Matrix Completion via Max-Norm Constrained Optimization

Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly when the underlying sampling scheme is non-uniform. In this paper we propose and analyze a max-norm constrained empirical risk minimization method for noisy matrix completion under a general sampling model. The optimal rate of convergence is established under the Frobenius norm loss in the context of approximately low-rank matrix reconstruction. It is shown that the max-norm constrained method is minimax rate-optimal and yields a unified and robust approximate recovery guarantee, with respect to the sampling distributions. The computational effectiveness of this method is also discussed, based on first-order algorithms for solving convex optimizations involving max-norm regularization.Comment: 33 page

A Conjoint Analysis of Wetland-Based Recreation: A Case Study of Louisiana Waterfowl Hunting.

Conjoint analysis is a recent evolution in mathematical psychology that has been employed extensively in the marketing environment. The technique is concerned with measuring the joint effect of two or more independent variables on the ordering of a dependent variable. Conjoint analysis relates an individual\u27s preferences to a set of prespecified attributes. The objective of conjoint analysis is to decomposed a set of responses to factorially designed stimuli in which the utility of each stimuli attribute can be inferred from the respondents\u27 evaluations of the stimuli. In addition, conjoint analysis and its economic foundations are developed in the context of conventional related market and non-market valuation approaches. Given the multiattribute nature of wetland based activities such as waterfowl hunting, conjoint analysis becomes an attractive approach in estimating the benefits and values derived from wetland based activities. An empirical and economic analysis is presented in which waterfowl hunters\u27 willingness-to-pay for various hunting trip attributes is derived from a rank-ordered logit specification of the indirect utility function. The hunting trip vignettes are developed according to seven different attributes with each attribute varying across three levels using a fractional factorial experiment. The data for the analysis were derived from questionnaires mailed to 7,500 randomly selected individuals who purchased 1990 Louisiana duck stamps. The statistical estimation technique employed in this research was rank-ordered logit via weighted least squares. Weighted least squares was chosen due to the presence of heteroskedasticity and uncertainty regarding the properties of the error term which masks the efficiency of the ordinary least squares regression. A Box-Cox transformation was also employed to test for specification of the functional form. The results indicated that the length of the hunting season, the daily duck bag limit, and the rate of congestion were three significant factors influencing waterfowl hunters\u27 trip rating preferences. In addition, conjoint analysis appears to be a viable technique for analysis of resource based multiattribute activities

Conflicting Objectives in Decisions

This book deals with quantitative approaches in making decisions when conflicting objectives are present. This problem is central to many applications of decision analysis, policy analysis, operational research, etc. in a wide range of fields, for example, business, economics, engineering, psychology, and planning. The book surveys different approaches to the same problem area and each approach is discussed in considerable detail so that the coverage of the book is both broad and deep. The problem of conflicting objectives is of paramount importance, both in planned and market economies, and this book represents a cross-cultural mixture of approaches from many countries to the same class of problem