Nonuniform subsampling methods are effective to reduce computational burden and maintain estimation efficiency for massive data. Existing methods mostly focus on subsampling with replacement due to its high computational efficiency. If the data volume is so large that nonuniform subsampling probabilities cannot be calculated all at once, then subsampling with replacement is infeasible to implement. This paper solves this problem using Poisson subsampling. We first derive optimal Poisson subsampling probabilities in the context of quasi-likelihood estimation under the A- and L-optimality criteria. For a practically implementable algorithm with approximated optimal subsampling probabilities, we establish the consistency and asymptotic normality of the resultant estimators. To deal with the situation that the full data are stored in different blocks or at multiple locations, we develop a distributed subsampling framework, in which statistics are computed simultaneously on smaller partitions of the full data. Asymptotic properties of the resultant aggregated estimator are investigated. We illustrate and evaluate the proposed strategies through numerical experiments on simulated and real data sets.

Stromal cell-derived factor-1 (SDF-1) and its receptor, CXCR4, are crucial for homing and migration of multiple stem cell types. Their potential role in mediating bone marrow-derived mesenchymal stem cell (BMSC) migration in areas of myocardial infarction (MI) has not been demonstrated. In this study, rat heart MI was created by left coronary artery ligation, and green fluorescent protein-labeled BMSCs were directly infused into the left ventricular cavity. Reverse transcriptase-polymerase chain reaction and Western blot analysis showed that SDF-1 was predominantly localized in the MI lesion, and its levels peaked by 3 to 7 days and were maintained at least 14 days. Additionally, this was matched with increased accumulation of BMSCs and an improvement in cardiac function. Furthermore, this effect was blocked by the phosphoinositide 3-kinase inhibitor, LY294002. In vitro experiments showed that CXCR4 expression by BMSCs was elevated during hypoxia and SDF-1 induced a concentration-dependent migration of BMSCs. This migration was CXCR4-dependent as confirmed by its total inhibition by AMD3100, a CXCR4-specific antagonist. Migration was also almost completely blocked by LY294002. Analysis showed that phosphorylated Akt was highly increased in SDF-1-treated BMSCs. Together these results demonstrated that SDF-1/CXCR4 may mediate the migration of BMSCs toward heart MI through activation of PI3K/Akt.

As a valid metric of metric-measure spaces, Gromov-Wasserstein (GW) distance has shown the potential for matching problems of structured data like point clouds and graphs. However, its application in practice is limited due to the high computational complexity. To overcome this challenge, we propose a novel importance sparsification method, called Spar-GW, to approximate GW distance efficiently. In particular, instead of considering a dense coupling matrix, our method leverages a simple but effective sampling strategy to construct a sparse coupling matrix and update it with few computations. The proposed Spar-GW method is applicable to the GW distance with arbitrary ground cost, and it reduces the complexity from $O(n^4)$ to $O(n^{2+\delta })$ for an arbitrary small $\delta >0$. Theoretically, the convergence and consistency of the proposed estimation for GW distance are established under mild regularity conditions. In addition, this method can be extended to approximate the variants of GW distance, including the entropic GW distance, the fused GW distance, and the unbalanced GW distance. Experiments show the superiority of our Spar-GW to state-of-the-art methods in both synthetic and real-world tasks. Supplementary materials for this article are available online.