Optimization problems on the Stiefel manifold, ranging from principal component analysis to enhancing neural network robustness, are ubiquitous in machine learning. The Landing algorithm avoids computationally expensive retraction operations on manifolds, making it highly competitive for large-scale problems. This paper extends this method to distributed settings, introducing EF-Landing, the first retraction-free and communication-efficient algorithm for distributed stochastic optimization on the Stiefel manifold. By incorporating communication compression and error feedback, EF-Landing ensures convergence and constraint feasibility while significantly reducing communication overhead. We provide sharp convergence guarantees, demonstrating that EF-Landing achieves the same asymptotic linear speedup convergence rate as existing methods without communication compression. Furthermore, our analysis is highly versatile, applying to both deterministic and stochastic settings and encompassing algorithms based on gradient descent or momentum-based gradient descent. We also generalize EF-Landing to operate on block-wise Stiefel manifolds, enabling greater flexibility for structured constraints. Extensive numerical experiments validate our theoretical results.

Although shadow banking widely exists in the financial systems of various countries, their definitions vary significantly due to specific economic and financial characteristics. This paper classifies Chinese shadow banking into six categories: securities, trust, private lending, banking, fund, and insurance. The AR-GARCH-DCC model is used to measure systemic risk spillover through from an industrial and institutional perspective. The network topology index is employed to analyze risk contagion and further explore influencing factors. Firstly, based on the results of the AR-GARCH-DCC, the estimated dynamic volatility (σ) indicates that shadow banking risk spillover is time-varying, especially in trust and securities. Second, according to the static risk spillover analysis, various institutions play different roles and can transform between risk spillovers and overflowers. Thirdly, eigenvector centrality, leverage, assets, CPI, and macroeconomic prosperity significantly impact shadow banking systemic risk spillover.