The set of all mappings between an input set of N elements to some output set with M elements has M^N elements. If you wanted to be able to represent in some w

The set of all mappings between an input set of N elements to some output set with M elements has M^N elements. If you wanted to be able to represent in some way any arbitrary mapping for given sets of input and output, then you would need at least log_2(M^N) = N x log_2(M) bits. In the case of an input set of 32x32 pixel images with 3 bytes per pixel (one for each channel) we have N = 2^8 x 2^8 x 2^8 x 2^5 x 2^5 = 2^34. In the case of an artificial neural network we have at the last level an output. There will be at least one node with at least one bit of output, so M >= 2. In general, to have anything else but the trivial map that maps every input to the same output, we always have M >= 2. So, we need at least 2^34 x log_2(2) = 2^34 bits to represent an arbitrary function between the input and the output. That is 2 gibibytes! Since the models don't need 2 gibibytes, something is going on. The magic here is that we are able to encode subsets of possible mappings very efficiently by using the execution logic of a computer. The compressed representation of the mappings in the restricted subset are the learned weights (the code to evaluate the model is also needed, but that requires less bits than what we save). We are, in a way, compressing functions, not data. Hence the "clamping of entropy of the formula". [0] The restricion of the set of possible functions will lead to new, interesting phenomena. Think of it as compression artifacts, however not on images or audio, but functions. To make a model resistant to attacks by someone knowledgeable about these artifacts, I would add noise to the input such that the artifacts are not predictable, hence not practically attackable. [0] The same basic phenomenon happens with block ciphers in cryptography. A block cipher on one block is just a permutation of the set of all different input blocks. If you have a blocksize of 64 bits, representing an arbitrary permutation would need log_2(2^64 !) bits, where the exclamation mark stands for the factorial. That number is huge, bigger than 2^69. We can't represent arbitrary permutations of blocks of 64 bits. Yet, block ciphers are permutations. What happens here is that once again we find subsets of the possible permutation we can represent efficiently. The compressed representation is the key.