Embedding Research Articles

Topological embedding and directional feature importance in ensemble classifiers for multi-class classification

Cancer is the second leading cause of disease-related death worldwide, and machine learning-based identification of novel biomarkers is crucial for improving early detection and treatment of various cancers. A key challenge in applying machine learning to high-dimensional data is deriving important features in an interpretable manner to provide meaningful insights into the underlying biological mechanismsWe developed a class-based directional feature importance (CLIFI) metric for decision tree methods and demonstrated its use for the The Cancer Genome Atlas proteomics data. The CLIFI metric was incorporated into four algorithms, Random Forest (RF), LAtent VAriable Stochastic Ensemble of Trees (LAVASET), and Gradient Boosted Decision Trees (GBDTs), and a new extension incorporating the LAVA step into GBDTs (LAVABOOST). Both LAVA methods incorporate topological information from protein interactions into the decision function.The different models' performance in classifying 28 cancers resulted in F1-scores of 92.6% (RF), 92.0% (LAVASET), 89.3% (LAVABOOST) and 85.7% (GBDT), with no method outperforming all others for individual cancer type prediction. The CLIFI metric enables visualisation of the model's decision-making functions. The resulting CLIFI value distributions indicated heterogeneity in the expression of several proteins (MYH11, ERα, BCL2) across different cancer types (including brain glioma, breast, kidney, thyroid and prostate cancer) aligning with the original raw expression data.In conclusion, we have developed an integrated, directional feature importance metric for multi-class decision tree-based classification models that facilitates interpretable feature importance assessment. The CLIFI metric can be combined with incorporating topological information into the decision functions of models to introduce inductive bias, enhancing interpretability.

Local Isometric Immersions and Breakdown of Manifolds Determined by Cauchy Problems of the Degasperis–Procesi Equation

Local Isometric Immersions and Breakdown of Manifolds Determined by Cauchy Problems of the Degasperis–Procesi Equation

Gating-Enhanced Hierarchical Structure Learning in Hyperbolic Space and Multi-scale Neighbor Topology Learning in Euclidean Space for Prediction of Microbe-Drug Associations.

Identifying drug-related microbes may help us explore how the microbes affect the functions of drugs by promoting or inhibiting their effects. Most previous methods for the prediction of microbe-drug associations focused on integrating the attributes and topologies of microbe and drug nodes in Euclidean space. The heterogeneous network composed of microbes and drugs has a hierarchical structure, and the hyperbolic space is helpful for reflecting the structure. However, the previous methods did not fully exploit the structure. We propose a multi-space feature learning enhanced microbe-drug association prediction method, MFLP, to fuse the hierarchical structure of microbe and drug nodes in hyperbolic space and the multiscale neighbor topologies in Euclidean space. First, we project the nodes of the microbe-drug heterogeneous network on the sphere in hyperbolic space and then construct a topology which implies hierarchical structure and forms a hierarchical attribute embedding. The node information from multiple types of neighbor nodes with the new topological structure in the tangent plane space of a sphere is aggregated by the designed gating-enhanced hyperbolic graph neural network. Second, the gate at the node feature level is constructed to adaptively fuse the hierarchical features of microbe and drug nodes from two adjacent graph neural encoding layers. Third, multiple neighbor topological embeddings for each microbe and drug node are formed by neighborhood random walks on the microbe-drug heterogeneous network, and they cover neighborhood topologies with multiple scales, respectively. Finally, as each scale of topological embedding contains its specific neighborhood topology, we establish an independent graph convolutional neural network for the topology and form the topological representations of microbe and drug nodes in Euclidean space. The comparison experiments based on cross validation showed that MFLP outperformed several advanced prediction methods, and the ablation experiments verified the effectiveness of MFLP's major innovations. The case studies on three drugs further demonstrated MFLP's ability in being applied to discover potential candidate microbes for the given drugs.

A multi-task prediction method based on neighborhood structure embedding and signed graph representation learning to infer the relationship between circRNA, miRNA, and cancer.

Research shows that competing endogenous RNA is widely involved in gene regulation in cells, and identifying the association between circular RNA (circRNA), microRNA (miRNA), and cancer can provide new hope for disease diagnosis, treatment, and prognosis. However, affected by reductionism, previous studies regarded the prediction of circRNA-miRNA interaction, circRNA-cancer association, and miRNA-cancer association as separate studies. Currently, few models are capable of simultaneously predicting these three associations. Inspired by holism, we propose a multi-task prediction method based on neighborhood structure embedding and signed graph representation learning, CMCSG, to infer the relationship between circRNA, miRNA, and cancer. Our method aims to extract feature descriptors of all molecules from the circRNA-miRNA-cancer regulatory network using known types of association information to predict unknown types of molecular associations. Specifically, we first constructed the circRNA-miRNA-cancer association network (CMCN), which is constructed based on the experimentally verified biomedical entity regulatory network; next, we combine topological structure embedding methods to extract feature representations in CMCN from local and global perspectives, and use denoising autoencoder for enhancement; then, combined with balance theory and state theory, molecular features are extracted from the point of social relations through the propagation and aggregation of signed graph attention network; finally, the GBDT classifier is used to predict the association of molecules. The results show that CMCSG can effectively predict the relationship between circRNA, miRNA, and cancer. Additionally, the case studies also demonstrate that CMCSG is capable of accurately identifying biomarkers across various types of cancer. The data and source code can be found at https://github.com/1axin/CMCSG.

Robertson’s conjecture and universal finite generation in the homology of graph braid groups

We formulate a categorification of Robertson’s conjecture analogous to the categorical graph minor conjecture of Miyata–Proudfoot–Ramos. We show that these conjectures imply the existence of a finite list of atomic graphs generating the homology of configuration spaces of graphs—in fixed degree, with a fixed number of particles, under topological embeddings. We explain how the simplest case of our conjecture follows from work of Barter and Proudfoot–Ramos, implying that the category of cographs is Noetherian, a result of potential independent interest.

Chiral topological light for detection of robust enantiosensitive observables

The topological response of matter to electromagnetic fields is a highly demanded property in materials design and metrology due to its robustness against noise and decoherence, stimulating recent advances in ultrafast photonics. Embedding topological properties into the enantiosensitive optical response of chiral molecules could therefore enhance the efficiency and robustness of chiral optical discrimination. Here we achieve such a topological embedding by introducing the concept of chiral topological light—a light beam which displays chirality locally, with an azimuthal distribution of its handedness described globally by a topological charge. The topological charge is mapped onto the azimuthal intensity modulation of the non-linear optical response, where enantiosensitivity is encoded into its spatial rotation. The spatial rotation is robust against intensity fluctuations and imperfect local polarization states of the driving field. Our theoretical results show that chiral topological light enables detection of percentage-level enantiomeric excesses in randomly oriented mixtures of chiral molecules, opening a way to new, extremely sensitive and robust chiro-optical spectroscopies with attosecond time resolution.

ECD-CDGI: An efficient energy-constrained diffusion model for cancer driver gene identification.

The identification of cancer driver genes (CDGs) poses challenges due to the intricate interdependencies among genes and the influence of measurement errors and noise. We propose a novel energy-constrained diffusion (ECD)-based model for identifying CDGs, termed ECD-CDGI. This model is the first to design an ECD-Attention encoder by combining the ECD technique with an attention mechanism. ECD-Attention encoder excels at generating robust gene representations that reveal the complex interdependencies among genes while reducing the impact of data noise. We concatenate topological embedding extracted from gene-gene networks through graph transformers to these gene representations. We conduct extensive experiments across three testing scenarios. Extensive experiments show that the ECD-CDGI model possesses the ability to not only be proficient in identifying known CDGs but also efficiently uncover unknown potential CDGs. Furthermore, compared to the GNN-based approach, the ECD-CDGI model exhibits fewer constraints by existing gene-gene networks, thereby enhancing its capability to identify CDGs. Additionally, ECD-CDGI is open-source and freely available. We have also launched the model as a complimentary online tool specifically crafted to expedite research efforts focused on CDGs identification.

ScLEGA: an attention-based deep clustering method with a tendency for low expression of genes on single-cell RNA-seq data.

Single-cell RNA sequencing (scRNA-seq) enables the exploration of biological heterogeneity among different cell types within tissues at a resolution. Inferring cell types within tissues is foundational for downstream research. Most existing methods for cell type inference based on scRNA-seq data primarily utilize highly variable genes (HVGs) with higher expression levels as clustering features, overlooking the contribution of HVGs with lower expression levels. To address this, we have designed a novel cell type inference method for scRNA-seq data, termed scLEGA. scLEGA employs a novel zero-inflated negative binomial (ZINB) loss function that fully considers the contribution of genes with lower expression levels and combines two distinct scRNA-seq clustering strategies through a multi-head attention mechanism. It utilizes a low-expression optimized denoising autoencoder, based on the novel ZINB model, to extract low-dimensional features and handle dropout events, and a GCN-based graph autoencoder (GAE) that leverages neighbor information to guide dimensionality reduction. The iterative fusion of denoising and topological embedding in scLEGA facilitates the acquisition of cluster-friendly cell representations in the hidden embedding, where similar cells are brought closer together. Compared to 12 state-of-the-art cell type inference methods on 15 scRNA-seq datasets, scLEGA demonstrates superior performance in clustering accuracy, scalability, and stability. Our scLEGA model codes are freely available at https://github.com/Masonze/scLEGA-main.

Equivariant embedding of finite-dimensional dynamical systems

AbstractWe prove an equivariant version of the classical Menger-Nöbeling theorem regarding topological embeddings: Whenever a group G acts on a finite-dimensional compact metric space X, a generic continuous equivariant function from X into $$([0,1]^r)^G$$ ( [ 0 , 1 ] r ) G is a topological embedding, provided that for every positive integer N the space of points in X with orbit size at most N has topological dimension strictly less than $$\frac{rN}{2}$$ rN 2 . We emphasize that the result imposes no restrictions whatsoever on the acting group G (beyond the existence of an action on a finite-dimensional space). Moreover if G is finitely generated then there exists a finite subset $$F\subset G$$ F ⊂ G so that for a generic continuous map $$h:X\rightarrow [0,1]^{r}$$ h : X → [ 0 , 1 ] r , the map $$ h^{F}:X\rightarrow ([0,1]^{r})^{F}$$ h F : X → ( [ 0 , 1 ] r ) F given by $$x\mapsto (f(gx))_{g\in F}$$ x ↦ ( f ( g x ) ) g ∈ F is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.

Enhancing Vehicle Tracking and Recognition Across Multiple Cameras with Multimodal Contrastive Domain Sharing GAN and Topological Embeddings

Using Multimodal Contrastive Domain Sharing Generative Adversarial Networks (GAN) and topological embeddings, this study shows a new way to improve car tracking and classification across multiple camera feeds. Different camera angles and lighting conditions can make it hard for current car tracking systems to work correctly. This study tries to solve these problems. Common Objects in Context (COCO) and ImageNet are two datasets that are used in this method for training. Multimodal Contrastive Domain Sharing GAN is used for detection and tracking. It makes cross-modal learning easier by letting you see things from different camera angles. This framework lets the model learn shared representations, which makes it better at recognizing vehicles in a wider range of visual domains. The Topological Information Embedded Convolutional Neural Network (TIE-CNN) is used to re-identify the car after it has been found and tracked. This network embeds the paths of vehicles into a continuous latent space, keeping the important spatial connections needed for accurate tracking. Real-world multi-camera datasets used for experimental confirmation show that tracking accuracy and recognition performance are much better than with standard methods. The suggested framework works great in tough situations like blocked views and sudden changes in lighting, showing that it are reliable in complicated surveillance settings. This study adds to the progress in multi-camera car tracking and identification by combining geometric data analysis with deep learning methods. This method uses Multimodal Contrastive Domain Sharing GAN and topological embeddings to improve the timing and spatial coherence of tracking results. It also sets the stage for future improvements in monitoring and self-driving systems.

Graph Representation Learning for Street-Level Crime Prediction

In contemporary research, the street network emerges as a prominent and recurring theme in crime prediction studies. Meanwhile, graph representation learning shows considerable success, which motivates us to apply the methodology to crime prediction research. In this article, a graph representation learning approach is utilized to derive topological structure embeddings within the street network. Subsequently, a heterogeneous information network that incorporates both the street network and urban facilities is constructed, and embeddings through link prediction tasks are obtained. Finally, the two types of high-order embeddings, along with other spatio-temporal features, are fed into a deep neural network for street-level crime prediction. The proposed framework is tested using data from Beijing, and the outcomes demonstrate that both types of embeddings have a positive impact on crime prediction, with the second embedding showing a more significant contribution. Comparative experiments indicate that the proposed deep neural network offers superior efficiency in crime prediction.

Fractional Sobolev isometric immersions of planar domains

We discuss $C^1$ regularity and developability of isometric immersions of flat domains into $\mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s, \frac2s}$ regularity for $2/3 \le s< 1 $, generalizing the known results on Sobolev and Holder regimes. Ingredients of the proof include analysis of the weak Codazzi-Mainardi equations of the isometric immersions and study of $W^{2,\frac2s}$ planar deformations with symmetric Jacobian derivative and vanishing distributional Jacobian determinant. On the way, we also show that the distributional Jacobian determinant, conceived as an operator defined on the Jacobian matrix, behaves like determinant of gradient matrices under products by scalar functions.

The geometry of C1,α flat isometric immersions

We show that any isometric immersion of a flat plane domain into ${\mathbb {R}}^3$ is developable provided it enjoys the little Hölder regularity $c^{1,2/3}$ . In particular, isometric immersions of local $C^{1,\alpha }$ regularity with $\alpha &gt;2/3$ belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss–Codazzi–Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge–Ampère equation analysed in [M. Lewicka and M. R. Pakzad. Anal. PDE 10 (2017), 695–727.].

Non-compactness results for the spinorial Yamabe-type problems with non-smooth geometric data

Let (M,g,σ) be an m-dimensional closed spin manifold, with a fixed Riemannian metric g and a fixed spin structure σ; let S(M) be the spinor bundle over M. The spinorial Yamabe-type problems address the solvability of the following equationDgψ=f(x)|ψ|g2m−1ψ,ψ:M→S(M),x∈M where Dg is the associated Dirac operator and f:M→R is a given function. The study of such nonlinear equation is motivated by its important applications in Spin Geometry: when m=2, a solution corresponds to a conformal isometric immersion of the universal covering M˜ into R3 with prescribed mean curvature f; meanwhile, for general dimensions and f≡constant≠0, a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant.The aim of this paper is to establish non-compactness results related to the spinorial Yamabe-type problems. Precisely, concrete analysis is made for two specific models on the manifold (Sm,g) where the solution set of the spinorial Yamabe-type problem is not compact: 1). the geometric potential f is constant (say f≡1) with the background metric g being a Ck perturbation of the canonical round metric gSm, which is not conformally flat somewhere on Sm; 2). f is a perturbation from constant and is of class C2, while the background metric g≡gSm.

A Short Survey on Biharmonic Riemannian Submersions

The study of biharmonic submanifolds, initiated by B. Y. Chen and G. Y. Jiang independently, has received a great attention in the past 30 years with many important progress (see the reference for some recent books with vast references therein). This note attempts to give a short survey on the study of biharmonic Riemannian submersions which are a dual concept of biharmonic submanifolds (i.e., biharmonic isometric immersions).

Isometric Immersions and the Waving of Flags.

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach, a flag is viewed as an isometric immersion from the square with values in satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. In the second part of the article we equip the space of isometric immersions with its natural kinetic energy and derive the corresponding equations of motion. This approach can be viewed in a spirit similar to Arnold's geometric picture for the motion of an incompressible fluid.

F-Biharmonic Submanifolds in Space Forms and f-Biharmonic Riemannian Submersions from 3-Manifolds

f-biharmonic maps are generalizations of harmonic maps and biharmonic maps. In this paper, we give some descriptions of f-biharmonic curves in a space form. We also obtain a complete classification of proper f-biharmonic isometric immersions of a developable surface in R3 by proving that a proper f-biharmonic developable surface exists only in the case where the surface is a cylinder. Based on this, we show that a proper biharmonic conformal immersion of a developable surface into R3 exists only in the case when the surface is a cylinder. Riemannian submersions can be viewed as a dual notion of isometric immersions (i.e., submanifolds). We also study f-biharmonicity of Riemannian submersions from 3-manifolds by using the integrability data. Examples are given of proper f-biharmonic Riemannian submersions and f-biharmonic surfaces and curves.

Multi-camera Vehicle Tracking and Recognition with Multimodal Contrastive Domain Sharing GAN and Topological Embeddings

Tracking vehicles across a city using a network of multiple cameras are pivotal for enhancing urban and traffic management systems. However, this task is riddled with challenges such as wide geographical coverage, frequent view obstructions, and the diverse appearances of vehicles from various angles.&#x0D; To address these complexities, the proposed solution, dubbed Overlapped Vehicle Detection and Tracking using Multimodal Contrastive Domain Sharing Generative Adversarial Network optimized with Efficient Multi-camera system (MCDS-GAN), leverages cutting-edge techniques from computer vision, image processing, machine learning, and sensor fusion. This advanced system detects and tracks vehicles even in scenarios where multiple camera views overlap, making it applicable across domains like traffic management, surveillance, and autonomous vehicles.&#x0D; The methodology involves utilizing datasets like Common Objects in Context and ImageNet for training. Detection and tracking are performed using the Multimodal Contrastive Domain Sharing Generative Adversarial Network, followed by vehicle re-identification facilitated by the Topological Information Embedded Convolution Neural Network (TIE-CNN).&#x0D; Moreover, optimization techniques are employed to ensure synchronization and efficiency within the system. Implemented in Python, the effectiveness of MCDS-GAN is rigorously evaluated using metrics such as Accuracy, Precision, Recall, Latency, Response Time, and Scalability. Simulation results showcase its superiority, achieving significantly higher accuracy rates compared to existing methods such as OC-MCT-OFOV, MT-MCT-VM-CLM, and TI-VRI.

DyGCN: Efficient Dynamic Graph Embedding With Graph Convolutional Network.

Graph embedding, aiming to learn low-dimensional representations (aka. embeddings) of nodes in graphs, has received significant attention. In recent years, there has been a surge of efforts, among which graph convolutional networks (GCNs) have emerged as an effective class of models. However, these methods mainly focus on the static graph embedding. In the present work, an efficient dynamic graph embedding approach is proposed, called dynamic GCN (DyGCN), which is an extension of the GCN-based methods. The embedding propagation scheme of GCN is naturally generalized to a dynamic setting in an efficient manner, which propagates the change in topological structure and neighborhood embeddings along the graph to update the node embeddings. The most affected nodes are updated first, and then their changes are propagated to further nodes, which in turn are updated. Extensive experiments on various dynamic graphs showed that the proposed model can update the node embeddings in a time-saving and performance-preserving way.

Rigidity properties of p-biharmonic maps and p-biharmonic submanifolds

We give some rigidity properties of a p-biharmonic map u:(M,g)→(N,h) between Riemannian manifolds (Mn,g) and (Nm,h). We first provide various sufficient conditions for p-biharmonic maps to be harmonic. Moreover, when the map u is an isometric immersion, by assuming that the Ln2-norm of the sectional curvature on M is sufficiently small or if the fundamental tone of the p-biharmonic submanifold is sufficiently big, it is proved that M is minimal.