Advances in Independent Component Analysis and Learning by Ella Bingham, Samuel Kaski, Jorma Laaksonen, Jouko Lampinen

By Ella Bingham, Samuel Kaski, Jorma Laaksonen, Jouko Lampinen

In honour of Professor Erkki Oja, one of many pioneers of autonomous part research (ICA), this publication studies key advances within the concept and alertness of ICA, in addition to its impact on sign processing, trend attractiveness, desktop studying, and knowledge mining.

Examples of themes that have built from the advances of ICA, that are lined within the ebook are:

  • A unifying probabilistic version for PCA and ICA
  • Optimization equipment for matrix decompositions
  • Insights into the FastICA algorithm
  • Unsupervised deep studying
  • Machine imaginative and prescient and photo retrieval
  • A assessment of advancements within the conception and purposes of self reliant part research, and its impression in very important components reminiscent of statistical sign processing, trend attractiveness and deep learning.
  • A varied set of software fields, starting from computer imaginative and prescient to technology coverage data.
  • Contributions from major researchers within the field.

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Extra resources for Advances in Independent Component Analysis and Learning Machines

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Letting p0 (θ) = 2/π for 0 ≤ θ ≤ pi/2, the expectation of ICIt is E{ICIt } = 2 π αt 0 a2 a−1 tan θ 2(3t ) dθ + a−2 π/2 αt (a cot θ )2(3 ) dθ . 135) 35 36 CHAPTER 1 The initial convergence rate of the FastICA algorithm Define 1 at = tan(αt ) = a · (a−1 ) 3t . 138) we can rewrite Eq. 135) as E{ICIt } = arctan(at ) 2 π 2(3t ) a2 a−1 tan θ dθ 0 +a arctan(a−1 t ) −2 (a tan θ )2(3 ) dθ . 139) 0 By the change of variables u = b−1 tan θ , one can show for any positive constants b t and bt = b · (b−1 )1/(3 ) that arctan(bt ) b−1 tan θ 2(3t ) t) (b−1 )1/(3 dθ = 0 0 u2(3 ) du.

00000423 E{ICI5 }. 4 ARBITRARY-KURTOSIS SOURCES CASE In the previous subsection, we focused on the behavior of FastICA applied to two-source mixtures assuming that the magnitudes of the source kurtosis were equal. In practice, the distributions of the sources will be unknown and different, and thus they will likely not have identical kurtosis magnitudes. The study in this subsection focuses on this scenario, deriving exact and limiting expressions for the average ICI in such cases. The starting point for our study is the expression for ICIt in Eq.

Clearly, we only need to consider stationary points of Eq. 32) for which any subset of the {ci,s } values for 1 ≤ i ≤ mp are nonzero. Call this subset of indices J . Then, for indices i ∈ J , we can simplify Eq. 32) to obtain |κi |c2i,s = κj2 c6j,s . 98) j∈J By dividing both sides of Eq. 98) by |κi | and summing across i ∈ J , we have c2i,s = i∈J i∈J 1 |κi | κj2 c6j,s . 99) j∈J Since cs must be of unit length, the left-hand side of Eq. 99) must be one, yielding the relation κj2 c6j,s = j∈J 1 . 100) Substituting Eq.

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