### Abstract

Signal denoising is closely related to function estimation from noisy samples. The same problem is also addressed in statistics (non-linear regression) and neural network learning. Vapnik-Chervonenkis (VC) theory has recently emerged as a general theory for estimation of dependencies from finite samples. This theory emphasizes model complexity control according to Structural Risk Minimization (SRM) inductive principle, which considers a nested set of models of increasing complexity (called a structure), and then selects an optimal model complexity providing minimum error for future samples. This paper applies the framework of VC-theory to signal estimation/denoising. There are three factors important for accurate signal estimation from finite samples: (1) the type of (orthogonal) basis functions used (i.e. Fourier basis, wavelets etc.). (2) The choice of a structure, i.e. ordering of the basis functions according to their 'importance' for accurate signal estimation. This corresponds to the choice of a 'structure' under SRM formulation. (3) Selecting an optimal number of terms (basis functions) from the ordered sequence of basis functions (2), aka model selection or complexity control (in statistics).We propose a methodology for specifying appropriate orderings (2) and an analytic expression for model selection (3) for signal processing applications. We also present empirical comparisons between the proposed methodology and current state-of-the-art wavelet thresholding methods for univariate signals. These comparisons suggest that the prudent choice of a structure (2) and the use of VC-based model selection (3) are critical for accurate signal estimation with finite samples.

Original language | English (US) |
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Pages (from-to) | 37-52 |

Number of pages | 16 |

Journal | Neural Networks |

Volume | 14 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2001 |

### Keywords

- Complexity control
- Model selection
- Orthonormal basis
- Signal denoising
- Sparse codes
- Structural risk minimization
- VC-theory
- Wavelet thresholding
- Wavelets

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## Cite this

*Neural Networks*,

*14*(1), 37-52. https://doi.org/10.1016/S0893-6080(00)00078-2