Learning spatiotemporal trajectories from manifold-valued longitudinal data
We propose a Bayesian mixed-effects model to learn typical scenarios of changes from longitudinal manifold-valued data, namely repeated measurements of the same objects or individuals at several points in time. The model allows to estimate a group-average trajectory in the space of measurements. Random variations of this trajectory result from spatiotemporal transformations, which allow changes in the direction of the trajectory and in the pace at which trajectories are followed. The use of the tools of Riemannian geometry allows to derive a generic algorithm for any kind of data with smooth constraints, which lie therefore on a Riemannian manifold. Stochastic approximations of the Expectation-Maximization algorithm is used to estimate the model parameters in this highly non-linear setting. The method is used to estimate a data-driven model of the progressive impairments of cognitive functions during the onset of Alzheimer's disease. Experimental results show that the model correctly put into correspondence the age at which each individual was diagnosed with the disease, thus validating the fact that it effectively estimated a normative scenario of disease progression. Random effects provide unique insights into the variations in the ordering and timing of the succession of cognitive impairments across different individuals.
The paper proposes a class of mixed-effects models for longitudinal manifold-valued data, as well as a stochastic EM method for a special case. This is applied to real data on Alzheimer's disease progression. Spatiotemporal modelling is a very timely topic. It's a hot topic in medical imaging, and very relevant to longitudinal clinical datasets (as here).Novelty: The methods here are related to those in [8] and [17], as the authors acknowledge. There are important differences, starting with the fact that the present paper applies to any Riemannian manifold, whereas [8] considers only diffeomorphism groups (appropriately, in an analysis of shape). More discussion of the relationship to previous work would be welcome. For example, lines 71-72 are puzzling: "Although the development of statistical models for manifold-valued data is a blooming topic (see [16], [17]), the construction of statistical models for longitudinal data on a manifold remains an open problem." (The title of [16] includes "longitudinal", as does [8].)Also, you say of [8] in Line 75 "the variance of shapes does not depend on time whereas it should adapt with the average scenario of shape changes". I don't understand this statement, and how your scheme does allow time-dependent variance. (Do you mean \epsilon in Eqn (1)?). The particular SAEM method seems somewhat novel to me, but the authors should discuss similarities and differences from [2].I was disappointed that the geometric framework, while starting in a general manifold setting, moved to a product of 1-D manifolds, all with the same metric. I am left wondering whether the paper might have reached a wider audience if presented on R^N with less jargon. On a related point, is the multivariate logistic curves model really necessary? There exists a mapping from (0,1) to \R that transforms the metric in Section 2.4 to the Euclidean one, after which geodesics are straight lines, thus simplifying the exposition and, unless I've missed something, the SAEM algorithm (see line 245 and lines 251 onward).Nonetheless, a general geometric framework can be very valuable. I like the idea of using parallel transport and exponentiation to parametrize the neighbourhood of a curve. There's a "tubular neighbourhood theorem" that says this is really a parametrization, at least if you stay close enough to the original curve; you should find and cite some version of it. The version I recall requires your vector w to be perpendicular to the curve, as you have required, though I presume more general versions exist. I am not convinced by your statements on lines 156-159. First: "The orthogonality condition ensures that a point on the parallel curve wi (, ·) moves at the same pace as in the average trajectory." I don't think that's correct, if "pace" means speed. E.g. if \gamma is the equator of a sphere, then the parallel curves are latitudes, all parametrised by longitude, but with speeds proportional to their radii. The same-pace claim is of course true in a flat metric, as in the motivating example, but orthogonality is not required in that case. Second, on line 159: "vectors w with the same orthogonal projection in (t0) lead to the same geometric parallel curve but with a different time parameterization." I don't think that's correct either (e.g. on a sphere). Nonetheless, I think the desired conclusion, on lines 160-161: "spatial and temporal transformations commute" is correct, by definition.The restriction of the typical disease progression to be a geodesic in \R^N, with the same metric on each dimension, seems quite restrictive. In the example, it means that (as the authors write in Section 2.3) "all biomarkers have on average the same dynamics but shifted in time" [and progressing more or less quickly]. That would seem an unreasonable assumption for many diseases. Nonetheless, it's a reasonable place to start, and has the great convenience of allowing a very low-dimensional parametrization of the mean dynamics. Note that the curve \gamma need not be a geodesic in the general geometric framework.The adapted SAEM algorithm impresses me. Even if it's not actually needed for this example (as I suspect, see above), it seems valuable. Could the authors please clarify its similarities to and differences from citation [2], and comment on its wider applicability?The experiment is interesting and topical. Fig 5 in particular suggests that the model is useful, demonstrating that the estimated time shifts correlate well with the age of conversion to AD, while the rate of progression doesn't.The English is very good but not perfect. Some typos and minor comments are noted below. p1 typo: "fourty"p2, line 89: shall - > should?p2, line 106: N-dim in R^N?? should be R^P for P > =Npp2/3: a citation to some standard reference on Riemannian geometry would be helpful, one that includes the parallel curve concept in Definition 1.p4, line 168: why choose a log-normal distribution? (no criticism, just a request for comment)p4, line 173: why choose a Laplace distribution?p4, line 181: could you choose another letter for \eta_i since \eta is used earlier for the parallel curve?p4, line 181: remove subscript k from \epsilonp5, line 231: where did A go?p5, line 244: "prevents from resorting ot" - > "prevents us from resorting to"p5, line 253: "shall write"? - > "may be written"?p8, line 378: "p_0=" should be followed by "0.3"Citations: capitalize Alzheimer's, EM algorithm, Riemannian ... and many others