The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case.

In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large.

The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.

https://faculty.washington.edu/mfazel/low-rank-v2.pdf

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

The idea of ensemble methodology is to build a predictive model by integrating multiple models. It is well-known that ensemble methods can be used for improving prediction performance. Researchers from various disciplines such as statistics and AI considered the use of ensemble methodology. This paper, review existing ensemble techniques and can be served as a tutorial for practitioners who are interested in building ensemble based systems.

https://www.ise.bgu.ac.il/faculty/liorr/AI.pdf

Ensemble-based classifiers

A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempting to interpret these sets as "real" communities, we employ approximation algorithms for the graph-partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be i...

/pdf/community-structure-in-large-networks-natural-cluster-sizes-3bjqfmupvi.pdf

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

A large body of work has been devoted to defining and identifying clusters or communities in social and information networks. We explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. We employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the "best" possible community--according to the conductance measure--over a wide range of size scales. We study over 100 large real-world social and information networks. Our results suggest a significantly more refined picture of community structure in large networks than has been appreciated previously. In particular, we observe tight communities that are barely connected to the rest of the network at very small size scales; and communities of larger size scales gradually "blend into" the expander-like core of the network and thus become less "community-like." This behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, it is exactly the opposite of what one would expect based on intuition from expander graphs, low-dimensional or manifold-like graphs, and from small social networks that have served as testbeds of community detection algorithms. We have found that a generative graph model, in which new edges are added via an iterative "forest fire" burning process, is able to produce graphs exhibiting a network community profile plot similar to what we observe in our network datasets.

In this paper, we present a nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form. The algorithm's distinguishing feature is a change of variables that replaces the symmetric, positive semidefinite variable X of the SDP with a rectangular variable R according to the factorization X=RR

 T
 . The rank of the factorization, i.e., the number of columns of R, is chosen minimally so as to enhance computational speed while maintaining equivalence with the SDP. Fundamental results concerning the convergence of the algorithm are derived, and encouraging computational results on some large-scale test problems are also presented.

/pdf/a-nonlinear-programming-algorithm-for-solving-semidefinite-5dvve5rurk.pdf

A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization

http://www.optimization-online.org/DB_FILE/2012/02/3378.pdf

Non-convex mixed-integer nonlinear programming: A survey

The low-rank semidefinite programming problem LRSDPr is a restriction of the semidefinite programming problem SDP in which a bound r is imposed on the rank of X, and it is well known that LRSDPr is equivalent to SDP if r is not too small. In this paper, we classify the local minima of LRSDPr and prove the optimal convergence of a slight variant of the successful, yet experimental, algorithm of Burer and Monteiro [5], which handles LRSDPr via the nonconvex change of variables X=RRT. In addition, for particular problem classes, we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm. Computational results are presented on a set of combinatorial optimization relaxations, including some of the largest quadratic assignment SDPs solved to date.

/pdf/local-minima-and-convergence-in-low-rank-semidefinite-lsm6c9sm59.pdf

Local Minima and Convergence in Low-Rank Semidefinite Programming

In this paper, we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices. This result can be viewed as an extension of earlier separate results, which have established the copositive representation of a small collection of NP-hard problems. A simplification, which reduces the dimension of the linear conic program, and an extension to complementarity constraints are established, and computational issues are discussed.

/pdf/on-the-copositive-representation-of-binary-and-continuous-3cijoofbqw.pdf

On the copositive representation of binary and continuous nonconvex quadratic programs

An ensemble is a group of learning models that jointly solve a problem. However, the ensembles generated by existing techniques are sometimes unnecessarily large, which can lead to extra memory usage, computational costs, and occasional decreases in effectiveness. The purpose of ensemble pruning is to search for a good subset of ensemble members that performs as well as, or better than, the original ensemble. This subset selection problem is a combinatorial optimization problem and thus finding the exact optimal solution is computationally prohibitive. Various heuristic methods have been developed to obtain an approximate solution. However, most of the existing heuristics use simple greedy search as the optimization method, which lacks either theoretical or empirical quality guarantees. In this paper, the ensemble subset selection problem is formulated as a quadratic integer programming problem. By applying semi-definite programming (SDP) as a solution technique, we are able to get better approximate solutions. Computational experiments show that this SDP-based pruning algorithm outperforms other heuristics in the literature. Its application in a classifier-sharing study also demonstrates the effectiveness of the method.

/pdf/ensemble-pruning-via-semi-definite-programming-z3dou72p0x.pdf

Samuel Burer

Papers

A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization

Non-convex mixed-integer nonlinear programming: A survey

Local Minima and Convergence in Low-Rank Semidefinite Programming

On the copositive representation of binary and continuous nonconvex quadratic programs

Ensemble Pruning Via Semi-definite Programming