Maps Of Manifolds Research Articles

Manifold Constraints for Imperceptible Adversarial Attacks on Point Clouds

Adversarial attacks on 3D point clouds often exhibit unsatisfactory imperceptibility, which primarily stems from the disregard for manifold-aware distortion, i.e., distortion of the underlying 2-manifold surfaces. In this paper, we develop novel manifold constraints to reduce such distortion, aiming to enhance the imperceptibility of adversarial attacks on 3D point clouds. Specifically, we construct a bijective manifold mapping between point clouds and a simple parameter shape using an invertible auto-encoder. Consequently, manifold-aware distortion during attacks can be captured within the parameter space. By enforcing manifold constraints that preserve local properties of the parameter shape, manifold-aware distortion is effectively mitigated, ultimately leading to enhanced imperceptibility. Extensive experiments demonstrate that integrating manifold constraints into conventional adversarial attack solutions yields superior imperceptibility, outperforming the state-of-the-art methods.

Predicting multiple observations in complex systems through low-dimensional embeddings

Forecasting all components in complex systems is an open and challenging task, possibly due to high dimensionality and undesirable predictors. We bridge this gap by proposing a data-driven and model-free framework, namely, feature-and-reconstructed manifold mapping (FRMM), which is a combination of feature embedding and delay embedding. For a high-dimensional dynamical system, FRMM finds its topologically equivalent manifolds with low dimensions from feature embedding and delay embedding and then sets the low-dimensional feature manifold as a generalized predictor to achieve predictions of all components. The substantial potential of FRMM is shown for both representative models and real-world data involving Indian monsoon, electroencephalogram (EEG) signals, foreign exchange market, and traffic speed in Los Angeles Country. FRMM overcomes the curse of dimensionality and finds a generalized predictor, and thus has potential for applications in many other real-world systems.

Multifidelity aerodynamic shape optimization for airfoil dynamic stall mitigation using manifold mapping

Aerodynamic shape optimization for dynamic stall mitigation is computationally challenging, requiring multiple costly CFD evaluations. This work proposes a multifidelity modeling technique to efficiently mitigate dynamic stall over an airfoil, particularly manifold mapping (MM) within a trust-region-based optimization framework to find an optimal shape defined by six PARSEC parameters. The high-fidelity (HF) responses are modeled using the unsteady Reynolds-averaged Navier–Stokes equations with Menter’s shear stress transport turbulence model at a Reynolds number of 135,000, Mach number of 0.1, and reduced frequency of 0.05. The low-fidelity responses are acquired from the Kriging regression (KR) trained over the design space. The MM with the pattern-search algorithm is utilized to find optimal designs in two different test cases. The result yields two distinct optimal shapes with a higher thickness, larger leading-edge radius, and more aft camber than the baseline airfoil (NACA 0012), indicating the presence of multiple valid designs. Both designs delay the dynamic stall angle over 3 deg while mitigating peak lift and pitching moment magnitudes. The current approach delivers a computational saving of over 70% compared to HF-KR and Cokriging regression techniques. These findings present an efficient multifidelity strategy with potential application to other computationally expensive optimization problems.

The Morse Index of Sacks–Uhlenbeck α‐Harmonic Maps for Riemannian Manifolds

In this paper, first we prove a nonexistence theorem for α‐harmonic mappings between Riemannian manifolds. Second, the instability of nonconstant α‐harmonic maps is studied with regard to the Ricci curvature criterion of their codomain. Then, we estimate the Morse index for measuring the degree of instability of some particular α‐harmonic maps. Furthermore, the notion of α‐stable manifolds and its applications are considered. Finally, we investigate the α‐stability of any compact Riemannian manifolds admitting a nonisometric conformal vector field and any Einstein Riemannian manifold under certain assumptions on the smallest positive eigenvalue of its Laplacian operator on functions.

Open-Set Cross-Domain Hyperspectral Image Classification Based on Manifold Mapping Alignment

Open-Set Cross-Domain Hyperspectral Image Classification Based on Manifold Mapping Alignment

Deep Geometric Learning With Monotonicity Constraints for Alzheimer's Disease Progression.

Alzheimer's disease (AD) is a devastating neurodegenerative condition that precedes progressive and irreversible dementia; thus, predicting its progression over time is vital for clinical diagnosis and treatment. For this, numerous studies have implemented structural magnetic resonance imaging (MRI) to model AD progression, focusing on three integral aspects: 1) temporal variability; 2) incomplete observations; and 3) temporal geometric characteristics. However, many pioneer deep learning-based approaches addressing data variability and sparsity have yet to consider inherent geometrical properties sufficiently. These properties are integral to modeling as they correlate with brain region size, thickness, volume, and shape in AD progression. The ordinary differential equation-based geometric modeling method (ODE-RGRU) has recently emerged as a promising strategy for modeling time-series data by intertwining a recurrent neural network (RNN) and an ODE in Riemannian space. Despite its achievements, ODE-RGRU encounters limitations when extrapolating positive definite symmetric matrices from incomplete samples, leading to feature reverse occurrences that are particularly problematic, especially within the clinical facet. Therefore, this study proposes a novel geometric learning approach that models longitudinal MRI biomarkers and cognitive scores by combining three modules: topological space shift, ODE-RGRU, and trajectory estimation. We have also developed a training algorithm that integrates the manifold mapping with monotonicity constraints to reflect measurement transition irreversibility. We verify our proposed method's efficacy by predicting clinical labels and cognitive scores over time in regular and irregular settings. Furthermore, we thoroughly analyze our proposed framework through an ablation study.

Manifold embedded global and local discriminative features selection for single-shot multi-categories clothing recognition and retrieval

PurposeSingle-shot multi-category clothing recognition and retrieval play a crucial role in online searching and offline settlement scenarios. Existing clothing recognition methods based on RGBD clothing images often suffer from high-dimensional feature representations, leading to compromised performance and efficiency.Design/methodology/approachTo address this issue, this paper proposes a novel method called Manifold Embedded Discriminative Feature Selection (MEDFS) to select global and local features, thereby reducing the dimensionality of the feature representation and improving performance. Specifically, by combining three global features and three local features, a low-dimensional embedding is constructed to capture the correlations between features and categories. The MEDFS method designs an optimization framework utilizing manifold mapping and sparse regularization to achieve feature selection. The optimization objective is solved using an alternating iterative strategy, ensuring convergence.FindingsEmpirical studies conducted on a publicly available RGBD clothing image dataset demonstrate that the proposed MEDFS method achieves highly competitive clothing classification performance while maintaining efficiency in clothing recognition and retrieval.Originality/valueThis paper introduces a novel approach for multi-category clothing recognition and retrieval, incorporating the selection of global and local features. The proposed method holds potential for practical applications in real-world clothing scenarios.

A Novel Intelligent Fault Diagnosis Method for Self-Priming Centrifugal Pumps.

The real-time diagnostic monitoring of self-priming centrifugal pumps is essential to ensure their safe operation. Nevertheless, owing to the intricate structure and complex operational conditions inherent in such pumps, existing fault diagnosis methods encounter challenges in effectively extracting crucial fault feature information and accurately identifying fault types. Consequently, this paper introduces an intelligent fault diagnosis method tailored for self-priming centrifugal pumps. The approach amalgamates refined time-shift multiscale fluctuation dispersion entropy, cosine pairwise-constrained supervised manifold mapping, and adaptive chaotic Aquila optimization support vector machine techniques. To begin with, refined time-shift multiscale fluctuation dispersion entropy is employed to extract fault-related features, adeptly mitigating concerns related to entropy domain deviations and instability. Subsequently, the application of cosine pairwise-constrained supervised manifold mapping serves to reduce the dimensionality of the extracted fault features, thereby bolstering the efficiency and precision of the ensuing identification process. Ultimately, the utilization of an adaptive chaotic Aquila optimization support vector machine facilitates intelligent fault classification, leading to enhanced accuracy in fault identification. The experimental findings unequivocally affirm the efficacy of the proposed method in accurately discerning among various fault types in self-priming centrifugal pumps, achieving an exceptional recognition rate of 100%. Moreover, it is noteworthy that the average correct recognition rate achieved by the proposed method surpasses that of five existing intelligent fault diagnosis techniques by a significant margin, registering a notable increase of 15.97%.

Map functions to facilitate situational awareness during emergency events

ABSTRACT Emergency events such as floods and wildfires are handled by various responders and at various levels: strategic, tactical, and operational. To facilitate situational awareness, emergency responders require customized map-based decision support systems that are tailored to specific needs depending on the responders’ organizational affiliation, role, objectives, and occupationally specific knowledge. As a result, the systems are equipped with manifold map functions. However, the diversity of map-based emergency tools in use impedes gaining common user skills among their target audiences and thus, requires a systematic overview. Through a multistep research process, this study was to: investigate the requirements for support from map-based tools expressed by various emergency responders in Norway, identify desired map functions, and categorize those functions to facilitate an overview. Six stages constituted our workflow: meetings with Norwegian emergency responders, survey on selected map-based tools, interviews with designers and users of tools, a table-top exercise, theoretical considerations, and validation with stakeholders. This study contributes to the state of the art by systematizing and structuring knowledge about map functions that facilitate situational awareness. In turn, it helps developing and optimizing functionality of map-based tools depending on needs of specific emergency responders.

Albanese map of special manifolds: a correction

Albanese map of special manifolds: a correction

Research on Network Behavior Risk Measurement Method Based on Traffic Analysis

At present, the network security problem is facing a serious threat, and network security events continue to occur. It has become an important link to prevent network attacks and ensure network security. According to the network security protection measures and security technical requirements, it has become an urgent need to establish appropriate security measurement methods and strengthen the monitoring and analysis of network security status. This study proposes a network behavior risk measurement method based on traffic analysis to accurately and objectively evaluate the security state of the network. Traffic is the most basic behavior of the network and the basis of security risk measurement. Firstly, we regard the traffic data as network behavior to build scenarios. Through differential manifold modeling, the traffic data and topology of the network system are semantically described to form a matrix. Then, after manifold dimensionality reduction, the objective risk assessment value can be obtained by manifold mapping and Riemann metric. In this study, the differential manifold theory is applied to network behavior risk measurement, and the innovation of differential manifold in the field of network behavior risk measurement is given. After giving the network behavior risk measurement theory, we first verify the effectiveness of the proposed method through the simulation experiments. Secondly, the public CIC-IDS-2017 data set is used for analysis and calculation to prove the accuracy of the proposed method.

Round fold maps of n-dimensional manifolds into (n-1)-dimensional Euclidean space

Round fold maps of n-dimensional manifolds into (n-1)-dimensional Euclidean space

An outlier detection algorithm based on segmentation and pruning of competitive network for glioma identification using Raman spectroscopy.

Raman spectroscopy is a promising diagnostic tool for brain gliomas, owing to its non-invasive and high information density properties. However, identifying patterns in glioma cancer tissue and healthy tissue in the brain is challenging, and outlier spectra resulting from operator error or changes in external conditions can compromise the model's robustness and generalizability to new data. Given the heterogeneity of glioma tissue, the within-group variance of data obtained by a portable Raman spectrometer is relatively high, and inconsistencies in instrument repeatability and experimental conditions can lead to an incompact distribution of non-outlier points, complicating outlier detection. Strict outlier criteria may result in the deletion of non-outlier points, leading to reduced sample utilization. To address these issues, we propose the SPCN outlier detection algorithm, which segments and prunes a competitive network to extract global outlier features, identifies topological errors, and divides initial outlier domains using the α-β region segmentation method. The algorithm also proposes a two-stage pruning method based on the characteristics of the manifold map and visualizes the outlier measure using a normalized histogram. Compared to traditional methods, SPCN is label-free and does not require an estimation of outlier distance threshold or data distribution density. We compared the accuracy of six outlier detection algorithms using Raman spectra collected from brain glioma tissues of 113 patients and examined changes in pattern recognition accuracy after removing the outliers, confirming the precision and robustness of SPCN. This method has the potential to enhance the accuracy and reliability of glioma diagnosis via Raman spectroscopy and can also be applied to outlier detection in other spectra such as near infrared and middle infrared.

Refined Composite Multiscale Fluctuation Dispersion Entropy and Supervised Manifold Mapping for Planetary Gearbox Fault Diagnosis

A novel fault diagnosis scheme was developed to address the difficulty of feature extraction for planetary gearboxes using refined composite multiscale fluctuation dispersion entropy (RCMFDE) and supervised manifold mapping. The RCMFDE was first utilized in this scheme to fully mine fault features from planetary gearbox signals under multiple scales. Subsequently, as a supervised manifold mapping method, supervised isometric mapping (S-Iso) was applied to decrease the dimensions of the original features and remove redundant information. Lastly, the marine predator algorithm-based support vector machine (MPA-SVM) classifier was employed to achieve intelligent fault diagnosis of planetary gearboxes. The suggested RCMFDE combines the composite coarse-grained construction and refined computing technology, overcoming unstable and invalid entropy in the traditional multiscale fluctuation dispersion entropy. Simulation experiments and fault diagnosis experiments from a real planetary gearbox drive system show that the complexity measure capability and feature extraction effectiveness of the proposed RCMFDE outperform the multiscale fluctuation dispersion entropy (MFDE) and multi-scale permutation entropy (MPE). The S-Iso’s visualization results and dimensionality reduction performance are better than principal components analysis (PCA), linear discriminant analysis (LDA), and isometric mapping (Isomap). Moreover, the suggested fault diagnosis scheme has an accuracy rate of 100% in identifying bearing and gear defects in planetary gearboxes.

Fold cobordisms and a Poincaré–Hopf-type theorem for the signature

We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were introduced in the author's earlier works. Secondly we take the induced stable partial framing on the source manifold whose cobordisms were studied in general by Koschorke. We show that these invariants describe completely the cobordism groups of framed fold maps into R^n. Then we are looking for dependencies between these invariants and we study fold maps of 4k-dimensional manifolds into R^2. We construct special fold maps which are representatives of the fold cobordism classes and we also compute cobordism groups. We obtain a Poincare-Hopf type formula, which connects local data of the singularities of a fold map of an oriented 4k-dimensional manifold M to the signature of M. We also study the unoriented case analogously and prove a similar formula about the twisting of the normal bundle of the fold singular set.

Short-Time Existence for Harmonic Map Heat Flow with Time-Dependent Metrics

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow $g(t)$ on $M$ into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that $|\text{Rm}(g(t))|\leq a/t$ and $g(t)$ is uniformly equivalent to $g(0)$.

A study of simulation-based surrogates in well-placement optimization for hydrocarbon production

An essential part of hydrocarbon field-development planning is the determination of well-drilling locations within a reservoir, which can be formulated as an optimization problem. Lending to the nonlinear nature and potentially high-dimensional search spaces, this problem is computationally intensive as it entails time-demanding reservoir flow simulations. To make matters worse, most practical decision-making scenarios in field development are subject to constraints on available computing resources (this also includes commercial-software licenses). In this paper, we study a surrogate-based well-placement optimization approach especially devised for these challenging scenarios. In this approach, surrogates are constructed either analytically (using a number of previous simulation runs) or by simplifying the underlying physics of the problem. The surrogates are iteratively corrected in a derivative-free, noninvasive (regarding the simulator) manner, through manifold mapping, a multi-fidelity technique. This is the first application of manifold mapping to well-placement optimization problems. We considered two optimization problems that involve benchmark models with different degrees of complexity. Multiple implementations of the methodology are compared with a single-level optimization method (i.e., that does not utilize any surrogate explicitly). The effect of surrogate accuracy and its impact on the quality of the optimized solutions is explored. We also discuss limitations of the analytical surrogates in the presence of complex optimization spaces and physical phenomena. It is shown here that physics-based surrogates are more suitable choices in these situations and still accelerate computations noticeably with respect to the single-level method. The promising findings in this study reflect how the optimization of well placement in hydrocarbon fields under (stringent) constraints on computing resources could be approached.

Metacell-2: a divide-and-conquer metacell algorithm for scalable scRNA-seq analysis

Scaling scRNA-seq to profile millions of cells is crucial for constructing high-resolution maps of transcriptional manifolds. Current analysis strategies, in particular dimensionality reduction and two-phase clustering, offer only limited scaling and sensitivity to define such manifolds. We introduce Metacell-2, a recursive divide-and-conquer algorithm allowing efficient decomposition of scRNA-seq datasets of any size into small and cohesive groups of cells called metacells. Metacell-2 improves outlier cell detection and rare cell type identification, as shown with human bone marrow cell atlas and mouse embryonic data. Metacell-2 is implemented over the scanpy framework for easy integration in any analysis pipeline.

Geodesic Mappings of Semi-Riemannian Manifolds with a Degenerate Metric

This article introduces the concept of geodesic mappings of manifolds with idempotent pseudo-connections. The basic equations of canonical geodesic mappings of manifolds with completely idempotent pseudo-connectivity and semi-Riemannian manifolds with a degenerate metric are obtained. It is proved that semi-Riemannian manifolds admitting concircular fields admit completely canonical geodesic mappings and form a closed class with respect to these mappings.

Passive Non-Line-of-Sight Imaging Using Optimal Transport

Passive non-line-of-sight (NLOS) imaging has drawn great attention in recent years. However, all existing methods are in common limited to simple hidden scenes, low-quality reconstruction, and small-scale datasets. In this paper, we propose NLOS-OT, a novel passive NLOS imaging framework based on manifold embedding and optimal transport, to reconstruct high-quality complicated hidden scenes. NLOS-OT converts the high-dimensional reconstruction task to a low-dimensional manifold mapping through optimal transport, alleviating the ill-posedness in passive NLOS imaging. Besides, we create the first large-scale passive NLOS imaging dataset, NLOS-Passive, which includes 50 groups and more than 3,200,000 images. NLOS-Passive collects target images with different distributions and their corresponding observed projections under various conditions, which can be used to evaluate the performance of passive NLOS imaging algorithms. It is shown that the proposed NLOS-OT framework achieves much better performance than the state-of-the-art methods on NLOS-Passive. We believe that the NLOS-OT framework together with the NLOS-Passive dataset is a big step and can inspire many ideas towards the development of learning-based passive NLOS imaging. Codes and dataset are publicly available (https://github.com/ruixv/NLOS-OT).