel integral operator?


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