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In this paper, we provide a framework based upon diffusion processes for finding meaningful geometric descriptions of data sets. We show that eigenfunctions of Markov matrices can be used to construct coordinates called diffusion maps that generate efficient representations of complex geometric structures. The associated family of diffusion distances,(More)
Adapted waveform analysis uses a library of or-thonormal bases and an efficiency functional to match a basis to a given signal or family of signals. It permits efficient compression of a variety of signals such as sound and images. The predefined libraries of modulated waveforms include orthogonal wavelet-packets, and localized trigonometric functions, have(More)
We provide a framework for structural multiscale geometric organization of graphs and subsets of R(n). We use diffusion semigroups to generate multiscale geometries in order to organize and represent complex structures. We show that appropriately selected eigenfunctions or scaling functions of Markov matrices, which describe local transitions, lead to(More)
This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adja-cency matrix of all points, we define a diffusion distance between any two data points and show that the low dimensional representation of the(More)
Data fusion and multicue data matching are fundamental tasks of high-dimensional data analysis. In this paper, we apply the recently introduced diffusion framework to address these tasks. Our contribution is three-fold: first, we present the Laplace-Beltrami approach for computing density invariant embeddings which are essential for integrating different(More)
We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green's function, in compressed form, and their fast application. Classes of operators satisfying these(More)
We describe an extension to the “best-basis” method to select an orthonormal basis suitable for signal/image classification problems from a large collection of orthonormal bases consisting of wavelet packets or local trigonometric bases. The original best-basis algorithm selects a basis minimizing entropy from such a “library of orthonormal bases” whereas(More)
The use of data-adapted kernels has been shown to lead to state-of-the-art results in machine learning tasks, especially in the context of semi-supervised and transductive learning. We introduce a general framework for analysis both of data sets and functions defined on them. Our approach is based on diffusion operators, adapted not only to the intrinsic(More)