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

In honour of Professor Erkki Oja, one of many pioneers of self sustaining part research (ICA), this e-book stories key advances within the concept and alertness of ICA, in addition to its impact on sign processing, trend reputation, computing device studying, and information mining.

Examples of subject matters that have constructed from the advances of ICA, that are lined within the booklet are:

- A unifying probabilistic version for PCA and ICA
- Optimization tools for matrix decompositions
- Insights into the FastICA algorithm
- Unsupervised deep studying
- Machine imaginative and prescient and photograph retrieval

- A overview of advancements within the conception and functions of autonomous part research, and its impact in very important parts reminiscent of statistical sign processing, development acceptance and deep learning.
- A different set of software fields, starting from desktop imaginative and prescient to technology coverage data.
- Contributions from best researchers within the field.

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**Additional info for Advances in Independent Component Analysis and Learning Machines**

**Example text**

90) m = clt cpt E{si (k)sj (k)sl (k)sp (k)}. 91) l=1 p=1 The expectation on the right-hand side of Eq. 91) can be evaluated using Eqs. 21) as E{si (k)sj (k)sl (k)sp (k)} = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 ⎪ ⎪ ⎪ ⎪ E{s4i (k)} ⎪ ⎪ ⎩ 0 if i = j = l = p or if i = l = j = p or if i = p = j = l . 92) Appendix Therefore, m m clt cpt E{si (k)sj (k)sl (k)sp (k)} = κi c2it + c 2 δij + 2cit cjt . 94) where diag[M] is a diagonal matrix whose diagonal entries are the diagonal elements of M. Therefore, E{y3t (k)s(k)} = E{s(k)y2t (k)sT (k)}ct = Kdiag ct cTt ct + 3 ct = Kf(ct ) + 3 ct 2 ct .

17). Given that we have an exact expression for E{ICIt } under a particular initial condition, it is interesting to see how accurate the approximate expression for the ICI in Eq. 53) is. For a uniformly distributed look direction for w0 or c0 in angular space, it is straightforward to show by setting u = 1 in Eq. 122) that K = 1/π , such that Eq. 53) predicts an average ICI at iteration t of approximately E{ICIt } ≈ 1 π 1 3 t . 60) Numerical evaluation of Eq. 60) indicates that it is quite accurate in predicting the value of E{ICIt } in Eq.

84). f. of {c1 , c2 , . . , cm } is p(c1 , c2 , . . , cm ) = m! 0 if 0 ≤ cm ≤ cm−1 ≤ · · · ≤ c1 ≤ 1 . 85) Based on this result, we can develop an exact expression for the average ICI of the single-unit FastICA algorithm with this particular initial condition prior. 23 24 CHAPTER 1 The initial convergence rate of the FastICA algorithm Theorem 12. Assume that the unnormalized initial combined system coefficient vector c0 has the distribution in Eq. 85). Then, if the source mixture contains equal-kurtosis sources, the average value of the ICI at iteration t is exactly E{ICIt } = m−1 .