TY - GEN

T1 - Multiscale wavelets on trees, graphs and high dimensional data

T2 - 27th International Conference on Machine Learning, ICML 2010

AU - Gavish, Matan

AU - Nadler, Boaz

AU - Coifman, Ronald R.

PY - 2010

Y1 - 2010

N2 - Harmonic analysis, and in particular the relation between function smoothness and approximate sparsity of its wavelet coefficients, has played a key role in signal processing and statistical inference for low dimensional data. In contrast, harmonic analysis has thus far had little impact in modern problems involving high dimensional data, or data encoded as graphs or networks. The main contribution of this paper is the development of a harmonic analysis approach, including both learning algorithms and supporting theory, applicable to these more general settings. Given data (be it high dimensional, graph or network) that is represented by one or more hierarchical trees, we first construct multi-scale wavelet-like orthonormal bases on it. Second, we prove that in analogy to the Euclidean case, function smoothness with respect to a specific metric induced by the tree is equivalent to exponential rate of coefficient decay, that is, to approximate sparsity. These results readily translate to simple practical algorithms for various learning tasks. We present an application to transductive semi-supervised learning.

AB - Harmonic analysis, and in particular the relation between function smoothness and approximate sparsity of its wavelet coefficients, has played a key role in signal processing and statistical inference for low dimensional data. In contrast, harmonic analysis has thus far had little impact in modern problems involving high dimensional data, or data encoded as graphs or networks. The main contribution of this paper is the development of a harmonic analysis approach, including both learning algorithms and supporting theory, applicable to these more general settings. Given data (be it high dimensional, graph or network) that is represented by one or more hierarchical trees, we first construct multi-scale wavelet-like orthonormal bases on it. Second, we prove that in analogy to the Euclidean case, function smoothness with respect to a specific metric induced by the tree is equivalent to exponential rate of coefficient decay, that is, to approximate sparsity. These results readily translate to simple practical algorithms for various learning tasks. We present an application to transductive semi-supervised learning.

UR - http://www.scopus.com/inward/record.url?scp=77956556554&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:77956556554

SN - 9781605589077

T3 - ICML 2010 - Proceedings, 27th International Conference on Machine Learning

SP - 367

EP - 374

BT - ICML 2010 - Proceedings, 27th International Conference on Machine Learning

Y2 - 21 June 2010 through 25 June 2010

ER -