Optimal Parameter Estimation with Homogeneous Entities and Arbitrary Constraints

Techniques in Computational Geometry

Topological sweep of the complete graph

This paper describes applications of the optimized pattern discovery framework to text and Web mining. In particular, we introduce a class of simple combinatorial patterns over phrases, called proximity phrase association patterns, and consider the problem of finding the patterns that optimize a given statistical measure within the whole class of patterns in a large collection of unstructured texts. For this class of patterns, we develop fast and robust text mining algorithms based on techniques in computational geometry and string matching. Finally, we successfully apply the developed text mining algorithms to the experiments on interactive document browsing in a large text database and keyword discovery from Web bases.

Computational algebraic geometry has experienced significant advancements in recent years, driven by both theoretical breakthroughs and practical applications comprehensive review of the latest techniques in computational algebraic geometry, highlighting their development and impact on various domains. We begin by discussing foundational methods, including Gröbner bases, resultants, and elimination theory, which have been pivotal in solving polynomial systems and understanding algebraic varieties emerging algorithms that leverage improvements in computational efficiency, such as homotropy continuation methods and numerical algebraic geometry techniques. the applications of these advancements across different fields. In robotics and computer vision, for example, algebraic geometry methods are applied to solve problems related to motion planning and object recognition. In cryptography, recent developments in computational algebraic geometry offer new approaches to designing secure cryptographic systems resistant to quantum attacks. We also discuss the integration of algebraic geometry with machine learning and data science, emphasizing its role in improving predictive modeling and pattern recognition.

Maze solving has direct applications like Robot path finding, emergency evacuation, and drug discovery. Traditional methods of maze solving treat the problem of maze as a directional graph by connecting the key nodes. Because of the directional graph treatment, which is applied on the simplified form of the original maze, the maze solving algorithms at present cannot be directly applied in the critical domains of emergency evacuation etc. The proposed method uses the original complex maze without simplifying and hence can be directly applied in any critical application related to path finding. Proposed approach solves the mazes in a fully automated mode by combining the techniques of image processing, computational geometry and computational fluid dynamics. Moreover the high amount of human intervention needed for generating graphs from mazes in the traditional algorithms is eliminated altogether.

The knowledge of channel covariance matrices is of paramount importance to the estimation of instantaneous channels and the design of beamforming vectors in multi-antenna systems. In practice, an abrupt change in channel covariance matrices may occur due to the change in the environment and the user location. Although several works have proposed efficient algorithms to estimate the channel covariance matrices after any change occurs, how to detect such a change accurately and quickly is still an open problem in the literature. In this paper, we focus on channel covariance change detection between a multi-antenna base station (BS) and a single-antenna user equipment (UE). To provide theoretical performance limit, we first propose a genie-aided change detector based on the log-likelihood ratio (LLR) test assuming the channel covariance matrix after change is known, and characterize the corresponding missed detection and false alarm probabilities. Then, this paper considers the practical case where the channel covariance matrix after change is unknown. The maximum likelihood (ML) estimation technique is used to predict the covariance matrix based on the received pilot signals over a certain number of coherence blocks, building upon which the LLR-based change detector is employed. Numerical results show that our proposed scheme can detect the change with low error probability even when the number of channel samples is small such that the estimation of the covariance matrix is not that accurate. This result verifies the possibility to detect the channel covariance change both accurately and quickly in practice.

Tangible cultural heritage has provided visually appealing material for those researching in computer graphics, computational geometry, and interactive techniques. However, experiments were initially of more benefit to computer scientists as tests for their novel techniques than of any practical benefit to cultural-heritage professionals. Recently, however, tools have matured considerably, and technology is now enabling novel applications for documenting and preserving tangible cultural heritage and is continuing to inspire computer graphics researchers. Nevertheless, there's still a long way to go.

We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F- and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential.

In many engineering applications, estimation of covariance and precision matrices is of great importance, helping researchers understand the dependency and conditional dependency between variables of interest. Among various matrix estimation methods, the modified Cholesky decomposition is a commonly used technique. It has the advantage of transforming the matrix estimation task into solving a sequence of regression models. Moreover, the sparsity on the regression coefficients implies certain sparse structure on the covariance and precision matrices. In this chapter, we first overview the Cholesky-based covariance and precision matrices estimation. It is known that the Cholesky-based matrix estimation depends on a prespecified ordering of variables, which is often not available in practice. To address this issue, we then introduce several techniques to enhance the Cholesky-based estimation of covariance and precision matrices. These approaches are able to ensure the positive definiteness of the matrix estimate and applicable in general situations without specifying the ordering of variables. The advantage of Cholesky-based estimation is illustrated by numerical studies and several real-case applications.

Journal Article Nonparametric estimation of large covariance matrices of longitudinal data Get access Wei Biao Wu, Wei Biao Wu Search for other works by this author on: Oxford Academic Google Scholar Mohsen Pourahmadi Mohsen Pourahmadi Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 90, Issue 4, December 2003, Pages 831–844, https://doi.org/10.1093/biomet/90.4.831 Published: 01 December 2003

A key component in most parametric classifiers is the estimation of an inverse covariance matrix. In hyperspectral images, the number of bands can be in the hundreds, leading to covariance matrices having tens of thousands of elements. Lately, the use of linear regression in estimating the inverse covariance matrix has been introduced in the time-series literature. This paper adopts and expands these ideas to ill-posed hyperspectral image classification problems. The results indicate that at least some of the approaches can give a lower classification error than traditional methods such as the linear discriminant analysis and the regularized discriminant analysis. Furthermore, the results show that, contrary to earlier beliefs, estimating long-range dependencies between bands appears necessary to build an effective hyperspectral classifier and that the high correlations between neighboring bands seem to allow differing sparsity configurations of the inverse covariance matrix to obtain similar classification results.

The estimation of a covariance matrix from an insufficient amount of data is one of the most common problems in fields as diverse as multivariate statistics, wireless communications, signal processing, biology, learning theory, and finance. In a joint work of Marzetta, Tucci and Simon, a new approach to handle singular covariance matrices was suggested. The main idea was to use dimensionality reduction in conjunction with an average over the Stiefel manifold. In this paper, we continue with this research and we consider some new approaches to handle this problem. One of the methods is called the mean conjugate estimator under Ewens measure and uses a randomization of the sample covariance matrix over all the permutation matrices with respect to the Ewens measure. The techniques used to attack this problem are broad and run from random matrix theory to combinatorics.

Estimation of the high-dimensional banded covariance matrix is widely used in multivariate statistical analysis. To ensure the validity of estimation, we aim to test the hypothesis that the covariance matrix is banded with a certain bandwidth under the high-dimensional framework. Though several testing methods have been proposed in the literature, the existing tests are only powerful for some alternatives with certain sparsity levels, whereas they may not be powerful for alternatives with other sparsity structures. The goal of this paper is to propose a new test for the bandedness of high-dimensional covariance matrix, which is powerful for alternatives with various sparsity levels. The proposed new test also be used for testing the banded structure of covariance matrices of error vectors in high-dimensional factor models. Based on these statistics, a consistent bandwidth estimator is also introduced for a banded high dimensional covariance matrix. Extensive simulation studies and an application to a prostate cancer dataset from protein mass spectroscopy are conducted for evaluating the effectiveness of the proposed adaptive tests blue and bandwidth estimator for the banded covariance matrix.

Inferences in panel data with interactive effects using large covariance matrices

Kalman filtering and smoothing algorithms are used in many areas, including tracking and navigation, medical applications, and financial trend filtering. One of the basic assumptions required to apply the Kalman smoothing framework is that error covariance matrices are known and given. In this paper, we study a general class of inference problems where covariance matrices can depend functionally on unknown parameters. In the Kalman framework, this allows modeling situations where covariance matrices may depend functionally on the state sequence being estimated. We present an extended formulation and generalized Gauss-Newton (GGN) algorithm for inference in this context. When applied to dynamic systems inference, we show the algorithm can be implemented to preserve the computational efficiency of the classic Kalman smoother. The new approach is illustrated with a synthetic numerical example.