Non-convex approaches for low-rank tensor completion under tubal sampling

Tensor completion is an important problem in modern data analysis. In this work, we investigate a specific sampling strategy, referred to as tubal sampling. We propose two novel non-convex tensor completion frameworks that are easy to implement, named tensor $L_1$-$L_2$ (TL12) and tensor completion via CUR (TCCUR). We test the efficiency of both methods on synthetic data and a color image inpainting problem. Empirical results reveal a trade-off between the accuracy and time efficiency of these two methods in a low sampling ratio. Each of them outperforms some classical completion methods in at least one aspect

Curvature-Aware Derivative-Free Optimization

The paper discusses derivative-free optimization (DFO), which involves minimizing a function without access to gradients or directional derivatives, only function evaluations. Classical DFO methods, which mimic gradient-based methods, such as Nelder-Mead and direct search have limited scalability for high-dimensional problems. Zeroth-order methods have been gaining popularity due to the demands of large-scale machine learning applications, and the paper focuses on the selection of the step size $\alpha_k$ in these methods. The proposed approach, called Curvature-Aware Random Search (CARS), uses first- and second-order finite difference approximations to compute a candidate $\alpha_{+}$. We prove that for strongly convex objective functions, CARS converges linearly provided that the search direction is drawn from a distribution satisfying very mild conditions. We also present a Cubic Regularized variant of CARS, named CARS-CR, which converges in a rate of $\mathcal{O}(k^{-1})$ without the assumption of strong convexity. Numerical experiments show that CARS and CARS-CR match or exceed the state-of-the-arts on benchmark problem sets.Comment: 31 pages, 9 figure

Towards Constituting Mathematical Structures for Learning to Optimize

Learning to Optimize (L2O), a technique that utilizes machine learning to learn an optimization algorithm automatically from data, has gained arising attention in recent years. A generic L2O approach parameterizes the iterative update rule and learns the update direction as a black-box network. While the generic approach is widely applicable, the learned model can overfit and may not generalize well to out-of-distribution test sets. In this paper, we derive the basic mathematical conditions that successful update rules commonly satisfy. Consequently, we propose a novel L2O model with a mathematics-inspired structure that is broadly applicable and generalized well to out-of-distribution problems. Numerical simulations validate our theoretical findings and demonstrate the superior empirical performance of the proposed L2O model.Comment: ICML 202

Modeling \u3cem\u3ep\u3c/em\u3eCO\u3csub\u3e2\u3c/sub\u3e Variability in the Gulf of Mexico

A three-dimensional coupled physical–biogeochemical model was used to simulate and examine temporal and spatial variability of sea surface pCO2 in the Gulf of Mexico (GoM). The model was driven by realistic atmospheric forcing, open boundary conditions from a data-assimilative global ocean circulation model, and observed freshwater and terrestrial nutrient and carbon input from major rivers. A 7-year model hindcast (2004–2010) was performed and validated against ship measurements. Model results revealed clear seasonality in surface pCO2 and were used to estimate carbon budgets in the Gulf. Based on the average of model simulations, the GoM was a net CO2 sink with a flux of 1.11 ± 0.84  ×  1012 mol C yr−1, which, together with the enormous fluvial inorganic carbon input, was comparable to the inorganic carbon export through the Loop Current. Two model sensitivity experiments were performed: one without biological sources and sinks and the other using river input from the 1904–1910 period as simulated by the Dynamic Land Ecosystem Model (DLEM). It was found that biological uptake was the primary driver making GoM an overall CO2 sink and that the carbon flux in the northern GoM was very susceptible to changes in river forcing. Large uncertainties in model simulations warrant further process-based investigations

Century-Long Increasing Trend and Variability of Dissolved Organic Carbon Export from the Mississippi River Basin Driven by Natural and Anthropogenic Forcing

There has been considerable debate as to how natural forcing and anthropogenic activities alter the timing and magnitude of the delivery of dissolved organic carbon (DOC) to the coastal ocean, which has ramifications for the ocean carbon budget, land-ocean interactions, and coastal life. Here we present an analysis of DOC export from the Mississippi River to the Gulf of Mexico during 1901–2010 as influenced by changes in climate, land use and management practices, atmospheric CO2, and nitrogen deposition, through the integration of observational data with a coupled hydrologic/biogeochemical land model. Model simulations show that DOC export in the 2000s increased more than 40% since the 1900s. For the recent three decades (1981–2010), however, our simulated DOC export did not show a significant increasing trend, which is consistent with observations by U.S. Geological Survey. Our factorial analyses suggest that land use and land cover change, including land management practices (LMPs: i.e., fertilization, irrigation, tillage, etc.), were the dominant contributors to the century-scale trend of rising total riverine DOC export, followed by changes in atmospheric CO2, nitrogen deposition, and climate. Decadal and interannual variations of DOC export were largely attributed to year-to-year climatic variability and extreme flooding events, which have been exacerbated by human activity. LMPs show incremental contributions to DOC increase since the 1960s, indicating the importance of sustainable agricultural practices in coping with future environmental changes such as extreme flooding events. Compared to the observational-based estimate, the modeled DOC export was 20% higher, while DOC concentrations were slightly lower. Further refinements in model structure and input data sets should enable reductions in uncertainties in our prediction of century-long trends in DOC