Optimal Singular Value Decomposition Research Articles

An efficient randomized QLP algorithm for approximating the singular value decomposition

The rank-revealing pivoted QLP decomposition approximates the computationally prohibitive singular value decomposition (SVD) via two consecutive column-pivoted QR (CPQR) decomposition. It furnishes information on all four fundamental subspaces of the matrix. In this paper, we introduce a randomized version of the QLP decomposition called Rand-QLP. Operating on a matrix A, Rand-QLP gives A=QLPT, where Q and P are orthonormal, and L is lower-triangular. Under the assumption that the rank of the input matrix is k, we derive several error bounds for Rand-QLP: bounds for the first k approximate singular values and for the trailing block of the middle factor L, which show that the decomposition is rank-revealing; bounds for the distance between approximate subspaces and the exact ones for all four fundamental subspaces of a given matrix; and bounds for the errors of low-rank approximations constructed by the columns of Q and P. Rand-QLP is able to effectively leverage modern computational architectures, due to the utilization of random sampling and the unpivoted QR decomposition, thus addressing a serious bottleneck associated with classical algorithms such as the SVD, CPQR and most recent matrix decomposition algorithms. We utilize a modern computing architecture to assess the performance behavior of different algorithms. In comparison to CPQR and the SVD, Rand-QLP respectively achieves a speedup of up to 5 times and 6.6 times on the CPU and up to 3.8 times and 4.4 times with the hybrid GPU architecture. In terms of quality of approximation, our results on synthetic and real data show that the approximations by Rand-QLP are comparable to those of pivoted QLP and the optimal SVD, and in most cases are considerably better than those of CPQR.

Channel Customization for Limited Feedback in RIS-Assisted FDD Systems

Reconfigurable intelligent surfaces (RISs) represent a pioneering technology to realize smart electromagnetic environments by reshaping the wireless channel. Jointly designing the transceiver and RIS relies on the channel state information (CSI), whose feedback has not been investigated in multi-RIS-assisted frequency division duplexing systems. In this study, the limited feedback of the RIS-assisted wireless channel is examined by capitalizing on the ability of the RIS in channel customization. By configuring the phase shifters of the surfaces using statistical CSI, we customize a sparse channel in rich-scattering environments, which significantly reduces the feedback overhead in designing the transceiver and RISs. Since the channel is customized in terms of singular value decomposition (SVD) with full-rank, the optimal SVD transceiver can be approached without a matrix decomposition and feeding back the complete channel parameters. The theoretical spectral efficiency (SE) loss of the proposed transceiver and RIS design is derived by considering the limited CSI quantization. To minimize the SE loss, a bit partitioning algorithm that splits the limited number of bits to quantize the CSI is developed. Extensive numerical results show that the channel customization-based transceiver with reduced CSI can achieve satisfactory performance compared with the optimal transceiver with full CSI. Given the limited number of feedback bits, the bit partitioning algorithm can minimize the SE loss by adaptively allocating bits to quantize the channel parameters.

Extraction of pipeline defect feature based on variational mode and optimal singular value decomposition

Because the magnetic signal information of pipeline defects obtained by magnetic flux leakage detection contains interference signals, it is difficult to accurately extract the features. Therefore, a novel pipeline defect feature extraction method based on VMD-OSVD (variational modal decomposition - optimal singular value decomposition) is proposed to promote the signal to noise ratio (SNR) and reduce aliasing in the frequency domain. By using the VMD method, the sampled magnetic signal is decomposed, and the optimal variational mode is selected according to the rate of relative change (VMK) of Shannon entropy (SE) to reconstruct the signal. After that, SVD algorithm is used to filter the reconstructed signal again, in which the H-matrix is optimized with the phase-space matrix to enhance SNR and decrease the frequency domain aliasing. The results show that the method has excellent denoising ability for defect magnetic signals, and SNR is increased by 21.01%, 24.04%, 0.96%, 32.14%, and 20.91%, respectively. The improved method has the best denoising effect on transverse mechanical scratches, but a poor denoising effect on spiral welding position. In the frequency domain, the characteristics of different defects are varied, and their corresponding frequency responses are spiral weld corrosion > transverse mechanical cracking > girth weld > deep hole > normal pipe. The high-frequency band is the spiral weld corrosion with f1 = 153.37 Hz. The low-frequency band is normal with f2 = 1 Hz. In general, the VMD-OSVD method is able to improve the SNR of the signal and characterize different pipe defects. And it has a certain guiding significance to the application of pipeline inspection in the field of safety in the future.

Optimal Singular Value Decomposition Based Big Data Compression Approach in Smart Grids

The smart grid is a fully automatic delivery grid for electricity power with a two-way reliable flow of electricity and information among different equipment on the grid. Smart meters and sensors monitoring the system provide a huge amount of data in various part of smart grid. To logically manage this trouble, a new lossy data compression approach for big data compression is proposed. The optimal singular value decomposition (SVD) is applied to a matrix that achieves the optimal number of singular values to the sending process, and the other ones will be neglected. This goal is done due to the quality of retrieved data and the compression ratio. In the presented scheme, to implement the optimization framework, various intelligent optimization methods are used to determine the number of optimal values in the elimination stage. The efficiency and capabilities of the proposed method are examined using a wide range of data types, from electricity market data to image processing benchmarks. The comparisons show that the compression level obtained by the proposed method can dominate the points given by the existing SVD rank reduction methods. Also, as the other finding of this article, the performance of the rank reduction methods depends on the application and data types. It means that a rank reduction method can reveal a good performance in one application and performs unacceptably for another purpose. So, the optimized rank reduction can pave the way toward a robust and reliable performance.

Optimal Singular Value Decomposition Based Pre-coding for Secret Key Extraction from Correlated Orthogonal Frequency Division Multiplexing Sub-channels

Secret key extraction is a crucial issue in physical layer security and a less complex and, at the same time, a more robust scheme for the next generation of 5G and beyond. Unlike previous works on this topic, in which Orthogonal Frequency Division Multiplexing (OFDM) sub-channels were considered to be independent, the effect of correlation between sub-channels on the secret key rate is addressed in this paper. As an assumption, a realistic model for dependency among sub-channels is considered. Benchmarked by simulation, the result shows that the key exchange rate may decline by up to 72% due to the correlation of sub-channels. A new approach for efficient key extraction is used in this study. To do this, a Singular Value Decomposition based (SVD-based) pre-coding is utilized to alleviate the sub-channels correlation and the channel noise. The low computational complexity of our proposed approach makes it a promising candidate for developing secure and high-speed networks. Results obtained through simulation indicate that applying pre-coding on the measured correlated data resulted in a minimum gain of 9 dB. In addition, the result also depicts the advantage of SVD versus other pre-coding techniques, namely PCA, DCT, and WT.

Adaptive and blind watermarking scheme based on optimal SVD blocks selection

To protect the ownership of the digital products, a novel adaptive and blind watermarking scheme is designed. Firstly, the underlying reason of image quality degradation in singular value decomposition (SVD)-based watermarking scheme is analyzed and the potential scenario causing visible destroy is pointed out. Then the optimal SVD blocks selection strategy is proposed to improve the imperceptibility. Different from other block selection rules devised by subjective evaluation means, our selection rule aims to retain the image quality as much as possible from the source. Furthermore, information entropy is utilized to achieve the purpose of adaptive embedding. In the experiment, the proposed watermarking scheme is tested under several attacks, such as noise attack, JPEG compression, blurring, sharping, and etc. Finally, the proposed watermarking scheme is compared with other existing schemes, and the experimental results demonstrate the robustness, imperceptibility and superior of the proposed watermarking scheme.

Spatially Sparse Beamforming Training for Millimeter Wave MIMO Systems

To realize high beamforming gain and thus sufficient link budget, millimeter wave (MMW) multiple-input multiple-output (MIMO) systems employ large antenna arrays at both the transmitter and receiver. However, due to the power and cost limitation, only a limited number of radio frequency (RF) chains are available for MMW antenna arrays. In other words, the number of RF chains is far smaller than that of antenna elements. This MMW MIMO setup poses a formidable challenge for channel estimation, which is conventionally required to realize the optimal singular value decomposition (SVD) beamforming. To circumvent the formidable channel estimation, this paper proposes two iterative antenna training schemes for the SVD beamforming in MMW MIMO systems. Relying on the channel reciprocity in time division duplex, the proposed schemes employ power iteration and Lanczos iteration, respectively, to gradually approach the SVD beamforming. During iterations, we only need to estimate several channel-related vectors instead of the MIMO channel matrix. Thanks to the spatial sparsity in MMW channels, the training overheads required by the proposed schemes are moderate. Simulations demonstrate that the proposed schemes outperform the counterpart in terms of performance and training overhead, and can achieve the performance very close to that of the perfect SVD beamforming.

Joint Antenna Beamforming, Multiuser Scheduling, and Power Allocation for Hierarchical Cellular Systems

Cognitive radio (CR) is concretely embodied in hierarchical cellular systems by deploying an underlying microcellular system to reuse the underutilized spectrum of a macrocellular system. One of the key challenges to the success of hierarchical cellular systems is to manage the intercell interference between the macrocell and the microcell and to maximize the spectrum efficiency. In this paper, antenna beamforming, power allocation, and multiuser scheduling are jointly designed to opportunistically utilize the macrocell's uplink spectrum in serving multiple secondary microcellular users concurrently. The joint design of antenna beamforming, power allocation, and scheduling with the objective of maximizing the sum rate is indeed a mixed-integer nonlinear programming NP-hard problem. The proposed simpler iterative subgradient projection and semidefinite programming approach can obtain better performance than the conventional zero-forcing beamforming. Furthermore, unlike the optimal singular value decomposition (SVD) beamforming that requires all users to have channel knowledge at the receiver for cooperation, the proposed joint design methodology requires no channel knowledge at the receiver and can outperform the SVD beamforming without user scheduling. When considering both implementation complexity and performance enhancement issues, the proposed joint power allocation, multiuser scheduling, and antenna beamforming technique can help provide important insights into the design of interference management techniques for hierarchical CR systems.

An adaptive watermarking approach based on weighted quantum particle swarm optimization

In this paper, we propose a novel optimal singular value decomposition (SVD)-based image watermarking approach that uses a new combination of weighted quantum particle swarm optimization (WQPSO) algorithm and a human visual system (HVS) model for both the hybrid discrete wavelet transform and discrete cosine transform (DCT). The proposed SVD-based watermarking approach initially decomposes the host image into sub-bands; afterwards, singular values of the DCT of the lower sub-band of the host image are quantized using a set of optimal quantization steps deduced from a combination of the WQPSO algorithm and the HVS model. To evaluate the performance of the proposed approach, we present tests on different images. The experimental results show that the proposed approach yields a watermarked image with good visual definition; at the same time, the embedded watermark was robust against a wide variety of common attacks, including JPEG compression, Gaussian noise, salt and pepper noises, Gaussian filters, median filters, image cropping, and image scaling. Moreover, the results of various experimental analyses demonstrated the superiority of the WQPSO approach over other optimization techniques, including classical PSO and QPSO in terms of local convergence speed, resulting in a better balance between global and local searches of the watermarking algorithm.

Adaptive Group Loading and Weighted Loading for MIMO OFDM Systems

Adaptive Bit Loading (ABL) in Multiple-Input Multiple-Output Orthogonal Frequency-Division Multiplexing (MIMO-OFDM) is often used to achieve the desired Bit Error Rate (BER) performance in wireless systems. In this paper, we discuss some of the bit loading algorithms, compare them in terms of the BER performance, and present an effective and concise Adaptive Grouped Loading (AGL) algorithm. Furthermore, we propose a “weight factor” for loading algorithm to converge rapidly to the final solution for various data rate with variable Signal to Noise Ratio (SNR) gaps. In particular, we consider the bit loading in near optimal Singular Value Decomposition (SVD) based MIMO-OFDM system. While using SVD based system, the system requires perfect Channel State Information (CSI) of channel transfer function at the transmitter. This scenario of SVD based system is taken as an ideal case for the comparison of loading algorithms and to show the actual enhancement achievable by our AGL algorithm. Irrespective of the CSI requirement imposed by the mode of the system itself, ABL demands high level of feedback. Grouped Loading (GL) would reduce the feedback requirement depending upon the group size. However, this also leads to considerable degradation in BER performance. In our AGL algorithm, groups are formed with a number of consecutive sub-channels belonging to the same transmit antenna, with individual gains satisfying predefined criteria. Simulation results show that the proposed “weight factor” leads a loading algorithm to rapid convergence for various data rates with variable SNR gap values and AGL requires much lesser CSI compared to GL for the same BER performance.

Gabor analysis of auditory midbrain receptive fields: spectro-temporal and binaural composition.

The spectro-temporal receptive field (STRF) is a model representation of the excitatory and inhibitory integration area of auditory neurons. Recently it has been used to study spectral and temporal aspects of monaural integration in auditory centers. Here we report the properties of monaural STRFs and the relationship between ipsi- and contralateral inputs to neurons of the central nucleus of cat inferior colliculus (ICC) of cats. First, we use an optimal singular-value decomposition method to approximate auditory STRFs as a sum of time-frequency separable Gabor functions. This procedure extracts nine physiologically meaningful parameters. The STRFs of approximately 60% of collicular neurons are well described by a time-frequency separable Gabor STRF model, whereas the remaining neurons exhibited obliquely oriented or multiple excitatory/inhibitory subfields that require a nonseparable Gabor fitting procedure. Parametric analysis reveals distinct spectro-temporal tradeoffs in receptive field size and modulation filtering resolution. Comparisons between an identical model used to study spatio-temporal integration areas of visual neurons further shows that auditory and visual STRFs share numerous structural properties. We then use the Gabor STRF model to compare quantitatively receptive field properties of contra- and ipsilateral inputs to the ICC. We show that most interaural STRF parameters are highly correlated bilaterally. However, the spectral and temporal phases of ipsi- and contralateral STRFs often differ significantly. This suggests that activity originating from each ear share various spectro-temporal response properties such as their temporal delay, bandwidth, and center frequency but have shifted or interleaved patterns of excitation and inhibition. These differences in converging monaural receptive fields expand binaural processing capacity beyond interaural time and intensity aspects and may enable colliculus neurons to detect disparities in the spectro-temporal composition of the binaural input.

Identification of Seismic Reflections Using Singular Value Decomposition

Singular value decomposition (SVD) is applied to the identification of seismic reflections by using two different models: the impulse response lOdel, where a seismic trace is assumed to consist of a known signal pulse convolved with a reflection coefficient series plus noise and the delayed pulse model, where the seismic signal is assumed to consist of a small number of delayed pulses of known shape and with unknown amplitudes and arrival times. SVD clearly shows how least-squares estimation of the reflection coefficients may become unstable, since a division by the singular values is required. Two methods for stabilizing this procedure are investigated. The inverse of the singular values may be replaced by zeros when they are less than a given threshold. This is called the SVD cut-off method. Alternatively, we may use ridge regression which in filter design corresponds to assuming white noise. Statistical methods are used to compute an optimal SVD cut-off level and also to compute an optimal weighting parameter in ridge regression. Numerical studies indicate that the use of SVD cut-off or ridge regression stabilizes the least-squares procedure, but that the results are inferior to maximum-likelihood estimation where the noise is assumed to be filtered white noise.more » For the delayed pulse model, we use a linearization procedure to iteratively update the estimates of both the reflection amplitudes and the arrival times. In each step, the optimal SVD cut-off method is used. Confidence regions for the estimated reflection amplitudes and arrival times are also computed. Synthetic data examples demonstrate the effectiveness of this method. In a real data example, the maximum-likelihood method assuming an impulse response model is first used to obtain initial estimates of the number of reflections and their amplitudes and traveltimes. Then the iterative procedure is used to obtain improved estimates of the reflection amplitudes and traveltimes.« less