Singular Value Decomposition Research Articles

Development and Evaluation of Pakistan Explorer: A Recommender Web Platform for Navigating Pakistani Culture

The study seeks to bridge the gap in existing cultural exploration resources by offering a personalized, technology-driven understanding of Pakistan’s diverse cultural heritage, which answers the significant void in tailored cultural exploration, particularly for Pakistani heritage. This project not only showcases the rich traditions, cuisine, and arts of Pakistan but also addresses the lack of tailored cultural information available for this region. The platform's advanced recommendation algorithms promote personalized user experiences, thus fostering cross-cultural understanding and appreciation. The backbone of this platform is a hybrid- recommendation system which combines both content-based filtering with collaborative-filtering through the innovative use of the 'Surprise' library's Singular Value Decomposition (SVD) algorithm, a method not typically applied in cultural context platforms. Additionally, the system integrates real-time data processing for dynamic content updates, including weather and local events, which are crucial for real-time travel and cultural recommendations. The platform has demonstrated a capability to offer personalized and dynamic cultural insights, substantially improving user engagement and satisfaction. The application of the SVD algorithm in a new domain—cultural recommendations—is a novel adaptation, showing promising results in enhancing the relevancy of content delivered to users. For future scalability, transitioning from embedded data within the Flask app to a more robust database system, such as SQL or NoSQL, is recommended. This would facilitate easier data management and richer feature implementation. The research could also benefit from focusing on integrating more AI-driven capabilities, such as natural language processing (NLP) to parse and categorize user reviews and feedback, which could further refine and personalize recommendations. The innovative use of SVD in cultural context platforms presents a new avenue for academic exploration and can be expanded to other cultural recommendation systems.

Enhanced FFT–Root–MUSIC Algorithm Based on Signal Reconstruction via CEEMD–SVD for Joint Range and Velocity Estimation for FMCW Radar

Frequency-modulated continuous-wave (FMCW) radar is used to extract range and velocity information from the beat signal. However, the traditional joint range–velocity estimation algorithms often experience significant performances degradation under low signal-to-noise ratio (SNR) conditions. To address this issue, this paper proposes a novel approach utilizing the complementary ensemble empirical mode decomposition (CEEMD) combined with singular value decomposition (SVD) to reconstruct the beat signal prior to applying the FFT-Root-MUSIC algorithm for joint range and velocity estimation. This results in a novel joint range–velocity estimation algorithm termed as the CEEMD-SVD-FFT-Root-MUSIC (CEEMD-SVD-FRM) algorithm. First, the beat signal contaminated with additive white Gaussian noise is decomposed using CEEMD, and an appropriate autocorrelation coefficient threshold is determined to select the highly correlated intrinsic mode functions (IMFs). Then, the SVD is applied to the selected highly correlated IMFs for denoising the beat signal. Subsequently, the denoised IMFs and signal residuals are combined to reconstruct the beat signal. Finally, the FFT-Root-MUSIC algorithm is applied to the reconstructed beat signal to estimate both the range and Doppler frequencies, which are then used to calculate the range and velocity estimates of the targets. The proposed CEEMD-SVD-FRM algorithm is validated though simulations and experiments, demonstrating significant improvement in the robustness and accuracy of range and velocity estimates for the FMCW radar due to the effective denoising of the reconstructed beat signal. Moreover, it substantially outperforms the traditional methods in low SNR environments.

Enhancing Clustering Stability and Efficiency: A Framework for Optimizing K-means, K-medoids, and K-shape with Intelligent Algorithms

Clustering methods like Kmeans often produce inconsistent results due to the random initialization of cluster centroids, even with optimizations such as Kmeans++, which improve centroid selection but fail to eliminate sensitivity to initialization randomness. Additionally, the choice of the number of clusters and distance metrics significantly impacts clustering performance. This paper proposes an enhanced framework combining intelligent optimization algorithms—Sparrow Search Algorithm (SSA), Dung Beetle Optimizer (DBO), and Sine Cosine Algorithm (SCA)—to optimize clustering outcomes for Kmeans, Kmedoids, and Kshape. The framework also incorporates dimensionality reduction techniques, including Principal Component Analysis (PCA), Non-Negative Matrix Factorization (NNMF), and Singular Value Decomposition (SVD), to address high-dimensional data challenges. Experimental results on benchmark datasets demonstrate the proposed framework’s effectiveness, with SSA achieving the highest silhouette score of 0.68 and reducing runtime by 35% compared to traditional methods. This approach enhances clustering stability and accuracy, offering a robust solution for diverse applications.

Observer‐Based Switching‐Like Adaptive‐Triggered Resilient Coordination Control of Discrete Singular Systems Under DI Attacks with Uncertain Occurrence Probabilities

ABSTRACTIn this paper, we investigate observer‐based switching‐like adaptive‐triggered resilient state feedback control for discrete singular systems under deception‐injection (DI) attacks. Considering state variables are not completely measurable, a delayed state observer is engineered to reconstruct system states and the occurrence of DI attacks is described by Bernoulli random variables. The upper bound method is used to deal with the DI attacks with jumping patterns and imprecise occurrence probabilities. Taking into account the sporadic nature of DI attacks, switching‐like adaptive‐triggered resilient control method is presented, and the collaboratively designed triggered mechanism and resilient control strategy can not only automatically adjust data transmission based on the system states, but also save communication resources and network bandwidth, and enhance the tolerance to hostile attacks so as to improve safety and reliability of networked singular systems. By utilizing singular value decomposition technique, the noncausal behavior of the singular system is avoided and by applying linear matrix inequality (LMI) technology, the desired gains of controller and observer are achieved concurrently, then neoteric conditions about stochastic admissibility of closed‐loop discrete singular systems are provided. Ultimately, the validity of the proposed resilient coordination control approach is verified via a direct current (DC) motor model.

Image Watermark Embedding Method Based on Security Service Secure

To solve the problem of privacy leakage and response latency in outsourced image watermark embedding in cloud computing, an efficient and privacy-preserving watermark embedding method for outsourced digital images was proposed by introducing edge computing technology. We had proposed a perturbing encryption method with homomorphism to ensure the information security and the correctness of discrete wavelet transformation in the encrypted domain. In addition, the framework was designed to guarantee the safety of singular value decomposition that edge server could not recover the original image matrix. The experimental results show that the proposed method is superior to similar secure watermarking schemes in terms of encryption/decryption time and ciphertext expansion. The proposed method enables the watermarking operation to be performed in an unsafe outsourced environment while achieving a watermarking effect similar to the plaintext equivalent

Exact parallelized dynamic mode decomposition with Hankel matrix for large-scale flow data

An exact parallel algorithm of dynamic mode decomposition (DMD) with Hankel matrices for large-scale flow data is proposed. The proposed algorithm enables the DMD and the Hankel DMD for large-scale data obtained by high-fidelity flow simulations, such as large-eddy simulations or direct numerical simulations using more than a billion grid points, on parallel computations without any approximations. The proposed algorithm completes the computations of the DMD by utilizing block matrices of XTX∈Rk×k (where X∈Rn×k is a large data matrix obtained by high-fidelity simulations, the number of snapshot data is n≳109 rsim 10^9$$\\end{document}]]>, and the number of snapshots is k≲O(103)) without any approximations: for example, the singular value decomposition of X is replaced by the eigenvalue decomposition of XTX. Then, the computation of XTX is parallelized by utilizing the domain decomposition often used in flow simulations, which reduces the memory consumption for each parallel process and wall-clock time in the DMD by a factor approximately equal to the number of parallel processes. The parallel computation with communication is performed only for XTX, allowing for high parallel efficiency under massively parallel computations. Furthermore, the proposed exact parallel algorithm is extended to the Hankel DMD without any additional parallel computations, realizing the Hankel DMD of large-scale data collected by over a billion grid points with comparable cost and memory to the DMD without Hankel matrices. Moreover, this study shows that the Hankel DMD, which has been employed to enrich information and augment rank, is advantageous for large-scale high-dimensional data collected by high-fidelity simulations in data reconstruction and predictions of future states (while prior studies have reported such advantages for low-dimensional data). Several numerical experiments using large-scale data, including laminar and turbulent flows around a cylinder and transonic buffeting flow around a full aircraft configuration, demonstrate that (i) the proposed exact parallel algorithm reproduces the existing non-parallelized Hankel DMD, (ii) the Hankel DMD for large-scale data consisting of over a billion grid points is feasible by using the proposed exact parallel algorithm with high parallel efficiency on more than 6 thousand CPU cores, and (iii) the Hankel DMD has advantages for high-dimensional data such as n≳109 rsim 10^9$$\\end{document}]]>.Graphical abstract

An Investigation of Extended-Dimension Embedded CKF-SLAM Based on the Akaike Information Criterion.

Simultaneous localization and mapping (SLAM) faces significant challenges due to high computational costs, low accuracy, and instability, which are particularly problematic because SLAM systems often operate in real-time environments where timely and precise state estimation is crucial. High computational costs can lead to delays, low accuracy can result in incorrect mapping and localization, and instability can make the entire system unreliable, especially in dynamic or complex environments. As the state-space dimension increases, the filtering error of the standard cubature Kalman filter (CKF) grows, leading to difficulties in multiplicative noise propagation and instability in state estimation results. To address these issues, this paper proposes an extended-dimensional embedded CKF based on truncated singular-value decomposition (TSVD-AECKF). Firstly, singular-value decomposition (SVD) is employed instead of the Cholesky decomposition in the standard CKF to mitigate the non-positive definiteness of the state covariance matrix. Considering the effect of small singular values on the stability of state estimation, a method is provided to truncate singular values by determining the truncation threshold using the Akaike information criterion (AIC). Furthermore, the system noise is embedded into the state variables, and an embedding volume criterion is used to improve the conventional CKF while extending the dimensionality. Finally, the proposed algorithm was validated and analyzed through both simulations and real-world experiments. The results indicate that the proposed method effectively mitigates the increase in localization error as the state-space dimension grows, enhancing time efficiency by 55.54%, and improving accuracy by 35.13% compared to the standard CKF algorithm, thereby enhancing the robustness and stability of mapping.

A Novel Self-Supervised Learning-Based Method for Dynamic CT Brain Perfusion Imaging.

Dynamic computed tomography (CT)-based brain perfusion imaging is a non-invasive technique that can provide quantitative measurements of cerebral blood flow (CBF), cerebral blood volume (CBV), and mean transit time (MTT). However, due to high radiation dose, dynamic CT scan with a low tube voltage and current protocol is commonly used. Because of this reason, the increased noise degrades the quality and reliability of perfusion maps. In this study, we aim to propose and investigate the feasibility of utilizing a convolutional neural network and a bi-directional long short-term memory model with an attention mechanism to self-supervisedly yield the impulse residue function (IRF) from dynamic CT images. Then, the predicted IRF can be used to compute the perfusion parameters. We evaluated the performance of the proposed method using both simulated and real brain perfusion data and compared the results with those obtained from two existing methods: singular value decomposition and tensor total-variation. The simulation results showed that the overall performance of parameter estimation obtained from the proposed method was superior to that obtained from the other two methods. The experimental results showed that the perfusion maps calculated from the three studied methods were visually similar, but small and significant differences in perfusion parameters between the proposed method and the other two methods were found. We also observed that there were several low-CBF and low-CBV lesions (i.e., suspected infarct core) found by all comparing methods, but only the proposed method revealed longer MTT. The proposed method has the potential to self-supervisedly yield reliable perfusion maps from dynamic CT images.

Deep learning-driven multi-omics sequential diagnosis with Hybrid-OmniSeq: Unraveling breast cancer complexity

Breast cancer results from an uncontrolled growth of breast tissue. Many methods of diagnosis are using multi-omics data to better understand the complexity of breast cancer. The new strategy laid out in this work, called “Hybrid-OmniSeq,” is a deep learning-based multi-omics data analysis technology that uses molecular subtypes of breast cancer gene to increase the precision and effectiveness of breast cancer diagnosis. For preprocessing, the BC-VM procedure is utilized, and for molecular subtype analysis, the BC-MSA procedure is utilized. The implementation of Deep Neural Network (DNN) technology in conjunction with Sequential Forward Floating Selection (SFFS) and Truncated Singular Value Decomposition (TSVD) entropy enable adaptive learning from multi-omics gene data. Five machine learning classifiers are used for classification purpose. Hybrid-OmniSeq uses a variety of machine learning classifiers in a thorough analytical process to achieve remarkable diagnostic accuracy. Deep Learning-based multi-omics sequential approach was evaluated using METABRIC RNA-seq data sets of intrinsic subtypes of breast cancer. According to test results, Logistic Regression (LR) had ER (Estrogen Receptor) status values of 94.51%, ER status values of 96.33%, and HER2 (Human Epidermal growth factor Receptor) status values of 92.3%; Random Forest (RF) had ER status values of 93.77%, ER status values of 95.23%, and HER2 status values of 93.4%. LR and RF increase the cancer detection accuracy for all subtypes when compared to alternative machine learning classifiers or the majority voting method, providing a comprehensive understanding of the underlying causes of breast cancer.

Optical coherence tomography image despeckling based on saliency enhancement and high-order singular value Marchenko-Pastur truncation

Abstract Optical Coherence Tomography (OCT) is a high-resolution imaging technique extensively used in various fields, including medical diagnosis. The presence of speckle noise significantly degrades the quality of OCT images. To address this, a novel denoising approach based on High-Order Singular Value Decomposition (HOSVD), along with saliency enhancement and Marchenko-Pastur (MP) truncation, has been proposed. Initially, the method leverages the weighted absolute distance based on variance and information entropy to locate non-local patches that are highly correlated with a given reference block. Following this, a three-dimensional tensor is decomposed via HOSVD and then contracted based on the principles from the MP theorem in random matrix theory. An iterative regularization technique, coupled with a saliency enhancement strategy, is also employed to improve the denoising performance. Experimental results show that the method is comparable to existing advanced denoising algorithms in terms of reducing speckle noise and preserving image details.

DeepTensor: Low-Rank Tensor Decomposition With Deep Network Priors.

DeepTensor is a computationally efficient framework for low-rank decomposition of matrices and tensors using deep generative networks. We decompose a tensor as the product of low-rank tensor factors (e.g., a matrix as the outer product of two vectors), where each low-rank tensor is generated by a deep network (DN) that is trained in a self-supervised manner to minimize the mean-square approximation error. Our key observation is that the implicit regularization inherent in DNs enables them to capture nonlinear signal structures (e.g., manifolds) that are out of the reach of classical linear methods like the singular value decomposition (SVD) and principal components analysis (PCA). Furthermore, in contrast to the SVD and PCA, whose performance deteriorates when the tensor's entries deviate from additive white Gaussian noise, we demonstrate that the performance of DeepTensor is robust to a wide range of distributions. We validate that DeepTensor is a robust and computationally efficient drop-in replacement for the SVD, PCA, nonnegative matrix factorization (NMF), and similar decompositions by exploring a range of real-world applications, including hyperspectral image denoising, 3D MRI tomography, and image classification. In particular, DeepTensor offers a 6 dB signal-to-noise ratio improvement over standard denoising methods for signal corrupted by Poisson noise and learns to decompose 3D tensors 60 times faster than a single DN equipped with 3D convolutions.

Scenario Superposition Method for Real‐Time Tsunami Prediction Using a Bayesian Approach

AbstractIn this study, we propose a scenario superposition method for real‐time tsunami wave prediction. In the offline phase, prior to actual tsunami occurrence, hypothetical tsunami scenarios are created, and their wave data are decomposed into spatial modes and scenario‐specific coefficients by the singular value decomposition. Then, once an actual tsunami event is observed, the proposed method executes an online phase, which is a novel contribution of this study. Specifically, the predicted waveform is represented by a linear combination of training scenarios consisting of precomputed tsunami simulation results. To make such a prediction, a set of weight parameters that allow for appropriate scenario superposition is identified by the Bayesian update process. At the same time, the probability distribution of the weight parameters is obtained as reference information regarding the reliability of the prediction. Then, the waveforms are predicted by superposition with the estimated weight parameters multiplied by the waveforms of the corresponding scenarios. To validate the performance and benefits of the proposed method, a series of synthetic experiments are performed for the Shikoku coastal region of Japan with the subduction zone of the Nankai Trough. All tsunami data are derived from numerical simulations and divided into a training data set used as scenario superposition components and a test data set for an unknown real event. The predicted waveforms at the synthetic gauges closest to the Shikoku Islands are compared to those obtained using our previous prediction method incorporating sequential Bayesian updating.

Reduced-Order Modeling for Subsurface Flow Simulation in Fractured Reservoirs

Summary Reservoir simulation for fractured reservoirs is often challenging and time-consuming due to the strong heterogeneity and complex flow dynamics introduced by fracture-matrix interactions. In this study, we introduce a novel reduced-order modeling procedure to speed up the flow simulation of fractured reservoirs. The reduced-order model (ROM) is developed based on proper orthogonal decomposition (POD) in conjunction with the embedded discrete fracture model (EDFM) that provides full-order simulation results. With the full-order training simulation, snapshots of reservoir pressure and saturation state at different timesteps are captured and assembled into separate data matrices. Singular value decomposition (SVD) is then applied to these data matrices to obtain a reduced set of orthogonal base vectors for pressure and saturation solutions, respectively. These base vectors enable the projection of high-dimensional linear equations into much lower-dimensional spaces, which significantly accelerates the process of solving nonlinear governing equations under the EDFM approach. The developed reduced-order modeling procedure is implemented in the MATLAB reservoir simulation toolbox (MRST) and tested via multiple cases for both 2D and 3D fractured reservoirs under different boundary and well control scenarios. In certain challenging cases, the use of multiple training simulations is explored and is shown to provide improved predictions. Overall, the proposed ROM approach is able to provide simulation results that are very consistent with those obtained from the full-order simulations while achieving computational speedups of about an order of magnitude for large-scale cases. These observations indicate that the proposed ROM exhibits satisfactory generalization performance, making it suitable for problems that require many flow simulations under different settings, such as production optimization.

A flexible class of priors for orthonormal matrices with basis function-specific structure

Statistical modeling of high-dimensional matrix-valued data motivates the use of a low-rank representation that simultaneously summarizes key characteristics of the data and enables dimension reduction. Low-rank representations commonly factor the original data into the product of orthonormal basis functions and weights, where each basis function represents an independent feature of the data. However, the basis functions in these factorizations are typically computed using algorithmic methods that cannot quantify uncertainty or account for basis function correlation structure a priori. While there exist Bayesian methods that allow for a common correlation structure across basis functions, empirical examples motivate the need for basis function-specific dependence structure. We propose a prior distribution for orthonormal matrices that can explicitly model basis function-specific structure. The prior is used within a general probabilistic model for singular value decomposition to conduct posterior inference on the basis functions while accounting for measurement error and fixed effects. We discuss how the prior specification can be used for various scenarios and demonstrate favorable model properties through synthetic data examples. Finally, we apply our method to two-meter air temperature data from the Pacific Northwest, enhancing our understanding of the Earth system’s internal variability.

Electrocardiography Denoising via Sparse Dictionary Learning from Small Datasets

Abstract Wearable electrocardiography monitors, e.g. embedded in textile shirts, offer new approaches in diagnosis but suffers upon limited computational capacities. Hence, we propose and evaluate a lightweight algorithm for electrocardiography denoising via sparse dictionary learning, targeting two types of noise: baseline wander and muscle artifacts. For each type of noise a dictionary is built using K-singular value decomposition. This iterative method alternates between finding a sparse representation for every training signal and then updating every atom of the dictionary on its own. A sparse representation is found using the orthogonal matching pursuit algorithm. The atoms are updated exploiting the properties of the singular value decomposition. For further sparse approximation, we use the basis pursuit denoising algorithm. Electrocardiography data stems from synthetically-generated signals as well as the freely-available Brno University of Technology ECG Quality Database. Noise is added to the signals using the MIT-BIH Noise Stress Database. Our results regarding baseline wander demonstrate that the algorithm outperforms the American Heart Association-recommended bandpass filter w.r.t. signal-to-noise ratio. Moreover, a small number of training data is sufficient for satisfying results which indicates the suitability of the method for wearable hardware with low memory and power specifications.

Color image restoration based on reduced biquaternion matrix singular value decomposition

Color image restoration based on reduced biquaternion matrix singular value decomposition

Improving Xenon-129 lung ventilation image SNR with deep-learning based image reconstruction.

To evaluate the feasibility and utility of a deep learning (DL)-based reconstruction for improving the SNR of hyperpolarized 129Xe lung ventilation MRI. 129Xe lung ventilation MRI data acquired from patients with asthma and/or chronic obstructive pulmonary disease (COPD) were retrospectively reconstructed with a commercial DL reconstruction pipeline at five different denoising levels. Quantitative imaging metrics of lung ventilation including ventilation defect percentage (VDP) and ventilation heterogeneity index (VHI) were compared between each set of DL-reconstructed images and alternative denoising strategies including: filtering, total variation denoising and higher-order singular value decomposition. Structural similarity between the denoised and original images was assessed. In a prospective study, the feasibility of using SNR gains from DL reconstruction to allow natural-abundance xenon MRI was evaluated in healthy volunteers. 129Xe ventilation image SNR was improved with DL reconstruction when compared with conventionally reconstructed images. In patients with asthma and/or COPD, DL-reconstructed images exhibited a slight positive bias in ventilation defect percentage (1.3% at 75% denoising) and ventilation heterogeneity index (˜1.4) when compared with conventionally reconstructed images. Additionally, DL-reconstructed images preserved structural similarity more effectively than data denoised using alternative approaches. DL reconstruction greatly improved image SNR (greater than threefold), to a level that 129Xe ventilation imaging using natural-abundance xenon appears feasible. DL-based image reconstruction significantly improves 129Xe ventilation image SNR, preserves structural similarity, and leads to a minor bias in ventilation metrics that can be attributed to differences in the image sharpness. This tool should help facilitate cost-effective 129Xe ventilation imaging with natural-abundance xenon in the future.

Detection and Classification of Malicious Websites Using Natural Language Processing (NLP) and Machine Learning (ML) Techniques

The internet, while offering extensive services and information, has also become a platform for malicious activities, particularly through harmful websites that threaten cybersecurity. Detecting and classifying these websites is crucial for protecting users from online threats. Traditional detection methods, primarily based on blacklists and signature-based techniques, struggle to match the pace with the dynamic evolving strategies of cybercriminals. Recent advancements in Machine Learning (ML) show promise, though they remain works in progress. This research addressed this challenge by exploring the usage of Natural Language Processing and Machine Learning techniques used to classify websites as benign or malicious. Unlike many existing studies that relied on URL features alone, this study incorporated a more comprehensive feature set, including URL, content, and additional web attributes, which enhanced classification accuracy. Using an imbalanced dataset skewed towards malicious sites, this study solved using SMOTE (Synthetic Minority Over-sampling Technique) the class imbalance problem, improving model performance. Utilized Hashing Vectorizer (HashingV) and TF-IDF (Term Frequency-Inverse Document Frequency), were adopted to transform textual features into their vector representations while PCA (Principal Component Analysis) and truncated Singular Value Decomposition (truncSVD), were then used to optimize feature representation across different dimensions. Five ML classifiers include Support Vector Machine (SVM), Decision Tree (DT), Random Forest (RF), K-Nearest Neighbors (KNN), and Logistic Regression (LR) were tested for classification, and performance was evaluated using metrics such as precision and recall, accuracy, F1-Score. The results revealed that Random Forest classifier utilizing HashingV recorded the best results, with accuracies of 99.9563% using truncSVD and 99.9561% with PCA.

Efficient and scalable wave function compression using corner hierarchical matrices.

The exponential scaling of complete active space and full configuration interaction (CI) calculations limits the ability of quantum chemists to simulate the electronic structures of strongly correlated systems. Herein, we present corner hierarchically approximated CI (CHACI), an approach to wave function compression based on corner hierarchical matrices (CH-matrices)-a new variant of hierarchical matrices based on block-wise low-rank decomposition. By application to dodecacene, a strongly correlated molecule, we demonstrate that CH matrix compression provides superior compression compared to truncated global singular value decomposition. The compression ratio is shown to improve with increasing active space size. By comparison of several alternative schemes, we demonstrate that superior compression is achieved by (a) using a blocking approach that emphasizes the upper-left corner of the CI vector, (b) sorting the CI vector prior to compression, and (c) optimizing the rank of each block to maximize information density.

On the origin of circular rolls in rotor-stator flow

Rotor-stator flows are known to exhibit instabilities in the form of circular and spiral rolls. While the spirals are known to emanate from a supercritical Hopf bifurcation, the origin of the circular rolls is still unclear. In the present work we suggest a quantitative scenario for the circular rolls as a response of the system to external forcing. We consider two types of axisymmetric forcing: bulk forcing (based on the resolvent analysis) and boundary forcing using direct numerical simulation. Using the singular value decomposition of the resolvent operator the optimal response is shown to take the form of circular rolls. The linear gain curve shows strong amplification at non-zero frequencies following a pseudo-resonance mechanism. The optimal energy gain is found to scale exponentially with the Reynolds number $Re$ (for $Re$ based on the rotation rate and interdisc spacing $H$ ). The results for both types of forcing are compared with former experimental works and previous numerical studies. Our findings suggest that the circular rolls observed experimentally are the effect of the high forcing gain together with the roll-like form of the leading response of the linearised operator. For high enough Reynolds number it is possible to delineate between linear and nonlinear responses. For sufficiently strong forcing amplitudes, the nonlinear response is consistent with the self-sustained states found recently for the unforced problem. The onset of such non-trivial dynamics is shown to correspond in state space to a deterministic leaky attractor, as in other subcritical wall-bounded shear flows.