Summary
This paper addresses the broad problem of learning interpretable dynamical systems from data; it builds upon existing approaches that use piecewise linear RNNs (PLRNNs), with one interesting twist: constraining the number of linear subregions to be much smaller than the "usual" $2^N$. This is achieved by allocating a small number P of units to have a ReLu activation function, every other unit being linear. This leads to $2^P$ linear subregions. The paper begins by proving a couple of theorems showing that these dynamical systems are amenable to interpretable symbolic analysis (though these results are generic to PLRNNs, not specific to the new version). The authors then fit AL-RNNs (this new breed of PLRNNs) to chaotic attractors, demonstrating that good dynamical reconstructions can indeed be obtained with relatively small P, providing a post-hoc justification for the approach. Finally, the approach is applied to ECG and fMRI data; in the ECG case, symbolic analysis returns a highly interpretable graph linking the different linear subregions.
Strengths
The core idea is neat; although the difference between existing PLRNNs and these new AL-RNNs is more a difference of degree than a difference of nature (just changing the number of units having a ReLu activation), it's great to have thought of taking PLRNNs into that regime and to show empirically that (i) it doesn't compromise DSR quality too much, yet (ii) it gives dynamics that are computationally more amenable to symbolic analysis.
Another strength (for me at least) is that this paper contributes to exposing an audience that has historically primarily cared about dynamical systems reconstruction (e.g. me) to the concept of symbolic dynamics -- indeed I'm glad I reviewed this paper and thus got a useful (even if rudimentary) primer on SD.
Weaknesses
I would very much like to see model recovery experiments; how easy are those amost-linear RNNs to fit, and to fit consistently? I could imagine that the model might settle into a suboptimal set of linear subregions early on during training and then have a hard time snapping out of it. I have to admit I don't have a good intuition for this, but this is something the authors could substantiate numerically by running simple model recovery experiments. On this note, what hard degeneracies do we expect here due to a majority of the state dimensions being unobserved? Can the authors use the symbolic dynamics grounding of section 3 (currently a little disconnected from the rest, I have to say) to derive meaningful measures of how well the ground truth system's topology / symbolic dynamics are recovered despite those degeneracies? (e.g. for small enough P it might be possible to look at all permutations of the transition matrix between linear subregions and conclude that the ground truth has been recovered?).
Re consistency, could the authors comment on whether the error bars (across training runs) in e.g. Figure 5d-f are to be considered small or large? The figure caption says "shows close agreement among different training runs" but judging from these whisker plots, the underlying coefficient of variation seems quite high (which actually triggered the concern I articulated above concerning potentially inconsistent recovery of ground truth PWL dynamics).
Questions
- You omitted to say that $\phi(\cdot)$ in Equation 1 is the ReLu function -- this is pretty critical!
- In equation 4, why are you over-parameterizing the linear part? It seems that the first $M-P$ columns of $W$ can be absorbed into the first $M-P$ columns of $A$. So you really just have the last $P$ columns of $W$ to learn.
- In Figure 2, why did you write "$P^2$ subregions / symbols"? Did you mean $2^P$? (and accordingly, should $P^4$ in fact read $2^{2P}$)?
- l.185: can you please define "hyperbolic AL-RNNs" (AL-RNNs in which fixed points are all hyperbolic?) and what that implies concretely for A and W?
- Theorems 1-3 make intuitive sense but appear a little ill-phrased to me -- for example, in Theorem 1, the specific $z^\star$ that appears in the first clause of the iif statement is not even referred to in the other clause (and indeed it cannot be uniquely identified from knowing “the corresponding symbolic sequence $a^\star$” -- you don't really say what you mean by “corresponding”, btw; do you mean the symbolic sequence associated with any state space trajectory that contains $z^\star$?). Perhaps this theorem could be rephrased as "if there exists a fixed point $z^\star$ in $U_e$, then $e^\infty$ is a fixed point of the shift map; conversely, if $e^\infty$ is a fixed point of the shift map for some $e$, $U_e$ must contain a fixed point of the $F_\theta$ map.” (?)
Same concern in theorem 2 and 3.
In theorem 2, I think the $k+p$ subscript should be modulo $p$?
- Should we be concerned by the fact that your measures of DSR accuracy in Figure 3 do not decrease mononotonically with the number of PWL units? Local minima in the teacher-forcing loss?
- For the ECG and fMRI datasets, I couldn't see a quantitative assessment of model performance; in particular, for the fMRI dataset, I think the authors should discuss whether and how much the addition of a categorical task-stage decoder impairs relevant DSR performance metrics (or perhaps even improves consistency across training runs?).
Limitations
Perhaps the authors could discuss the extent to which it really is easier to analyze $2^P$ subregions rather than $2^N$ subregions -- quantitatively I understand that this is a lot fewer subregions, but when $2^P$ is beyond a handful, whether it's 50 or 5 million, it's unclear to me how "easy" it is to analyze/understand these things (or indeed what that even means...).
*post-rebuttal EDIT*:
Having read the rebuttals to all 3 revieweres, I am raising my score to an 8; strong paper likely to have an impact.