We consider singular integral operators of the form (a)$Z$_{1}L^{−1}Z_{2}, (b)$Z$_{1}Z_{2}L^{−1}, and (c)$L$^{−1}Z_{1}Z_{2}, where$Z$_{1}and$Z$_{2}are nonzero right-invariant vector fields, and$L$is the$L$^{2}-closure of a canonical Laplacian. The operators (a) are shown to be bounded on$L$^{p}for all$p$∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type ($p, p$) for any$p$∈[1, ∞).

Nonconvex reformulations via low-rank factorization for stochastic convex semidefinite optimization problem have attracted arising attention due to their empirical efficiency and scalability. Compared with the original convex formulations, the nonconvex ones typically involve much fewer variables, allowing them to scale to scenarios with millions of variables. However, it opens a new challenge that under what conditions the nonconvex stochastic algorithms may find the population minimizer within the optimal statistical precision despite their empirical success in applications. In this paper, we provide an answer that the stochastic gradient descent (SGD) method can be adapted to solve the nonconvex reformulation of the original convex problem, with a global linear convergence when using a fixed step size, i.e., converging exponentially fast to the population minimizer within an optimal statistical precision in the restricted strongly convex case. If a diminishing step size is adopted, the bad effect caused by the variance of gradients on the optimization error can be eliminated but the rate is dropped to be sublinear. The core of our treatment relies on a novel second-order descent lemma, which is more general than the existing best result in the literature and improves the analysis on both online and batch algorithms. The developed theoretical results and effectiveness of the suggested SGD are also verified by a series of experiments.

Short integer solution (SIS) and learning with errors (LWE) are two hard lattice problems. These two problems are believed having huge potential in application of cryptography. In 2012, Ding et al. [5] introduced the first provably secure key exchange based on LWE problem. On the other hand, we believe that it is very difficult to do key exchange on SIS problem only. In 2014, Wang et al. [6] did an attempt, but it was not successful. Mao et al. [7] broke the protocol by an attack based on CBi-SIS problem in 2016. However, their attack is not efficient. In this paper, we present a extremely straightforward and simple attack to Wang’s key exchange and then we will construct a key exchange based on SIS and LWE problems.

No abstract uploaded!

No abstract uploaded!