# Inverting a matrix using a convolutional neural network

Just for a fun exercise, I am trying to invert a matrix, say size 28x28 (or even 5x5) with a neural network. The way I approached this (quite naively) is as follows:

1. I built a fully convolutional network with some 8 layers and ReLU activations (not sure if this is the right choice)

2. I put in an input $$X$$ and get an output $$Y$$. Here $$Y=NN(X)$$ where $$X$$ and $$Y$$ are both of same dimension, say $$n\times n$$. Assume NN is the conv net.

3. Now, I write a custom loss function, which is the $$MSE (YX-I)$$, where $$I$$ is identity, and $$YX$$ is the matrix multiplication of input and output and MSE is the mean squared error, where mean is taken across the batches.

4. For training, I generate 1000 random matrices and put them as input. There's no output label since the loss function doesn't need it. Shouldn't this ideally work? I can't seem to get the loss to converge. Is there a math flaw here? Is MSE not the right metric here ?

My custom loss function in TF-keras looks like this :

def custom_loss(I):

def loss(y_true, y_pred):

return keras.backend.mean(keras.backend.square(tensorflow.matmul(y_true,y_pred) - I ),axis=-1)
return loss
$$$$
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• Convolutions feel wrong for this problem, especially if the kernels are smaller than the matrix dimensions. The nonlinearities might be hurting you too since everything is linear. As an easier(?) intermediate step maybe try having it learn the Gram-Schmidt process. Aug 6 at 13:01
• As well, it's probably worth checking that the random matrices you generate are even invertible. Aug 6 at 13:15