Learning spatiotemporal piecewise-geodesic trajectories from longitudinal manifold-valued data
We introduce a hierarchical model which allows to estimate a group-average piecewise-geodesic trajectory in the Riemannian space of measurements and individual variability. This model falls into the well defined mixed-effect models. The subject-specific trajectories are defined through spatial and temporal transformations of the group-average piecewise-geodesic path, component by component. Thus we can apply our model to a wide variety of situations. Due to the non-linearity of the model, we use the Stochastic Approximation ExpectationMaximization algorithm to estimate the model parameters. Experiments on synthetic data validate this choice. The model is then applied to the metastatic renal cancer chemotherapy monitoring: we run estimations on RECIST scores of treated patients and estimate the time they escape from the treatment. Experiments highlight the role of the different parameters on the response to treatment.
The paper proposes a mixed effects dynamical model which describes individual trajectories as being generated from transformations of an average behaviour. The work is motivated by modelling of cancer treatment, but the proposed method is more generally applicable.
The paper is well-written and -structured, and it considers an interesting application. I have limited knowledge of Riemannian geometry, but the methodology was not hard to follow and appears sound (to the extent that I can judge); that said, I cannot comment on the novelty of the approach. I generally found the presentation to be clear (despite some inconsistencies in the notation). The evaluation is fairly comprehensive and explained well in the text, and the model seems to perform well, especially with regards to capturing a variety of behaviours. It is interesting that the error does not monotonically increase as the sample size increases, but I suppose its variation is not that great anyway.
Overall, I believe this is a strong paper that is worthy of acceptance. Most of my comments have to do with minor language or notation issues and are given below. I only have two "major" observations. The first concerns Section 2.2, where the notation changes from the previous section. Specifically, the time warping transformation (previously \psi) is now denoted with \phi, while the space warping (previously \phi) is now strangely \phi(1) (line 154). This is pretty confusing, although presumably an oversight. Secondly, in Figure 1 and its description in Section 4.1, there is no explanation of what the vertical axis represents - presumably the number of cancerous cells, or something similar?
Other than that, I think it would be useful to slightly expand the explanation in a couple of places. The first of these is the introduction of the transformations (Section 2.1), where I personally would have welcomed a more intuitive explanation of the time and space warping (although Figure 1b helps with that some). The second place is Section 2.2: here, various decisions are said to be appropriate for the target application (eg lines 140, 155, 166), but without explaining why that is; some details of how these quantities and decisions correspond to the application might be useful. Some of the terminology could also be better explained, especially relating to the treatment, e.g. "escaping", "rupture". Finally, to better place the work in context, the paper could perhaps include a more concrete description of what the proposed model allows (for example, in this particular application) that could not be done with previous approaches (Section 4.3 mentions this very briefly). 
Detailed comments:
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l21: Actually -> Currently?
l22: I don't think "topical" is the appropriate word here; perhaps "contested"?
l23: [Mathematics] have -> has
l24: [their] efficiency -> its
l29: the profile are fitting -> the profiles are fitted?
l36: [amounts] to estimate -> to estimating
l45: in the aim to grant -> with the aim of granting?
l50: temporary [efficient] -> temporarily
l54: little [assumptions] -> few
l54: to deal with -> (to be) dealt with
l55: his -> its
l56: i.e -> i.e.
Equation between lines 81-82: t_R^m in the subscript in the final term should be t_R^{m-1} (similarly in Equation (*))
l91: I count 2m-2 constraints, not 2m+1, but may be mistaken
l93: constrain -> constraint (also repeated a couple of times further down)
l106: "we decree the less constrains possible": I am not sure of the meaning here; perhaps "we impose/require the fewest constraints possible"?
l118: in the fraction, I think t_{R,i}^{l-1} should be t_R^{l-1} (based on the constraints above)
l131: used to chemotherapy monitoring -> either "used for/in chemotherapy monitoring" or "used to [do something] in chemotherapy monitoring"
l134: n = 2 -> m = 2 (if I understand right)
l139: especially -> effectively / essentially (also further down)
l154: \phi_i^l (1), as mentioned earlier in the comments
l156: *** -> fin?
l160: I think (but may be wrong) that the composition should be the other way around, i.e. \gamma_i^1 = \gamma_0^1 \circ \phi_i^1, if \phi is the time warping here
l167: [interested] to -> in
l177: independents -> independent
l183-185: is it not the case that, given z and \theta, the measurements are (conditionally) independent?
l241: the table 1 -> Table 1
Figure 2: there is a thick dark green line with no observations - is that intentional?
Table 1: I found the caption a little hard to understand, but I am not sure what to suggest as an improvement
l245: the biggest [...], the best -> the bigger [...], the better
l257: Symetric -> Symmetric
l260: is 14, 50 intended to be 14.50?
l260: Figure 3a, -> no comma
l264: early on [the progression] -> early in / early on in
l264: to consider with quite reserve -> to be considered with some reserve
Figure 3: (last line of caption) large scale -> large range?
l280: "we have explicit it": I'm not sure what you mean here
l284: Last -> Lastly