reviews: 
Comments: The paper introduces regularisation to the standard jump model, akin to the same modification in regularised k-means. The model is calibrated in the usual way. Hyperparameter tuning is via preferring clustering stability. Simulation results compare the regularised jump model to the standardised and sparse jump model.
Strengths: The contribution of the paper is clearly presented.
Weaknesses: Presentational Clarity:
There is a general lack of presentational clarity throughout. The reader has to guess what is intended much of the time (and if you're relying on this, then don't be unhappy if they can't follow):
* p.1. how does \bold{Y} relate to \bold{y}_t?
* p.2.\bar{\bold{\mu}} seems to be undefined.
* p.2. what mathematical object a cluster C_k entails is not clear.
* p.3 is \underline{\bold{\mu}} related to the \bold{\mu}_k? (c.f. line 2).
* p.3. Is \bold{\mu}_{\cdot,j} the same as or different to \bold{\mu}_j? It looks like there is different notation for the same object. (Same comment for \bold{y}_t and \bold{y}_{\cdot,j} in Appendix B.)
* p.6: It's not clear from what is written exactly what Step 3 in Algorithm actually means. Maybe it means (just guessing...) that one locates the largest (by some norm?) cluster centre magnitudes, take those observations allocated to those clusters, and then perform K-means clustering on these observations only? This detail (or whatever the detail actually is) really needs to be clearer.
* p.7 para 3: The  meaning of this notation is not clear. \psi takes a single p-dimensional vector X and returns a cluster ID integer. What is \Psi in relation to this? And why does \Psi have some parameter \boldysymbol{\theta} as a condition and \psi does not?
* Following from the previous point: this becomes more confusing when the clustering distance function d can seemingly take both \psi and \Psi as inputs (definitions 3.1 and 3.2), when these are clearly different objects.
* d in Definition 3.1 seems to be dependent on X and Y, which are random variables. This makes d a random variable. Is this correct and the paper's intention? Distances are typically deterministic. 
* This is related to the expectation in (3.1). What is this expectation with respect to? Z_1 and Z_2 are draws from F. Once drawn and put into (3.1) they are fixed. So what is the expectation with respect to?
* Proposition B1: is Y here related to \bold{Y} in the main text? Is M related to \Psi in the main text?


Contribution:
a) The overall contribution seems fairly incremental. The paper adds a standard regularisation penalisation term to the jump model objective function in exactly the same way that a standard regularisation penalisation term is added to k-means. To be clear: there's nothing wrong with this, but it's a fairly low contribution (and not ICLR standard), and the simulations presented were not particularly convincing in support of this contribution (see more below).
b) There is no real motivation presented at the start of the paper for why the proposed work is needed or has value. Typically one would expect to see a strong justification to solve the problem, and a literature review discussing what has currently been done, and their strengths and weaknessess. This then allows the paper to present the justification for the work it contains in a clear and as strong as possible light, and against the current state of the art. This is not present in the current paper, which just lists things that were done.
c) Conclusion paragraph 2: Given the paper's incremental contribution, it would have been helpful to include the mentioned theoretical results "future work" in the current paper, though these results would likely have been standard extensions of existing results.


General Comments:
a) Section 2.2: S is introduced as a soft thresholding operator. S is also used as the vector of states (notation clash).
b) Section 2.2: Presumably Algorithm 2 should be referenced in here somewhere? 
c) Regardless, algorithm 2 (sparse jump model calibration) is not really used in the development of the proposed regularised jump model (nor its implementation), so its not clear why it should be included on p.6 taking up lots of space that could be used  for improving the current paper. Suggest it is moved to supporting information at best.
d) Section 2.3: The strategy in publication [11] of repeating the optimisation with 7 initial state sequences seems alarmingly ad hoc and situation specific/dependent, and one might be wary of following such a strategy in general, as well as recommending it as a basis for new work. It should also be made clearer how this initialisation strategy "incorporates potential sparsity in the initial cluster centres", and indeed why there is any hope that this is likely to get the user anywhere close to the global optimum of the target function. (Note that "most optimal" in this paragraph should just be "optimal".)
e) P.6. Algorithm 3: does Step 3 include (with the "100% largest d_j") an exact repeat of Step 1?
f) Clustering stability (definition 3.2) seems to prioritise the tendency of producing similar clusters over (say) parsimony. Is it clear that this is the best approach? (E.g. an algorithm that gives clusters of (1,1,1, ..., 1) deterministically each time without variation will be preferred over any other.) It would be good to understand a little more (via a discussion) why this prioritisation is aligned with the goals of the analysis.


Simulation Study:
It was difficult to take much from this section, as written. It would benefit from a complete re-write (and expansion).
a) The simulation study details are all in another paper [9]. The reader is not going to go and read that paper so that they can understand what has been done in this paper. All relevant detail should be contained in the paper it is used in. As such, it was difficult for this reader to take much from this simulation study. 
b) This includes algorithm/simulation settings etc. Was K fixed at the true value of K each time? How would it perform if it was not? etc. We have no idea.
c) Section 4.1 para 1: BAC is not defined (or at least, this reader couldn't find it), and so can't be understood. \mu here is not bold face. Previously \mu was \underline{\bold{\mu}} was the matrix of the cluster centres \bold{\mu}_k. Are these related? \mu seems to be a single integer in Tables 2 and 3. This really all makes no sense.
d) Section 4.1 para 4: \mu seems to impact algorithm performance, which seems incredible if it is a cluster centre.
e) The captions in the tables are not sufficient for the reader to know what the content of the tables is.
f) p.9 para 2: Its not clear what calculation was performed here.
g) Much of the discussion in Section 4.1 is of the form of empirical observations on tables of numbers, without attempting real understanding of what this means as general principles of algorithm performance. That is, we only have very specific, empirical observations.
Recommendation: 3
Reviewer confidence: 4
Soundness/correctness: 3
Clarity: 2