Graph Manifold Research Articles

Sparse-Laplace hybrid graph manifold method for fluorescence molecular tomography

Objective.Fluorescence molecular tomography (FMT) holds promise for early tumor detection by mapping fluorescent agents in three dimensions non-invasively with low cost. However, since ill-posedness and ill-condition due to strong scattering effects in biotissues and limited measurable data, current FMT reconstruction is still up against unsatisfactory accuracy, including location prediction and morphological preservation.Approach.To strike the above challenges, we propose a novel Sparse-Laplace hybrid graph manifold (SLHGM) model. This model integrates a hybrid Laplace norm-based graph manifold learning term, facilitating a trade-off between sparsity and preservation of morphological features. To address the non-convexity of the hybrid objective function, a fixed-point equation is designed, which employs two successive resolvent operators and a forward operator to find a converged solution.Main results.Through numerical simulations andin vivoexperiments, we demonstrate that the SLHGM model achieves an improved performance in providing accurate spatial localization while preserving morphological details.Significance.Our findings suggest that the SLHGM model has the potential to advance the application of FMT in biological research, not only in simulation but also inin vivostudies.

Embedding-Alignment Fusion-Based Graph Convolution Network With Mixed Learning Strategy for 4D Medical Image Reconstruction.

In recent years, 4D medical image involving structural and motion information of tissue has attracted increasing attention. The key to the 4D image reconstruction is to stack the 2D slices based on matching the aligned motion states. In this study, the distribution of the 2D slices with the different motion states is modeled as a manifold graph, and the reconstruction is turned to be the graph alignment. An embedding-alignment fusion-based graph convolution network (GCN) with a mixed-learning strategy is proposed to align the graphs. Herein, the embedding and alignment processes of graphs interact with each other to realize a precise alignment with retaining the manifold distribution. The mixed strategy of self- and semi-supervised learning makes the alignment sparse to avoid the mismatching caused by outliers in the graph. In the experiment, the proposed 4D reconstruction approach is validated on the different modalities including Computed Tomography (CT), Magnetic Resonance Imaging (MRI), and Ultrasound (US). We evaluate the reconstruction accuracy and compare it with those of state-of-the-art methods. The experiment results demonstrate that our approach can reconstruct a more accurate 4D image.

Volume of Seifert representations for graph manifolds and their finite covers

AbstractFor any closed orientable 3‐manifold, there is a volume function defined on the space of all Seifert representations of the fundamental group. The maximum absolute value of this function agrees with the Seifert volume of the manifold due to Brooks and Goldman. For any Seifert representation of a graph manifold, the authors establish an effective formula for computing its volume, and obtain restrictions to the representation as analogous to the Milnor–Wood inequality (about transversely projective foliations on Seifert fiber spaces). It is shown that the Seifert volume of any graph manifold is a rational multiple of . Among all finite covers of a given nongeometric graph manifold, the supremum ratio of the Seifert volume over the covering degree can be a positive number, and can be infinite. Examples of both possibilities are discovered, and confirmed with the explicit values determined for the finite ones.

Graph manifold learning with non-gradient decision layer

Generally, Graph convolution network (GCN) utilizes the graph convolution operators and the softmax to extract the deep representation and make the prediction, respectively. Although GCN successfully represents the connectivity relationship among the nodes by aggregating the information on the graph, the softmax-based decision layer may result in suboptimal performance in semi-supervised learning with less label support due to ignoring the inner distribution of the graph nodes. Besides, the gradient descent will take thousands of interaction for optimization. To address the referred issues, we propose a novel graph deep model with a non-gradient decision layer for graph mining. Firstly, manifold learning is unified with label local-structure preservation to capture the topological information and make accurate predictions with limited label support. Moreover, it is theoretically proven to have analytical solutions and acts as a non-gradient decision layer in graph convolution networks. Particularly, a joint optimization method is designed for this graph model, which extremely accelerates the convergence of the model. Finally, extensive experiments show that the proposed model has achieved excellent performance compared to the current models.

A stability result for translating spacelike graphs in Lorentz manifolds

A stability result for translating spacelike graphs in Lorentz manifolds

A novel adaptive safe semi-supervised learning framework for pattern extraction and classification

<p>Manifold regularization semi-supervised learning is a powerful graph-based semi-supervised learning method. However, the performance of semi-supervised learning methods based on manifold regularization depends to some extent on the quality of the manifold graph and unlabeled samples. Intuitively speaking, the quality of the graph directly affects the final classification performance of the model. In response to the above problems, this paper first proposed an adaptive safety semi-supervised learning framework. The framework implements the weight assignment of the self-similarity graph during the model learning process. In order to adapt to the learning needs, accelerate the learning speed, and avoid the impact of the curse of dimensionality, the framework also optimizes the features of each sample point through an automatic weighting mechanism to extract effective features and eliminate redundant information in the learning task. In addition, the framework defines an adaptive risk measurement mechanism for the uncertainty and potential risks of unlabeled samples to determine the degree of risk of unlabeled samples. Finally, a new adaptive safe semi-supervised extreme learning machine was proposed. Comprehensive experimental results across various class imbalance scenarios demonstrated that our proposed method outperforms other methods in terms of classification accuracy, and other critical performance metrics.</p>

Regularized reconstruction based on joint smoothly clipped absolute deviation regularization and graph manifold learning for fluorescence molecular tomography

Objective. Fluorescence molecular tomography (FMT) is an optical imaging modality that provides high sensitivity and low cost, which can offer the three-dimensional distribution of biomarkers by detecting the fluorescently labeled probe noninvasively. In the field of preclinical cancer diagnosis and treatment, FMT has gained significant traction. Nonetheless, the current FMT reconstruction results suffer from unsatisfactory morphology and location accuracy of the fluorescence distribution, primarily due to the light scattering effect and the ill-posed nature of the inverse problem. Approach. To address these challenges, a regularized reconstruction method based on joint smoothly clipped absolute deviation regularization and graph manifold learning (SCAD-GML) for FMT is presented in this paper. The SCAD-GML approach combines the sparsity of the fluorescent sources with the latent manifold structure of fluorescent source distribution to achieve more accurate and sparse reconstruction results. To obtain the reconstruction results efficiently, the non-convex gradient descent iterative method is employed to solve the established objective function. To assess the performance of the proposed SCAD-GML method, a comprehensive evaluation is conducted through numerical simulation experiments as well as in vivo experiments. Main results. The results demonstrate that the SCAD-GML method outperforms other methods in terms of both location and shape recovery of fluorescence biomarkers distribution. Siginificance. These findings indicate that the SCAD-GML method has the potential to advance the application of FMT in in vivo biological research.

Ollivier Curvature of Random Geometric Graphs Converges to Ricci Curvature of Their Riemannian Manifolds

Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier–Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. Here the scaling of the average degree, as a function of the graph size, can range from nearly logarithmic to nearly linear.

Structure Evolution on Manifold for Graph Learning.

Graph has been widely used in various applications, while how to optimize the graph is still an open question. In this paper, we propose a framework to optimize the graph structure via structure evolution on graph manifold. We first define the graph manifold and search the best graph structure on this manifold. Concretely, associated with the data features and the prediction results of a given task, we define a graph energy to measure how the graph fits the graph manifold from an initial graph structure. The graph structure then evolves by minimizing the graph energy. In this process, the graph structure can be evolved on the graph manifold corresponding to the update of the prediction results. Alternatively iterating these two processes, both the graph structure and the prediction results can be updated until converge. It achieves the suitable structure for graph learning without searching all hyperparameters. To evaluate the performance of the proposed method, we have conducted experiments on eight datasets and compared with the recent state-of-the-art methods. Experiment results demonstrate that our method outperforms the state-of-the-art methods in both transductive and inductive settings.

Row-Column Overcomplete Structured Dictionary Learning for Enhanced Fault Detection and Isolation

To improve the monitoring performance of dictionary learning based methods, this article proposes a row-column overcomplete structured dictionary learning (RCOSDL) method for fault detection and isolation of industrial processes. Unlike conventional dictionary learning approaches which are overcomplete column-wise, the proposed method involves a dictionary that is overcomplete both row- and column-wise. The introduction of row-column overcomplete dictionary results in monitoring statistics that are more sensitive to incipient faults. In order to incorporate structured information, two graph Laplacian regularization terms, namely, manifold graph Laplacian and full graph Laplacian are considered. Whilst the inherent local geometric structure in the data is preserved in the sparse representation by the manifold graph Laplacian term, the correlation structure between process variables is preserved by using the full graph Laplacian term. Hence violation in the geometric or correlation structure will be promptly detected. To pinpoint faulty variables, a fault isolation method is developed by imposing the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{1}$</tex-math></inline-formula> / <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{2,1}$</tex-math></inline-formula> -norm constraint on the sparse coefficients. In addition, theoretical property involving the condition for guaranteed fault isolation is presented in Theorem 1. The contributions of this article include the introduction of monitoring statistics based on RCOSDL that are suitable for incipient faults, a new fault isolation scheme as well as theoretical analysis on the condition of guaranteed fault isolation. The better performance of the proposed method is illustrated by applications to numerical studies and practical industrial cases.

Adaptive Manifold Graph representation for Two-Dimensional Discriminant Projection

Two-dimensional (2D) local discriminant analysis is one of the popular techniques for image representation and recognition. Conventional 2D methods extract features of images relying on the affinity relationships among samples. However, affinity graph learning and dimensionality reduction are two separate processes, and the predefined affinity graph is sensitive to noisy and irrelevant features. To this end, we propose a graph embedding model, named as Two-Dimensional Discriminant Projection with Adaptive Manifold Graph (2DDP-AMG). In our model, the local manifold structure of data is characterized by the k-nearest-neighbor (kNN) graph within each class. More importantly, the above graph is updated adaptively in the process of dimensionality reduction to eliminate the interference of irrelevant features and noise. Furthermore, F-norm is introduced to measure the distance between homogeneous samples and improve the robustness of the model to outliers. To solve this model, an efficient algorithm is developed to update each variable iteratively. Extensive experiments conducted on benchmark face image data sets show the superiority and robustness of our model compared to state-of-the-art alternatives.

Cosmological BF Theory on Topological Graph Manifold with Seifert Fibered Homology Spheres

Cosmological BF Theory on Topological Graph Manifold with Seifert Fibered Homology Spheres

TMIC-58. PATTERNS OF CELLULAR SUBPOPULATION COHABITATION DEFINE GLIOBLASTOMA STATES

Abstract Characterizing intra- and inter-tumoral heterogeneity of glioblastoma has historically relied on discrete classifications of malignant cell populations leaving immune and other cell populations, known to exist admixed with the malignant tumor cells, relatively neglected. Manifold learning algorithms can manage deconvolving multiple cell populations and are often used to track cell state transitions in single cell transcriptomics. We applied a manifold learning approach to TCGA microarray data (Nf525) and bulk transcriptomics of 134 image localized biopsies across 30 patients with primary and 9 with recurrent glioblastoma to further elucidate how to organize biopsies across a spectrum of possible tissue states. The algorithm revealed a low-dimensional manifold graph for which each biopsy lives across 3 polarizing tissue states - one that is associated with diffusely invaded brain, one that is enriched in mesenchymal genes, and one that is enriched in classical proliferative tumor signatures. We deconvolved the bulk transcriptomics of the image-localized biopsies to reveal the relative abundance of 18 malignant, immune, and other cell subpopulations in each biopsy. Overlaying the cellular decomposition onto the manifold graph visualizing the tissue state distributions revealed that transitions between states correlate with changes in cellular cohabitation composition. The tumor cellular cohabitation ecologies have the lowest diversity, as inferred by ecological measures such as Shannon entropy and evenness, at the distal poles of the graph when compared to the transitional arms. Further, we found that the relationship between imaging appearance of contrast enhancement on T1-weighted MRI and the biopsy cellular composition varies with sex and primary vs recurrent biopsy status. The limited spectrum of possible tissue states revealed by the manifold learning is suggestive of a limited continuum along which tumor and non-tumoral cell populations can cohabitate. Such a limited low-dimensional biological space may constrain the dynamics of tumor biology in a predictable manner.

Sparse multi-label feature selection via dynamic graph manifold regularization

Sparse multi-label feature selection via dynamic graph manifold regularization

On the existence of supporting broken book decompositions for contact forms in dimension 3

We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. As an application we obtain that on a closed 3-manifold, every nondegenerate Reeb vector field has either two or infinitely many periodic orbits, and two periodic orbits are possible only on the tight sphere or on a tight lens space. Moreover we get that if M is a closed oriented 3-manifold that is not a graph manifold, for example a hyperbolic manifold, then every nondegenerate Reeb vector field on M has positive topological entropy.

Multi-task manifold learning for partial label learning

In partial label learning (PLL), each instance is associated with a candidate label set, and only one label is ground-truth. PLL aims to identify the ground-truth label out of these candidate labels. Most of the existing PLL approaches focus on single-task PLL, and ignore the auxiliary information of the related tasks. This paper puts forward a novel multi-task manifold learning method for partial label learning (MT-PLL), which learns multiple PLL tasks jointly, and incorporates the auxiliary information of the related tasks to improve the performance of PLL classifiers. MT-PLL assumes that the graph manifold structure guides the generation of labeling confidence for instances in each task. In addition, the information of related tasks can be used to boost the performance of the overall classification model. Then, a heuristic framework is used to optimize the objective function. Numerical experiments have demonstrated that MT-PLL can deliver better performance than state-of-the-art single-task PLL methods.

Total Variation Constrained Graph Manifold Learning Strategy for Cerenkov Luminescence Tomography.

Harnessing the power and flexibility of radiolabeled molecules, Cerenkov luminescence tomography (CLT) provides a novel technique for non-invasive visualisation and quantification of viable tumour cells in a living organism. However, owing to the photon scattering effect and the ill-posed inverse problem, CLT still suffers from insufficient spatial resolution and shape recovery in various preclinical applications. In this study, we proposed a total variation constrained graph manifold learning (TV-GML) strategy for achieving accurate spatial location, dual-source resolution, and tumour morphology. TV-GML integrates the isotropic total variation term and dynamic graph Laplacian constraint to make a trade-off between edge preservation and piecewise smooth region reconstruction. Meanwhile, the tetrahedral mesh-Cartesian grid pair method based on the k-nearest neighbour, and the adaptive and composite Barzilai-Borwein method, were proposed to ensure global super linear convergence of the solution of TV-GML. The comparison results of both simulation experiments and in vivo experiments further indicated that TV-GML achieved superior reconstruction performance in terms of location accuracy, dual-source resolution, shape recovery capability, robustness, and in vivo practicability. Significance: We believe that this novel method will be beneficial to the application of CLT for quantitative analysis and morphological observation of various preclinical applications and facilitate the development of the theory of solving inverse problem.

Iterative joint classifier and domain adaptation for visual transfer learning.

Current available supervised classifiers cannot generalize across various domains due to distribution mismatch among them. Domain adaptation and transfer learning algorithms are proposed to tackle domain shift problem that originates from different data collection conditions. In this paper, we propose a transfer learning framework called iterative joint classifier and domain adaptation for visual transfer learning (ICDAV), which utilizes the balanced maximum mean discrepancy to better transfer knowledge across domains. Also, for learning a robust classifier against domain shift, a set of graph manifold regularizer and modified joint probability maximum mean discrepancy are simultaneously exploited to capture the domain structures and adapt the distribution of projected samples during the model learning process. Variety of experiments on several public datasets indicates that our approach achieves remarkable performance on visual domain adaptation and transfer learning tasks.

Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

Time-Dependent Lagrangian Energy Systems on Supermanifolds with Graph Bundles

The aim of this article is firstly to improve time-dependent Lagrangian energy equations using the super jet bundles on supermanifolds. Later, we adapted this study to the graph bundle. Thus, we created a graph bundle by examining the graph manifold structure in superspace. The geometric structures obtained for the mechanical energy system with superbundle coordinates were reexamined with the graph bundle coordinates. Thus, we were able to calculate the energy that occurs during the motion of a particle when we examine this motion with graph points. The supercoordinates on the superbundle structure of supermanifolds have been given for body and soul and also even and odd dimensions. We have given the geometric interpretation of this property in coordinates for the movement on graph points. Lagrangian energy equations have been applied to the presented example, and the advantage of examining the movement with graph points was presented. In this article, we will use the graph theory to determine the optimal motion, velocity, and energy of the particle, due to graph points. This study showed a physical application and interpretation of supervelocity and supertime dimensions in super-Lagrangian energy equations utilizing graph theory.