ADMM for Nonsmooth Composite Optimization under Orthogonality Constraints

1. Could the authors point out the core technical difficulties in this paper?
2. Could the authors compare OADMM with the ManPG (by linearizing the concave part) in the experiments?
3. Note that the sparse PCA is not weakly convex, it is not rigorious to use the Riemannian subgradient method in the experiment. That is partly why subgradient methods do not work. See Line 473.
4. The applications are rather limited. The authors only provide the sparse PCA example. Could the authors find other applications with the linear operator A not being the identity operator?
5. There is some mistake in Line 602 in the reference part.
6. I do not understand why (a) in Lemma 2.2 holds. Could the authors explain it?
7. The paper claims that the iteration complexity improves from $O(\epsilon^{-4})$ to $O(\epsilon^{-3})$ due to the decreasing step size in lines 142-143. However, in lines 255–259, the paper also claims that the over-relaxation step size $\sigma$ for the dual variable and the Nesterov extrapolation parameter $\alpha$ accelerate the convergence. This is somewhat confusing. My understanding is that the complexity might remain the same if we set $\alpha = 0$, $\sigma = 1$, and $\theta > 1$ (in this case, OADMM would reduce to the classical linearized proximal ADMM). In other words, $\alpha$ and $\sigma$ primarily enhance numerical performance rather than complexity. Could the authors clarify this point to make the mechanism behind the complexity improvement more explicit?
8. The paper does not include a convergence rate analysis for cases where the KL exponent $\tilde{\sigma} < 1/4$. Could the authors provide an explanation for this omission?
9. The description of SPGM as a penalty method (lines 100–103) seems inconsistent with the approach in the original paper, where the problem is solved by taking a gradient step with respect to the smoothed problem $\min_{x \in M} f(x) + h_\mu(x)$ . Are these descriptions equivalent? Clarification on this equivalence would be helpful.
10. It seems that $\mathcal{A}$ could be a smooth mapping with a bounded operator norm and Lipschitz continuous gradient to ensure that the function $S$ is smooth. Is the assumption that $\mathcal{A}$ is a linear mapping is due to practical computation considerations?

Here’s a refined version of each point to improve clarity and readability:

1. Definition 2.7: Since the subgradient is a set, how is Crit calculated?
2. Definition 2.7: The defined stationarity differs from that in the existing literature because the problem is more complex. Please validate this notion of stationarity with established concepts for problem (1).
3. Algorithm 1: Only one iteration is performed for each $X$ update. Is the BB step size truly beneficial, given that the gradients $G^{t-1}$ and $G^t$ are computed based on different objectives?
4. Algorithm 1: In classical ADMM, the penalty parameter is typically fixed. Does a constant $\beta^t$ ensure the same convergence guarantee?
5. The connection between Crit and $e^t$ is unclear. From Lemmas 4.7 and 4.12, it appears that the average of $e^t$ converges. However, the definition of $e^t$ involves the optimality of subproblems, such as $x^t - x^{t-1}$ and $y^t - y^{t-1}$. Careful elaboration on $e^t$ and the stationarity of the original problem is needed.
6. Assumption 5.1: How are the KL assumptions on $L$ connected to problem (1)? Currently, it seems standard to show the rate under the KL of $\Theta$.
7. Line 453-454: There are mismatches in the descriptions.
8. The RADMM algorithm cannot handle nonconvex nonsmooth terms in the objectives. How is it implemented?
9. SPGM-EP: It is unclear how these two algorithms are implemented. Is $\|x\|\_1 - \|x\|\_{[k]}$ weakly convex, and is its Moreau envelope smoothing used?
10. Line 471-478: The description of RADMM is missing.
11. Abstract: What does "O" in OADMM stand for?
12. Line 132: "OADMM-EP" should be "OADMM-RR." Are these abbreviations necessary here?

• The paper discusses the convergence rate of the proposed method, though this might be somewhat standard, as it has been addressed in prior work on nonconvex and nonsmooth ADMM. Moreover, based on the K\L\ framework, when the parameter $\tilde \sigma \in (0, 1/2]$, the convergence rate is linear. Why, then, is the range $(1/4, 1/2]$ in this paper?

• The authors claim that they can use a larger dual step size, $\sigma \in [1,2)$; however, they do not clarify how this value is chosen. The same issue arises for the parameter $\theta$.

• Although the authors note that OADMM-RR uses the BB step size, only a simple constant step size $b^t \equiv 1$ is employed in the numerical experiments.

• Several parameters in the proposed algorithm require careful tuning. For instance, the authors select $\xi$ from a range---are the results in Figures 1 and 2 based on the optimal $\xi$?

• The performance of OADMM-RR and RADMM-II is very close. Could RADMM-II achieve better results with careful selection of $\beta^t$?

Is the approach extendable to general submanifolds? The current approach seems to be applicable only to compact manifolds. For instance, in the primal update, it is challenging to control boundedness, which implies that the lower boundedness of the potential function may not be guaranteed. Could you comment on this?