Blind Signal Separation in the Presence of Gaussian Noise

Mikhail Belkin, Luis Rademacher, James Voss
Proceedings of the 26th Annual Conference on Learning Theory, PMLR 30:270-287, 2013.

Abstract

A prototypical blind signal separation problem is the so-called cocktail party problem, with n people talking simultaneously and n different microphones within a room. The goal is to recover each speech signal from the microphone inputs. Mathematically this can be modeled by assuming that we are given samples from a n-dimensional random variable X=AS, where S is a vector whose coordinates are independent random variables corresponding to each speaker. The objective is to recover the matrix A^-1 given random samples from X. A range of techniques collectively known as Independent Component Analysis (ICA) have been proposed to address this problem in the signal processing and machine learning literature. Many of these techniques are based on using the kurtosis or other cumulants to recover the components. In this paper we propose a new algorithm for solving the blind signal separation problem in the presence of additive Gaussian noise, when we are given samples from X=AS+η, where ηis drawn from an unknown, not necessarily spherical n-dimensional Gaussian distribution. Our approach is based on a method for decorrelating a sample with additive Gaussian noise under the assumption that the underlying distribution is a linear transformation of a distribution with independent components. Our decorrelation routine is based on the properties of cumulant tensors and can be combined with any standard cumulant-based method for ICA to get an algorithm that is provably robust in the presence of Gaussian noise. We derive polynomial bounds for sample complexity and error propagation of our method.

Cite this Paper

BibTeX
@InProceedings{pmlr-v30-Belkin13, title = {Blind Signal Separation in the Presence of Gaussian Noise}, author = {Belkin, Mikhail and Rademacher, Luis and Voss, James}, booktitle = {Proceedings of the 26th Annual Conference on Learning Theory}, pages = {270--287}, year = {2013}, editor = {Shalev-Shwartz, Shai and Steinwart, Ingo}, volume = {30}, series = {Proceedings of Machine Learning Research}, address = {Princeton, NJ, USA}, month = {12--14 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v30/Belkin13.pdf}, url = {https://proceedings.mlr.press/v30/Belkin13.html}, abstract = {A prototypical blind signal separation problem is the so-called cocktail party problem, with n people talking simultaneously and n different microphones within a room. The goal is to recover each speech signal from the microphone inputs. Mathematically this can be modeled by assuming that we are given samples from a n-dimensional random variable X=AS, where S is a vector whose coordinates are independent random variables corresponding to each speaker. The objective is to recover the matrix A^-1 given random samples from X. A range of techniques collectively known as Independent Component Analysis (ICA) have been proposed to address this problem in the signal processing and machine learning literature. Many of these techniques are based on using the kurtosis or other cumulants to recover the components. In this paper we propose a new algorithm for solving the blind signal separation problem in the presence of additive Gaussian noise, when we are given samples from X=AS+η, where ηis drawn from an unknown, not necessarily spherical n-dimensional Gaussian distribution. Our approach is based on a method for decorrelating a sample with additive Gaussian noise under the assumption that the underlying distribution is a linear transformation of a distribution with independent components. Our decorrelation routine is based on the properties of cumulant tensors and can be combined with any standard cumulant-based method for ICA to get an algorithm that is provably robust in the presence of Gaussian noise. We derive polynomial bounds for sample complexity and error propagation of our method.} }
Endnote
%0 Conference Paper %T Blind Signal Separation in the Presence of Gaussian Noise %A Mikhail Belkin %A Luis Rademacher %A James Voss %B Proceedings of the 26th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2013 %E Shai Shalev-Shwartz %E Ingo Steinwart %F pmlr-v30-Belkin13 %I PMLR %P 270--287 %U https://proceedings.mlr.press/v30/Belkin13.html %V 30 %X A prototypical blind signal separation problem is the so-called cocktail party problem, with n people talking simultaneously and n different microphones within a room. The goal is to recover each speech signal from the microphone inputs. Mathematically this can be modeled by assuming that we are given samples from a n-dimensional random variable X=AS, where S is a vector whose coordinates are independent random variables corresponding to each speaker. The objective is to recover the matrix A^-1 given random samples from X. A range of techniques collectively known as Independent Component Analysis (ICA) have been proposed to address this problem in the signal processing and machine learning literature. Many of these techniques are based on using the kurtosis or other cumulants to recover the components. In this paper we propose a new algorithm for solving the blind signal separation problem in the presence of additive Gaussian noise, when we are given samples from X=AS+η, where ηis drawn from an unknown, not necessarily spherical n-dimensional Gaussian distribution. Our approach is based on a method for decorrelating a sample with additive Gaussian noise under the assumption that the underlying distribution is a linear transformation of a distribution with independent components. Our decorrelation routine is based on the properties of cumulant tensors and can be combined with any standard cumulant-based method for ICA to get an algorithm that is provably robust in the presence of Gaussian noise. We derive polynomial bounds for sample complexity and error propagation of our method.
RIS
TY - CPAPER TI - Blind Signal Separation in the Presence of Gaussian Noise AU - Mikhail Belkin AU - Luis Rademacher AU - James Voss BT - Proceedings of the 26th Annual Conference on Learning Theory DA - 2013/06/13 ED - Shai Shalev-Shwartz ED - Ingo Steinwart ID - pmlr-v30-Belkin13 PB - PMLR DP - Proceedings of Machine Learning Research VL - 30 SP - 270 EP - 287 L1 - http://proceedings.mlr.press/v30/Belkin13.pdf UR - https://proceedings.mlr.press/v30/Belkin13.html AB - A prototypical blind signal separation problem is the so-called cocktail party problem, with n people talking simultaneously and n different microphones within a room. The goal is to recover each speech signal from the microphone inputs. Mathematically this can be modeled by assuming that we are given samples from a n-dimensional random variable X=AS, where S is a vector whose coordinates are independent random variables corresponding to each speaker. The objective is to recover the matrix A^-1 given random samples from X. A range of techniques collectively known as Independent Component Analysis (ICA) have been proposed to address this problem in the signal processing and machine learning literature. Many of these techniques are based on using the kurtosis or other cumulants to recover the components. In this paper we propose a new algorithm for solving the blind signal separation problem in the presence of additive Gaussian noise, when we are given samples from X=AS+η, where ηis drawn from an unknown, not necessarily spherical n-dimensional Gaussian distribution. Our approach is based on a method for decorrelating a sample with additive Gaussian noise under the assumption that the underlying distribution is a linear transformation of a distribution with independent components. Our decorrelation routine is based on the properties of cumulant tensors and can be combined with any standard cumulant-based method for ICA to get an algorithm that is provably robust in the presence of Gaussian noise. We derive polynomial bounds for sample complexity and error propagation of our method. ER -
APA
Belkin, M., Rademacher, L. & Voss, J.. (2013). Blind Signal Separation in the Presence of Gaussian Noise. Proceedings of the 26th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 30:270-287 Available from https://proceedings.mlr.press/v30/Belkin13.html.

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