Before closing, we would like to make some additional comments on the multiscale method and suggest some possible extensions.

In a pattern-recognition problem such as character recognition, the two-dimensional spatial structure of the problem is important. The multiscale method preserves this structure so that ``reasonable'' feature extractors are produced. An obvious extension to the present work is to increase the number of hidden units as one boosts the MLP to higher resolution. This corresponds to adding completely new feature extractors. We did not do this in the present case since 32 hidden units were sufficient-the problem of recognizing upper-case Roman characters is too easy. For a more challenging problem such as Chinese characters, adding hidden units will probably be necessary. We should mention that incrementally adding hidden units is easy to do and works well-we have used it to achieve perfect convergence of a back-propagation network for the problem of tic-tac-toe.

When boosting, the weights are scaled down by a factor of four and so it is important to also scale down the learning rate (in the back-propagation algorithm) by a factor of four.

We defined our ``blocking,'' or coarsening, procedure to be a simple, grey scale averaging of blocks. There are many other possibilities, well known in the field of real-space renormalization in physics. Other interesting blocking procedures include: using a scale factor, , different from two; using majority rule averaging; simple decimation; and so on.

Multiscale methods work well in cases where spatial locality or smoothness is relevant (otherwise, the interpolation approximation is bad). Another way of thinking about this is that we are decomposing the problem onto a set of spatially local basis functions such as gaussians. In other problems, a different set of basis functions may be more appropriate and hence give better performance.

The multiscale method uses results from a small net to help in the training
of a large network. The different-sized networks are related by the
rescaling or dilation operator. A variant of this general approach would be
to use the translation operator to produce a pattern matcher for the game of
Go. The idea is that at least *some* of the complexity of Go is
concerned with local strategies. Instead of training an MLP to learn this on
the full board of Go, do the training on a ``mini-Go'' board of
or . The appropriate way to relate these networks to
the full-sized one is not through dilations, but via the translations: The
same local strategies are valid everywhere on the board.

Steve Otto had the original idea for the MultiScale training technique. Otto and Ed Felten and Olivier Martin developed the method. Jim Hutchinson contributed by supplying the original back-propagation program.

Wed Mar 1 10:19:35 EST 1995