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We propose a kernel-spectral embedding algorithm for learning low-dimensionalnonlinear structures from high-dimensional and noisy observations, where thedatasets are assumed to be sampled from an intrinsically low-dimensionalmanifold and corrupted by high-dimensional noise. The algorithm employs anadaptive bandwidth selection procedure which does not rely on prior knowledgeof the underlying manifold. The obtained low-dimensional embeddings can befurther utilized for downstream purposes such as data visualization, clusteringand prediction. Our method is theoretically justified and practicallyinterpretable. Specifically, we establish the convergence of the finalembeddings to their noiseless counterparts when the dimension and size of thesamples are comparably large, and characterize the effect of thesignal-to-noise ratio on the rate of convergence and phase transition. We alsoprove convergence of the embeddings to the eigenfunctions of an integraloperator defined by the kernel map of some reproducing kernel Hilbert spacecapturing the underlying nonlinear structures. Numerical simulations andanalysis of three real datasets show the superior empirical performance of theproposed method, compared to many existing methods, on learning variousmanifolds in diverse applications.

Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach