- The assumption that a set of analytical derivative functions is available is a very strong hypothesis so the number of cases where this method can be applied seems limited. - The high dimensional tensor can be also compactly represented by the set of derivative functions avoiding the curse of dimensionality, so it is not clear what is the advantage of replacing the original compact representation by the TT representation. Maybe the reason is that in TT-format many operations can be implemented more efficiently. The paper gives not a clear explanation about the necessity of the TT representation in this case. - It is not clear in which cases the minimum rank is achieved by the proposed method. Is there a way to check it? - In the paper it is mentioned that the obtained core tensors can be rounded to smaller ranks with a given accuracy by clustering the values of the domain sets or imposing some error decision epsilon if the values are not discrete. It is not clear what is, in theory, the effect on the approximation in the full tensor error. Is there any error bound in terms of epsilon? - The last two bullets in the list of main contributions and advantages of the proposed approach are not clear to me (Page 2). - The method is introduced by an application example using the P_step function (section 2.2). I found this example difficult to follow and maybe not relevant from the point of view of an application. I think, a better option would be to use some problem easier to understand, for example, one application to game theory as it is done later in the paper. - Very relevant ideas and results are not included in the main paper and referred instead to the Appendix, which makes the paper not well self-contained. - The obtained performance in terms of complexity for the calculation of the permanent of a matrix is not better than standard algorithms as commented by the authors (Hamilton walks obtained the result with half of the complexity). It is not clear what is the advantage of the proposed new method for this application. - The comparison with the TT-cross method is not clear enough. What is the number of samples taken in the TT-cross method? What is the effect to increase the number of samples in the TT-cross method. I wonder if the accuracy of the TT-cross method can be improved by sampling more entries of the tensor.
Minor issues: - Page 2: “an unified approach” -> “a unified approach” - Page 2: “and in several examples is Appendix” -> “and in several examples in the Appendix” - In page 3, “basic vector e” is not defined. I think the authors refers to different elements of the canonical base, i.e., vectors containing all zeros except one “1” in a different location. This should be formally introduced somewhere in the paper. - Page 9: “as an contraction” -> “as a contraction”
- In the paper it is mentioned that the obtained core tensors can be rounded to smaller ranks with a given accuracy by clustering the values of the domain sets or imposing some error decision epsilon if the values are not discrete. It is not clear what is, in theory, the effect on the approximation in the full tensor error. Is there any error bound in terms of epsilon?