Further discussion
We thank the reviewer for the prompt reply.
- __Q1__ We agree that Galerkin projection interpretation seems more convenient from an expository standpoint. So, based on the reviewer’s feedback, we propose to modify the content in lines 185-191, by briefly explaining why Galerkin projection should be done in energy space $\cal{H}^\eta_\pi$ instead of $\cal{L}^2_\pi$ space. Then, since the statistical learning risk formulation is the key to understand how to generalizes w.r.t. (unobserved) true distribution $\pi$ from biased simulations, we briefly discuss its equivalence with the Galerkin viewpoint. We believe that this change can help the reader to better grasp the main idea behind the energy, and connect it to the material that follows. Does the reviewer find this proposal adequate?
- __Q2: about the loss:__ To answer this question, let us compare (19) to the two losses used in the most related works, [24] and [46]. While the former one (only tested on unbiased dynamics of a toy system) depends on one hyperparameter, the latter loss has $1+m$ hyperparameters, where $m$ is the latent dimension. Importantly, __both losses suffer from statistically biased estimation__ of gradients when optimized over the sample distributions, which may negatively impact the optimization. Moreover, as reported in [46] (see also lines 287-289) additional $m$ hyperparameters in loss of [46] are delicate to tune. __In sharp contrast__, our empirical loss, as stated in Theorem 2, __is an unbiased estimate__ of the true loss in Equation (18), and we didn’t experience any particular difficulty in tuning our only hyperparameter $\alpha>0$. Note also that, in principle $\alpha>0$ is optional to speed up the training, and we may as well use $\alpha=0$. Concerning the computation, we respectfully disagree that the loss is hard to compute. Indeed, for two size $n$ batches of data, it just relies on computation of covariances of features $C$ and their gradients $W$, which is of the order $\cal{O}(n m^2 d)$. Since in the molecular dynamics setting $m$ is typically small, computing the loss is very efficient, similarly to losses in self-supervised learning based on canonical correlations, see e.g. [A]. The only computational bottleneck lies in using the gradients for $W$. However, we believe that this is a necessary price to pay to be able to generalize properly from biased simulations, as motivated in Sec. 3. Finally, we remark that optimization of our loss follows interpretable dynamics, see Figure 3 of the Appendix where plateaus reveal discovery of new relevant features, helping practitioners to decide when to stop the training.
- __Q2: about the contribution:__ As the reviewer rightfully recognizes, introducing loss (19) for representation learning is an important contribution of this work. However, our main focus is on __biased simulations__, or, in other words, __learning dynamics that ML algorithm did not observe__. With this in mind, our main contributions are summarized in lines 67-74. For the reviewer's convenience we rephrase and slightly expand them here. To the best of our knowledge, for the first time
- __(1)__ we __formalize the problem of learning__ the spectral decomposition of the infinitesimal generator __from biased simulations__ as a regression,
- __(2)__ we __derive empirical estimators with guaranteed statistical consistency__ (Theorem 1), and
- __(3)__ we propose an end-to-end __representation learning + regression__ pipeline that efficiently scales to large problems.
- __(4)__ we __empirically validate our methodology__ on molecular dynamics datasets of increasing complexity. That said, __we will follow reviewer’s suggestion and emphasize better the important role of representation learning__ for the generator regression.
Once more we thank the reviewer, and if any other doubt remains or clarification is needed, we are happy to discuss in more detail.
Ref.
[A] Balestriero et al., A Cookbook of Self-Supervised Learning. Arxiv:2304.12210 (2023).