Convex regression and its extensions to learning a Bregman divergence and difference of convex functions

Nonparametric convex regression has been extensively studied over the last two decades. It has been shown any Lipschitz convex function can be approximated with arbitrarily accuracy with a max of linear functions. Using this framework, in this thesis we generalize convex regression to learning an arbitrary Bregman divergence and learning a difference of convex functions. We provide approximation guarantees and sample complexity bounds for both these extensions. Furthermore, we provide algorithms to solve the resulting optimization problems based on 2-block alternative direction method of multipliers (ADMM). For this algorithm, we provide convergence guarantees with iteration complexity of O(n√d/) for a dataset X ℝ^n,d and arbitrary positive . Finally we provide experiments for both the Bregman divergence learning and difference of convex functions learning based on UCI datasets that demonstrate the state of the art on regression and classification datasets

Learning to Approximate a Bregman Divergence

Bregman divergences generalize measures such as the squared Euclidean distance and the KL divergence, and arise throughout many areas of machine learning. In this paper, we focus on the problem of approximating an arbitrary Bregman divergence from supervision, and we provide a well-principled approach to analyzing such approximations. We develop a formulation and algorithm for learning arbitrary Bregman divergences based on approximating their underlying convex generating function via a piecewise linear function. We provide theoretical approximation bounds using our parameterization and show that the generalization error $O_p(m^{-1/2})$ for metric learning using our framework matches the known generalization error in the strictly less general Mahalanobis metric learning setting. We further demonstrate empirically that our method performs well in comparison to existing metric learning methods, particularly for clustering and ranking problems.Comment: 19 pages, 4 figure

Piecewise linear regression via a difference of convex functions

We present a new piecewise linear regression methodology that utilizes fitting a difference of convex functions (DC functions) to the data. These are functions f that may be represented as the difference _1- _2 for a choice of convex functions _1,_2. The method proceeds by estimating piecewise-liner convex functions, in a manner similar to max-affine regression, whose difference approximates the data. The choice of the function is regularised by a new seminorm over the class of DC functions that controls the _∞ Lipschitz constant of the estimate. The resulting methodology can be efficiently implemented via Quadratic programming even in high dimensions, and is shown to have close to minimax statistical risk. We empirically validate the method, showing it to be practically implementable, and to have comparable performance to existing egression/classification methods on real-world datasets.http://proceedings.mlr.press/v119/siahkamari20a/siahkamari20a.pdfPublished versio

Faster algorithms for learning convex functions

W911NF2110246 - Department of Defense/ARO; CCF-2007350 - National Science Foundation; CCF-1955981 - National Science Foundationhttps://proceedings.mlr.press/v162/siahkamari22a/siahkamari22a.pdfFirst author draf

Spatial prediction of flood-susceptible areas using frequency ratio and maximum entropy models

Modelling the flood in watersheds and reducing the damages caused by this natural disaster is one of the primary objectives of watershed management. This study aims to investigate the application of the frequency ratio and maximum entropy models for flood susceptibility mapping in the Madarsoo watershed, Golestan Province, Iran. Based on the maximum entropy and frequency ratio methods as well as analysis of the relationship between the flood events belonging to training group and the factors affecting on the risk of flooding, the weight of classes of each factor was determined in a GIS environment. Finally, prediction map of flooding potential was validated using receiver operating characteristic (ROC) curve method. ROC curve estimated the area under the curve for frequency ratio and the maximum entropy models as 74.3% and 92.6%, respectively, indicating that the maximum entropy model led to better results for evaluating flooding potential in the study area

A Review on Recent Development of Cooling Technologies for Photovoltaic Modules

When converting solar energy to electricity, a big proportion of energy is not converted for electricity but for heating PV cells, resulting in increased cell temperature and reduced electrical efficiency. Many cooling technologies have been developed and used for PV modules to lower cell temperature and boost electric energy yield. However, little crucial review work was proposed to comment cooling technologies for PV modules. Therefore, this paper has provided a thorough review of the up-to-date development of existing cooling technologies for PV modules, and given appropriate comments, comparisons and discussions. According to the ways or principles of cooling, existing cooling technologies have been classified as fluid medium cooling (air cooling, water cooling and nanofluids cooling), optimizing structural configuration cooling and phase change materials cooling. Potential influential factors and sub-methods were collected from the review work, and their contributions and impact have been discussed to guide future studies. Although most cooling technologies reviewed in this paper are matured, there are still problems need to be solved, such as the choice of cooling fluid and its usability for specific regions, the fouling accumulation and cleaning of enhanced heat exchangers with complex structures, the balance between cooling cost and net efficiency of PV modules, the cooling of circulating water in tropical areas and the freezing of circulating water in cold areas. To be advocated, due to efficient heat transfer and spectral filter characters, nanofluids can promote the effective matching of solar energy at both spectral and spatial scales to achieve orderly energy utilization