The Scaling Limit of High-Dimensional Online Independent Component Analysis
We analyze the dynamics of an online algorithm for independent component analysis in the high-dimensional scaling limit. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measure of the target feature vector and the estimates provided by the algorithm will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numerical solutions of this PDE, which involves two spatial variables and one time variable, can be efficiently obtained. These solutions provide detailed information about the performance of the ICA algorithm, as many practical performance metrics are functionals of the joint empirical measures. Numerical simulations show that our asymptotic analysis is accurate even for moderate dimensions. In addition to providing a tool for understanding the performance of the algorithm, our PDE analysis also provides useful insight. In particular, in the high-dimensional limit, the original coupled dynamics associated with the algorithm will be asymptotically "decoupled", with each coordinate independently solving a 1-D effective minimization problem via stochastic gradient descent. Exploiting this insight to design new algorithms for achieving optimal trade-offs between computational and statistical efficiency may prove an interesting line of future research.
The paper studies the high-dimensional scaling limit of a stochastic update algorithm for online Independent Component Analysis.
The main result of the paper is an exact characterization of the evolution of the joint empirical distribution of the estimates
output by the algorithm and the signal to be recovered, when the number of observations scales linearly with the dimension of the problem.
The authors argue that in the limit, this joint distribution is the unique solution to a certain partial differential equation,
from which the performance of the algorithm can be predicted, in accordance with the provided simulations.
1- Overall, the result is fairly novel, and provides interesting insight into the behavior of stochastic algorithms in this non-convex problem.
The paper is written in a fairly clear manner and is (mostly) easy to read.
My main concern is that the mathematics are written in a very informal way, so that it is not clear what
the authors have actually proved. There are no theorem or proposition statements.
E.g., under what conditions does the empirical measure have a weak limit?
Under what (additional) conditions does this limit have a density?
And when is this density the unique solution to the PDE?
The heuristic justification in the appendix is not very convincing either (see bullet 4).
2- A very interesting insight of the theory is that the p variables of the problem decouple in the limit,
where each variable obeys a diffusion equation independently of the others.
The only global aspect retained by these local dynamics is via the order parameters R and Q. 
This phenomenon is standard in mean-field models of interacting particle systems, where the dynamics of the particles decouple when the order parameters converge to a limit.
The authors should probably draw this connection explicitly; the relevant examples are the Sherrington-Kirkpatrick model, rank-one estimation in spiked models, compressed sensing, high-dim. robust estimation…
3- The connection to the SDE line 175 should be immediate given that the PDE (5) is its associated Fokker-Planck equation. 
The iteration (14) is easier seen as a discretization of the SDE rather than as some approximate dynamics solving the PDE. Therefore, the SDE should come immediately after the statement of the PDE, then the iteration.
4- The derivations in the appendix proceed essentially by means of a cavity (or a leave-one-out) method,
but end abruptly without clear conclusion. The conclusion seems to be the decoupled iteration (14), not the actual PDE.
This should be made clear, or otherwise, the derivation should be conducted further to the desired conclusion.
The derivation contain some errors and typos; I couldn't follow the computations
(e.g., not clear how line 207 was obtained, a factor c_k should appear next to Q_k^p on many lines...)
  
5-There are several imprecisions/typos in the notation:
line 36: support recover*y*
eq 2: sign inconsistency in front of \tau_k (- gradient, not +)
line 100: x_k should be x
line 133: u should be \xi (in many later occurrences)
eq 8: \tau not defined (is it the limit of the step sizes \tau_k?)
line 116: t = [k/p] should be k = [tp]