Planar 3-dimensional assignment problems with Monge-like cost arrays

Given an $n\times n\times p$ cost array $C$ we consider the problem $p$-P3AP which consists in finding $p$ pairwise disjoint permutations $\varphi_1,\varphi_2,\ldots,\varphi_p$ of $\{1,\ldots,n\}$ such that $\sum_{k=1}^{p}\sum_{i=1}^nc_{i\varphi_k(i)k}$ is minimized. For the case $p=n$ the planar 3-dimensional assignment problem P3AP results. Our main result concerns the $p$-P3AP on cost arrays $C$ that are layered Monge arrays. In a layered Monge array all $n\times n$ matrices that result from fixing the third index $k$ are Monge matrices. We prove that the $p$-P3AP and the P3AP remain NP-hard for layered Monge arrays. Furthermore, we show that in the layered Monge case there always exists an optimal solution of the $p$-3PAP which can be represented as matrix with bandwidth $\le 4p-3$. This structural result allows us to provide a dynamic programming algorithm that solves the $p$-P3AP in polynomial time on layered Monge arrays when $p$ is fixed.Comment: 16 pages, appendix will follow in v

New special cases of the quadratic assignment problem with diagonally structured coefficient matrices

We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time

Motion-capture-based hand gesture recognition for computing and control

This dissertation focuses on the study and development of algorithms that enable the analysis and recognition of hand gestures in a motion capture environment. Central to this work is the study of unlabeled point sets in a more abstract sense. Evaluations of proposed methods focus on examining their generalization to users not encountered during system training. In an initial exploratory study, we compare various classification algorithms based upon multiple interpretations and feature transformations of point sets, including those based upon aggregate features (e.g. mean) and a pseudo-rasterization of the capture space. We find aggregate feature classifiers to be balanced across multiple users but relatively limited in maximum achievable accuracy. Certain classifiers based upon the pseudo-rasterization performed best among tested classification algorithms. We follow this study with targeted examinations of certain subproblems. For the first subproblem, we introduce the a fortiori expectation-maximization (AFEM) algorithm for computing the parameters of a distribution from which unlabeled, correlated point sets are presumed to be generated. Each unlabeled point is assumed to correspond to a target with independent probability of appearance but correlated positions. We propose replacing the expectation phase of the algorithm with a Kalman filter modified within a Bayesian framework to account for the unknown point labels which manifest as uncertain measurement matrices. We also propose a mechanism to reorder the measurements in order to improve parameter estimates. In addition, we use a state-of-the-art Markov chain Monte Carlo sampler to efficiently sample measurement matrices. In the process, we indirectly propose a constrained k-means clustering algorithm. Simulations verify the utility of AFEM against a traditional expectation-maximization algorithm in a variety of scenarios. In the second subproblem, we consider the application of positive definite kernels and the earth mover\u27s distance (END) to our work. Positive definite kernels are an important tool in machine learning that enable efficient solutions to otherwise difficult or intractable problems by implicitly linearizing the problem geometry. We develop a set-theoretic interpretation of ENID and propose earth mover\u27s intersection (EMI). a positive definite analog to ENID. We offer proof of EMD\u27s negative definiteness and provide necessary and sufficient conditions for ENID to be conditionally negative definite, including approximations that guarantee negative definiteness. In particular, we show that ENID is related to various min-like kernels. We also present a positive definite preserving transformation that can be applied to any kernel and can be used to derive positive definite EMD-based kernels, and we show that the Jaccard index is simply the result of this transformation applied to set intersection. Finally, we evaluate kernels based on EMI and the proposed transformation versus ENID in various computer vision tasks and show that END is generally inferior even with indefinite kernel techniques. Finally, we apply deep learning to our problem. We propose neural network architectures for hand posture and gesture recognition from unlabeled marker sets in a coordinate system local to the hand. As a means of ensuring data integrity, we also propose an extended Kalman filter for tracking the rigid pattern of markers on which the local coordinate system is based. We consider fixed- and variable-size architectures including convolutional and recurrent neural networks that accept unlabeled marker input. We also consider a data-driven approach to labeling markers with a neural network and a collection of Kalman filters. Experimental evaluations with posture and gesture datasets show promising results for the proposed architectures with unlabeled markers, which outperform the alternative data-driven labeling method

The Econometrics of Matching Models

In October 2012 the Nobel prize was attributed to Al Roth and Lloyd Shapley for their work on matching. Both the seminal Gale-Shapley (1962) paper and most of Roth’s work were concerned with allocation mechanisms when prices or other transfers cannot be used—what we will call non-transferable utility (NTU) in this survey. Gale and Shapley used college admissions, marriage, and roommate assignments as examples; and Roth’s fundamental work in market design has led to major improvements in the National Resident Matching Program (Roth and Peranson 1999) and to the creation of a mechanism for kidney exchange (Roth, Sönmez and Ünver 2004.) The resulting insights have been applied to a host of issues, including the allocation of students to schools, the marriage market with unbalanced gender distributions, the role of marital prospects in human capital investment decisions, the social impact of improved birth control technologies and many others. The econometrics of matching models have recently been reconsidered, from different and equally innovative perspectives. The goal of the present project will be to survey these methodological advances. We shall describe the main difficulties at stake, the various answers provided so far, and the issues that remain open