Because of the significant increase in size and complexity of the networks, the distributed computation of eigenvalues and eigenvectors of graph matrices has become very challenging and yet it remains as important as before. In this paper we develop efficient distributed algorithms to detect, with higher resolution, closely situated eigenvalues and corresponding eigenvectors of symmetric graph matrices. We model the system of graph spectral computation as physical systems with Lagrangian and Hamiltonian dynamics. The spectrum of Laplacian matrix, in particular, is framed as a classical spring-mass system with Lagrangian dynamics. The spectrum of any general symmetric graph matrix turns out to have a simple connection with quantum systems and it can be thus formulated as a solution to a Schr "odinger-type differential equation. Taking into account the higher resolution requirement in the spectrum computation and the related stability issues in the numerical solution of the underlying differential equation, we propose the application of symplectic integrators to the calculation of eigenspectrum. The effectiveness of the proposed techniques is demonstrated with numerical simulations on real-world networks of different sizes and complexities.
The general idea of using mechanical oscillatory behaviour for the detection of eigenvalues has appeared in a few previous works, see e.g., @cite_17 @cite_7 . Though the technique in @cite_17 is close to ours, our methods differ by focussing on a Schr "odinger-type equation and numerical integrators specific to it. Moreover, we demonstrate the efficiency and stability of the methods in real-world networks of varying sizes, in contrast to a small size synthetic network considered in @cite_17 , and our methods can be used to estimate eigenvectors as well.
@cite_7 @cite_17
2155421090 2777654136 2056358962 1988833430
The detection and estimation of signals in noisy, limited data is a problem of interest to many scientific and engineering communities. We present a mathematically justifiable, computationally simple, sample-eigenvalue-based procedure for estimating the number of high-dimensional signals in white noise using relatively few samples. The main motivation for considering a sample-eigenvalue-based scheme is the computational simplicity and the robustness to eigenvector modelling errors which can adversely impact the performance of estimators that exploit information in the sample eigenvectors. There is, however, a price we pay by discarding the information in the sample eigenvectors; we highlight a fundamental asymptotic limit of sample-eigenvalue-based detection of weak or closely spaced high-dimensional signals from a limited sample size. This motivates our heuristic definition of the effective number of identifiable signals which is equal to the number of ldquosignalrdquo eigenvalues of the population covariance matrix which exceed the noise variance by a factor strictly greater than . The fundamental asymptotic limit brings into sharp focus why, when there are too few samples available so that the effective number of signals is less than the actual number of signals, underestimation of the model order is unavoidable (in an asymptotic sense) when using any sample-eigenvalue-based detection scheme, including the one proposed herein. The analysis reveals why adding more sensors can only exacerbate the situation. Numerical simulations are used to demonstrate that the proposed estimator, like Wax and Kailath's MDL-based estimator, consistently estimates the true number of signals in the dimension fixed, large sample size limit and the effective number of identifiable signals, unlike Wax and Kailath's MDL-based estimator, in the large dimension, (relatively) large sample size limit. We introduce scGAN, a novel extension of conditional Generative Adversarial Networks (GAN) tailored for the challenging problem of shadow detection in images. Previous methods for shadow detection focus on learning the local appearance of shadow regions, while using limited local context reasoning in the form of pairwise potentials in a Conditional Random Field. In contrast, the proposed adversarial approach is able to model higher level relationships and global scene characteristics. We train a shadow detector that corresponds to the generator of a conditional GAN, and augment its shadow accuracy by combining the typical GAN loss with a data loss term. Due to the unbalanced distribution of the shadow labels, we use weighted cross entropy. With the standard GAN architecture, properly setting the weight for the cross entropy would require training multiple GANs, a computationally expensive grid procedure. In scGAN, we introduce an additional sensitivity parameter w to the generator. The proposed approach effectively parameterizes the loss of the trained detector. The resulting shadow detector is a single network that can generate shadow maps corresponding to different sensitivity levels, obviating the need for multiple models and a costly training procedure. We evaluate our method on the large-scale SBU and UCF shadow datasets, and observe up to 17 error reduction with respect to the previous state-of-the-art method. In this paper, we focus on the problem of motion tracking in unknown environments using visual and inertial sensors. We term this estimation task visual-inertial odometry (VIO), in analogy to the well-known visual-odometry problem. We present a detailed study of extended Kalman filter (EKF)-based VIO algorithms, by comparing both their theoretical properties and empirical performance. We show that an EKF formulation where the state vector comprises a sliding window of poses (the multi-state-constraint Kalman filter (MSCKF)) attains better accuracy, consistency, and computational efficiency than the simultaneous localization and mapping (SLAM) formulation of the EKF, in which the state vector contains the current pose and the features seen by the camera. Moreover, we prove that both types of EKF approaches are inconsistent, due to the way in which Jacobians are computed. Specifically, we show that the observability properties of the EKF's linearized system models do not match those of the underlying system, which causes the filters to underestimate the uncertainty in the state estimates. Based on our analysis, we propose a novel, real-time EKF-based VIO algorithm, which achieves consistent estimation by (i) ensuring the correct observability properties of its linearized system model, and (ii) performing online estimation of the camera-to-inertial measurement unit (IMU) calibration parameters. This algorithm, which we term MSCKF 2.0, is shown to achieve accuracy and consistency higher than even an iterative, sliding-window fixed-lag smoother, in both Monte Carlo simulations and real-world testing. We present an outlier detection using indegree number (ODIN) algorithm that utilizes k-nearest neighbour graph. Improvements to existing kNN distance-based method are also proposed. We compare the methods with real and synthetic datasets. The results show that the proposed method achieves reasonable results with synthetic data and outperforms compared methods with real data sets with small number of observations.