Training neural network to implement discrete Fourier transform (DFT/FFT)

readme.md

My second neural network experiment (first was FIR filter). DFT output is just a linear combination of inputs, so it should be implementable by a single layer with no activation function.

Animation of weights being trained:

Red are positive, blue are negative. The black squares (2336 out of 4096) are unused, and could be pruned out to save computation time (if I knew how to do that).

Even with pruning, it would be less efficient than an FFT, so if the FFT output is useful, probably best to do it externally and provide it as separate inputs?

This at least demonstrates that neural networks can figure out frequency content on their own, though, if it's useful to the problem.

The loss goes down for a while but then goes up. I don't know why:

DFT_ANN.py

"""
Train a neural network to implement the discrete Fourier transform
"""
from keras.models import Sequential
from keras.layers import Dense
import numpy as np
import matplotlib.pyplot as plt

N = 32
batch = 10000

# Generate random input data and desired output data
sig = np.random.randn(batch, N) + 1j*np.random.randn(batch, N)
F = np.fft.fft(sig, axis=-1)

# First half of inputs/outputs is real part, second half is imaginary part
X = np.hstack([sig.real, sig.imag])
Y = np.hstack([F.real, F.imag])

# Create model with no hidden layers, same number of outputs as inputs.
# No bias needed.  No activation function, since DFT is linear.
model = Sequential([Dense(N*2, input_dim=N*2, use_bias=False)])
model.compile(loss='mean_squared_error', optimizer='adam')
model.fit(X, Y, epochs=100, batch_size=100)

# Confirm that it works
data = np.arange(N)


def ANN_DFT(x):
    if len(x) != N:
        raise ValueError(f'Input must be length {N}')
    pred = model.predict(np.hstack([x.real, x.imag])[np.newaxis])[0]
    result = pred[:N] + 1j*pred[N:]
    return result


ANN = ANN_DFT(data)
FFT = np.fft.fft(data)
print(f'ANN matches FFT: {np.allclose(ANN, FFT)}')

# Heat map of neuron weights
plt.imshow(model.get_weights()[0], vmin=-1, vmax=1, cmap='coolwarm')