Fast Transform Layers for Neural Networks

In a fast transform layer you replace the weight matrix with a fast transform matrix (which is enacted by a fast algorithm.)

Dense Layer - A neuron projects a pattern into the next layer (encoded in the neuron's outgoing weight vector.)   Fast Transform Layer - the pattern projected into the next layer is encoded in the fast transform algorithm.


In both dense layers and fast-transform layers, each neuron’s activation is essentially a coefficient that gets multiplied by a pattern into the next layer’s neurons:

  • Dense layer:

    • The pattern is given by that neuron’s outgoing weight vector.

    • These patterns are learned, so the network can tailor them to the task.

    • Each neuron can develop a very specialized “projection” pattern that routes its information selectively.

  • Fast-transform layer (WHT, FFT, etc.):

    • The pattern is fixed by the transform’s structure — e.g., Walsh–Hadamard patterns of ±1, sine/cosine waves for FFT.

    • A neuron’s activation just scales one of these fixed patterns, so it “lights up” the next layer in a predetermined way.

    • Adjustability has to come from switching, permutation, or activation parameterization, not from learning the projection pattern itself.

Why that difference matters for width

Since fast-transform networks can’t tailor each neuron’s projection pattern through learned weights, they need:

  • More channels (wider layers) so enough different fixed patterns are available to encode diverse features.

  • Extra nonlinearity or switching to combine patterns in richer ways over multiple layers.

In short:

Both systems send patterns forward; dense layers learn them, fast transforms predefine them. Wider fast-transform layers compensate for the lost flexibility in learned patterns.

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