We have omitted a number of complicated formulas in the main text to provide clear intuition and concise proof sketch. We will list all mentioned formulas here for readers' reference. We consider the case where U = V = Aand Σ is symmetric and full-rank, and we use gradient flow. We can derive the dynamics of S = AA>as S:= (Σ S)S+ S(Σ S), which is a quadratic ordinary differential equation and it is hard to solve directly. For simplicity, define X:= X Σ 1. Then X = XΣ ΣX. (24) Solving this equation and we have And it is interesting to verify that S(t) + P(t) Σ by using the following lemma.

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Bounds on the smallest eigenvalue of the neural tangent kernel (NTK) are a key ingredient in the analysis of neural network optimization and memorization. However, existing results require distributional assumptions on the data and are limited to a high-dimensional setting, where the input dimension $d_0$ scales at least logarithmically in the number of samples $n$. In this work we remove both of these requirements and instead provide bounds in terms of a measure of distance between data points: notably these bounds hold with high probability even when $d_0$ is held constant versus $n$. We prove our results through a novel application of the hemisphere transform.

While initial breakthroughs on the convergence of gradient optimization in neural networks (Li & Liang, 2018; Du et al., 2019a; Allen-Zhu et al., 2019) required unrealistic conditions on the

Stein V ariational Gradient Descent (SVGD) is a nonparametric particle-based deterministic sampling algorithm.

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We thank the reviewers for their appreciative and thoughtful feedback. Reviewer 1. "However, the authors fail to bring the result to their impact of the current state OT, or any novel stochastic optimization algorithm designed to compute it faster. We will further emphasize these aspects. " A measure with 0 mean. Reviewer 2. "If the paper could show the formula for that case [TV] that would be Figure 1: Large dimensions need more samples to approximate the moments of the unbalanced optimal transport plan. Reviewer 3. ""Figure 1 illustrates the convergence"... the convergence of what?" "Figure 2 is also difficult to understand.

Langevin dynamics (LD) has been proven to be a powerful technique for optimizing a non-convex objective as an efficient algorithm to find local minima while eventually visiting a global minimum on longer time-scales. LD is based on the first-order Langevin diffusion which is reversible in time. We study two variants that are based on non-reversible Langevin diffusions: the underdamped Langevin dynamics (ULD) and the Langevin dynamics with a non-symmetric drift (NLD). Adopting the techniques of Tzen et al. (2018) for LD to non-reversible diffusions, we show that for a given local minimum that is within an arbitrary distance from the initialization, with high probability, either the ULD trajectory ends up somewhere outside a small neighborhood of this local minimum within a recurrence time which depends on the smallest eigenvalue of the Hessian at the local minimum or they enter this neighborhood by the recurrence time and stay there for a potentially exponentially long escape time. The ULD algorithm improves upon the recurrence time obtained for LD in Tzen et al. (2018) with respect to the dependency on the smallest eigenvalue of the Hessian at the local minimum. Similar results and improvements are obtained for the NLD algorithm. We also show that non-reversible variants can exit the basin of attraction of a local minimum faster in discrete time when the objective has two local minima separated by a saddle point and quantify the amount of improvement. Our analysis suggests that non-reversible Langevin algorithms are more efficient to locate a local minimum as well as exploring the state space.