1. Equivariant Adaptive and Blind Source Separation
Arindam Bose

2. Motivation
European Spatial Agency (ESA) released the new reference map of the Cosmic Microwave Background (CMB)

3. Motivation (cont.)

4. Applications Processing of real data sets: Cosmic microwave background
Mobile telephony
Airport radar
Bio-medical signals (ECG, EEG, multi-electrode neural recordings)

5. Source Separation and Equivariance
Source separation model is a transformation model
It leads to the notion of ‘serial updating’ by following the relative gradient by which efficient adaptive algorithms can be derived.
Adaptive Algorithms: Relative Gradient Algorithms
Equivariant algorithms
Stochastic relative gradient algorithms

6. Equivariant Adaptive Source Separation
𝒙 𝑡 =𝑨 𝒔 𝑡 𝒚 𝑡 = 𝑩 𝑡 𝒙 𝑡 = 𝒔 𝑡
𝑪 𝑡 ≝𝑩 𝑡 𝑨

7. Outline
The main point of this project is to introduce and study ‘serial updating’ algorithms.
Defining an algorithm consists in specifying an n x n matrix-valued function 𝒚→𝐻(𝒚) which is used for updating 𝐵 𝑡 according to

8. The EASI algorithms
A stationary point for a serial updating algorithm is any matrix B such that EH(y) = 0
To any component-wise non-linear function 𝑔 is associated a corresponding EASI algorithm:

9. Numerical Experiments

10. Normalized Results

11. Summary and Conclusion
A class of equivariant adaptive algorithms for the blind separation of 𝑛 independent sources was introduced.
The very desirable property of ‘uniform performance’ is guaranteed by the simple device of serial updating.
To obtain uniformly good performance, the (relative) gradient of an orthogonal contrast functions is computed and the resulting form was then generalized into a of matrixvalued functions.

12. References
J. F. Cardoso and B. H. Laheld, "Equivariant adaptive source separation," in IEEE Transactions on Signal Processing, vol. 44, no. 12, pp , Dec 1996.
B. H. Laheld and J. F. Cardoso, “Adaptive Source Separation With Uniform Performance,” In Proc. EUSIPCO,  pp , 1994.
J. F. Cardoso, "Blind signal separation: statistical principles," in Proceedings of the IEEE, vol. 86, no. 10, pp , Oct 1998.
Planck Collaboration, P. A. R. Ade et. al., “Planck 2013 results. XII. Component separation”.
J. F. Cardoso, M. Martin, J. Delabrouille, M. Betoule and G. Patanchon, “Component separation with flexible models. Application to the separation of astrophysical emissions.”
J. F. Cardoso, "Precision Cosmology with the Cosmic Microwave Background.”