A Least Squares Approach to Linear Discriminant Analysis

The protection of sensitive and confidential data become a challenging task in the present scenario as more and more digital data is stored and transmitted between the end users. The privacy is vitally necessary in case of medical data, which contains the important information of the patients. In this article, a novel biometric inspired medical encryption technique is proposed based on newly introduced parameterized all phase orthogonal transformation (PR-APBST), singular value, and QR decomposition. The proposed technique utilizes the biometrics of the patient/owner to generate a key management system to obtain the parameters involved in the proposed technique. The medical image is then encrypted employing PR-APBST, QR and singular value decomposition and is ready for secure transmission or storage. Finally, a reliable decryption process is employed to reconstruct the original medical image from the encrypted image. The validity and feasibility of the proposed framework have been demonstrated using an extensive experiments on various medical images and security analysis.

This paper is an updated and extended version of the paper "The QR Decomposition and the Singular Value Decomposition in the Symmetrized Max-Plus Algebra" (B. De Schutter and B. De Moor, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 378--406). The max-plus algebra, which has maximization and addition as its basic operations, can be used to describe and analyze certain classes of discrete-event systems, such as flexible manufacturing systems, railway networks, and parallel processor systems. In contrast to conventional algebra and conventional (linear) system theory, the max-plus algebra and the max-plus-algebraic system theory for discrete-event systems are at present far from fully developed, and many fundamental problems still have to be solved. Currently, much research is going on to deal with these problems and to further extend the max-plus algebra and to develop a complete max-plus-algebraic system theory for discrete-event systems. In this paper we address one of the remaining gaps in the max-plus algebra by considering matrix decompositions in the symmetrized max-plus algebra. The symmetrized max-plus algebra is an extension of the max-plus algebra obtained by introducing a max-plus-algebraic analogue of the $-$-operator. We show that we can use well-known linear algebra algorithms to prove the existence of max-plus-algebraic analogues of basic matrix decompositions from linear algebra such as the QR decomposition, the singular value decomposition, the Hessenberg decomposition, the LU decomposition, and so on. These max-plus-algebraic matrix decompositions could play an important role in the max-plus-algebraic system theory for discrete-event systems.

Content based image retrieval represents images as N- dimensional feature vectors. The k images most similar to a target image, i.e., those closest to its feature vector, are determined by applying a k-nearest-neighbor (k-NN) query. A sequential scan of the feature vectors for k-NN queries is costly for a large number of images when N is high. The search space can be reduced by indexing the data, but the effectiveness of multidimensional indices is poor for high dimensional data. Building indices on dimensionality reduced data is one method to improve indexing efficiency. We utilize the Singular Value Decomposition (SVD) method to attain dimensionality reduction (DR) with minimum information loss for static data. Clustered SVD (CSVD) combines clustering with SVD to attain a lower normalized mean square error (NMSE) by taking advantage of the fact that most real-world datasets exhibit local rather than global correlations. The Local Dimensionality Reduction (LDR) method differs from CSVD in that it uses an SVD-friendly clustering method, rather than the k-means clustering method. We propose a hybrid method which combines the clustering method of LDR with the DR method of CSVD, so that the vector of the number of retained dimensions of the clusters is determined by varying the NMSE. We build SR-tree indices based on the vectors in the clusters to determine the number of accessed pages for exact k-NN queries (Thomasian et al., Inf Process Lett - IPL 94(6):247---252, 2005) (see Section A.3 versus the NMSE. Varying the NMSE a minimum cost can be found, because the lower cost of accessing a smaller index is offset with the higher postprocessing cost resulting from lower retrieval accuracy. Experimenting with one synthetic and three real-world datasets leads to the conclusion that the lowest cost is attained at NMSE???0.03 and between 1/3 and 1/2 of the number of dimensions are retained. In one case doubling the number of dimensions cuts the number of accessed pages by one half. The Appendix provides the requisite background information for reading this paper.

In this paper we survey a number of linear algebra algorithms that are used in many modern real-time high-throughput applications. We focus in particular on algorithms that are used in applications as diverse as adaptive filtering/beamforming, communications, signal/array processing, control, etc. We first consider least-squares estimation, QR decomposition, singular value decomposition, recursive least-squares estimation, and Kalman filtering algorithms. In particular, the QR decomposition which is used in least-squares solutions of linear system of equations, also forms one component of more complex algorithms such as Kalman filtering and the singular value decomposition. Systolic arrays, due to their modularity and regularity, are perfect matches for all the algorithms considered in this paper. All these problems are defined and described alongside the systolic algorithms and the architectures that implement them.

We propose a novel approach for multi-view object detection in 3D scenes reconstructed from RGB-D sensor. We utilize shape based representation using local shape context descriptors along with the voting strategy which is supported by unsupervised object proposals generated from 3D point cloud data. Our algorithm starts with a single-view object detection where object proposals generated in 3D space are combined with object specific hypotheses generated by the voting strategy. To tackle the multi-view setting, the data association between multiple views enabled view registration and 3D object proposals. The evidence from multiple views is combined in simple bayesian setting. The approach is evaluated on the Washington RGB-D scenes datasets [1], [2] containing several classes of objects in a table top setting. We evaluated our approach against the other state-of-the-art methods and demonstrated superior performance on the same dataset.

The paper presents a novel image (feature) interpolation method to reinforce the adaptive object recognition system. The system deals with texture images in a sequence according to changing perceptual conditions. When it recognizes several classes of objects under variable conditions, a fundamental problem is that two or more classes are overlapped on the feature space. This interpolation method is useful to resolve the overlapping issue.KeywordsInterpolation MethodTexture ImageModel ModificationPerceptual ConditionTexture RecognitionThese 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.

For each pixel, the values of the unwrapped phase produced by interferometric pairs can be parameterized and phase components, such as height, atmospheric path delay difference for the dates of every interferometric set image acquisition, and unwrapping error can be estimated through the solution of a rank deficient system of equations. There are several state of the art methods for solving rank deficient systems of equations, such as the Lower-Upper method, QR factorization and Singular Value Decomposition (SVD). In a previous work, the improved SVD method, which enables the accurate estimation of phase components, has been proposed. In this study, one alternative method for the estimation of the differential interferometry parameters, the improved QR, is proposed. The method relies on the reorganization of the system of equations by using external meteorological data which correspond to the acquisition dates that are less present in the interferometric SAR data set. All the aforementioned methods were tested and compared. The improved SVD and improved QR methods yield the most satisfactory results for all the phase components. The former estimates the heights by achieving almost GPS accuracy, whereas the latter is more robust, producing results of almost the same accuracy for all the estimated atmospheric path delays. However, the requirement of known values for at least two phase components makes the improved QR method less operational.

Non-negative Matrix Factorization (NMF) and Singular Value Decomposition (SVD) are widely recognized as pivotal dimensionality reduction techniques in the literature, particularly for deep learning applications involving large and high-dimensional datasets like SEM images. This study systematically evaluates the impact of SVD and NMF on the performance, efficiency, and energy consumption of four deep learning architectures: GoogleNet, AlexNet, ResNet, and SqueezeNet. By applying these techniques to reduce dataset dimensions, we observed that SVD excelled in computational efficiency, achieving up to 35% faster processing times compared to raw datasets. NMF, on the other hand, provided superior feature interpretability, which proved beneficial for tasks requiring meaningful pattern extraction. Energy consumption analysis revealed that SVD led to a 28% reduction in computational energy cost on average, making it a practical choice for resource-constrained environments. Among the evaluated models, ResNet consistently delivered the highest classification accuracy after dimensionality reduction, showing an improvement of 4-6% over models trained on non-reduced data. These findings underscore the critical role of dimensionality reduction in enhancing the scalability, energy efficiency, and classification accuracy of deep learning models, offering valuable insights for optimizing high-dimensional data applications in both academic and industrial contexts.

The article investigates the use of linear algebra methods for query optimisation in modern database management systems. The relevance of the study is due to the growth of data volumes and the need for more efficient methods of their processing. The proposed approach is to formalise tabular data structures as numerical matrices, which allows us to interpret relational operations (selection, projection, join) as linear algebraic transformations, which opens up opportunities for the use of powerful mathematical tools that optimise computations. The main methods of linear algebra for query optimisation are considered: matrix factorisation (LU, QR decomposition) for structuring complex calculations; singular value decomposition (SVD) for reducing dimensionality and identifying hidden dependencies; matrix modelling of JOIN operations as an efficient block multiplication; systems of linear equations for formalising data filtering. It is noted that matrix factorisation helps to optimise the execution of complex analytical queries, SVD is effective in aggregation and dimensionality reduction tasks, the matrix representation of JOIN provides high performance when processing large amounts of data, and linear systems unify the filtering logic with numerical processing. To empirically verify the proposed methods, an experimental study was conducted on a synthetic dataset simulating a transactional table with a million records. The experiment was carried out using the NumPy and PyTorch libraries with the use of GPU acceleration. We evaluated such indicators as query execution time, RAM usage, and scalability. The results showed that JOIN matrix modelling was the fastest, which is especially important for parallel computing. LU/QR factorisation demonstrated stability and versatility, while SVD proved to be resource-intensive but valuable for data compression. Linear equation methods showed balanced performance. It is concluded that linear algebra is a powerful formalisation framework for next-generation DBMS architectures capable of providing mathematical consistency, computational efficiency and scalability.

While data-dependent dimensionality reduction has dominated in many applications of hyperspectral imagery, there is increasing interest in data-independent strategies—such as random projections—due to their promise for reduced computational complexity as well as their demonstrated ability to preserve application-important information. Such random-projection-based dimensionality reduction is investigated in the specific context of supervised hyperspectral classification. Both Hadamard- and Gaussian-based random projections are considered, applied alone as well as incorporated into a fast approximate singular value decomposition (SVD). Experimental results reveal that the proposed Hadamard-based random projection with the fast SVD (FSVD) offers a computationally attractive alternative to not only traditional SVD but also Gaussian-based FSVD for dimensionality reduction in hyperspectral classification.

Feature selection using singular value decomposition and QR factorization with column pivoting for text-independent speaker identification

In this paper we present QR based principal component analysis (PCA) method. Similar to the singular value decomposition (SVD) based PCA method this method is numerically stable. We have carried out analytical comparison as well as numerical comparison (on Matlab software) to investigate the performance (in terms of computational complexity) of our method. The computational complexity of SVD based PCA is around $$ 14dn^{2} $$ flops (where d is the dimensionality of feature space and n is the number of training feature vectors); whereas the computational complexity of QR based PCA is around $$ 2dn^{2} \, + \,2dth $$ flops (where t is the rank of data covariance matrix and h is the dimensionality of reduced feature space). It is observed that the QR based PCA is more efficient in terms of computational complexity.

Similarity of signal processing effect between Hankel matrix-based SVD and wavelet transform and its mechanism analysis

A new precoder that reduces the feedback overhead in an OFDM-based relay network is proposed. A joint Singular Value Decomposition (SVD) and QR Decomposition is employed at the source-destination link and source-relay-destination link. Using the proposed precoder, only one precoding matrix needs to be fed back, independent of the total number of subcarriers. The link achievable rate using the reduced complexity precoder is evaluated via simulations and compared with that of full SVD precoders and for varying codebook size.

Long-term EEG monitoring (LTM) in acute neurology generates massive data volumes. We investigated whether data-analytic techniques could reduce LTM data size yet conserve their visual diagnostic features. LTM exemplars from 50 patients underwent singular value decomposition (SVD). High-variance SVD components were transformed using discrete cosine transform (DCT), and significant elements run-length encoded. Two regimes were tested: (I) SVD and DCT compression ratio (CR) of 1.7 and 12, and (II) CR of 3.7 and 5.7; each achieved an overall CR of ≈20. Compressed data were reconstructed alongside uncompressed originals, to create a total of 200 recordings that were scored by two blinded reviewers. Scores of original and reconstructed data were statistically analyzed. Score differences between original recordings were smaller than comparisons involving reconstructions using the first regime but did not differ significantly from reconstructions using the second regime. Raw LTM EEG has sufficient redundancy to undergo extreme (20-fold) data compression without compromising visual diagnostic information. A balanced mix of SVD and DCT appears to be a suitable data-analytic pipeline for achieving such compression. Dimension reduction is a significant goal in managing big biomedical data. Our results suggest a pathway for archival of meaningful representations of entire LTM datasets. The latent space suggests new lines of data-scientific inquiry of the EEG in acute neurological illness.