We would like to thank the Reviewer for the careful evaluation of our work and for the valuable suggestions for improvements. In the following, we provide responses and clarifications to the weaknesses or questions raised by the Reviewer.

Q1: Tackling monomial vs. general polynomial activations: This is a good point and we totally agree with the Reviewer, however, we also emphasize that, in all papers starting from the paper by Kileel, Trager, and Bruna (reference [7]), the term "polynomial neural networks" refers to neural networks with monomial activations, and it is a standard terminology used in the literature. Nevertheless, we will clarify that the results focus on monomial activations instead of general polynomial activations when describing the setup of the revised paper, which should better contextualize our contribution, and also in the discussion of the limitations of the paper. We would also like to emphasize that the study of the monomial case is a fundamental first step in tackling more general polynomial activations, as further clarified in the answer to the following question.

[7] J. Kileel et al. On the expressive power of deep polynomial neural networks. Advances in neural information processing systems, 32, 2019.

Q2: Empirical impact and generalization to arbitrary polynomial activations: We thank the Reviewer for the suggestion. We agree that an extension to consider general polynomial activations instead of monomial ones would greatly improve the practical impact of our results. We do believe such an extension is possible. In fact, the monomial case is the key to address the general polynomial case. For instance, the very recent paper [Shahverdi et al., 2025] (published May 17th, 2025, after the submission of our paper) considers more general polynomial activations and they first rely on results for the monomial case. This is an exciting subject which we plan to investigate in a followup work.

[Shahverdi et al., 2025] V. Shahverdi et al. Learning on a Razor's Edge: the Singularity Bias of Polynomial Neural Networks. arXiv preprint arXiv:2505.11846. (2025).

Q3: Typo in line 271: Thank you very much for pointing this out, we will fix it in the revised paper.

Q4: Local identifiability rather than global identifiability in [1, 2, 3]: The reviewer is right, this was an incorrect wording from our side. Indeed, the proof of Theorem 18 in [11] contains a result on global identifiability (for a high degree though, so weaker than our results, as we mentioned in the introduction), but the statement of the theorem is about local identifiability (more precisely, about dimension of the neurovariety, which implies that the parameterization map is finite-to-one, subject to scaling ambiguities). References [8] and [9] indeed also contain results about global identifiability, but for other/particular architectures (convolutional and attention-like layers) and thus are not applicable to general PNNs. What we meant is that identifiability is not the main focus of these papers, but we believe it is a central concept that is key to understanding the geometry of PNNs. We do apologize for this incorrect wording and we will correct it in the revised paper.

Q5: Kruskal rank condition and MLPs without inactive neurons: Thank you for the insightful question. The Kruskal rank conditions used in our results requires that, for a 2-layer network with weight matrices $W_1$ and $W_2$ for the first and second layer, respectively, $W_1$ has no collinear rows, and $W_2$ has no zero columns. This in turn corresponds to networks that are irreducible in the sense that the activations in the hidden layer are linearly independent, which is slightly stronger than being "active"/nonzero.

In fact, we have that for high enough degrees, uniqueness is achieved if and only if the weight matrices satisfy these Kruskal rank constraints (these necessary conditions become sufficient). This is also true for the case of deep (L-layers) networks due to the localization theorem. We will improve the discussion of the implication of the Kruskal rank conditions in the revised version of the paper.

We would like to thank the Reviewer for the careful evaluation of our work and for the valuable suggestions for improvements. In the following, we provide responses and clarifications to the weaknesses or questions raised by the Reviewer.

Response to weaknesses: First we would like to emphasize that the proposed techniques constitute a crucial first step in the study of the identifiability of more general architectures, such as general polynomial networks (which can provide good approximations to activations commonly used by practitioners), see the answer to Q2 by Reviewer Ux2R for more details. Moreover, our results also address networks of arbitrary depth. Furthermore, we also address the question of uniqueness of a specific set of NN parameters (Theorem 11), which is a much stronger property than identifiability (Corollary 12).

We also agree that approximate or probabilistic identifiability are of great interest in ML theory and applied statistics. However, a first stepping stone to these analysis requires understanding the identifiability in the noiseless case, which is the aim of our paper. We do provide an a clear and easy-to-digest answer to this question and advance the understanding of their neurovarieties in a wide range of settings. Thus, while approximate or probabilistic identifiability are are exciting and challenging topics, these will be better suited to be addressed in a dedicated future work.

Q1: Role of Bézout's theorem: We thank the Reviewer for this very interesting remark. Bézout's Theorem is indeed is very classical and fundamental result in algebraic geometry. Briefly, Bézout's theorem asserts that a general system of $n$ polynomial equations in $n$ variables has a finite number of solutions. However, for several reasons, it is not applicable in our situation.

• The uniqueness property can indeed be written as a system of polynomial equations. Nevertheless to fulfill the requirements of Bézout's theorem, we would need to have more independent equations than the number of variables, that is, a "small" number of neurons in the hidden layers compared to the others parameters (degree, extremal widths). This is generally not the case in applications.
• Bézout's theorem holds for general polynomials, i.e. chosen out of a set of zero Lebesgue measure. However, this set is not explicit: in other words, Bézout does not provide an explicit manner to check whether our uniqueness property (or even identifiability) holds for the equations arising from a given PNN.
• It is also important to notice that for monomial activations, as we explained in the paper, we can only have uniqueness/finiteness up to the trivial ambiguities, and there always exist infinitely many equivalent decompositions. Results like Bézout's theorem do not apply directly in this context, and we would need more advanced algebro-geometric techniques to tackle this very important question.

Q2: Relevance and extension to more common (real analytic or piecewise-linear activations) activations: On the one hand, our proof techniques are considerably different than those of previous works focused on special cases of analytic functions such as tanh or sigmoidal activations; thus, directly generalizing our proofs to general real analytic functions appears to be highly nontrivial. On the other hand, considering monomial activation functions is a necessary and crucial step to tackle the case of PNNs with general polynomial activations. In a future work, we plan to tackle this general polynomial case, which is of significant practical interest, by extending our proof techniques. We will include a discussion about possible research directions to extend our results to NNs with more general polynomial activation functions in the revised version of the paper.

Q3: Experiment or example showing what identifiability looks like in practice: Thank you for the valuable suggestion. In order to better illustrate identifiable and non-identifiable networks in practice, we consider will include additional examples in the revised paper. In particular, we will include an example to compare PNNs that are 1) unique and thus identifiable, 2) identifiable but non-unique (i.e., with pathological weight matrices, such as collinear rows), and 3) non-identifiable (and thus, also not unique).

Q4: Implications of identifiability on learning or optimization: We thank the Reviewer for this interesting question. In fact, identifiability has many important links to learning and optimization of PNNs. In particular, identifiability provides important information on the dimension of their associated neurovariety and neuromanifold, which, in turn, has deep implications on their expressivity, sample complexity, and on the geometry of their learning landscapes, as noted in previous works [7] and [9]. Moreover, a key result from singular learning theory (see [Watanabe, 2009] for background) is that non-identifiable networks are precisely singular points of the neurovariety, and in fact, learning algorithms are biased towards such solutions [Shahverdi et al., 2025]. Thus, it is very important to characterize which networks are non-identifiable [Shahverdi et al., 2025]. We will comment on the implications of identifiability to learning in the revised version of the paper.

[Shahverdi et al., 2025] Shahverdi, V., Marchetti, G. L., and Kohn, K. (2025). Learning on a Razor's Edge: the Singularity Bias of Polynomial Neural Networks. arXiv preprint arXiv:2505.11846.

[Watanabe, 2009] S. Watanabe, "Algebraic Geometry and Statistical Learning Theory", Cambridge University Press, 2009.

[7] J. Kileel et al. On the expressive power of deep polynomial neural networks. Advances in neural information processing systems, 32, 2019.

[9] N. W Henry et al. Geometry of lightning self-attention: Identifiability and dimension. ICLR 2025, 2025. arXiv preprint arXiv:2408.17221.

We would like to thank the Reviewer for the careful evaluation of our work and for the valuable suggestions for improvements. In the following, we provide responses and clarifications to the weaknesses or questions raised by the Reviewer.

Q1: Lack of numerical experiments: We agree that the paper would benefit from numerical experiments. However, our main goal to clearly present the uniqueness/identifiability result and its proof in a digestible form, which we really hope will be very useful for community. We emphasize, however, that the constructive nature of our proofs make it feasible to develop algorithms based on linear algebra to recover the parameters of an (h)PNN. We will include a discussion on numerical algorithms and how they can be derived from the proposed proofs in the revised version of the manuscript.

Q2: Computationally intensive to compute Kruskal ranks: Thank you for this important question. We note that we do not actually need to compute Kruskal ranks to verify several architectures of practical interest. For various architectures, for example, those with sufficiently high degrees, we just need to check whether the weight matrices have collinear rows/columns. Moreover, in many cases (for high degrees or numbers of neurons not being too large) the corresponding tensor representation of a PNN can be decomposed using efficient large-scale algorithms based on linear algebra, such as the one in [Batselier et al., 2016]. This is a topic we plan to study in a followup work. We will include a discussion about the development of computationally efficient algorithms in the revised version of the paper.

[Batselier et al., 2016] K. Batselier et al. Symmetric tensor decomposition by an iterative eigendecomposition algorithm. Journal of Computational and Applied Mathematics, 308, 69-82. (2016)

Q3: Expand on the relevance to practitioners: We thank the Reviewer for this interesting question. As we already mentioned in the answer to Question 1 of Reviewer Mfr2, interpretability of neural networks is tightly linked to uniqueness of its parameters. Our work could be also a guide for the design of pruning strategies, aiming at building identifiable architectures from a non-identifiable given one while preserving its original functional response. This work can be an useful to guide for compression.

Furthermore, as also mentioned in the answer to Question 1 from Reviewer Mfr2, identifiability plays an important role in model comparison, "stitching" or averaging. This has been used in several recent works [34], [Ito et al., 2025], [Ainsworth et al., 2022], where one finds a permutation (and possibly rescaling) that best aligns the hidden representations of different pretrained models to perform model comparison, averaging, uncertainty quantification, among other tasks. For this procedure to yield meaningful results, identification is essential.

Finally, identifiability/uniqueness also has direct implications on the geometry of PNNs, since it is directly linked to the dimension of its associated neurovariety as discussed in Appendix B of our paper.

We will emphasize the role of identification in the revised manuscript, as well as the influence of the equivalence class of parameters.

[34] C. Godfrey et al. On the symmetries of deep-learning models and their internal representations. Advances in Neural Information Processing Systems, 35:11893–11905, 2022

[Ito et al., 2025] A. Ito et al. (2025). Linear Mode Connectivity between Multiple Models modulo Permutation Symmetries. In Forty-second International Conference on Machine Learning.

[Ainsworth et al., 2022] S. K. Ainsworth et al. . Git re-basin: Merging models modulo permutation symmetries. The Eleventh International Conference on Learning Representations. (2022)

We would like to thank the Reviewer for the careful evaluation of our work and for the valuable suggestions for improvements. In the following, we provide responses and clarifications to the weaknesses or questions raised by the Reviewer.

Q1: How interpretability relates to identification and size of the equivalence class: Thank you for the insightful question. In fact, the equivalence class of the parameters of a neural network (i.e., a set of parameter ambiguities which will lead to a representation of the same function) depends on choice of the activation function. In a more general setting, this has been elegantly formulated through the notion of "intertwiner groups" studied in [34], which characterizes classes of equivalent parameters for various activations.

For the case of polynomial neural networks with monomial activation, the equivalence class described in Lemma 4 consists of permutations of neurons and rescaling of the input and the output of the activation function. If a PNN is unique/identifiable, then this equivalence class describes all possible representations of the same network with the same number of neurons. For the case of the polynomial activation function of general shape (not just a monomial), the equivalence class reduces to the set of permutations of neurons (which is unavoidable in any MLP), and no rescaling is possible [Shahverdi et al., 2025].

From a practical perspective, if we assume the weight matrices to be properly normalized to fix the scaling ambiguity, identification is linked to interpretability in the sense that for an identifiable network, any alternative representation will be a permuted version of the original one, whereas for any non-identifiable model the parameters and representations could be completely different. Moreover, identifiability plays an important role in model comparison, "stitching" or averaging, which has been used in several recent works [34], [Ito et al., 2025], [Ainsworth et al., 2022]. In such works, one finds a permutation (and possibly rescaling) that best aligns the hidden representations of different pretrained models in order to properly perform model comparison, averaging, uncertainty quantification, among other tasks. For this procedure to yield meaningful results, identification is essential.

We will emphasize the role of identification in the revised manuscript, as well as the influence of the equivalence class of parameters.

[Shahverdi et al., 2025] Shahverdi, V., Marchetti, G. L., and Kohn, K. (2025). Learning on a Razor's Edge: the Singularity Bias of Polynomial Neural Networks. arXiv preprint arXiv:2505.11846.

[34] Charles Godfrey, Davis Brown, Tegan Emerson, and Henry Kvinge. On the symmetries of deep-.- learning models and their internal representations. Advances in Neural Information Processing Systems, 35:11893–11905, 2022

[Ito et al., 2025] Ito, A., Yamada, M., and Kumagai, A. (2025) Linear Mode Connectivity between Multiple Models modulo Permutation Symmetries. In Forty-second International Conference on Machine Learning.

[Ainsworth et al., 2022] Ainsworth, S. K., Hayase, J., and Srinivasa, S. (2022). Git re-basin: Merging models modulo permutation symmetries. The Eleventh International Conference on Learning Representations.

Q2: Novelty of the results vs. building on prior work, existing results for other activation functions: We thank the Reviewer for this important question. The main novelty of the paper is on two key results, namely, Theorem 11 (our localization theorem) about the uniqueness of deep PNNs, and Corollary 12 on their identifiability, as well as the key results which are used in their proofs. The uniqueness of homogeneous PNNs in the 2-layer case is already known in the literature, but, to the best of our knowledge, there exists no previous localization results that allowed one to leverage them in the study of deep PNNs. In particular, all results on the uniqueness of deep homogeneous PNNs and (shallow and deep) PNNs with biases are completely novel.

Regarding the relevance of polynomial activations, the main advantage is that this is the case of deep NNs with nonlinear activations that are amenable to analysis with powerful methods of algebraic geometry. This is a very active research direction, as witnessed by a spark of recent results on geometry of the related neuromanifolds, expressivity of the networks, optimization landscape and properties of critical points [37]. For our paper, polynomial (monomial) activations are crucial for key steps in the proofs. Moreover, these techniques can serve as a key first step in the study of general polynomial activations, see the recent preprint that appeared after submission of this article [Shahverdi et al., 2025]. We also believe that an extension to other activation functions (for example, analytic) is feasible, for example, by using polynomial approximations, but this would require additional work and major changes in the proof techniques.

Finally, we highlight that some identifiability results for deep networks also exist for other architectures, for example, for networks with ReLU [29,30] or tanh [23] activation functions. However, the proof techniques used in these works are completely different, and do not apply to the case of polynomial activation functions. Moreover, an important characteristic of our proofs is that they are constructive and can be exploited as a first step in the development of algorithms to learn the weight matrices of a PNN. We will also emphasize this distinction in the related work section of the revised manuscript.

[7] Joe Kileel, Matthew Trager, and Joan Bruna. On the expressive power of deep polynomial neural networks. Advances in neural information processing systems, 32, 2019.

[11] Bella Finkel, Jose Israel Rodriguez, Chenxi Wu, and Thomas Yahl. Activation thresholds and expressiveness of polynomial neural networks. arXiv preprint arXiv:2408.04569, 2024.

[23] Charles Fefferman. Reconstructing a neural net from its output. Revista Matemática Iberoamericana, 10(3):507–555, 1994.

[29] Pierre Stock and Rémi Gribonval. An embedding of ReLU networks and an analysis of their identifiability. Constructive Approximation, pages 1–47, 2022.

[30] Joachim Bona-Pellissier, François Malgouyres, and François Bachoc. Local identifiability of deep ReLU neural networks: the theory. Advances in neural information processing systems, 35:27549–27562, 2022.

[37] Giovanni Luca Marchetti, Vahid Shahverdi, Stefano Mereta, Matthew Trager, and Kathlén Kohn. An invitation to neuroalgebraic geometry. arXiv preprint arXiv:2501.18915, 2025

Q3: Why identifiability definitions exclude a set of measure zero: In the definition of identifiability/generic uniqueness (Definition 8), the requirement that for a network to be identifiable uniqueness must hold for all points except possibly for some subset of measure zero serves to eliminate some pathological cases which would appear had the definition allowed the weights to be completely arbitrary. A special case of parameters to exclude is the subnetworks (i.e., networks representable with fewer neurons in at least one layer). This corresponds to weight matrices that have zero rows or columns, or collinear rows. In fact, such networks corresponds to the singular points of the underlying neurovarieties [Shahverdi et al., 2025].

However, we emphasize that our paper does not only treat generic uniqueness. Definition 6 concerns only a single set of parameters, and says whether a specific network representation is unique. The main theorem (Theorem 11) also applies to specific uniqueness, and does not assume genericity of the parameters. We will clarify these definitions and their implications in the revised version of the paper.

Q4: Relate proposition 13 to results in [7] and rank condition: Thank you for this comment and for the suggestion for clarifying the relation between this proposition and previous work and the link to the recurrent example. Indeed, Proposition 13 is related to the identifiability of 2-layer homogeneous PNNs, which was known in the literature (the link between 2-layer homogeneous PNNs and a CP tensor decomposition was already established in [7]). Thus, Proposition 13 (and Proposition 38, corresponding to the case of fixed weight matrices) are direct consequences of the uniqueness of this CP decomposition, which is known in the literature (in fact, stronger conditions are available for identifiability of homogeneous PNNs in the 2-layer case, see Remark 17). We emphasize that the novelty in our paper lies in the constructive proof for the uniqueness and identifiability of deep PNNs (as well as of shallow and deep PNNs with biases), which comes from the uniqueness of the CP decomposition, see the proof of Proposition 38. This is totally novel and, in turn, supports the development of algorithms, since learning deep PNNs can then be performed by computing the CPD of its corresponding tensor. This will be clarified in the revised version of the paper. We will also relate this proposition to the running example, to show how it applies to its corresponding architecture in the revised paper.

Q5: Typos: Thank you very much for pointing this out, we will fix this and other typos in the revised paper.

We would like to thank the Reviewer for the careful evaluation of our work and for the valuable suggestions for improvements. In the following, we provide responses and clarifications to the weaknesses or questions raised by the Reviewer.

Q1: Relationship to the argument present in Fefferman (1994): We agree with the Reviewer that the property proved by Fefferman is similar to our localization result for polynomial networks. We want nonetheless to emphasize the difference between the objects in consideration. Fefferman's paper indeed focuses on the case of a very specific analytic function (the tanh activation), which is far from being polynomial, and makes use of clever analytic techniques (study of singularities), which only work, though, for activation functions with specific properties (real analytic, finite limit at infinity, meromorphic extension to $\mathbb C$). Also note that Fefferman's arguments would not work in our situation, since monomial activations do not fulfill his requirements (having finite limit at infinity in one direction of the complex plane). In contrast, our proof is based on purely algebraic arguments, well suited to tackle the monomial case. We would also like to add that an exciting future research direction is to tackle the analytic activation case using general (non-monomial) polynomial activations. This would indeed encompass Fefferman's result, and our results on monomial polynomial activations constitute a crucial first step in this analysis.

We also would like to emphasize that the very powerful Fefferman's result provides only a sufficient condition for identifiability. Theorem 9 in Fefferman's work [23] excludes a lot of parameters (though a discrete set) and is not very practical. In our case, not only the proofs are constructive, but also they provide if and only if condition for uniqueness, provided the degree of the activation is high enough (no collinear rows in weight matrices and no zero columns).

We would like to thank the reviewer for mentioning this important paper. Indeed, although Fefferman's paper was already cited in the introduction of the original paper (reference [23]), there is not enough credit given to it, and we will correct this in the revised version.

[23] C. Fefferman. Reconstructing a neural net from its output. Revista Matemática Iberoamericana, 10(3):507–555, 1994.

Q2: Novelty of the homogenization procedure: We agree with the Reviewer that homogeneization itself is a fairly standard technique in machine learning and mathematics. We highlight, as the Reviewer correctly notes, that the contribution in our paper lies in leveraging it to demonstrate the uniqueness of the resulting deep PNN architecture by reducing it to a homogeneous PNN. The challenge comes from the fact that neither the weight matrices nor the input signals and hidden activations can be assumed to be distributed according to an absolutely continuous measure, which precludes the use of various standard arguments. This difficulty brought on considerable work to tackle NNs with biases even in the case of 2-layer NNs (see for example [68]). Nonetheless, we appreciate the Reviewer's remark that the current phrasing be potentially misleading in the original paper, and we will therefore revise it in the final manuscript to make this distinction clear to the reader.

[68] P. Awasthi et al. Efficient algorithms for learning depth-2 neural networks with general ReLU activations. Adv. Neur. Inf. Proc. Syst., 34:13485–13496, 2021.

Q3: Limited scope of application: We note that considering PNNs with monomial activations in fact is a crucial first step in tackling general polynomial activations -- which in turn can approximate all classical activation functions over compact sets. This is a direction we will investigate for a follow up work. For more details, see also the answer to Q5 just below, and also the answer to Q2 by Reviewer Ux2R.

Q4: Reliance on a general configuration of weights: We highlight that our paper include both uniqueness (fixed architecture) and identifiability (generic uniqueness) results. The conditions for uniqueness of a given architecture are in fact reasonably weak in general: we only require that (for high enough degrees) the weight matrices (with weaker conditions for the first and last layers) have no collinear (or, in the more pathological case, zero) rows or columns, otherwise the network would have redundant neurons that could be suppressed without changing its response. The generic identifiability conditions require the matrices to be sampled from any distribution that is absolutely continuous with respect to the Lebesgue measure. This requirement is fairly weak, and serves to exclude weights that fall in such pathological cases (e.g., some weight matrices with zero/collinear rows/columns). The conditions of uniqueness at fixed architecture are in fact helpful to diagnose the resulting trained network: if it can accurately represent the training data, and yet its weight matrices do not satisfy simple conditions such as having no collinear columns, then this means that the considered architecture could be potentially reduced to a smaller one. If we properly train a PNN to approximate data representable by an identifiable PNN architecture, then the weights should not converge networks with such "degenerate" weight matrices because otherwise the trained network would not be able to represent the data coming from the original identifiable one. These reasons lead us to believe that these conditions are indeed reasonably mild. We will include a more detailed discussion of the implication of these assumptions in the revised version of the paper.

Q5: Generalizing the results to non-homogeneous activations (GeLU, tanh, etc): On the one hand, our proof techniques are considerably different than those of previous works focused on tanh or sigmoidal activations; thus, directly generalizing them for such activation functions appears to be highly nontrivial. On the other hand, considering monomial activation functions is a necessary and crucial step to tackle the PNNs with general polynomial activations. This general polynomial case, which is of significant practical interest, can be potentially tackled with an extension of our proof techniques, where our identifiability results for the case of monomial activations should be an essential part of the analysis. See also the answer to Q2 by Reviewer Ux2R.

Q6: Extension to Hermite polynomials or other bases: This is a very interesting question. This is related to the study of the identifiability of PNNs whose activation functions are general polynomials. As mentioned in the previous response, our results on monomial activations serve as a crucial first step to tackle the uniqueness of PNNs with general polynomial activations, which can encompass Hermite polynomial bases. This is a topic we are interested in pursuing in a future work. We will clarify this perspective in the revised version of our paper.

Q7: Complexity aspects: Thank you for this insightful question. An important and distinctive aspect of our proofs is that they are constructive, and the parameters of a PNN can be computed from the canonical polyadic decomposition of its associated tensor. While the order of this tensor can be potentially high, under some conditions there exist very efficient algorithms based on linear algebra to compute the factor matrices (see, for example, [Batselier et al., 2016]). While the question of sample complexity is also of great practical interest, our analysis concerns identifiability in the noiseless setting, which is a first stepping stone to study the finite sample case. We find these questions very exciting and plan to tackle algorithmic and sample complexity issues in upcoming work, for instance, following an approach similar to the one used in [Oymak et al., 2021], which leverage the calculation of moment tensors from empirical data.

[Batselier et al., 2016] K. Batselier et al. Symmetric tensor decomposition by an iterative eigendecomposition algorithm. Journal of Computational and Applied Mathematics, 308, 69-82. (2016)

[Oymak et al., 2021] S. Oymak et al. Learning a deep convolutional neural network via tensor decomposition. Information and Inference: A Journal of the IMA, 10(3), 1031-1071 (2021).

Q8: Link between our proof and the one in Fefferman (1994): Please see the response to Q1 just above.

From my reading of the paper, I had the strong impression that there is a very fundamental limitation of applying the ideas in present paper to an activation that is not homogenous. As noted in your response here, even adding a bias term was quite challenging. Is my initial understanding that extension of current results to non-homogeneous PNN's will be a major leap, and presumably would require substantial new technical innovations? Or do you believe such a step is a natural next step? are authors claiming that the present results on homogeneous PNNs can be naturally extended for identifiability of a general PNNs?