text: Summary: This paper presents a derivation which links a DNN to recursive application of maximum entropy model fitting.  The mathematical notation is unclear,  and in one cases the lemmas are circular (i.e. two lemmas each assume the other is correct for their proof).  Additionally the main theorem requires complete independence,  but the second theorem provides pairwise independence,  and the two are not the same. Major comments: - The second condition of the maximum entropy equivalence theorem requires that all T are conditionally independent of Y.  This statement is unclear,  as it could mean pairwise independence, or it could mean jointly independent (i.e. for all pairs of non-overlapping subsets A & B of T I(T_A;T_B|Y) = 0). This is the same as saying the mapping X->T is making each dimension of T orthogonal, as otherwise it would introduce correlations.  The proof of the theorem assumes that pairwise independence induces joint independence  and this is not correct. - Section 4.1 makes an analogy to EM,  but gradient descent is not like this process as all the parameters are updated at once, and only optimised by a single (noisy) step.  The optimisation with respect to a single layer is conditional on all the other layers remaining fixed,  but the gradient information is stale  (as it knows about the previous step of the parameters in the layer above).  This means that gradient descent does all 1..L steps in parallel,  and this is different to the definition given. - The proofs in Appendix C which are used for the statement I(T_i;T_j) >=I(T_i;T_j|Y) are incomplete,  and in generate this statement is not true,  so requires proof. - Lemma 1 appears to assume Lemma 2, and Lemma 2 appears to assume Lemma 1. Either these lemmas are circular or the derivations of both of them are unclear. - In Lemma 3 what is the minimum taken over for the left hand side?  Elsewhere the minimum is taken over T, but T does not appear on the left hand side. Explicit minimums help the reader to follow the logic,  and implicit ones should only be used when it is obvious what the minimum is over. - In Lemma 5, what does "T is only related to X" mean?  The proof states that Y -> T -> X forms a Markov chain,  but this implies that T is a function of Y, not X. Minor comments:- I assume that the E_{P(X,Y)} notation is the expectation of that probability distribution,  but this notation is uncommon, and should be replaced with a more explicit one. - Markov is usually romanized with a "k" not a "c". - The paper is missing numerous prepositions and articles,  and contains multiple spelling mistakes & typos.