U-Statistic Reduction: Higher-Order Accurate Risk Control and Statistical-Computational Trade-Off, with Application to Network Method-of-Moments

U-statistics play central roles in many statistical learning tools but face the haunting issue of scalability. Significant efforts have been devoted into accelerating computation by U-statistic reduction. However, existing results almost exclusively focus on power analysis, while little work addresses risk control accuracy -- comparatively, the latter requires distinct and much more challenging techniques. In this paper, we establish the first statistical inference procedure with provably higher-order accurate risk control for incomplete U-statistics. The sharpness of our new result enables us to reveal how risk control accuracy also trades off with speed for the first time in literature, which complements the well-known variance-speed trade-off. Our proposed general framework converts the long-standing challenge of formulating accurate statistical inference procedures for many different designs into a surprisingly routine task. This paper covers non-degenerate and degenerate U-statistics, and network moments. We conducted comprehensive numerical studies and observed results that validate our theory's sharpness. Our method also demonstrates effectiveness on real-world data applications

Higher-order accurate two-sample network inference and network hashing

Two-sample hypothesis testing for comparing two networks is an important yet difficult problem. Major challenges include: potentially different sizes and sparsity levels; non-repeated observations of adjacency matrices; computational scalability; and theoretical investigations, especially on finite-sample accuracy and minimax optimality. In this article, we propose the first provably higher-order accurate two-sample inference method by comparing network moments. Our method extends the classical two-sample t-test to the network setting. We make weak modeling assumptions and can effectively handle networks of different sizes and sparsity levels. We establish strong finite-sample theoretical guarantees, including rate-optimality properties. Our method is easy to implement and computes fast. We also devise a novel nonparametric framework of offline hashing and fast querying particularly effective for maintaining and querying very large network databases. We demonstrate the effectiveness of our method by comprehensive simulations. We apply our method to two real-world data sets and discover interesting novel structures

Chemical Structure and Anticoagulant Property of a Novel Sulfated Polysaccharide from the Green Alga Cladophora oligoclada

Marine macroalgae are efficient producers of sulfated polysaccharides. The algal sulfated polysaccharides possess diverse bioactivities and peculiar chemical structures, and represent a great potential source to be explored. In the present study, a heparinoid-active sulfated polysaccharide was isolated from the green alga Cladophora oligoclada. Results of chemical and spectroscopic analyses indicated that the sulfated polysaccharide was composed of →6)-β-d-Galp-(1→, β-d-Galp-(1→, →6)-α-d-Glcp-(1→ and →3)-β-d-Galp-(1→ units with sulfate esters at C-2/C-4 of →6)-β-d-Galp-(1→, C-6 of →3)-β-d-Galp-(1→ and C-3 of →6)-α-d-Glcp-(1→ units. The branches consisting of β-d-Galp-(1→ and →6)-β-d-Galp-(1→ units were located in C-3 of →6)-β-d-Galp-(1→ units. The sulfated polysaccharide exhibited potent anticoagulant activity in vitro and in vivo as evaluated by activated partial thromboplastin time (APTT), thrombin time, and the fibrinogen level. For the APTT, the signal for clotting time was more than 200 s at 100 μg/mL in vitro and at 15 mg/kg in vivo. The obvious thrombolytic activity of the sulfated polysaccharide in vitro was also found. The mechanism analysis of anticoagulant action demonstrated that the sulfated polysaccharide significantly inhibited the activities of all intrinsic coagulation factors, which were less than 1.0% at 50 μg/mL, but selectively inhibited common coagulation factors. Furthermore, the sulfated polysaccharide strongly stimulated the inhibition of thrombin by potentiating antithrombin-III (AT-III) or heparin cofactor-II, and it also largely promoted the inhibition of factor Xa mediated by AT-III. These results revealed that the sulfated polysaccharide from C. oligoclada had potential to become an anticoagulant agent for prevention and therapy of thrombotic diseases