Smoothed analysis of the low-rank approach for smooth semidefinite programs

We consider semidefinite programs (SDPs) of size n with equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size $n$ by $k$ such that $X = YY^*$ is the SDP variable. The advantages of such formulation are twofold: the dimension of the optimization variable is reduced and positive semidefiniteness is naturally enforced. However, the problem in Y is non-convex. In prior work, it has been shown that, when the constraints on the factorized variable regularly define a smooth manifold, provided k is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading to the question: are such points also approximately optimal? To this end, and under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with k scaling like the square root of the number of constraints (up to log factors). Moreover, we bound the optimality gap at the approximate solution of the perturbed problem with respect to the original problem. We particularize our results to an SDP relaxation of phase retrieval

A scalable mining of frequent quadratic concepts in d-folksonomies

Folksonomy mining is grasping the interest of web 2.0 community since it represents the core data of social resource sharing systems. However, a scrutiny of the related works interested in mining folksonomies unveils that the time stamp dimension has not been considered. For example, the wealthy number of works dedicated to mining tri-concepts from folksonomies did not take into account time dimension. In this paper, we will consider a folksonomy commonly composed of triples and we shall consider the time as a new dimension. We motivate our approach by highlighting the battery of potential applications. Then, we present the foundations for mining quadri-concepts, provide a formal definition of the problem and introduce a new efficient algorithm, called QUADRICONS for its solution to allow for mining folksonomies in time, i.e., d-folksonomies. We also introduce a new closure operator that splits the induced search space into equivalence classes whose smallest elements are the quadri-minimal generators. Carried out experiments on large-scale real-world datasets highlight good performances of our algorithm

On the Range of the Fourier Transform Associated with the Spherical Mean Operator

We characterize the range of some spaces of functions by the Fourier transform associated with the spherical mean operator R and we give a new description of the Schwartz spaces. Next, we prove a Paley-Wiener and a Paley-Wiener-Schawrtz theorems

DATABASE APPROACH FOR MULTIPLE-CRITERIA DECISION SUPPORT SYSTEMS

This paper focuses on data management aspects of computerized decision support systems which use interactive multiple criteria decision methods. In this context, we point out the technical requirements for such systems and the importance of the data management tool to MCDSS. After a discussion of candidate data models (i.e. relational, hierarchical, and network), we examine the criteria to use in choosing the data model for MCDSS. In the last part of this paper, we review some database management services which support data definition, data manipulation, and data integrity within the multiple-criteria decision making framework. These services guide us when designing the appropriate architecture for the MCDSSâs data component.Information Systems Working Papers Serie

MEDIATOR: TOWARDS A NEGOTIATION SUPPORT SYSTEM

MEDIATOR is a negotiation support system (NSS) based on evolutionary systems design (ESD) and database-centered implementation. It supports negotiations by consensus seeking through exchange of information and, where consensus is incomplete, by compromise. The negotiation problem is shown --graphically or as relational data in matrix form-- in three spaces as a mapping from control space to goal space (and through marginal utility functions) to utility space. Within each of these spaces the negotiation process is characterized by adaptive change, i.e., mappings of group target and feasible sets by which these sets are redefined in seeking a solution characterized by a single-point intersection between them. This concept is being implemented in MEDIATOR, a data-based micro-mainframe NSS intended to support the players and a human mediator in multi-player decision situations. Each player employs private and shared database views, using his/her own micro-computer decision support system enhanced with a communications manager to interact with the mediator DSS. Sharing of views constitutes exchange of information which can lead towards consensus. The human mediator can support compromise, as needed, through use of solution concepts and/or concession-making procedures in the NSS model base. As a concrete example, we demonstrate the use of the system for group car buying decisions.Information Systems Working Papers Serie

SOLVING MULTI-CRITERIA ALLOCATION PROBLEMS: A DECISION SUPPORT SYSTEM APPROACH

MCADSS is a multi-criteria allocation decision support system for assisting in the task of partitioning a set of individuals into groups. Based upon multiple criteria, MCADSSâs goal is to maximize the diversity of members within groups, while minimizing the average differences between groups. (The project may be viewed from several perspectives: as a multi-criteria decision making problem, as a "reverse" clustering problem, or as a personnel assignment problem). The system is currently being used to allocate MBA students into sections and study teams at INSEAD, a leading European business school. This paper describes the rationale for MCADSS, design criteria, system methodology, and application results. It also suggests how the approach outlined here might be used for further applications.Information Systems Working Papers Serie