The Scientist and Engineer's Guide to
Digital Signal Processing
By Steven W. Smith, Ph.D.

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The weights required to make a neural network carry out a particular task are found by a learning algorithm, together with examples of how the system should operate. For instance, the examples in the sonar problem would be a database of several hundred (or more) of the 1000 sample segments. Some of the example segments would correspond to submarines, others to whales, others to random noise, etc. The learning algorithm uses these examples to calculate a set of weights appropriate for the task at hand. The term learning is widely used in the neural network field to describe this process; however, a better description might be: determining an optimized set of weights based on the statistics of the examples. Regardless of what the method is called, the resulting weights are virtually impossible for humans to understand. Patterns may be observable in some rare cases, but generally they appear to be random numbers. A neural network using these weights can be observed to have the proper input/output relationship, but why these particular weights work is quite baffling. This mystic quality of neural networks has caused many scientists and engineers to shy away from them. Remember all those science fiction movies of renegade computers taking over the earth?

In spite of this, it is common to hear neural network advocates make statements such as: "neural networks are well understood." To explore this claim, we will first show that it is possible to pick neural network weights through traditional DSP methods. Next, we will demonstrate that the learning algorithms provide better solutions than the traditional techniques. While this doesn't explain why a particular set of weights works, it does provide confidence in the method.

In the most sophisticated view, the neural network is a method of labeling the various regions in parameter space. For example, consider the sonar system neural network with 1000 inputs and a single output. With proper weight selection, the output will be near one if the input signal is an echo from a submarine, and near zero if the input is only noise. This forms a parameter hyperspace of 1000 dimensions. The neural network is a method of assigning a value to each location in this hyperspace. That is, the 1000 input values define a location in the hyperspace, while the output of the neural network provides the value at that location. A look-up table could perform this task perfectly, having an output value stored for each possible input address. The difference is that the neural network calculates the value at each location (address), rather than the impossibly large task of storing each value. In fact, neural network architectures are often evaluated by how well they separate the hyperspace for a given number of weights.

This approach also provides a clue to the number of nodes required in the hidden layer. A parameter space of N dimensions requires N numbers to specify a location. Identifying a region in the hyperspace requires 2N values (i.e., a minimum and maximum value along each axis defines a hyperspace rectangular solid). For instance, these simple calculations would indicate that a neural network with 1000 inputs needs 2000 weights to identify one region of the hyperspace from another. In a fully interconnected network, this would require two hidden nodes. The number of regions needed depends on the particular problem, but can be expected to be far less than the number of dimensions in the parameter space. While this is only a crude approximation, it generally explains why most neural networks can operate with a hidden layer of 2% to 30% the size of the input layer.

A completely different way of understanding neural networks uses the DSP concept of correlation. As discussed in Chapter 7, correlation is the optimal way of detecting if a known pattern is contained within a signal. It is carried out by multiplying the signal with the pattern being looked for, and adding the products. The higher the sum, the more the signal resembles the pattern. Now, examine Fig. 26-5 and think of each hidden node as looking for a specific pattern in the input data. That is, each of the hidden nodes correlates the input data with the set of weights associated with that hidden node. If the pattern is present, the sum passed to the sigmoid will be large, otherwise it will be small.

The action of the sigmoid is quite interesting in this viewpoint. Look back at Fig. 26-1d and notice that the probability curve separating two bell shaped distributions resembles a sigmoid. If we were manually designing a neural network, we could make the output of each hidden node be the fractional probability that a specific pattern is present in the input data. The output layer repeats this operation, making the entire three-layer structure a correlation of correlations, a network that looks for patterns of patterns.

Conventional DSP is based on two techniques, convolution and Fourier analysis. It is reassuring that neural networks can carry out both these operations, plus much more. Imagine an N sample signal being filtered to produce another N sample signal. According to the output side view of convolution, each sample in the output signal is a weighted sum of samples from the input. Now, imagine a two-layer neural network with N nodes in each layer. The value produced by each output layer node is also a weighted sum of the input values. If each output layer node uses the same weights as all the other output nodes, the network will implement linear convolution. Likewise, the DFT can be calculated with a two layer neural network with N nodes in each layer. Each output layer node finds the amplitude of one frequency component. This is done by making the weights of each output layer node the same as the sinusoid being looked for. The resulting network correlates the input signal with each of the basis function sinusoids, thus calculating the DFT. Of course, a two-layer neural network is much less powerful than the standard three layer architecture. This means neural networks can carry out nonlinear as well as linear processing.

Suppose that one of these conventional DSP strategies is used to design the weights of a neural network. Can it be claimed that the network is optimal? Traditional DSP algorithms are usually based on assumptions about the characteristics of the input signal. For instance, Wiener filtering is optimal for maximizing the signal-to-noise ratio assuming the signal and noise spectra are both known; correlation is optimal for detecting targets assuming the noise is white; deconvolution counteracts an undesired convolution assuming the deconvolution kernel is the inverse of the original convolution kernel, etc. The problem is, scientist and engineer's seldom have a perfect knowledge of the input signals that will be encountered. While the underlying mathematics may be elegant, the overall performance is limited by how well the data are understood.

For instance, imagine testing a traditional DSP algorithm with actual input signals. Next, repeat the test with the algorithm changed slightly, say, by increasing one of the parameters by one percent. If the second test result is better than the first, the original algorithm is not optimized for the task at hand. Nearly all conventional DSP algorithms can be significantly improved by a trial-and-error evaluation of small changes to the algorithm's parameters and procedures. This is the strategy of the neural network.