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## Validity of the Approximations

The posterior pdf of all the unknown variables was approximated with a Gaussian density with diagonal covariance, which means that the variables were assumed independent given the observations. The Gaussianity assumption is not severe since the parametrisation is chosen so as to make the posterior close to Gaussian. If the hidden neurons were linear, the posterior of the sources, weights and biases would, in fact, be exactly Gaussian. Gaussian approximation therefore penalises strong nonlinearities to some extent.

The posterior independence seems to be the most unrealistic assumption. It is probable that a change in one of the weights can be compensated by changing the values of other weights and sources, which means that they have posterior dependencies.

In general, the cost function tries to make the approximation of the posterior more accurate, which means that during learning the posterior will also try to be more independent. In PCA, the mapping has a degeneracy which will be used by the algorithm to do exactly this. In linear PCA the mapping is such that the sources are independent a posteriori. In the nonlinear factor analysis, the dependencies of the sources are different in different parts of the latent space and it would be reasonable to model these dependencies. Computational load would not increase significantly since the Jacobian matrix computed in the algorithm can be used also for estimating the posterior interdependencies of the sources. For the sake of simplicity, the derivations were not included here.

It should be possible to do the same for the nonlinear independent factor analysis, but it would probably be necessary to assume the different Gaussians of each source to be independent. Otherwise the posterior approximation of Q(M | X) would be computationally too intensive.

The other approximation was done when approximating the nonlinearities of the hidden neurons by Taylor's series expansions. For small variances this is valid and it is therefore good to check that the variances of the inputs for the hidden neurons are not outside the range where the approximation is valid. In the computation of the gradients, some terms were neglected to discourage the network from adapting itself to areas of parameter space where the approximation is inaccurate. Experiments have proven that this seems to be working. For the network which minimised the cost function in Fig. 8, for instance, the maximum variance of the input for a hidden neurons was 0.06. Even this maximum value is safely below the values where the approximation could become too inaccurate.

Next: Initial Inversion by Auxiliary Up: Discussion Previous: Discussion
Harri Lappalainen
2000-03-03