Flatten image using Neural network and matrix transpose

Question

I have read a lecture note of Prof. Andrew Ng. There was something about data normalization like how can we flatten an image of (64x64x3) into a (64x64x3)*x1 vector. After that there is pictorial representation of flatten

As per the picture height, length and width of the picture is 64 , 64, 3. I think nx is a row vector which is then transpose to a column vector. If there is 3 pictures I think nx contains {64,64,3,64,64,3,64,64,3}. Am I right?

To use a 64x64x3 image as an input to our neuron, we need to flatten the image into a (64x64x3)x1 vector. And to make Wᵀx + b output a single value z, we need W to be a (64x64x3)x1 vector: (dimension of input)x(dimension of output), and b to be a single value. With N number of images, we can make a matrix X of shape (64x64x3)xN. WᵀX + b outputs Z of shape 1xN containing z’s for every single sample, and by passing Z through a sigmoid function we get final ŷ of shape 1xN that contains predictions for every single sample. We do not have to explicitly create a b of 1xN with the same value copied N times, thanks to Python broadcasting.

As per my understanding, Wᵀ = nx and x= nxᵀ.

Is it Wᵀ= [64,64,3,64,64,3,64,64,3] and x = [64,64,3,64,64,3,64,64,3]ᵀ?

In that case there product will be a symmetry matrix.

Is there any significance of symmetry matrix?

I just messed up all the things while flatten the image. If anyone has any idea please share with me.

Thank you in advance.

Answer 1

Yes, if you have 3 images (and by images I assume you mean samples) the flatten layer will be of the shape $12288*3$ ($64*64*3=12288$). The size of $W$ however does not change, and nor does the size of $b$ as these are parameters and are independent of the amount of samples passed through the network.

ETA: I only answered the "Am I right?" part of your question because that's the only part of your questions that's actually a question. I don't know what you're trying to ask in the second half of your question