This paper introduces an efficient multi-linear nonparametric (kernel-based) approximation framework for data regression and imputation, and its application to dynamic magnetic-resonance imaging (dMRI). Data features are assumed to reside in or close to a smooth manifold embedded in a reproducing kernel Hilbert space. Landmark points are identified to describe concisely the point cloud of features by linear approximating patches which mimic the concept of tangent spaces to smooth manifolds. The multi-linear model effects dimensionality reduction, enables efficient computations, and extracts data patterns and their geometry without any training data or additional information. Numerical tests on dMRI data under severe under-sampling demonstrate remarkable improvements in efficiency and accuracy of the proposed approach over its predecessors, popular data modeling methods, as well as recent tensor-based and deep-image-prior schemes.
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We propose fork-join and task-based hybrid implementations of four classical linear algebra iterative methods (Jacobi, Gauss-Seidel, conjugate gradient and biconjugate gradient stabilised) as well as variations of them. Algorithms are duly documented and the corresponding source code is made publicly available for reproducibility. Both weak and strong scalability benchmarks are conducted to statistically analyse their relative efficiencies. The weak scalability results assert the superiority of a task-based hybrid parallelisation over MPI-only and fork-join hybrid implementations. Indeed, the task-based model is able to achieve speedups of up to 25% larger than its MPI-only counterpart depending on the numerical method and the computational resources used. For strong scalability scenarios, hybrid methods based on tasks remain more efficient with moderate computational resources where data locality does not play an important role. Fork-join hybridisation often yields mixed results and hence does not present a competitive advantage over a much simpler MPI approach.
A Gaussian process (GP)-based methodology is proposed to emulate complex dynamical computer models (or simulators). The method relies on emulating the short-time numerical flow map of the system, where the flow map is a function that returns the solution of a dynamical system at a certain time point, given initial conditions. In order to predict the model output times series, a single realisation of the emulated flow map (i.e., its posterior distribution) is taken and used to iterate from the initial condition ahead in time. Repeating this procedure with multiple such draws creates a distribution over the time series whose mean and variance serve as the model output prediction and the associated uncertainty, respectively. However, since there is no known method to draw an exact sample from the GP posterior analytically, we approximate the kernel with random Fourier features and generate approximate sample paths. The proposed method is applied to emulate several dynamic nonlinear simulators including the well-known Lorenz and van der Pol models. The results suggest that our approach has a high predictive performance and the associated uncertainty can capture the dynamics of the system accurately. Additionally, our approach has potential for ``embarrassingly" parallel implementations where one can conduct the iterative predictions performed by a realisation on a single computing node.
We deal with the reduced four-equation model for dynamics of the heterogeneous compressible binary mixtures with the stiffened gas equations of state. We study its further reduced form, with the excluded volume concentrations and a quadratic equation for the common pressure of the components, that can be called a quasi-homogeneous form. We prove new properties of this equation, derive a simple formula for the squared speed of sound, give an alternative proof for a formula that relates it to the squared Wood speed of sound, and a short derivation of the pressure balance equation. For the first time, we introduce regularizations of the heterogeneous model (in the quasi-homogeneous form). In the 1D case, we construct the corresponding explicit two-level in time and symmetric three-point in space finite-difference schemes without limiters and present numerical results for various flows with shock waves.
Diffusion models (DMs) have recently been introduced in image deblurring and exhibited promising performance, particularly in terms of details reconstruction. However, the diffusion model requires a large number of inference iterations to recover the clean image from pure Gaussian noise, which consumes massive computational resources. Moreover, the distribution synthesized by the diffusion model is often misaligned with the target results, leading to restrictions in distortion-based metrics. To address the above issues, we propose the Hierarchical Integration Diffusion Model (HI-Diff), for realistic image deblurring. Specifically, we perform the DM in a highly compacted latent space to generate the prior feature for the deblurring process. The deblurring process is implemented by a regression-based method to obtain better distortion accuracy. Meanwhile, the highly compact latent space ensures the efficiency of the DM. Furthermore, we design the hierarchical integration module to fuse the prior into the regression-based model from multiple scales, enabling better generalization in complex blurry scenarios. Comprehensive experiments on synthetic and real-world blur datasets demonstrate that our HI-Diff outperforms state-of-the-art methods. Code and trained models are available at //github.com/zhengchen1999/HI-Diff.
Wearable devices permit the continuous monitoring of biological processes, such as blood glucose metabolism, and behavior, such as sleep quality and physical activity. The continuous monitoring often occurs in epochs of 60 seconds over multiple days, resulting in high dimensional longitudinal curves that are best described and analyzed as functional data. From this perspective, the functional data are smooth, latent functions obtained at discrete time intervals and prone to homoscedastic white noise. However, the assumption of homoscedastic errors might not be appropriate in this setting because the devices collect the data serially. While researchers have previously addressed measurement error in scalar covariates prone to errors, less work has been done on correcting measurement error in high dimensional longitudinal curves prone to heteroscedastic errors. We present two new methods for correcting measurement error in longitudinal functional curves prone to complex measurement error structures in multi-level generalized functional linear regression models. These methods are based on two-stage scalable regression calibration. We assume that the distribution of the scalar responses and the surrogate measures prone to heteroscedastic errors both belong in the exponential family and that the measurement errors follow Gaussian processes. In simulations and sensitivity analyses, we established some finite sample properties of these methods. In our simulations, both regression calibration methods for correcting measurement error performed better than estimators based on averaging the longitudinal functional data and using observations from a single day. We also applied the methods to assess the relationship between physical activity and type 2 diabetes in community dwelling adults in the United States who participated in the National Health and Nutrition Examination Survey.
Supervised dimension reduction (SDR) has been a topic of growing interest in data science, as it enables the reduction of high-dimensional covariates while preserving the functional relation with certain response variables of interest. However, existing SDR methods are not suitable for analyzing datasets collected from case-control studies. In this setting, the goal is to learn and exploit the low-dimensional structure unique to or enriched by the case group, also known as the foreground group. While some unsupervised techniques such as the contrastive latent variable model and its variants have been developed for this purpose, they fail to preserve the functional relationship between the dimension-reduced covariates and the response variable. In this paper, we propose a supervised dimension reduction method called contrastive inverse regression (CIR) specifically designed for the contrastive setting. CIR introduces an optimization problem defined on the Stiefel manifold with a non-standard loss function. We prove the convergence of CIR to a local optimum using a gradient descent-based algorithm, and our numerical study empirically demonstrates the improved performance over competing methods for high-dimensional data.
We present an effective framework for improving the breakdown point of robust regression algorithms. Robust regression has attracted widespread attention due to the ubiquity of outliers, which significantly affect the estimation results. However, many existing robust least-squares regression algorithms suffer from a low breakdown point, as they become stuck around local optima when facing severe attacks. By expanding on the previous work, we propose a novel framework that enhances the breakdown point of these algorithms by inserting a prior distribution in each iteration step, and adjusting the prior distribution according to historical information. We apply this framework to a specific algorithm and derive the consistent robust regression algorithm with iterative local search (CORALS). The relationship between CORALS and momentum gradient descent is described, and a detailed proof of the theoretical convergence of CORALS is presented. Finally, we demonstrate that the breakdown point of CORALS is indeed higher than that of the algorithm from which it is derived. We apply the proposed framework to other robust algorithms, and show that the improved algorithms achieve better results than the original algorithms, indicating the effectiveness of the proposed framework.
For solving linear inverse problems, particularly of the type that appear in tomographic imaging and compressive sensing, this paper develops two new approaches. The first approach is an iterative algorithm that minimizers a regularized least squares objective function where the regularization is based on a compound Gaussian prior distribution. The Compound Gaussian prior subsumes many of the commonly used priors in image reconstruction, including those of sparsity-based approaches. The developed iterative algorithm gives rise to the paper's second new approach, which is a deep neural network that corresponds to an "unrolling" or "unfolding" of the iterative algorithm. Unrolled deep neural networks have interpretable layers and outperform standard deep learning methods. This paper includes a detailed computational theory that provides insight into the construction and performance of both algorithms. The conclusion is that both algorithms outperform other state-of-the-art approaches to tomographic image formation and compressive sensing, especially in the difficult regime of low training.
Affected by the massive amount of parameters, ViT usually suffers from serious overfitting problems with a relatively limited number of training samples. In addition, ViT generally demands heavy computing resources, which limit its deployment on resource-constrained devices. As a type of model-compression method,model binarization is potentially a good choice to solve the above problems. Compared with the full-precision one, the model with the binarization method replaces complex tensor multiplication with simple bit-wise binary operations and represents full-precision model parameters and activations with only 1-bit ones, which potentially solves the problem of model size and computational complexity, respectively. In this paper, we find that the decline of the accuracy of the binary ViT model is mainly due to the information loss of the Attention module and the Value vector. Therefore, we propose a novel model binarization technique, called Group Superposition Binarization (GSB), to deal with these issues. Furthermore, in order to further improve the performance of the binarization model, we have investigated the gradient calculation procedure in the binarization process and derived more proper gradient calculation equations for GSB to reduce the influence of gradient mismatch. Then, the knowledge distillation technique is introduced to alleviate the performance degradation caused by model binarization. Experiments on three datasets with limited numbers of training samples demonstrate that the proposed GSB model achieves state-of-the-art performance among the binary quantization schemes and exceeds its full-precision counterpart on some indicators.
Modern neural network training relies heavily on data augmentation for improved generalization. After the initial success of label-preserving augmentations, there has been a recent surge of interest in label-perturbing approaches, which combine features and labels across training samples to smooth the learned decision surface. In this paper, we propose a new augmentation method that leverages the first and second moments extracted and re-injected by feature normalization. We replace the moments of the learned features of one training image by those of another, and also interpolate the target labels. As our approach is fast, operates entirely in feature space, and mixes different signals than prior methods, one can effectively combine it with existing augmentation methods. We demonstrate its efficacy across benchmark data sets in computer vision, speech, and natural language processing, where it consistently improves the generalization performance of highly competitive baseline networks.
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