# Mathematical Perspectives on Neural Networks

By Paul Smolensky; Michael C. Mozer et al. | Go to book overview

with

given by
(121)

The M step of training is simple. Assemble the results of the E step to form estimates of the k mean vectors and the k covariance matrices of the model and extract the required regression coefficients βj and residual variances

. Compute the neural net weights and thresholds wij after the last iteration.

8.3. The n-Class Problem

Suppose that the bitstring encoding Z ∈ {0, 1}m is given, and (X, Z) has some joint distribution. Define the complete data model by
(X, Y, J) X ∈ Rd, Y ∈ Rm, J ∈ {1, . . ., k}(122)
and set Zi = H(Yi) i = 1, . . .,m. In this case conditionally on J = j the density of (X, Y) is chosen to be d + m-dimensional Gaussian. For convenience in computing P ( I = iX = x) ≡ P ( Z = zX = x) we take the conditional covariance matrix of Y given both X = x and J = j to be diagonal. Then P(Z =zX=x) is given by

(123)
where
pij(x) = p(Yi > 0 ∣ X = x, J = j)(124)
The EM algorithm is again applicable; the only new object is the conditional covariance between components of Y given both X and J. Although this is zero by construction, enforcing this constraint in the M-step requires some care.

APPENDIX: ON OPTIMIZATION CRITERIA FOR TRAINING HMMs18

This appendix compares various optimization criteria; in particular, the maximum likelihood criterion for training HMMs is generalized. These ideas first appeared in a speech recognition context, and we continue here to present them in that way. In speech recognition, the HMMs will be used to decode the speech signal and the idea is to use training methods that emphasize discriminatory power rather than the usual parsimonious description of the data. The generalizations are constructed by considering weighted linear combinations of the logarithms of the likelihoods of words, of acoustics, and of (word, acoustic) pairs. The utility of various patterns of weights are examined.

____________________
18
This appendix is based on the work of Gopalakrishnan et al. [21].

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