Eigenvector Synchronization, Graph Rigidity and the Molecule Problem

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend on previous work and propose the 3D-ASAP algorithm, for the graph realization problem in $\mathbb{R}^3$, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in $\mathbb{R}^3$, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms.Comment: 49 pages, 8 figure

Deep Adaptive Feature Embedding with Local Sample Distributions for Person Re-identification

Person re-identification (re-id) aims to match pedestrians observed by disjoint camera views. It attracts increasing attention in computer vision due to its importance to surveillance system. To combat the major challenge of cross-view visual variations, deep embedding approaches are proposed by learning a compact feature space from images such that the Euclidean distances correspond to their cross-view similarity metric. However, the global Euclidean distance cannot faithfully characterize the ideal similarity in a complex visual feature space because features of pedestrian images exhibit unknown distributions due to large variations in poses, illumination and occlusion. Moreover, intra-personal training samples within a local range are robust to guide deep embedding against uncontrolled variations, which however, cannot be captured by a global Euclidean distance. In this paper, we study the problem of person re-id by proposing a novel sampling to mine suitable \textit{positives} (i.e. intra-class) within a local range to improve the deep embedding in the context of large intra-class variations. Our method is capable of learning a deep similarity metric adaptive to local sample structure by minimizing each sample's local distances while propagating through the relationship between samples to attain the whole intra-class minimization. To this end, a novel objective function is proposed to jointly optimize similarity metric learning, local positive mining and robust deep embedding. This yields local discriminations by selecting local-ranged positive samples, and the learned features are robust to dramatic intra-class variations. Experiments on benchmarks show state-of-the-art results achieved by our method.Comment: Published on Pattern Recognitio

Passenger-Centric Urban Air Mobility: Fairness Trade-Offs and Operational Efficiency

Urban Air Mobility (UAM) has the potential to revolutionize transportation. It will exploit the third dimension to help smooth ground traffic in densely populated areas. To be successful, it will require an integrated approach able to balance efficiency and safety while harnessing common resources and information. In this work we focus on future urban air-taxi services, and present the first methods and algorithms to efficiently operate air-taxi at scale. Our approach is twofold. First, we use a passenger-centric perspective which introduces traveling classes, and information sharing between transport modes to differentiate quality of services. This helps smooth multimodal journeys and increase passenger satisfaction. Second, we provide a flight routing and recharging solution which minimizes direct operational costs while preserving long term battery life through reduced energy-intense recharging. Our methods, which surpass the performance of a general state-of-the-art commercial solver, are also used to gain meaningful insights on the design space of the air-taxi problem, including solutions to hidden fairness issues.Comment: Submitted to Transportation Research Part C: Emerging Technologie

Efficient Methods For Large-Scale Empirical Risk Minimization

Empirical risk minimization (ERM) problems express optimal classifiers as solutions of optimization problems in which the objective is the sum of a very large number of sample costs. An evident obstacle in using traditional descent algorithms for solving this class of problems is their prohibitive computational complexity when the number of component functions in the ERM problem is large. The main goal of this thesis is to study different approaches to solve these large-scale ERM problems. We begin by focusing on incremental and stochastic methods which split the training samples into smaller sets across time to lower the computation burden of traditional descent algorithms. We develop and analyze convergent stochastic variants of quasi-Newton methods which do not require computation of the objective Hessian and approximate the curvature using only gradient information. We show that the curvature approximation in stochastic quasi-Newton methods leads to faster convergence relative to first-order stochastic methods when the problem is ill-conditioned. We culminate with the introduction of an incremental method that exploits memory to achieve a superlinear convergence rate. This is the best known convergence rate for an incremental method. An alternative strategy for lowering the prohibitive cost of solving large-scale ERM problems is decentralized optimization whereby samples are separated not across time but across multiple nodes of a network. In this regime, the main contribution of this thesis is in incorporating second-order information of the aggregate risk corresponding to samples of all nodes in the network in a way that can be implemented in a distributed fashion. We also explore the separation of samples across both, time and space, to reduce the computational and communication cost for solving large-scale ERM problems. We study this path by introducing a decentralized stochastic method which incorporates the idea of stochastic averaging gradient leading to a low computational complexity method with a fast linear convergence rate. We then introduce a rethinking of ERM in which we consider not a partition of the training set as in the case of stochastic and distributed optimization, but a nested collection of subsets that we grow geometrically. The key insight is that the optimal argument associated with a training subset of a certain size is not that far from the optimal argument associated with a larger training subset. Based on this insight, we present adaptive sample size schemes which start with a small number of samples and solve the corresponding ERM problem to its statistical accuracy. The sample size is then grown geometrically and use the solution of the previous ERM as a warm start for the new ERM. Theoretical analyses show that the use of adaptive sample size methods reduces the overall computational cost of achieving the statistical accuracy of the whole dataset for a broad range of deterministic and stochastic first-order methods. We further show that if we couple the adaptive sample size scheme with Newton\u27s method, it is possible to consider subsequent doubling of the training set and perform a single Newton iteration in between. This is possible because of the interplay between the statistical accuracy and the quadratic convergence region of these problems and yields a method that is guaranteed to solve an ERM problem by performing just two passes over the dataset