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Federated learning (FL) has found many important applications in smart-phone-APP based machine learning applications. Although many algorithms have been studied for FL, to the best of our knowledge, algorithms for FL with nonconvex constraints have not been studied. This paper studies FL over Riemannian manifolds, which finds important applications such as federated PCA and federated kPCA. We propose a Riemannian federated SVRG (RFedSVRG) method to solve federated optimization over Riemannian manifolds. We analyze its convergence rate under different scenarios. Numerical experiments are conducted to compare RFedSVRG with the Riemannian counterparts of FedAvg and FedProx. We observed from the numerical experiments that the advantages of RFedSVRG are significant. 

Riemannian optimization has drawn a lot of attention due to its wide applications in practice. Riemannian stochastic first-order algorithms have been studied in the literature to solve large-scale machine learning problems over Riemannian manifolds. However, most of the existing Riemannian stochastic algorithms require the objective function to be differentiable, and they do not apply to the case where the objective function is nonsmooth. In this paper, we present two Riemannian stochastic proximal gradient methods for minimizing nonsmooth function over the Stiefel manifold. The two methods, named R-ProxSGD and R-ProxSPB, are generalizations of proximal SGD and proximal SpiderBoost in Euclidean setting to the Riemannian setting. Analysis on the incremental first-order oracle (IFO) complexity of the proposed algorithms is provided. Specifically, the R-ProxSPB algorithm finds an ϵ -stationary point with O(ϵ−3) IFOs in the online case, and O(n+n‾√ϵ−2) IFOs in the finite-sum case with n being the number of summands in the objective. Experimental results on online sparse PCA and robust low-rank matrix completion show that our proposed methods significantly outperform the existing methods that use Riemannian subgradient information. 

Protein transfer into nanoscale compartments is critical for many cellular/life processes, yet there are few reports on how compartment properties impact the protein orientation during a transfer. Such a knowledge gap limits a deeper understanding of the protein transfer mechanism, which could be bridged using nanoporous materials. Here, we use a mesoporous silica, a covalent organic framework, and a metal-organic framework with charged, hydrophobic, and neutral surfaces, respectively, to elucidate the impact of channel properties on the transfer of a model protein, lysozyme. Using site-directed spin labeling and time-resolved electron paramagnetic resonance spectroscopy, we reveal that the transfer can be a multi-step process depending on channel properties and depict the relative orientation changes of lysozyme upon transfer into each channel. To the best of our knowledge, this is the first structural insight into protein orientation upon transfer into different compartments, meaningful for the rational design of synthetic materials to host enzymes or mimic the cellular compartments.