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8.2 Approximation of the data distribution

Two strategies were pursued: either the distribution of sensorimotor data was approximated by a mixture of ellipsoids (chapter 3), or the distribution's mapping into a single higher-dimensional space was approximated by a hyper-plane (chapter 5). The first was accomplished by a mixture of local principal component analyzers (PCA). Here, a single PCA operates on a region within the distribution. For each ellipsoid, the resulting principal components are the axes' directions, and the corresponding eigenvalues are the squared axes' lengths. To obtain the free parameters of the mixture model (the ellipsoids' centers, axes' directions and lengths) three new methods were presented:

These algorithms were demonstrated to work on synthetic data and on the classification of hand-written digits (section 3.4). On classification tasks, however, mixture models are not the most suitable. Here, feed-forward networks and support-vector machines (Cortes and Vapnik, 1995) can do better (LeCun et al., 1998).

On most tasks, the NGPCA variants gave about the same performance than MPPCA-ext, though some differences were visible. The NGPCA variants were more sensitive than MPPCA-ext on the training parameters (section 3.5 and 6.3). Further, the dependence on these parameters is not understood.

On the other hand, the NGPCA variants could cope with arbitrarily many dimensions; MPPCA-ext fails on high-dimensional data (observed for 676 and 784 dimensions, see section 4.4.2 and 3.4). This problem arises from the Gaussian function, which vanishes for the large distances that the high-dimensional space provides (section 3.4.2). Thus, probably, this problem can be also found in other Gaussian mixture models (Albrecht et al., 2000; Meinicke and Ritter, 2001; Tipping and Bishop, 1999). These studies did not report tests on such high-dimensional data.

In the robot experiments, NGPCA-constV was observed to be better than NGPCA (section 6.3 and 7.3.2). This performance difference is possibly linked to the assignment of the data points among the units. For NGPCA-constV, this assignment was more balanced. In particular, more training patterns were assigned to the unit with the fewest patterns (section 6.3 and 7.3.2).

The mixture model parameters, the number of units and the number of principal components were defined before training. The first was set by trial and error. Using more units improved the performance until the point where the number of training patterns per unit was not sufficient for a robust training (section 4.5.2 and 5.3.2). The number of principal components was estimated from the local dimensionality of the distribution. This dimensionality was accessible experimentally by computing a PCA in the neighborhood of each data point (section 4.5.2, 6.2.5, and 7.2.5).

In the second strategy, which uses a hyper-plane to approximate the data distribution, the data were mapped into a higher-dimensional space, a so-called feature space. The algorithm `kernel PCA' Schölkopf et al. (1998b) extracted the principal components of the data's mapping in this feature space (section 2.4). This algorithm does not require that the mapping is actually carried out; instead, all computation is done in the original space. The principal components in feature space spanned the hyper-plane that approximated the data.

next up previous contents
Next: 8.3 Pattern association Up: 8. Conclusions Previous: 8.1 Data collection and
Heiko Hoffmann