Manifold In Space Research Articles

Shifts in structural connectome organization in the limbic and sensory systems of patients with episodic migraine

Migraine is a complex neurological condition characterized by recurrent headaches, which is often accompanied by various neurological symptoms. Magnetic resonance imaging (MRI) is a powerful tool for investigating whole-brain connectivity patterns; however, systematic assessment of structural connectome organization has rarely been performed. In the present study, we aimed to examine the changes in structural connectivity in patients with episodic migraines using diffusion MRI. First, we computed structural connectivity using diffusion MRI tractography, after which we applied dimensionality reduction techniques to the structural connectivity and generated three low-dimensional eigenvectors. We subsequently calculated the manifold eccentricity, defined as the Euclidean distance between each data point and the center of the data in the manifold space. We then compared the manifold eccentricity between patients with migraines and healthy controls, revealing significant between-group differences in the orbitofrontal cortex, temporal pole, and sensory/motor regions. Between-group differences in subcortico-cortical connectivity further revealed significant changes in the amygdala, accumbens, and caudate nuclei. Finally, supervised machine learning effectively classified patients with migraines and healthy controls using cortical and subcortical structural connectivity features, highlighting the importance of the orbitofrontal and sensory cortices, in addition to the caudate, in distinguishing between the groups. Our findings confirmed that episodic migraine is related to the structural connectome changes in the limbic and sensory systems, suggesting its potential utility as a diagnostic marker for migraine.

Stream label distribution learning processing via broad learning system

Label distribution learning (LDL) is designed for label ambiguity problem which widely exists in tasks like image classification and sentiment analysis. Nevertheless, most existing LDL methods adopt a batch-wise processing strategy that is not applicable to stream data issues, like video semantic segmentation and object tracking in auto-driving applications. For these scenarios where dynamic new data is generated continuously, stream LDL processing can be applied to reflect the changes in time and improve the accuracy of online prediction. This paper proposes a novel broad learning system (BLS) based online label distribution learning framework (SLD_BLS). Broad learning system was adopted as the baseline model because its incremental weight updating scheme, excellent efficiency, and outstanding performance can process streaming data in dynamically changing environments. However, there are several challenges to adapt BLS for LDL: 1) BLS's update rule cannot process LDL data; 2) BLS cannot show the mapping relationship between feature space and label space; 3) BLS neglects the inter-label relationships. To tackle these challenges, 1) a new weight update rule for LDL is designed; 2) a manifold regularization is applied to exploit the feature space manifolds; 3) an explicit label collaborating approach is introduced to positively bootstrapping the model training. Extensive experiments on 13 label distribution datasets indicated the performance of SLD_BLS outperforms 9 state-of-the-art LDL models on most datasets with significant reductions in training time across 6 metrics.

Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems

While many phenomena in physics and engineering are formally high-dimensional, their long-time dynamics often live on a lower-dimensional manifold. The present work introduces an autoencoder framework that combines implicit regularization with internal linear layers and L 2 regularization (weight decay) to automatically estimate the underlying dimensionality of a data set, produce an orthogonal manifold coordinate system, and provide the mapping functions between the ambient space and manifold space, allowing for out-of-sample projections. We validate our framework’s ability to estimate the manifold dimension for a series of datasets from dynamical systems of varying complexities and compare to other state-of-the-art estimators. We analyze the training dynamics of the network to glean insight into the mechanism of low-rank learning and find that collectively each of the implicit regularizing layers compound the low-rank representation and even self-correct during training. Analysis of gradient descent dynamics for this architecture in the linear case reveals the role of the internal linear layers in leading to faster decay of a ‘collective weight variable’ incorporating all layers, and the role of weight decay in breaking degeneracies and thus driving convergence along directions in which no decay would occur in its absence. We show that this framework can be naturally extended for applications of state-space modeling and forecasting by generating a data-driven dynamic model of a spatiotemporally chaotic partial differential equation using only the manifold coordinates. Finally, we demonstrate that our framework is robust to hyperparameter choices.

A class of closed manifolds in nonseparable Hilbert spaces

A class of closed manifolds in nonseparable Hilbert spaces

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.

EEG functional connectivity as a Riemannian mediator: An application to malnutrition and cognition.

Mediation analysis assesses whether an exposure directly produces changes in cognitive behavior or is influenced by intermediate "mediators". Electroencephalographic (EEG) spectral measurements have been previously used as effective mediators representing diverse aspects of brain function. However, it has been necessary to collapse EEG measures onto a single scalar using standard mediation methods. In this article, we overcome this limitation and examine EEG frequency-resolved functional connectivity measures as a mediator using the full EEG cross-spectral tensor (CST). Since CST samples do not exist in Euclidean space but in the Riemannian manifold of positive-definite tensors, we transform the problem, allowing for the use of classic multivariate statistics. Toward this end, we map the data from the original manifold space to the Euclidean tangent space, eliminating redundant information to conform to a "compressed CST." The resulting object is a matrix with rows corresponding to frequencies and columns to cross spectra between channels. We have developed a novel matrix mediation approach that leverages a nuclear norm regularization to determine the matrix-valued regression parameters. Furthermore, we introduced a global test for the overall CST mediation and a test to determine specific channels and frequencies driving the mediation. We validated the method through simulations and applied it to our well-studied 50+-year Barbados Nutrition Study dataset by comparing EEGs collected in school-age children (5-11 years) who were malnourished in the first year of life with those of healthy classmate controls. We hypothesized that the CST mediates the effect of malnutrition on cognitive performance. We can now explicitly pinpoint the frequencies (delta, theta, alpha, and beta bands) and regions (frontal, central, and occipital) in which functional connectivity was altered in previously malnourished children, an improvement to prior studies. Understanding the specific networks impacted by a history of postnatal malnutrition could pave the way for developing more targeted and personalized therapeutic interventions. Our methods offer a versatile framework applicable to mediation studies encompassing matrix and Hermitian 3D tensor mediators alongside scalar exposures and outcomes, facilitating comprehensive analyses across diverse research domains.

C2FMI: Corse-to-Fine Black-Box Model Inversion Attack

Privacy-preserving machine learning requires that models do not reveal any private information about their training data. However, model inversion attacks (MIAs), which aim to recover the features of training data, pose a huge threat to the security of AI models. Most existing MIAs assume that the target model is white-box, but most models deployed in reality are black-box, and these models can only be accessed like an oracle. There are a few studies for black-box scenarios, but their performance is limited. In this paper, we firstly formulate the MIA problem completely in Bayesian perspective. Second, we propose a novel two-stage MIA approach, the Coarse-to-Fine Model Inversion Attack (C2FMI), which efficiently addresses the MIA problem in the black-box scenario. In stage I of C2FMI, we design a reverse network that constrains the recovered images (also named attacked images) to fall near the manifold space of the training data. In stage II, we design a black-box oriented strategy which further facilitates the attacked images to approach the training data. Empirically, C2FMI achieves a performance that even surpasses existing white-box attack methods. Furthermore, we design the stability analysis method for analyzing the stability of C2FMI along with existing MIAs. Finally, we explore the potential countermeasures which could defend against our attacks.

Redundant Non-Serial Implicit Manipulator Kinematics and Dynamics

Abstract Redundant non-serial manipulators that include a spectrum of parallel and non-parallel heavy load bearing construction and material handling equipment are treated, using foundations of differential geometry. Kinematics of this category of manipulator are defined in manipulator configuration space by algebraic equations in input and output coordinates that cannot be explicitly solved for either as a function of the other. New sets called assembly components of manipulator configuration space are defined that partition the space into maximal, path-connected, disjoint, topological components. All configurations within an assembly component can be connected by one or more continuous paths within that component, but configurations in different assembly components cannot be connected by continuous paths. Forward and inverse kinematically singular configurations are characterized by criteria that partition each assembly component into path-connected, singularity-free assembly components in which equations of kinematics and dynamics are well behaved. It is shown that a generalized inverse velocity-based kinematic formulation that is problematic for serial manipulators is likewise plagued with problems for non-serial implicit manipulators that can be avoided using the methods presented. Singularity-free differentiable manipulator configuration space components are defined and parameterized by both input and operational coordinates, leading to well-posed ordinary differential equations of manipulator dynamics in both input and operational coordinates. Three typical applications and associated model problems are studied throughout the paper to illustrate the methods and results presented.

Manifold-based multi-graph embedding for semi-supervised classification

Graph semi-supervised learning (GSSL) plays an important role in semi-supervised classification by leveraging the similarity of graph topology and convex optimization with Laplacian-based regularization. Current approaches mainly focus on data representation in the Euclidean space and solely rely on a single graph structure, ignoring the underlined manifold-valued structure of the data. To this end, we propose a novel manifold-based multi-graph embedding (M2GE), which explicitly learns representations on both Riemannian manifold and Euclidean space in the embedding learning process. Incorporating multi-graph structure, the rich and varied features are used by M2GE to more fully capture categorical information, where the heterogeneous spatial representations from manifold are complementary. Extensive experimental results on three popular benchmarks verify demonstrate the superiority of our method over state-of-the art competitors.

Integrability of rank-two web models

We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [1,2]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [3]. These models are based on the A2, G2 and B2 Lie algebras, and their local vertex configurations are intertwiners of the corresponding q-deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of B2 also on the square lattice. For specific root-of-unity choices of q, we show the equivalence to a number of three- and four-state spin models on the dual lattice.The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For q on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix.In the A2 case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics. Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the G2 case we identify four regimes numerically. The B2 case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra.

Combinatorial CNN-Transformer Learning with Manifold Constraints for Semi-supervised Medical Image Segmentation

Semi-supervised learning (SSL), as one of the dominant methods, aims at leveraging the unlabeled data to deal with the annotation dilemma of supervised learning, which has attracted much attentions in the medical image segmentation. Most of the existing approaches leverage a unitary network by convolutional neural networks (CNNs) with compulsory consistency of the predictions through small perturbations applied to inputs or models. The penalties of such a learning paradigm are that (1) CNN-based models place severe limitations on global learning; (2) rich and diverse class-level distributions are inhibited. In this paper, we present a novel CNN-Transformer learning framework in the manifold space for semi-supervised medical image segmentation. First, at intra-student level, we propose a novel class-wise consistency loss to facilitate the learning of both discriminative and compact target feature representations. Then, at inter-student level, we align the CNN and Transformer features using a prototype-based optimal transport method. Extensive experiments show that our method outperforms previous state-of-the-art methods on three public medical image segmentation benchmarks.

An adaptive fault detection model based on variational auto-encoders and unsupervised transfer learning

Aiming at the problem of insufficient generalization of fault detection in traditional machine learning, an SDN controller fault detection method based on unsupervised transfer learning is proposed. The method mainly includes two parts. (1) A Gaussian mixture variational autoencoder based on the autoregressive flow is proposed. First, the encoder and decoder of variational autocoding are improved with gated recurrent units, and the improved variational autocoding can process time series data. Secondly, the gated recurrent unit is improved by using the gravitational search algorithm, which speeds up the search of the weight of the gated recurrent unit. Further, considering that the latent space of the variational autoencoder is a single Gaussian distribution, and the complex data in reality is often too simple to be represented by a single Gaussian distribution. (2) Aiming at the problem of poor generalization of fault detection models in practical scenarios, a domain adaptive fault detection algorithm based on multi-kernel maximum mean difference and intra-class distance constraints is proposed. Map the features into the manifold space to eliminate the distortion of the features in the original space. After mapping, the distance between fields needs to be measured, and the maximum mean difference of a single kernel cannot determine which kernel function is more suitable for the current task in practical applications. Therefore, the maximum mean difference based on multi-core is introduced to measure between the two fields. The experimental results show that the algorithm proposed improves the accuracy about 5% compared with the previous algorithm.

The training process of many deep networks explores the same low-dimensional manifold

We develop information-geometric techniques to analyze the trajectories of the predictions of deep networks during training. By examining the underlying high-dimensional probabilistic models, we reveal that the training process explores an effectively low-dimensional manifold. Networks with a wide range of architectures, sizes, trained using different optimization methods, regularization techniques, data augmentation techniques, and weight initializations lie on the same manifold in the prediction space. We study the details of this manifold to find that networks with different architectures follow distinguishable trajectories, but other factors have a minimal influence; larger networks train along a similar manifold as that of smaller networks, just faster; and networks initialized at very different parts of the prediction space converge to the solution along a similar manifold.

Life on the edge: subcritical pipe flow transition as a spectral submanifold

Subcritical pipe flow transition has received a great deal of attention over the past decades, as it constitutes a quintessential bifurcation process between two metastable fluid states: the laminar and turbulent solutions. Coherent lower-branch structures, forming flow states that facilitate between these two attracting equilibria, have been proposed that together form an edge manifold in phase space separating relaminarizing from transitioning perturbations. Typically, direct numerical simulations or low-dimensional model equations have been used to study this edge manifold with bisection methods. In the article by Kaszás & Haller (J. Fluid Mech., vol. 979, 2024, A48), an effective nonlinear invariant-manifold technique has been applied to extract a low-dimensional, global representation of the phase-space dynamics directly from simulation data. It allows the computation of the intersection of the edge manifold with a low-dimensional surface that is strikingly accurate in predicting the long-term dynamics of perturbations about the lower-branch solution and thus provides an accessible parameterization of the edge manifold for subcritical pipe flow transition.

A data-driven implicit deep adaptive neuro-fuzzy inference system capable of manifold learning for function approximation

Fuzzy Neural Networks (FNN) have the ability of decision-making based on constructing semi-ellipsoidal clusters in the input space as the antecedent parts of their fuzzy rules. To determine the output value for each input instance, FNNs consider its membership degree to different sub-regions of the input space. However, forming such meaningful sub-regions is not possible in all applications due to the nonlinear interactions among input variables and their low information gain. Indeed, the samples could be distributed on a manifold in the input space. Therefore, to cover the input space, we need lots of rules, each representing a small region of input space. This issue decreases the generalization ability of the model along with its explainability. Consequently, to efficiently form fuzzy rules, first, it is necessary to unfold the manifold by mapping the samples to an appropriate embedding space. Next, the fuzzy rules in the form of semi-ellipsoidal regions should be constructed in this extracted feature space. Deep Fuzzy Neural Networks address this problem by representation learning through stacking multiple cascade mapping layers. In this paper, we propose a novel approach for nonlinear function approximation and time-series prediction problems, based on using the kernel trick to implicitly learn the mapping function to the new feature space. Moreover, to initialize the fuzzy rules, a KNN-based method using the kernel trick is proposed. A hierarchical Levenberg–Marquardt approach is applied to learn the model’s parameters. The performance and structure of the proposed method are studied and compared with some other relevant methods in synthetic and real-world benchmarks. Based on these experiments, the proposed method has the best performance with the most parsimonious architecture. According to these experiments, the test RMSE of the proposed method is 0.002 for Mc-Glass chaotic time-series prediction, 0.015 for a Nonlinear dynamic system identification, 0.0345 for Box–Jenkins nonlinear system identification, 0.0609 for Fuel consumption prediction of automobiles, 10.24 for Sydney stock price tracking, and 0.595 for California housing.

Upper bounds for the critical values of homology classes of loops

In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of compact manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a compact and simply-connected n-dimensional manifold of positive Ricci curvature Ric≥n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Ric}\\ge n-1$$\\end{document} has length ≤nπ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le n \\pi .$$\\end{document} This improves the bound 8π(n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi (n-1)$$\\end{document} given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022).

A Stabilisation System Synthesis for Motion along a Preset Trajectory and Its Solution by Symbolic Regression

The problem of a stabilisation system synthesis for the motion of a control object along a given spatial trajectory is considered. The complexity of the problem is that the preset trajectory is defined in the state subspace and not in time. This paper describes a stabilisation system synthesis for motion along a trajectory specified in time and along a trajectory specified in the form of a manifold in a state space. In order to construct a stabilisation system, it is necessary to determine a distance between an object and the given trajectory at each moment in time. For trajectories that are not given in time, the determination of this distance can be ambiguous. An object may be exactly on a trajectory but at a different time. This paper proposes some approaches to solve the problem. One of the approaches is to transform a given trajectory in a state subspace into a trajectory given in time. A description of a universal method to perform this transformation is presented. In order to solve the synthesis problem automatically, without having to analyse the mathematical model of the control object, it is suggested that machine learning control by symbolic regression is used. In computational experiments, examples of stabilisation system syntheses for quadcopter motion along a given spatial trajectory are presented.

A fault diagnosis method based on label-wise density-domain space learning

Abstract Nonlinear space learning of fault samples is a category of common fault diagnosis methods, which usually use Euclidean distances to describe manifold structures among fault samples. However, in nonlinear space, Euclidean distances lead to a potential manifold loss problem. Aiming these issues, we propose a novel fault diagnosis method based on label-wise density-domain space learning. The label-wise density-domain space learns more intrinsic manifold structures from four density-constrained order graphs. Density-constrained order graphs constructed by our method integrate different discriminative relationships from original fault samples with the help of density-domain information, and the density-domain information can effectively capture potential density information and global structure between fault samples. By density Laplacian of the graphs, we further construct a label-wise density-domain manifold space learning model, and the analytical solutions of space projections can be obtained by solving the model. Fault features directly obtained by the space projections possess good class separability. Extensive experiments on the Case Western Reserve University fault dataset and a roll-bearing fault dataset from our roll-bearing test platform show the effectiveness and robustness of our method.

Pointwise density estimation on metric spaces and applications in seismology

AbstractWe are studying the problem of estimating density in a wide range of metric spaces, including the Euclidean space, the sphere, the ball, and various Riemannian manifolds. Our framework involves a metric space with a doubling measure and a self-adjoint operator, whose heat kernel exhibits Gaussian behaviour. We begin by reviewing the construction of kernel density estimators and the related background information. As a novel result, we present a pointwise kernel density estimation for probability density functions that belong to general Hölder spaces. The study is accompanied by an application in Seismology. Precisely, we analyze a globally-indexed dataset of earthquake occurrence and compare the out-of-sample performance of several approximated kernel density estimators indexed on the sphere.

MGFKD: A semi-supervised multi-source domain adaptation algorithm for cross-subject EEG emotion recognition

Currently, most models rarely consider the negative transfer problem in the research field of cross-subject EEG emotion recognition. To solve this problem, this paper proposes a semi-supervised domain adaptive algorithm based on few labeled samples of target subject, which called multi-domain geodesic flow kernel dynamic distribution alignment (MGFKD). It consists of three modules: 1) GFK common feature extractor: projects the feature distribution of source and target subjects to the Grassmann manifold space, and obtains the latent common features of the two feature distributions through GFK method. 2) Source domain selector: obtains pseudo-labels of the target subject through weak classifier, finds "golden source subjects" by using few known labels of target subjects. 3) Label corrector: uses a dynamic distribution balance strategy to correct the pseudo-labels of the target subject. We conducted comparison experiments on the SEED and SEED-IV datasets, and the results show that MGFKD outperforms unsupervised and semi-supervised domain adaptation algorithms, achieving an average accuracy of 87.51±7.68% and 68.79±8.25% on the SEED and SEED-IV datasets with only one labeled sample per video for target subject. Especially when the number of source domains is set as 6 and the number of known labels is set as 5, the accuracy increase to 90.20±7.57% and 69.99±7.38%, respectively. The above results prove that our proposed algorithm can efficiently and quickly improve the cross-subject EEG emotion classification performance. Since it only need a small number of labeled samples of new subjects, making it has strong application value in future EEG-based emotion recognition applications.