Nonlinear Projection Research Articles

A nonlinear manifold model order reduction approach for thermo‐mechanically coupled plasticity

AbstractThis study employs a nonlinear manifold mode order reduction (MOR) approach to address the high computational cost inherent in multi‐physical nonlinear models, which are discretized by the finite element method (FEM), with a particular focus on thermo‐mechanically coupled plasticity under finite strain conditions. Utilizing nodal snapshots obtained from a full‐order simulation, a projection matrix is constructed through singular value decomposition (SVD), facilitating the construction of a reduced system within a lower‐dimensional space. This reduction process entails the multiplication of the discretized residual vector and stiffness matrix by a linear projection matrix, complemented by the incorporation of a nonlinear projection component to ensure accurate reconstruction within the nonlinear part. Subsequently, a comprehensive three‐dimensional benchmark test is conducted to assess the accuracy and efficacy of the reduced‐order model.

Water–carbon–economy multivariate spatial–temporal collaborative relationships and nonlinear projections in urban agglomerations

A complex dynamic coupling process exists among water resource consumption, carbon emissions, and economic growth; however, their intricate collaborative mechanisms remain unclear. This study proposes a new barycenter formula-based multivariate spatial–temporal collaborative relationship model for identifying the complex relationships and interactions among the water footprint (WF), carbon footprint (CF), and gross domestic product (GDP) in the three largest urban agglomerations in the Yangtze River Economic Belt (YREB) of China. A nonlinear grey Bernoulli model is then advanced for illustrating the future evolution in WF–CF–GDP and their multivariate collaborative relationship. Results reveal that WF and CF of all urban agglomerations increase continuously, with annual growth rates of 0.59% and 5.63%, respectively. WF, CF, and GDP are significantly positively correlated, and they exhibit similar spatial clustering characteristics and centroid migration directions along the main course of the Yangtze River. Moreover, their multivariate temporal collaborative relationship (0.79) significantly surpasses their spatial (0.63), while the spatial–temporal (0.77) is intermediate. WF has the greatest impact on the binary spatial–temporal collaborative relationship between CF and GDP. In contrast, GDP has the least impact on the relationship between CF and WF. The projected WF, CF, and GDP indicate that the YREB will face more severe water and carbon pressures as its economy grows. Their multivariate spatial–temporal collaborative relationships are projected to decrease by 10.26% in 2050. These findings contribute to promoting collaborative and sustainable development of inter-regional socio-economic and ecological environment.

Generalized nonlinear Langevin equation from quantum nonlinear projection operator

Generalized nonlinear Langevin equation from quantum nonlinear projection operator

Gauss-Seidel Type Iterative Algorithm for a Generalized System of Extended Non-linear Variational Inequalities

This study is focused on a new generalized system of extended non-linear variational inequalities (GSENVI, for short) involving 3k-distinct non-linear relaxed cocoercive operators. We give the equivalent formulation of GSENVI in a more convenient form by using auxiliary principal technique. Through the projection technique, we demonstrate that the non-linear projection equations are analogous to equivalent form of GSENVI. By the use of alternative fixed point formulations, we proposed the k-steps Gauss-Seidel type iterative algorithms to obtain an approximate solution of the GSENVI. Further, we discuss the convergence of proposed k-step Gauss Seidel type iterative algorithms. Several special cases of GSENVI are discussed for the reliability of our findings.

Model reduction on manifolds: A differential geometric framework

Using nonlinear projections and preserving structure in model order reduction (MOR) are currently active research fields. In this paper, we provide a novel differential geometric framework for model reduction on smooth manifolds, which emphasizes the geometric nature of the objects involved. The crucial ingredient is the construction of an embedding for the low-dimensional submanifold and a compatible reduction map, for which we discuss several options. Our general framework allows capturing and generalizing several existing MOR techniques, such as structure preservation for Lagrangian- or Hamiltonian dynamics, and using nonlinear projections that are, for instance, relevant in transport-dominated problems. The joint abstraction can be used to derive shared theoretical properties for different methods, such as an exact reproduction result. To connect our framework to existing work in the field, we demonstrate that various techniques for data-driven construction of nonlinear projections can be included in our framework.

Functional Oil Price Expectations Shocks and Inflation

ABSTRACTThis paper investigates the inflation effects of oil price expectations shocks constructed as functional shocks, that is, as shifts in the entire oil futures term structure (both standard and risk‐adjusted). The latter are then included in a vector autoregressive model with exogenous variables (VARX) to examine the US case. Counterfactual analysis is also carried out to investigate second‐round effects on inflation through the inflation expectations channel. These are found to be significant, in contrast to earlier studies based on standard oil price shocks. Additional nonlinear local projections including a shock decomposition exercise show that inflation and inflation expectations are primarily driven by changes in the curvature (level and slope) factor when the latter are anchored (unanchored). These findings provide useful information to policymakers concerning the impact of oil price expectations on inflation and inflation expectations.

Toward a theory of perspective perception in pictures.

This paper reviews projection models and their perception in realistic pictures, and proposes hypotheses for three-dimensional (3D) shape and space perception in pictures. In these hypotheses, eye fixations, and foveal vision play a central role. Many past theories and experimental studies focus solely on linear perspective. Yet, these theories fail to explain many important perceptual phenomena, including the effectiveness of nonlinear projections. Indeed, few classical paintings strictly obey linear perspective, nor do the best distortion-avoidance techniques for wide-angle computational photography. The hypotheses here employ a two-stage model for 3D human vision. When viewing a picture, the first stage perceives 3D shape for the current gaze. Each fixation has its own perspective projection, but, owing to the nature of foveal and peripheral vision, shape information is obtained primarily for a small region of the picture around the fixation. As a viewer moves their eyes, the second stage continually integrates some of the per-gaze information into an overall interpretation of a picture. The interpretation need not be geometrically stable or consistent over time. It is argued that this framework could explain many disparate pictorial phenomena, including different projection styles throughout art history and computational photography, while being consistent with the constraints of human 3D vision. The paper reviews open questions and suggests new studies to explore these hypotheses.

Extreme weather shocks and state-level inflation of the United States

This study investigates the impact of a metric of extreme weather shocks on 32 state-level inflation rates of the United States (US) over the quarterly period of 1989:01 to 2017:04. In this regard, we first utilize a dynamic factor model with stochastic volatility (DFM-SV) to filter out the national factor from the local components of overall, non-tradable and tradable inflation rates, to ensure that the effect of regional climate risks is not underestimated, given the derived sizeable common component. Second, using impulse responses derived from linear and nonlinear local projections models, we find statistically significant increases in the state (and national) factor of overall inflation rates, with the aggregate effect being driven by the tradable sector relative to the non-tradable one, particularly across the agricultural states in comparison to the non (less)-agricultural ones. Our findings have important policy implications.

Spherical random projection

Abstract We propose a new method for dimension reduction of high-dimensional spherical data based on the nonlinear projection of sphere-valued data to a randomly chosen subsphere. The proposed method, spherical random projection, leads to a probabilistic lower-dimensional mapping of spherical data into a subsphere of the original. In this paper, we investigate some properties of spherical random projection, including expectation preservation and distance concentration, from which we derive an analogue of the Johnson–Lindenstrauss Lemma for spherical random projection. Clustering model selection is discussed as an application of spherical random projection, and numerical experiments are conducted using real and simulated data.

Cell Spatial Analysis in Crohn's Disease: Unveiling Local Cell Arrangement Pattern with Graph-based Signatures.

Crohn's disease (CD) is a chronic and relapsing inflammatory condition that affects segments of the gastrointestinal tract. CD activity is determined by histological findings, particularly the density of neutrophils observed on Hematoxylin and Eosin stains (H&E) imaging. However, understanding the broader morphometry and local cell arrangement beyond cell counting and tissue morphology remains challenging. To address this, we characterize six distinct cell types from H&E images and develop a novel approach for the local spatial signature of each cell. Specifically, we create a 10-cell neighborhood matrix, representing neighboring cell arrangements for each individual cell. Utilizing t-SNE for non-linear spatial projection in scatter-plot and Kernel Density Estimation contour-plot formats, our study examines patterns of differences in the cellular environment associated with the odds ratio of spatial patterns between active CD and control groups. This analysis is based on data collected at the two research institutes. The findings reveal heterogeneous nearest-neighbor patterns, signifying distinct tendencies of cell clustering, with a particular focus on the rectum region. These variations underscore the impact of data heterogeneity on cell spatial arrangements in CD patients. Moreover, the spatial distribution disparities between the two research sites highlight the significance of collaborative efforts among healthcare organizations. All research analysis pipeline tools are available at https://github.com/MASILab/cellNN.

Fiber optic computing using distributed feedback

The widespread adoption of machine learning and other matrix intensive computing algorithms has renewed interest in analog optical computing, which has the potential to perform large-scale matrix multiplications with superior energy scaling and lower latency than digital electronics. However, most optical techniques rely on spatial multiplexing, requiring a large number of modulators and detectors, and are typically restricted to performing a single kernel convolution operation per layer. Here, we introduce a fiber-optic computing architecture based on temporal multiplexing and distributed feedback that performs multiple convolutions on the input data in a single layer. Using Rayleigh backscattering in standard single mode fiber, we show that this technique can efficiently apply a series of random nonlinear projections to the input data, facilitating a variety of computing tasks. The approach enables efficient energy scaling with orders of magnitude lower power consumption than GPUs, while maintaining low latency and high data-throughput.

Recognizing and detecting COVID-19 in chest X-ray images using constrained multi-view spectral clustering

AbstractMachine learning, particularly classification algorithms, has been widely employed for diagnosing COVID-19 cases. However, these methods typically rely on labeled datasets and analyze a single data view. With the vast amount of patient data available without labels, this paper addresses the novel challenge of unsupervised COVID-19 diagnosis. The goal is to harness the abundant data without labels effectively. In recent times, multi-view clustering has garnered considerable attention in the research community. Spectral clustering, known for its robust theoretical framework, is a key focus. However, traditional spectral clustering methods generate only nonlinear data projections, necessitating additional clustering steps. The quality of these post-processing steps can be influenced by various factors, such as initialization procedures and outliers. This paper introduces an enhanced version of the recent “Multiview Spectral Clustering via integrating Nonnegative Embedding and Spectral Embedding” method. While retaining the benefits of the original technique, the proposed model integrates two essential constraints: (1) a constraint for ensuring the consistent smoothness of the nonnegative embedding across all views and (2) an orthogonality constraint imposed on the columns of the nonnegative embedding matrix. The effectiveness of this approach is demonstrated using COVIDx datasets. Additionally, the method is evaluated on other image datasets to validate its suitability for this study.

Encoding scheme design for gradient-free, nonlinear projection imaging using Bloch-Siegert RF spatial encoding in a low-field, open MRI system

Eliminating conventional pulsed B0-gradient coils for magnetic resonance imaging (MRI) can significantly reduce the cost of and increase access to these devices. Phase shifts induced by the Bloch-Siegert shift effect have been proposed as a means for gradient-free, RF spatial encoding for low-field MR imaging. However, nonlinear phasor patterns like those generated from loop coils have not been systematically studied in the context of 2D spatial encoding. This work presents an optimization algorithm to select an efficient encoding trajectory among the nonlinear patterns achievable with a given hardware setup. Performance of encoding trajectories or projections was evaluated through simulated and experimental image reconstructions. Results show that the encodings schemes designed by this algorithm provide more efficient spatial encoding than comparison encoding sets, and the method produces images with the predicted spatial resolution and minimal artifacts. Overall, the work demonstrates the feasibility of performing 2D gradient-free, low-field imaging using the Bloch-Siegert shift which is an important step towards creating low-cost, point-of-care MR systems.

Optical Extreme Learning Machines with Atomic Vapors

Extreme learning machines explore nonlinear random projections to perform computing tasks on high-dimensional output spaces. Since training only occurs at the output layer, the approach has the potential to speed up the training process and the capacity to turn any physical system into a computing platform. Yet, requiring strong nonlinear dynamics, optical solutions operating at fast processing rates and low power can be hard to achieve with conventional nonlinear optical materials. In this context, this manuscript explores the possibility of using atomic gases in near-resonant conditions to implement an optical extreme learning machine leveraging their enhanced nonlinear optical properties. Our results suggest that these systems have the potential not only to work as an optical extreme learning machine but also to perform these computations at the few-photon level, paving opportunities for energy-efficient computing solutions.

On the Dimension of Exceptional Parameters for Nonlinear Projections, and the Discretized Elekes-Rónyai Theorem

On the Dimension of Exceptional Parameters for Nonlinear Projections, and the Discretized Elekes-Rónyai Theorem

Investigating the Surrogate Modeling Capabilities of Continuous Time Echo State Networks

Continuous Time Echo State Networks (CTESNs) are a promising yet under-explored surrogate modeling technique for dynamical systems, particularly those governed by stiff Ordinary Differential Equations (ODEs). A key determinant of the generalization accuracy of a CTESN surrogate is the method of projecting the reservoir state to the output. This paper shows that of the two common projection methods (linear and nonlinear), the surrogates developed via the nonlinear projection consistently outperform those developed via the linear method. CTESN surrogates are developed for several challenging benchmark cases governed by stiff ODEs, and for each case, the performance of the linear and nonlinear projections is compared. The results of this paper demonstrate the applicability of CTESNs to a variety of problems while serving as a reference for important algorithmic and hyper-parameter choices for CTESNs.

An adaptive grouping sonar-inertial odometry for underwater navigation

This paper presents a sonar-inertial navigation system with an adaptive grouping optimization framework, which tightly couples inertial measurements with acoustic data from forward-looking sonar (FLS). The system’s front-end uses the sonar-inertial constraint-based feature matching (SICFM) to increase the accuracy and robustness of data association. SICFM is designed by incorporating two novel measures. Firstly, it combines prior motion information to refine the feature search region. Then, a Gaussian-gamma mixture model is formulated to reduce the impact of speckle noise on the selection of matching points. In the back-end, an adaptive grouping factor graph optimization framework is established. A frame manager is designed using the solvability of motion estimation to dynamically group sonar frames, thereby ensuring stable feature tracking for factor graph optimization. Pre-integration-based inertial measurement factor and reprojection-based sonar measurement factor are constructed as constraints in the optimization, which reduces the dependence on prior knowledge of the scene. Furthermore, an elevation angle parameterization is introduced, aiming at the nonlinear projection of sonar features. The evaluation is presented in simulation and real-world experiments, which validate our proposed algorithm in terms of robust localization with high accuracy.

Nonlinear Locality-Preserving Projections With Dynamic Graph Learning

Nonlinear Locality-Preserving Projections With Dynamic Graph Learning

Parallel Solution of Nonlinear Projection Equations in a Multitask Learning Framework.

Nonlinear projection equations (NPEs) provide a unified framework for addressing various constrained nonlinear optimization and engineering problems. However, when it comes to solving multiple NPEs, traditional numerical integration methods are not efficient enough. This is because traditional methods solve each NPE iteratively and independently. In this article, we propose a novel approach based on multitask learning (MTL) for solving multiple NPEs. The solution procedure is outlined as follows. First, we model each NPE as a system of ordinary differential equations (ODEs) using neurodynamic optimization. Second, for each ODE system, we use a physics-informed neural network (PINN) as the solution. Third, we use a multibranch MTL framework, where each branch corresponds to a PINN model. This allows us to solve multiple NPEs in parallel by training a single neural network model. Experimental results show that our approach has superior computational performance, especially when the number of NPEs to be solved is large.

Global supply chain inflationary pressures and monetary policy in Mexico

In this paper, we examine the impact of stress in the global supply chains on inflation and monetary policy in Mexico, a representative emerging market economy. Using non-linear local projections, we estimate the degree of monetary policy tightening required in a high-stress supply chain environment and compare it to that in a low-stress environment. We instrument the monetary policy shocks with shocks to the federal funds rate. Results suggest that in a high-stress regime, the effect of an increase in the monetary policy interest rate on inflation over a one-year period is reduced considerably. We argue that this reduction is due to the slow response of inflation expectations to a monetary policy tightening in a high-stress regime. Furthermore, raising the interest rate has an effect on producer price inflation, a channel that is absent in a low-stress regime. This finding highlights the role of monetary policy in stabilizing inflation when facing supply shocks that are not necessarily permanent.