We propose a general method for optimally approximating an arbitrary matrix by a structured matrix (circulant, Toeplitz/Hankel, etc.) and examine its use for estimating the spectra of genomic linkage disequilibrium matrices. This application is prototypical of a variety of genomic and proteomic problems that demand robustness to incomplete biosequence information. We perform a simulation study and corroborative test of our method using real genomic data from the Mouse Genome Database (Bult et al., 2019). The results confirm the predicted utility of the method and provide strong evidence of its potential value to a wide range of bioinformatics applications. Our optimal general matrix approximation method is expected to be of independent interest to an even broader range of applications in applied mathematics and engineering.

We introduce a new consistency-based approach for defining and solving nonnegative/positive matrix and tensor completion problems. The novelty of the framework is that instead of artificially making the problem well-posed in the form of an application-arbitrary optimization problem, e.g., minimizing a bulk structural measure such as rank or norm, we show that a single property/constraint: preserving unit-scale consistency, guarantees the existence of both a solution and, under relatively weak support assumptions, uniqueness. The framework and solution algorithms also generalize directly to tensors of arbitrary dimensions while maintaining computational complexity that is linear in problem size for fixed dimensiond. In the context of recommender system (RS) applications, we prove that tworeasonableproperties that should be expected to hold for any solution to the RS problem are sufficient to permit uniqueness guarantees to be established within our framework. This is remarkable because it obviates the need for heuristic-based statistical or AI methods despite what appear to be distinctly human/subjective variables at the heart of the problem. Key theoretical contributions include a general unit-consistent tensor-completion framework with proofs of its properties, e.g., consensus-order and fairness, and algorithms with optimal runtime and space complexities, e.g.,O(1) term-completion with preprocessing complexity that is linear in the number of known terms of the matrix/tensor. From a practical perspective, the seamless ability of the framework to generalize to exploit high-dimensional structural relationships among key state variables, e.g., user and product attributes, offers a means for extracting significantly more information than is possible for alternative methods that cannot generalize beyond direct user-product relationships. Finally, we propose our consensus ordering property as an admissibility criterion for any proposed RS method.

Data retrieval serves a critical role in the development of multimedia applications. However, due to the exponential growth of multimedia data, high-speed and efficient indexing is becoming more and more difficult than ever. In this paper, we propose a novel approach to speed up the retrieval process by adopting a distributed computing paradigm through the Apache Spark framework. Utilizing search trees in a Big Data ecosystem leads to fast and cost-effective media database retrievals by caching indexing structures into memory and aggregating ranked results with flexibilities for users to specify the importance of search cues. We conducted computational experiments on large-scaled vector files for remote sensing image database and synthesized pollen image database to demonstrate the effectiveness and scalability of our system with reasonably high accuracy.

In this paper we develop novel quantum algorithms based on Quantum Multi-Object Search (QMOS) for convex hulls and general object intersection reporting, with applications to computer graphics. These algorithms are developed and described using standard concepts from computer science by encapsulating the physics of quantum computation within black-box subroutines.

In this paper we consider the use of Covariance Union (CU) with multi-hypothesis techniques (MHT) and Gaussian mixture models (GMMs) to generalize the conventional mean and covariance representation of information. More specifically, we address the representation of multi-modal information using multiple mean and covariance estimates. A significant challenge is to define a rigorous fusion algorithm that can bind the complexity of the filtering process. This requires a mechanism for subsuming subsets of modes into single modes so that the complexity of the representation satisfies a specified upper bound. We discuss how this can be accomplished using CU. The practical challenge is to develop efficient implementations of the CU algorithm. Because of the novelty of the CU algorithm, there are no existing real-time codes for use in real applications. In this paper we address this deficiency by considering a general-purpose implementation of the CU algorithm based on general nonlinear optimization techniques. Computational results are reported

The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.

The state of the art in unscented techniques for nonlinear estimation is surveyed. The process noise covariance matrix used on each filter is not the same as the process noise used to drive the motion of the true projectile in the simulation. All the Jacobian matrices for the extended Kalman filter (EKF) are calculated numerically using a central difference scheme and a step size of 10-3. It is observed that EKF yields an inconsistent estimate, while the unscented Kalman filter yields a consistent estimate.

A novel image interpolation method for resolution enhancement of still images is proposed. The approach consists of a nonlinear mapping from pixel index space to color space based on the low-resolution image so that nonintegral pixel indices in a super-sampling can be mapped to colors, i.e., interpolated. This nonlinear mapping is based on an extensor transformation of 2D pixel indices to an, N-dimensional vector space that encodes the Euclidean proximity interrelationships among a neighborhood of N pixels. Experimental results indicate that extensor-based interpolation yields results that are qualitatively superior to classical bilinear and bicubic interpolation, that has important advantages in comparison to edge-directed and optimal-recovery interpolation methods, and that has comparable or lower computational cost.

Simultaneous localisation and map building (SLAM) is one of the most important and challenging areas of mobile robotics. Unfortunately, the optimal Kalman filter solution incurs computational costs that scale quadratically with the number of beacons, which is prohibitive for many real time and large scale applications. Consequently, there is a significant practical need for more efficient approaches. The challenge is to develop methods that are both efficient and mathematically rigorous. In this paper we show that the full SLAM problem can be decomposed into two distinct mathematical operations. One is the maintenance of global state information for both the vehicle and the beacons, and the other is the maintenance of relative state information. These operations are distinct because the former is an unobserservable estimation problem while the latter is not. We argue that solutions to these two problems can be applied as scaffolding for the development of a wide variety of specialized SLAM algorithms. As a practical demonstration of the power of the two operations when applied as a generic solution to the SLAM problem, we provide empirical results for a scenario requiring the real-time construction and maintenance of a map containing 1.1 million beacons.

The paper analyzes the properties of the full covariance simultaneous localization and map building problem (SLAM). We prove that, even for the special case of a stationary vehicle (with no process noise) which uses a range-bearing sensor and has non-zero angular uncertainty, the full covariance SLAM algorithm always yields an inconsistent map. We also show, through simulations, that these conclusions appear to extend to a moving vehicle with process noise. However, these inconsistencies only become apparent after several hundred beacon updates.

This paper develops a simultaneous localisation and map building (SLAM) algorithm which utilises the split covariance intersection (SCI) update rule. This algorithm decomposes estimates into a correlated component (whose precise structure is unknown) and a lower bound on an independent component. For a map of n beacons, the storage is O(n) and the computational costs are constant irrespective of map size. In a simple simulation example we show that the SCI algorithm, through exploiting a lower bound on independent information, performs substantially better than the traditional covariance intersection SLAM algorithm.

The paper describes a computer modeling and simulation system that supports computational steering, which is an effort to make the typical simulation workflow more efficient. Our system provides an interface that allows scientists to perform all of the steps in the simulation process in parallel and online. It uses a standard network flow visualization package, which has been extended to display graphical output in an immersive virtual environment such as a CAVE. Our system allows scientists to interactively manipulate simulation parameters and observe the results. It also supports inverse steering, where the user specifies the desired simulation result, and the system searches for the simulation parameters that achieve this result. Taken together, these capabilities allow scientists to more efficiently and effectively understand model behavior, as well as to search through simulation parameter space. The paper is also a case study of applying our system to the problem of simulating microwave interactions with missile bodies. Because these interactions are difficult to study experimentally, and have important effects on missile electronics, there is a strong desire to develop and validate simulation models of this phenomena.