Network Lasso: Clustering and Optimization in Large Graphs

Convex optimization is an essential tool for modern data analysis, as it provides a framework to formulate and solve many problems in machine learning and data mining. However, general convex optimization solvers do not scale well, and scalable solvers are often specialized to only work on a narrow class of problems. Therefore, there is a need for simple, scalable algorithms that can solve many common optimization problems. In this paper, we introduce the \emph{network lasso}, a generalization of the group lasso to a network setting that allows for simultaneous clustering and optimization on graphs. We develop an algorithm based on the Alternating Direction Method of Multipliers (ADMM) to solve this problem in a distributed and scalable manner, which allows for guaranteed global convergence even on large graphs. We also examine a non-convex extension of this approach. We then demonstrate that many types of problems can be expressed in our framework. We focus on three in particular - binary classification, predicting housing prices, and event detection in time series data - comparing the network lasso to baseline approaches and showing that it is both a fast and accurate method of solving large optimization problems

The structure of algebraic covariant derivative curvature tensors

We use the Nash embedding theorem to construct generators for the space of algebraic covariant derivative curvature tensors

Differentially Private Federated Clustering over Non-IID Data

In this paper, we investigate federated clustering (FedC) problem, that aims to accurately partition unlabeled data samples distributed over massive clients into finite clusters under the orchestration of a parameter server, meanwhile considering data privacy. Though it is an NP-hard optimization problem involving real variables denoting cluster centroids and binary variables denoting the cluster membership of each data sample, we judiciously reformulate the FedC problem into a non-convex optimization problem with only one convex constraint, accordingly yielding a soft clustering solution. Then a novel FedC algorithm using differential privacy (DP) technique, referred to as DP-FedC, is proposed in which partial clients participation and multiple local model updating steps are also considered. Furthermore, various attributes of the proposed DP-FedC are obtained through theoretical analyses of privacy protection and convergence rate, especially for the case of non-identically and independently distributed (non-i.i.d.) data, that ideally serve as the guidelines for the design of the proposed DP-FedC. Then some experimental results on two real datasets are provided to demonstrate the efficacy of the proposed DP-FedC together with its much superior performance over some state-of-the-art FedC algorithms, and the consistency with all the presented analytical results.Comment: 31 pages, 4 figures, 1 tabl

Improved lattice QCD with quarks: the 2 dimensional case

QCD in two dimensions is investigated using the improved fermionic lattice Hamiltonian proposed by Luo, Chen, Xu, and Jiang. We show that the improved theory leads to a significant reduction of the finite lattice spacing errors. The quark condensate and the mass of lightest quark and anti-quark bound state in the strong coupling phase (different from t'Hooft phase) are computed. We find agreement between our results and the analytical ones in the continuum.Comment: LaTeX file (including text + 10 figures

Privacy-preserving Federated Primal-dual Learning for Non-convex and Non-smooth Problems with Model Sparsification

Federated learning (FL) has been recognized as a rapidly growing research area, where the model is trained over massively distributed clients under the orchestration of a parameter server (PS) without sharing clients' data. This paper delves into a class of federated problems characterized by non-convex and non-smooth loss functions, that are prevalent in FL applications but challenging to handle due to their intricate non-convexity and non-smoothness nature and the conflicting requirements on communication efficiency and privacy protection. In this paper, we propose a novel federated primal-dual algorithm with bidirectional model sparsification tailored for non-convex and non-smooth FL problems, and differential privacy is applied for strong privacy guarantee. Its unique insightful properties and some privacy and convergence analyses are also presented for the FL algorithm design guidelines. Extensive experiments on real-world data are conducted to demonstrate the effectiveness of the proposed algorithm and much superior performance than some state-of-the-art FL algorithms, together with the validation of all the analytical results and properties.Comment: 30 pages, 8 figure

Sedimentary mechanisms of a modern banded iron formation on Milos Island, Greece

An Early Quaternary shallow submarine hydrothermal iron formation (IF) in the Cape Vani sedimentary basin (CVSB) on Milos Island, Greece, displays banded rhythmicity similar to Precambrian banded iron formation (BIF). Sedimentary, stratigraphic reconstruction, biogeochemical analysis and micro-nanoscale mineralogical characterization confirms the Milos rocks as modern Precambrian BIF analogues. Spatial coverage of the BIF-type rocks in relation to the economic grade Mn ore that brought prominence to the CVSB implicates tectonic activity and changing redox in the deposition of the BIF-type rocks. Field-wide stratigraphic and biogeochemical reconstruction demonstrates two temporal and spatially isolated iron deposits in the CVSB with distinct sedimentological character. Petrographic screening suggest the previously described photoferrotrophic-like microfossil-rich IF (MFIF), accumulated on basement andesite in a ~ 150 m wide basin, in the SW margin of the basin. A strongly banded non-fossiliferous IF (NFIF) caps the Mn-rich sandstones at the transition to the renowned Mn-rich formation. Geochemical evidence relates the origin of the NFIF to periodic submarine volcanism and water column oxidation of released Fe(II) in conditions apparently predominated by anoxia, similar to the MFIF. This is manifested in the lack of shale-normalized Ce anomalies. Raman spectroscopy pairs hematite-rich grains in the NFIF with relics of a carbonaceous material carrying an average δ13Corg signature of ~ −25 ‰. However, a similar δ13Corg signature in the MFIF is not directly coupled to hematite by mineralogy. The NFIF, which post dates large-scale Mn deposition in the CVSB, is composed primarily of amorphous Si (opal-SiO2 · nH2O) while crystalline quartz (SiO2) predominates the MFIF. An intricate interaction between tectonic processes, changing redox, biological activity and abiotic Si precipitation, formed the unmetamorphosed BIF-type deposits

Time-dependent density-functional theory for open systems

By introducing the self-energy density functionals for the dissipative interactions between the reduced system and its environment, we develop a time-dependent density-functional theory formalism based on an equation of motion for the Kohn-Sham reduced single-electron density matrix of the reduced system. Two approximate schemes are proposed for the self-energy density functionals, the complete second order approximation and the wide-band limit approximation. A numerical method based on the wide-band limit approximation is subsequently developed and implemented to simulate the steady and transient current through various realistic molecular devices. Simulation results are presented and discussed.Comment: 16 pages, 12 figure

Higher order Jordan Osserman Pseudo-Riemannian manifolds

We study the higher order Jacobi operator in pseudo-Riemannian geometry. We exhibit a family of manifolds so that this operator has constant Jordan normal form on the Grassmannian of subspaces of signature (r,s) for certain values of (r,s). These pseudo-Riemannian manifolds are new and non-trivial examples of higher order Osserman manifolds