Undercover Boolean Matrix Factorization with MaxSAT

Boolean matrices, i.e., matrices over a Boolean algebra, do not satisfy many of the fundamental properties of matrices over the real or complex field. Algebraic aspects of the Boolean pseudoinverse matrix concept are investigated. Maps operating on Boolean vector spaces by analogy with those on linear spaces are defined. Some properties of them are obtained. Most of these properties remain the same in the case of linear spaces and they are essential for solving linear systems. In particular orthogonal projections in linear spaces receive their algebraic analogue in Boolean vector spaces. The maps defined are useful for solving equations in the Boolean algebra, if no pseudoinverse matrix to a given matrix exists. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra. Some examples for computations with binary relations and Boolean matrices using computer algebra systems are given.

Principal Component Analysis (PCA) is one of the most important unsupervised methods to handle high-dimensional data. However, due to the high computational complexity of its eigen-decomposition solution, it is hard to apply PCA to the large-scale data with high dimensionality, e.g., millions of data points with millions of variables. Meanwhile, the squared L2-norm based objective makes it sensitive to data outliers. In recent research, the L1-norm maximization based PCA method was proposed for efficient computation and being robust to outliers. However, this work used a greedy strategy to solve the eigenvectors. Moreover, the L1-norm maximization based objective may not be the correct robust PCA formulation, because it loses the theoretical connection to the minimization of data reconstruction error, which is one of the most important intuitions and goals of PCA. In this paper, we propose to maximize the L21-norm based robust PCA objective, which is theoretically connected to the minimization of reconstruction error. More importantly, we propose the efficient non-greedy optimization algorithms to solve our objective and the more general L21-norm maximization problem with theoretically guaranteed convergence. Experimental results on real world data sets show the effectiveness of the proposed method for principal component analysis.

We give necessary and sufficient conditions for cofactor expansibility of determinants along a row or column for Boolean square matrices over an arbitrary Boolean algebra. First of all we define a natural decomposition of an arbitrary Boolean matrix by interior, exterior, and determinate parts. The introduced notions allow us to establish the main result of this paper. It is shown that the formulas of the cofactor expansion along a row (column) of determinants of an arbitrary square Boolean matrix hold if and only if the formulas of the cofactor expansion along the corresponding row (column) hold for determinants of its interior part.

The design of optimal power complementary filters for multiresolution signal decomposition is considered. As a new performance measure in statistical optimization of filter banks the norm of cross-correlation (all subbands including all delays) will be used for optimization. The obtained filters will be compared to filters having minimum reconstruction error when the high-pass branches are dropped. In the paraunitary case, which is considered here, this reconstruction error is closely related to the coding gain of PCM in subbands over direct PCM. While the KL-transform combines both minimum reconstruction error and perfect decorrelation, filter banks do not. It is clear that the norm of cross-correlation becomes very important when better techniques than PCM are considered for the coding of subband signals. >

Background: Using Boolean algebra for differential diagnosis of neurological disorders involves treating neurologic findings and disease entities such as Boolean variables and combining them via OR (∨) or other logical operations. Aim: To demonstrate how Boolean algebra (mathematical logic) and Boolean matrices can be applied to support differential diagnosis in neurology. Results: Each neurologic finding was encoded as a Boolean variable (1 = present, 0 = absent). Boolean variables and Boolean matrices reflect mutual interdependencies of neurologic findings across diseases, enabling reverse inference (i.e., inferring possible diseases from a pattern of findings). Conclusion: Boolean algebra offers a formal, rigorous framework to represent and manipulate differential diagnostic relationships in neurology; it may complement clinical reasoning and computational decision support.

Regression analysis between features of high-dimension is receiving attention in environmental learning of mobile robot. In this paper, we propose a novel framework, namely General regression neural network (GRNN), for approximating the functional relationship between high-dimensional map features and robot’s states. We firstly adopt PCA to preprocess images taken from omnidirenctional vision. The method extracts map features optimally and reduces the correlated features while keeping the minimum reconstruction error. Then, the robot states and corresponding features of the training panoramic snapshots are used to train the given neural network. This enables robot to memorize the environmental features as well as to predict available scene given its location information. Experimental results are shown finally.KeywordsMobile RobotTraining ImageReference LocationGeneral Regression Neural NetworkRegression Neural NetworkThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Non-negative matrix factorization (NMF) was introduced by Paatero and Tapper in 1994 and it was a general way of reducing the dimension of the matrix with non-negative entries. Non-negative matrix factorization is very useful in many data analysis applications such as character recognition, text mining, and others. This paper aims to study the application in Chinese character recognition using non-negative matrix factorization. Python was used to carry out the LU factorization and non-negative matrix factorization of a Chinese character in Boolean Matrix. Preliminary analysis confirmed that the data size of and and are chosen for the NMF of the Boolean matrix. In this project, one hundred printed Chinese characters were selected, and all the Chinese characters can be categorized into ten categories according to the number of strokes , for . The Euclidean distance between the Boolean matrix of a Chinese character and the matrix after both LU factorization and NMF is calculated for further analysis. Paired t-test confirmed that the factorization of Chinese characters in the Boolean matrix using NMF is better than the LU factorization. Finally, ten handwritten Chinese characters were selected to test whether the program is able to identify the handwritten and the printed Chinese characters. Experimental results showed that 70% of the characters can be recognized via the least Euclidean distance obtained. NMF is suitable to be applied in Chinese character recognition since it can reduce the dimension of the image and the error between the original Boolean matrix and after NMF is less than 5%.

Principal component analysis (PCA) is one of the most widely used unsupervised dimensionality reduction algorithms, but it is very sensitive to outliers because the squared l2 -norm is used as distance metric. Recently, many scholars have devoted themselves to solving this difficulty. They learn the projection matrix from minimum reconstruction error or maximum projection variance as the starting point, which leads them to ignore a serious problem, that is, the original PCA learns the projection matrix by minimizing the reconstruction error and maximizing the projection variance simultaneously, but they only consider one of them, which imposes various limitations on the performance of model. To solve this problem, we propose a novel robust principal component analysis via joint reconstruction and projection, namely, RPCA-RP, which combines reconstruction error and projection variance to fully mine the potential information of data. Furthermore, we carefully design a discrete weight for model to implicitly distinguish between normal data and outliers, so as to easily remove outliers and improve the robustness of method. In addition, we also unexpectedly discovered that our method has anomaly detection capabilities. Subsequently, an effective iterative algorithm is explored to solve this problem and perform related theoretical analysis. Extensive experimental results on several real-world datasets and RGB large-scale dataset demonstrate the superiority of our method.

Classifying action videos became challenging problem in computer vision community. In this work, action videos are represented by dictionaries which are learned by online dictionary learning (ODL). Here, we have used two simple measures to classify action videos, reconstruction error and projection. Sparse approximation algorithm LASSO is used to reconstruct test video and reconstruction error is calculated for each of the dictionaries. To get another discriminative measure projection, the test vector is projected onto the atoms in the dictionary. Minimum reconstruction error and maximum projection give information regarding the action category of the test vector. With action bank as a feature vector, our best performance is 59.3% on UCF50 (benchmark is 57.9%), 97.7% on KTH (benchmark is 98.2%)and 23.63% on HMDB51 (benchmark is 26.9%).

Let n be a positive integer, and C n (r) the set of all n × n r-circulant matrices over the Boolean algebra B = {0, 1}, $$G_n = \bigcup\limits_{r = 0}^{n - 1} {C_n (r)} $$ . For any fixed r-circulant matrix C (C ≠ 0) in G n , we define an operation “*” in G n as follows: A * B = ACB for any A, B in G n , where ACB is the usual product of Boolean matrices. Then (G n , *) is a semigroup. We denote this semigroup by G n (C) and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix C. Let F be an idempotent element in G n (C) and M(F) the maximal subgroup in G n (C) containing the idempotent element F. In this paper, the elements in M(F) are characterized and an algorithm to determine all the elements in M(F) is given.

We present a new distributed computing algorithm, Parallel Pattern Discovery (PPD), for constrained Non-negative Matrix Factorization (NMF). Our implementation offers the ability to constrain a specific pattern for optimization of the data while minimizing reconstruction error. Parallel Pattern Discovery operates within a distributed environment using a message passing interface. Distribution of the PPD algorithm provides better scalability and allows operation in single- or multiple-system environments. The algorithm was tested on a set of time-series, dose-dependent mRNA gene expression data. Parallel Pattern Discovery was found to accurately identify patterns within the data and reconstruct the original matrices. Our NMF algorithm found a smaller reconstruction error when compared against standard NMF algorithms. Development focused on running PPD as part of a system which identifies significantly contributing genes. Parallel Pattern Discovery is first run to find patterns from biological data. It is followed by Gene Set Enrichment (GSE) which takes the pattern data and relates it back to genetic pathways.

Motion capture (MoCap) technology, essential for biomechanics and motion analysis, faces challenges from data loss due to occlusions and technical issues. Traditional recovery methods, based on inter-marker relationships or independent marker treatment, have limitations. This study introduces a novel U-net-inspired bi-directional long short-term memory (U-Bi-LSTM) autoencoder-based technique for recovering missing MoCap data across multi-camera setups. Leveraging multi-camera and triangulated 3D data, this method employs a sophisticated U-shaped deep learning structure with an adaptive Huber regression layer, enhancing outlier robustness and minimizing reconstruction errors, proving particularly beneficial for long-term data loss scenarios. Our approach surpasses traditional piecewise cubic spline and state-of-the-art sparse low rank methods, demonstrating statistically significant improvements in reconstruction error across various gap lengths and numbers. This research not only advances the technical capabilities of MoCap systems but also enriches the analytical tools available for biomechanical research, offering new possibilities for enhancing athletic performance, optimizing rehabilitation protocols, and developing personalized treatment plans based on precise biomechanical data.

Solutions to optimal control problems obtained via direct methods are usually of low resolution. As the desired resolution becomes higher, the number of parameters in the resulting parameter optimization problem increases nonlinearly. A concatenated approach to trajectory optimization previously introduced by the authors to obtain high resolution optimal solutions with low storage requirements is applicable only to a specific class of problems, such as minimum time and minimum energy optimal control problems. The hodograph ballooning approach introduced in this paper generalizes the concatenated approach to the most general optimal control problem. Introduction The development of powerful nonlinear programming software [1, 2] has made direct approaches to trajectory optimization the method of choice for most practical applications [3-5]. The discretization of an optimal control problem to a finite dimensional parameter optimization problem avoids many of the troublesome pathologies that are typical for the optimization over infinite dimensional function spaces [6-8]. Furthermore, direct approaches usually do not. require guessing the switching structure, and the dynamical system is devoid of the nonintuitive costate variables. However, to achieve good resolution it is usually necessary to refine some type of discretization mesh, and, ultimately, to use a large number of design parameters. For typical gradient search methods, this increase in the dimension of the optimization problem has nearly always a negative effect on the convergence rate and the robustness. Additionally, CPU time and RAM requirements for each iteration increase with the second power of the number of parameters (third power for non-sparse solvers) so that direct optimization approaches can become quite expensive. In [9], a method was introduced that achieves excellent resolution without solving high-dimensional nonlinear programming problems. This was achieved by lo'Senior Engineer, Member AIAA. t Supervising Engineer, Member AIAA. 117 Research Drive, Hampton, VA 23666. Copyright ©1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved cally optimizing the trajectory on an iterative sequence of short, partly overlapping subarcs. However, this approach is not applicable to the most general problem formulation. Loosely sneaking, it was required in [9] that the cost state does not appear explicitly in the right-hand side of the equations of motion. The present paper introduces a method through which the approach introduced in [9] becomes applicable to the most general problem formulation. Application of this method, however, requires a priori knowledge of the optimal cost. In practice it is observed that the optimal cost can be obtained with very high precision with reasonably low order discretizations. Once a good guess for the optimal cost is obtained presented in this paper can be applied to refine the trajectory, or, if the optimal cost value is not known a priori with sufficient precision, it can be applied iteratively with the prescribed cost value adjusted in an outer loop. Problem Formulation Let us consider the following simple optimal control problem stated in Mayer form: ^ 4 ( x ( t f ) , t f ) (1 )

In order to improve synthetic aperture radar (SAR) target recognition performance, this paper proposes a novel method using multi-level deep features. The multi-level deep features are learned by the convolutional neural network (CNN), which are capable of describing the target characteristics from different aspects. In order to make full use of the discrimination contained in the multi-level deep features, the joint sparse representation (JSR) is used as the basic classifier, which performs the multi-task learning to jointly classify the multi-level deep features. It could not only represent each feature properly but also consider the correlations between different levels of features. Based on the solutions from JSR, the target label is classified as the training class with the minimum reconstruction error. By fully exploiting the discriminative information contained in the multi-level deep features, the proposed method could effectively enhance SAR target recognition performance. The moving and stationary target acquisition and recognition (MSTAR) dataset is employed in the experiments. The results show that the proposed method could achieve a significantly high recognition rate of 99.38% for classifying 10 classes of targets under the standard operating condition (SOC), which is higher than those from some reference methods drawn from current literatures. Under different types of extended operating conditions (EOCs), the overall performance of the proposed method keeps superior over the reference methods. In addition, the outlier rejection capability of the proposed method is also better than the compared methods. All these experimental results validate the high effectiveness of the proposed method.

We describe theoretical and experimental results for a new class of optimal features for feature-specific imaging (FSI). In this paper, we theoretically solve the reconstruction problem without noise, and find a more general solution than principle component analysis (PCA). We present a generalized framework to find FSI projection matrices. Using Stochastic Tunneling, we find an optimal solution in the presence of noise and under an energy conservation constraint. We also show that a non-negativity requirement does not significantly reduce system performance. Finally, we propose an experimental system for FSI using a polarization-based optical pipeline processor.