Local Manifold Research Articles

Deformations of some local Calabi-Yau manifolds

We study deformations of certain crepant resolutions of isolated rational Gorenstein singularities. After a general discussion of the deformation theory, we specialize to dimension $3$ and consider examples which are good (log) resolutions as well as the case of small resolutions. We obtain some partial results on the classification of canonical threefold singularities that admit good crepant resolutions. Finally, we study a noncrepant example, the blowup of a small resolution whose exceptional set is a smooth curve.

Bifurcation and stability analysis of a Leslie–Gower diffusion predator–prey model with prey refuge and Beddington–DeAngelis functional response

In this paper, we consider the spatiotemporal dynamics behaviors of a Leslie–Gower diffusion predator–prey system with prey refuge and Beddington–DeAnglis (B‐D) functional response. By using the Poincaré inequality and topological degree theory, we first investigate the Turing instability of the reaction–diffusion system and prove the existence of nonconstant positive steady‐state solutions. Then we discuss the steady‐state bifurcation and the direction to Hopf bifurcation of the PDE model by the local bifurcation theorem and center manifold theory. Finally, some numerical simulations are presented to supplement the analytic results in one dimension which indicates that changes in prey refuge and diffusion coefficient can increase the complexity of the system.

Novel modified weights and cosine similarity based maximum marginal projection and its application in fault diagnosis

Abstract Facing the problem that the data generated in industrial processes have few labeled samples and the local manifold learning dimensionality reduction method ignores the local spatial structure of sample points and the distance relationship in constructing different weights. To solve the above problems, this paper presents a novel modified weights and cosine similarity based maximum marginal projection named MCMMP. In MCMMP, cosine similarity is used to consider the space feature of sample points, which enhances the performance of dimensionality reduction. The new modified weights are applied to measure the between-class and the within-class sample points, which enhance the divisibility of sample points. After MCMMP dimensionality reduction, the classifier is used to classify the dimensionality reduction sample points. Finally, the proposed new method is used in two cases Tennessee Eastman Process (TEP) and Three-phase Flow Facility (TFF) to test the fault diagnosis performance. The results of the simulation process indicated that the new fault diagnosis method based on MCMMP, compared with other related diagnosis methods, has good performance.

BundleMoCap++: Efficient, robust and smooth motion capture from sparse multiview videos

Producing smooth and accurate motions from sparse videos without requiring specialized equipment and markers is a long-standing problem in the research community. Most approaches typically involve complex processes such as temporal constraints, multiple stages combining data-driven regression and optimization techniques, and bundle solving over temporal windows. These increase the computational burden and introduce the challenge of hyperparameter tuning for the different objective terms. In contrast, BundleMoCap++ offers a simple yet effective approach to this problem. It solves the motion in a single stage, eliminating the need for temporal smoothness objectives while still delivering smooth motions without compromising accuracy. BundleMoCap++ outperforms the state-of-the-art without increasing complexity. Our approach is based on manifold interpolation between latent keyframes. By relying on a local manifold smoothness assumption and appropriate interpolation schemes, we efficiently solve a bundle of frames using two or more latent codes. Additionally, the method is implemented as a sliding window optimization and requires only the first frame to be properly initialized, reducing the overall computational burden. BundleMoCap++’s strength lies in achieving high-quality motion capture results with fewer computational resources. To do this efficiently, we propose a novel human pose prior that focuses on the geometric aspect of the latent space, modeling it as a hypersphere, allowing for the introduction of sophisticated interpolation techniques. We also propose an algorithm for optimizing the latent variables directly on the learned manifold, improving convergence and performance. Finally, we introduce high-order interpolation techniques adapted for the hypersphere, allowing us to increase the solving temporal window, enhancing performance and efficiency.

The Wong-Zakai approximations of invariant manifolds for retarded partial differential equations with multiplicative white noise

We focus on the dynamics and Wong-Zakai approximation for a class of retarded partial differential equations subjected to multiplicative white noise. We show that when restricted to a local region and under certain conditions, there exists a unique global solution for the truncated system driven by either the white noise or the approximation noise. Such solution generates a random dynamical system, and the solutions of Wong-Zakai approximations are convergent to solutions of the stochastic retarded differential equation. We also show that there exist invariant manifolds for the truncated system driven by either the white noise or the approximation noise, which are then the local manifolds for the untruncated systems, and prove that such invariant manifolds of the Wong-Zakai approximations converge to those of the stochastic retarded differential equation.

An adaptive imbalance robust graph embedding broad learning system fault diagnosis for imbalanced batch processes data

Modern batch processes develop toward higher levels, and devices usually work at normal conditions with generally very few faults. Therefore, fewer fault data are collected than normal data, which makes the normal and fault modes imbalanced, and the fault diagnosis model is incapable focusing on the minority of fault samples as much as on most normal samples, consequently leading to insufficient generalization ability for the model. For that, we propose an adaptive imbalance-robust graph embedding broad learning system (AI-RGEBLS) in this paper. Firstly, it achieves adaptive correction of unbalanced samples by niche technique and synthetic minority over-sampling technique (SMOTE) with improved Mahalanobis distance. Then, broad learning system (BLS) is regularized by graph embedding, and further introduced the L2,1 norm constraint, which makes it possible to enhance the robustness of the model while considering the local manifold information of the data in the feature extraction process. Finally, the incremental learning approach is applied to the model to avoid the disastrous forgetting problem caused by training the whole model from zero. The effectiveness of the proposed method is verified by penicillin fermentation process and semiconductor etching process. It can effectively improve the speed of model training and provide better fault diagnosis for imbalanced batch processes compared with the existing methods.

Extending discrete geometric singular perturbation theory to non-hyperbolic points

Abstract We extend the recently developed discrete geometric singular perturbation theory to the non-normally hyperbolic regime. Our primary tool is the Takens embedding theorem, which provides a means of approximating the dynamics of particular maps with the time-1 map of a formal vector field. First, we show that the so-called reduced map, which governs the slow dynamics near slow manifolds in the normally hyperbolic regime, can be locally approximated by the time-1 map of the reduced vector field which appears in continuous-time geometric singular perturbation theory. In the non-normally hyperbolic regime, we show that the dynamics of fast-slow maps with a unipotent linear part can be locally approximated by the time-1 map induced by a fast-slow vector field in the same dimension, which has a nilpotent singularity of the corresponding type. The latter result is used to describe (i) the local dynamics of two-dimensional fast-slow maps with non-normally singularities of regular fold, transcritical and pitchfork type, and (ii) dynamics on a (potentially high-dimensional) local center manifold in n-dimensional fast-slow maps with regular contact or fold submanifolds of the critical manifold. In general, our results show that the dynamics near a large and important class of singularities in fast-slow maps can be described via the use of formal embedding theorems which allow for their approximation by the time-1 map of a fast-slow vector field featuring a loss of normal hyperbolicity.

ANS-SCMC: A matrix completion method based on adaptive neighbourhood similarity and sparse constraints for predicting microbe-disease associations.

The use of matrix completion methods to predict the association between microbes and diseases can effectively improve treatment efficiency. However, the similarity measures used in the existing methods are often influenced by various factors such as neighbourhood size, choice of similarity metric, or multiple parameters for similarity fusion, making it challenging. Additionally, matrix completion is currently limited by the sparsity of the initial association matrix, which restricts its predictive performance. To address these problems, we propose a matrix completion method based on adaptive neighbourhood similarity and sparse constraints (ANS-SCMC) for predict microbe-disease potential associations. Adaptive neighbourhood similarity learning dynamically uses the decomposition results as effective information for the next learning iteration by simultaneously performing local manifold structure learning and decomposition. This approach effectively preserves fine local structure information and avoids the influence of weight parameters directly involved in similarity measurement. Additionally, the sparse constraint-based matrix completion approach can better handle the sparsity challenge in the association matrix. Finally, the algorithm we proposed has achieved significantly higher predictive performance in the validation compared to several commonly used prediction methods proposed to date. Furthermore, in the case study, the prediction algorithm achieved an accuracy of up to 80% for the top 10 microbes associated with type 1 diabetes and 100% for Crohn's disease respectively.

Yosida distance and existence of invariant manifolds in the infinite-dimensional dynamical systems

We consider the existence of invariant manifolds to evolution equations u ′ ( t ) = A u ( t ) u’(t)=Au(t) , A : D ( A ) ⊂ X → X A:D(A)\subset \mathbb {X}\to \mathbb {X} near its equilibrium A ( 0 ) = 0 A(0)=0 under the assumption that its proto-derivative ∂ A ( x ) \partial A(x) exists and is continuous in x ∈ D ( A ) x\in D(A) in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators U U and V V in a Banach space X \mathbb {X} is defined as d Y ( U , V ) ≔ lim sup μ → + ∞ ‖ U μ − V μ ‖ d_Y(U,V)≔\limsup _{\mu \to +\infty } \| U_\mu -V_\mu \| , where U μ U_\mu and V μ V_\mu are the Yosida approximations of U U and V V , respectively. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of ∂ A \partial A is continuous in the sense of Yosida distance. The Yosida distance approach allows us to generalize the well-known results with possible applications to larger classes of partial differential equations and functional differential equations. The obtained results seem to be new.

Fast Discriminant Analysis With Adaptive Reconstruction Structure Preserving.

Neighborhood reconstruction methods have been widely applied to feature engineering. Existing reconstruction-based discriminant analysis methods normally project high-dimensional data into a low-dimensional space while preserving the reconstruction relationships among samples. However, there are three limitations: 1) the reconstruction coefficients are learned based on the collaborative representation of all sample pairs, which requires the training time to be the cube of the number of samples; 2) these coefficients are learned in the original space, ignoring the interference of the noise and redundant features; and 3) there is a reconstruction relationship between heterogeneous samples; this will enlarge the similarity of heterogeneous samples in the subspace. In this article, we propose a fast and adaptive discriminant neighborhood projection model to tackle the above drawbacks. First, the local manifold structure is captured by bipartite graphs in which each sample is reconstructed by anchor points derived from the same class as that sample; this can avoid the reconstruction between heterogeneous samples. Second, the number of anchor points is far less than the number of samples; this strategy can reduce the time complexity substantially. Third, anchor points and reconstruction coefficients of bipartite graphs are updated adaptively in the process of dimensionality reduction, which can enhance the quality of bipartite graphs and extract discriminative features simultaneously. An iterative algorithm is designed to solve this model. Extensive results on toy data and benchmark datasets show the effectiveness and superiority of our model.

Global-local manifold embedding broad graph convolutional network for hyperspectral image classification

Graph convolutional neural networks (GCNs) with domain-specific feature aggregation capabilities have unique advantages in hyperspectral image (HSI) classification. However, current GCN-based approaches frequently encounter the issue of node characteristics being over-smoothed while aggregating in higher-order domains. Furthermore, GCN linear classifiers focus solely on sample separability and ignore the potential manifold information of graph features, resulting in a failure to fully investigate extracted features. To address these problems, we propose a global-local manifold embedding broad graph convolutional network (GLMBG) for HSI classification. In GLMBG, we designed two modules from feature extraction and classification perspectives: The graph convolutional edge feature fusion extractor (GEFF) and the broad classifier of global-local manifold embedding (BGLME). GEFF is designed to learn graph node and local edge features from HSI through GCN and recursive filtering, combining them in a weighted manner to construct fused graph features. BGLME is designed to replace traditional linear classifiers with broad learning classifiers through manifold regularized embedding, fully utilizing the global and local manifold discriminant information of graph node features. The combination of GEFF and BGLME effectively reduces over-smoothing of graph node features while maximizing the utilization of manifold discriminant information, hence improving model feature discriminative ability. Experimental evaluations of three commonly used hyperspectral datasets show that our method surpasses state-of-the-art methods.

Local-manifold-distance-based regression: an estimation method for quantifying dynamic biological interactions with empirical time series.

Quantifying species interactions based on empirical observations is crucial for ecological studies. Advancements in nonlinear time-series analyses, particularly S-maps, are promising for high-dimensional and non-equilibrium ecosystems. S-maps sequentially perform a local linear model fitting to the time evolution of neighbouring points on the reconstructed attractor manifold, and the coefficients can approximate the Jacobian elements corresponding to interaction effects. However, despite that the advantages in nonlinear forecasting with noise-contaminated data, these methodologies have a limitation in the Jacobian estimation accuracy owing to non-equidistantly stretched local manifolds in the state space. Herein, we therefore introduced a local manifold distance (LMD) concept, a non-equidistant measure based on the multi-faceted state dependency. By integrating LMD with advanced computation techniques, we presented a robust and efficient analytical method, LMD-based regression (LMDr). To validate its advantages in prediction and Jacobian estimation, we analysed synthetic time series of model ecosystems with different noise levels and applied it to an experimental protozoan predator-prey system with established biological information. The robustness to noise was the highest for LMDr, which also showed a better correspondence to expected predator-prey interactions in the protozoan system. Thus, LMDr can be applied to study complex ecological networks under dynamic conditions.

Joint Projected Fuzzy Neighborhood Preserving C-means Clustering with Local Adaptive Learning

Graph-based dimensionality reduction approaches are extensively applied for preserving the intrinsic structures of high-dimensional data. To capture discriminative information from data clusters that lack a clear manifold structure, previous studies have combined clustering algorithms with local manifold structure preservation. These methods assume that two neighbor data points share the same cluster label. However, the assumption may not hold for Fuzzy C-Means which may assign two nearby data points to different clusters depending on the distance between data points and cluster centroids. This inconsistency degrades the performance in clustering and dimensionality reduction. To tackle this issue, we propose a novel approach for dimensionality reduction, termed joint Projected Fuzzy Neighborhood Preserving C-means Clustering with Local Adaptive Learning (PFNPCM). PFNPCM learns cluster centroids, sparse membership grades, embedded subspaces, and structured graphs simultaneously and extracts both global and local structures of data. Through clustering data that preserves the local neighborhood structure in a low-dimensional space instead of the original one, PFNPCM can effectively capture both global and local information from the inherent manifold structure and cluster structure. The employment of ℓ2,1-norm as the distance metric in the clustering and the preservation of neighborhood structure further ensure the robustness of clustering results against noise or outliers. To achieve the consistency of the captured global and local discriminative information, PFNPCM adopts a novel fuzzy local similarity measure based on the manifold structure, and the manifold assumption thus holds when partitioning unlabeled samples into subgroups. Massive experiments on diverse well-known datasets demonstrate that PFNPCM is superior to some state-of-the-art methods.

Manifold embedding stationary subspace analysis for nonstationary process monitoring with industrial applications

Industrial processes frequently exhibit nonstationary characteristics due to factors like load fluctuations and external interference. Accurate monitoring of nonstationary industrial processes is of vital importance in ensuring production stability and safety. Unfortunately, most existing monitoring methods overlook the manifold structure presented in nonstationary data due to nonstationary features, causing the loss of critical information and poor interpretability. As a consequence, monitoring performance is compromised. To address this issue, this paper proposes a manifold embedding stationary subspace analysis (MESSA) algorithm. By embedding a neighborhood preservation term into the objective function of SSA, MESSA effectively mitigates the impact of nonstationarity on manifold structure. The extracted features incorporate both global stationarity and local manifold characteristics, facilitating a more comprehensive reconstruction of the intricate underlying mechanisms in industrial processes. This contributes to a substantial enhancement in the accuracy and reliability of process monitoring. A set of nonstationary swiss-roll dataset is designed to visually demonstrate the capability of MESSA in extracting manifold structure. Case studies including a numerical case, a continuous stirred tank reactor system and a real industrial roasting process validate the superior monitoring performance of the proposed method.

Dual space-based fuzzy graphs and orthogonal basis clustering for unsupervised feature selection

Unsupervised feature selection (UFS) takes an important position because gaining the class labels is laborious or even impossible. In the domain of UFS, clustering is a major means to exploit label information. The existing methods either could not model a clear clustering structure or could not utilize the local structure of data. Consequently, a new clustering method in combination with graph learning is proposed in this work. Specifically, for clustering, orthogonal basis clustering is introduced, where orthogonal constraints are imposed on the cluster center matrix and the clustering matrix. The clustering indicator matrix is also imposed by a non-negative constraint. A clearer clustering structure and more independent clustering centers are obtained through these constraints. For local preservation, given that traditional graphs for keeping the local manifold are faced with the problem of imbalanced neighbors, the fuzzy graph is introduced to acquire a robust structure, which is applied to both data space and feature space. The topological structure in these spaces can be well maintained. For the choice of salient features, ℓ2,0-norm regularization is imposed on the projection matrix. The object function is solved alternately. Then, a feature selection algorithm is designed. Experiments are designed and performed on nine real-world data sets. The results attest to the effectiveness of the proposed algorithm compared with other relevant algorithms.

Revealing the hidden structure of disordered materials by parameterizing their local structural manifold

Durable interest in developing a framework for the detailed structure of glassy materials has produced numerous structural descriptors that trade off between general applicability and interpretability. However, none approach the combination of simplicity and wide-ranging predictive power of the lattice-grain-defect framework for crystalline materials. Working from the hypothesis that the local atomic environments of a glassy material are constrained by enthalpy minimization to a low-dimensional manifold in atomic coordinate space, we develop a generalized distance function, the Gaussian Integral Inner Product (GIIP) distance, in connection with agglomerative clustering and diffusion maps, to parameterize that manifold. Applying this approach to a two-dimensional model crystal and a three-dimensional binary model metallic glass results in parameters interpretable as coordination number, composition, volumetric strain, and local symmetry. In particular, we show that a more slowly quenched glass has a higher degree of local tetrahedral symmetry at the expense of cyclic symmetry. While these descriptors require post-hoc interpretation, they minimize bias rooted in crystalline materials science and illuminate a range of structural trends that might otherwise be missed.

An upper bound for the box dimension of the hyperbolic dynamics via unstable topological pressure

In this paper, we utilize the sub-additive unstable topological pressure to give an upper bound for the upper box dimension of the C1 hyperbolic set on local unstable manifold. As a by-product, we give a new expression of the topological pressure on symbolic spaces. This work is inspired by Barreira (1996), Feng and Simon (2023) and Zhang et al. (2022).

Class-driven nonnegative matrix factorization with manifold regularization for data clustering

Nonnegative matrix factorization (NMF) is an effective technique to extract the underlying low-dimensional structure of data by utilizing its parts-based representation, which has been widely used in feature extraction and machine learning. However, NMF is an unsupervised learning algorithm without utilizing the discriminative prior information. In this paper, we put forward a new class-driven NMF with manifold regularization (MCDNMF) algorithm, which incorporates both the local manifold regularization and the label information of data into the NMF model. Specifically, MCDNMF not only encodes the local geometrical structure of data space by using the manifold regularization, but also takes the available label information by introducing the class-driven constraint. This class-driven constraint forces the new representations of data points to be more similar within the same class while different between other classes. Therefore, the discriminative abilities of clustering are greatly boosted. Experimental results on several datasets validate the effectiveness of proposed MCDNMF in comparison with the other state-of-the-art methods.

Manifold regularized deep canonical variate analysis with interpretable attribute guidance for three-phase flow process monitoring

Oil–gas–water three-phase flow has multiple flow states, exhibiting dynamic, nonlinear, and instantaneous behaviors. Monitoring and analysis of flow state are crucial for ensuring safe operation of industrial processes. However, the absence of a universal definition for three-phase flow states, owing to the complex flow characteristics and structures, leads to that labeling three-phase flow states by categories is impractical. For comprehensive monitoring and analysis, the flow states should be described from both global (e.g., the dominant phase) and local (e.g., the interphase structure) aspects. Given the aforementioned challenges, the work aims to address the accurate identification of typical three-phase flow states and to meticulously monitor and analyze the continuous evolutionary process of three-phase flow. To achieve it, the manifold regularized deep canonical variate analysis with attribute guidance (Ag-MRDCVA) is proposed, which not only introduces an innovative monitoring paradigm that designs state attributes for process description but also facilitates knowledge transfer from typical flow states to transition states. For enhancing the model’s feature embedding capabilities, an attribute guidance module is introduced to increase supervision information at both class and attribute levels. Then, convolutional neural network (CNN) backbones are developed with a dual objective of global serial correlation and local manifold regularization, which capture spatiotemporal information at both global and local scales, facilitating effective attribute encoding and characterization of flow states. Finally, the attribute evolution heat map and a monitoring metric (MM) collectively offer a compelling and comprehensive analysis of the three-phase flow process. Extensive experiments demonstrate that Ag-MRDCVA surpasses existing methods, showcasing its ability to minutely monitor the process while providing reasonable explanations.

Sparse semi-supervised multi-label feature selection based on latent representation

With the rapid development of the Internet, there are a large number of high-dimensional multi-label data to be processed in real life. To save resources and time, semi-supervised multi-label feature selection, as a dimension reduction method, has been widely used in many machine learning and data mining. In this paper, we design a new semi-supervised multi-label feature selection algorithm. First, we construct an initial similarity matrix with supervised information by considering the similarity between labels, so as to learn a more ideal similarity matrix, which can better guide feature selection. By combining latent representation with semi-supervised information, a more ideal pseudo-label matrix is learned. Second, the local manifold structure of the original data space is preserved by the manifold regularization term based on the graph. Finally, an effective alternating iterative updating algorithm is applied to optimize the proposed model, and the experimental results on several datasets prove the effectiveness of the approach.