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Preconditoned ICA for Real Data

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This repository hosts code of the Preconditioned ICA for Real Data (Picard) and Picard-O algorithms.

See the documentation.


Picard is an algorithm for maximum likelihood independent component analysis. It solves the same problem as Infomax, faster. It uses a preconditioned L-BFGS strategy, resulting in a very fast convergence.

Picard-O uses an adaptation of that strategy to solve the same problem under the constraint of whiteness of the signals. It solves the same problem as FastICA, but faster.


To install the package, the simplest way is to use pip to get the latest release:

$ pip install python-picard

or to get the latest version of the code:

$ pip install git+


To get started, you can build a synthetic signals matrix:

>>> import numpy as np
>>> N, T = 3, 1000
>>> S = np.random.laplace(size=(N, T))
>>> A = np.random.randn(N, N)
>>> X =, S)

And then feed Picard with it:

>>> from picard import picard
>>> Y, W = picard(X)

or Picard-O:

>>> from picard import picardo
>>> Y, W = picardo(X)

Picard or Picard-O output the estimated sources, Y, and estimated unmixing matrix, W.


These are the dependencies to use Picard:

  • numpy (>=1.8)
  • matplotlib (>=1.3)
  • numexpr (>= 2.0)

These are the dependencies to run the EEG example:

  • mne (>=0.14)
  • scikit-learn (>=0.18)
  • scipy (>=0.19)


If you use this code in your project, please cite:

Pierre Ablin, Jean-Francois Cardoso, and Alexandre Gramfort
Faster independent component analysis by preconditioning with Hessian approximations
ArXiv Preprint, June 2017

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