Higher Dimensional Manifolds Research Articles

Geometry of positive scalar curvature on complete manifold

Abstract In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. On three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. On higher-dimensional manifold, we obtain a volume growth with a stronger condition.

Soft-margin classification of object manifolds.

A neural population responding to multiple appearances of a single object defines a manifold in the neural response space. The ability to classify such manifolds is of interest, as object recognition and other computational tasks require a response that is insensitive to variability within a manifold. Linear classification of object manifolds was previously studied for max-margin classifiers. Soft-margin classifiers are a larger class of algorithms and provide an additional regularization parameter used in applications to optimize performance outside the training set by balancing between making fewer training errors and learning more robust classifiers. Here we develop a mean-field theory describing the behavior of soft-margin classifiers applied to object manifolds. Analyzing manifolds with increasing complexity, from points through spheres to general manifolds, a mean-field theory describes the expected value of the linear classifier's norm, as well as the distribution of fields and slack variables. By analyzing the robustness of the learned classification to noise, we can predict the probability of classification errors and their dependence on regularization, demonstrating a finite optimal choice. The theory describes a previously unknown phase transition, corresponding to the disappearance of a nontrivial solution, thus providing a soft version of the well-known classification capacity of max-margin classifiers. Furthermore, for high-dimensional manifolds of any shape, the theory prescribes how to define manifold radius and dimension, two measurable geometric quantities that capture the aspects of manifold shape relevant to soft classification.

Ion channel model reduction using manifold boundaries.

Mathematical models of voltage-gated ion channels are used in basic research, industrial and clinical settings. These models range in complexity, but typically contain numerous variables representing the proportion of channels in a given state, and parameters describing the voltage-dependent rates of transition between states. An open problem is selecting the appropriate degree of complexity and structure for an ion channel model given data availability. Here, we simplify a model of the cardiac human Ether-à-go-go related gene (hERG) potassium ion channel, which carries cardiac IKr, using the manifold boundary approximation method (MBAM). The MBAM approximates high-dimensional model-output manifolds by reduced models describing their boundaries, resulting in models with fewer parameters (and often variables). We produced a series of models of reducing complexity starting from an established five-state hERG model with 15 parameters. Models with up to three fewer states and eight fewer parameters were shown to retain much of the predictive capability of the full model and were validated using experimental hERG1a data collected in HEK293 cells at 37°C. The method provides a way to simplify complex models of ion channels that improves parameter identifiability and will aid in future model development.

A peeling theorem for the Weyl tensor in higher dimensions

A peeling theorem for the Weyl tensor in higher dimensional Lorentzian manifolds is presented. We obtain it by generalizing a proof from the four dimensional case. We derive a generic behavior, discuss interesting subcases and retrieve the four dimensional result.

Unsupervised Learning Discriminative MIG Detectors in Nonhomogeneous Clutter

Principal component analysis (PCA) is a commonly used pattern analysis method that maps high-dimensional data into a lower-dimensional space maximizing the data variance, that results in the promotion of separability of data. Inspired by the principle of PCA, a novel type of learning discriminative matrix information geometry (MIG) detectors in the unsupervised scenario are developed, and applied to signal detection in nonhomogeneous environments. Hermitian positive-definite (HPD) matrices can be used to model the sample data, while the clutter covariance matrix is estimated by the geometric mean of a set of secondary HPD matrices. We define a projection that maps the HPD matrices in a high-dimensional manifold to a low-dimensional and more discriminative one to increase the degree of separation of HPD matrices by maximizing the data variance. Learning a mapping can be formulated as a two-step mini-max optimization problem in Riemannian manifolds, which can be solved by the Riemannian gradient descent algorithm. Three discriminative MIG detectors are illustrated with respect to different geometric measures, i.e., the Log-Euclidean metric, the Jensen–Bregman LogDet divergence and the symmetrized Kullback–Leibler divergence. Simulation results show that performance improvements of the novel MIG detectors can be achieved compared with the conventional detectors and their state-of-the-art counterparts within nonhomogeneous environments.

Decoupling Decorations on Moduli Spaces of Manifolds

AbstractWe study moduli spaces of d-dimensional manifolds with embedded particles and discs, which we refer to as decorations. These spaces admit a model in which points are unparametrised d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs constrained to it. We compare this to the space of d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs that are no longer constrained, i.e. the decorations are decoupled. We show that under certain conditions these spaces cannot be distinguished by homology groups within a range. This generalises work by Bödigheimer–Tillmann for oriented surfaces to different tangential structures and also to higher dimensional manifolds. We also extend this result to moduli spaces with more general submanifolds as decorations and specialise in the case of decorations being embedded circles.

Empirical analysis of latent space encodings for submerged small target acoustic backscattering data

With future sights on specialized classification methods, we work to generalize the acoustic backscattering data from sonar measurements of small targets submerged in water by learning a non-invertible mapping (encoding) to a low-dimensional vector space (ℝn). Finding the optimal dimensionality of this latent space is an important task. The encoding is accomplished by utilizing modality agnostic convolutional machine learning methods that have seen success in other signal and image processing domains. We have explored the autoencoder and its variants, the sparse autoencoder, and the variational autoencoder. Autoencoders encode input samples from a high-dimensional manifold to a lower latent vector space and then reverse the lossy mapping back to a high-dimensional manifold similar to the initial domain. The sparse autoencoder induces sparsity in the components of the latent vectors, and the variational autoencoder learns an encoding to n-dimensional gaussian distributions instead of n-dimensional vectors. The TREX13 data are the primary dataset used for training the networks used in experiments. We evaluate the change in models’ accuracies as the latent dimensionality is increased as well as the models’ ability to generalize to unseen data. Additionally, the PONDEX09 and PONDEX10 data are used to evaluate the models’ cross-domain efficacy.

3D Deep Heterogeneous Manifold Network for Behavior Recognition

With the broadening of application scenarios for Internet of Things, intelligent behavior recognition task has attracted more and more attention. Since human behavior is nonrigid motion with strong spatiotemporal topological association, modeling it directly with traditional Euclidean space-based methods may destroy its underlying nonlinearity. Based on the advantages of Riemannian manifold in describing 3D motion, we propose an end-to-end 3D behavior manifold feature learning framework composed of deep heterogeneous networks. This heterogeneous architecture aims to leverage the graph construction to guide manifold backbone network to mine more discriminative nonlinear spatiotemporal features. Therefore, we first model the nonlinear spatiotemporal co-occurrence of 3D behavior in the high-dimensional Riemannian manifold space. Secondly, we implement a non-Euclidean heterogeneous architecture on the Riemannian manifold so that the backbone network can learn deep spatiotemporal features while preserving the manifold topology. Finally, an end-to-end deep graph similarity-guided learning optimization mechanism is introduced to enable the overall model to fully utilize the complex similarity relationship between manifold features. We have verified our 3D deep heterogeneous manifold network on popular skeleton behavior datasets and achieved competitive results.

Construction of Networks by Associating with Submanifolds of Almost Hermitian Manifolds

The idea that data lies in a non-linear space has brought up the concept of manifold learning as a part of machine learning and such notion is one of the most important research fields of today. The main idea here is to design the data as a submanifold model embedded in a high-dimensional manifold. On the other hand, graph theory is one of the most important research areas of applied mathematics and computer science. As a result, many researchers investigate new methods for machine learning on graphs. From the above information, it is seen that the theory of submanifolds and graph theory have become two important concepts in machine learning and nowadays, the geometric deep learning research area using these two concepts has emerged. By combining these two fields, this article aims to present the relationships between submanifolds of complex manifolds with the help of graphs. In this paper, we build some directed networks by identifying with submanifolds of almost Hermitian manifolds. Moreover, we give some results and relations among holomorphic submanifolds, totally real submanifolds, CR-submanifolds, slant submanifolds, semi-slant submanifolds, hemi-slant submanifolds, and bi-slant submanifolds in almost Hermitian manifolds in terms of graph theory.

Mod-two APS index and domain-wall fermion

We show a mathematical relation between the mod-two Atiyah–Patodi–Singer (APS) index of a massless Dirac operator and massive domain-wall fermion determinant. The domain-wall fermion is given on a closed manifold, which is extended from the original manifold with boundary, where we instead give a fermion mass term changing its sign at the location of the original boundary. This new setup does not need the APS boundary condition, which is non-local. A mathematical proof of equivalence between the two different formulations is given by two different evaluations of the same index of a Dirac operator on a higher-dimensional manifold. The domain-wall fermion allows us to separate the edge and bulk mode contributions in a natural but not in a gauge invariant way, which offers a straightforward description of the global anomaly inflow.

Geodesic Normal Coordinate-Based Manifold Filtering for Target Detection

Recently, the matrix information geometry (MIG) detector, which characterizes sample data as a Hermitian positive definite (HPD) matrix located on the HPD manifold, was rapidly developed and demonstrated extraordinary performance in numerous applications, especially in heterogeneous clutter backgrounds. In this paper, the geodesic normal coordinate (GNC)-based manifold filter is proposed to improve the detection performance of the MIG detector in strong clutter backgrounds. Using the GNC system, the distribution of target echoes and clutter on the high-dimensional manifold can be visualized and analyzed. Moreover, by exploiting the information concerning the distribution of matrices, the manifold filter is proposed to enhance target echoes and suppress strong clutter. Then, the manifold-filter-based MIG detector is designed, and its superiority is theoretically analyzed. The actual clutter data is utilized to verify the effectiveness of the proposed method. The results show that the proposed manifold filter achieves a signal-to-clutter ratio improvement of more than 5 dB over the existing MIG detectors.

Embedding non-arithmetic hyperbolic manifolds

This paper shows that many hyperbolic manifolds obtained by glueing arithmetic pieces embed into higher-dimensional hyperbolic manifolds as codimension-one totally geodesic submanifolds. As a consequence, many Gromov--Pyatetski-Shapiro and Agol--Belolipetsky--Thomson non-arithmetic manifolds embed geodesically. Moreover, we show that the number of commensurability classes of hyperbolic manifolds with a representative of volume $\leq v$ that bounds geometrically is at least $v^{Cv}$, for $v$ large enough.

Classification of audio signals using spectrogram surfaces and extrinsic distortion measures

Representation of one-dimensional (1D) signals as surfaces and higher-dimensional manifolds reveals geometric structures that can enhance assessment of signal similarity and classification of large sets of signals. Motivated by this observation, we propose a novel robust algorithm for extraction of geometric features, by mapping the obtained geometric objects into a reference domain. This yields a set of highly descriptive features that are instrumental in feature engineering and in analysis of 1D signals. Two examples illustrate applications of our approach to well-structured audio signals: Lung sounds were chosen because of the interest in respiratory pathologies caused by the coronavirus and environmental conditions; accent detection was selected as a challenging speech analysis problem. Our approach outperformed baseline models under all measured metrics. It can be further extended by considering higher-dimensional distortion measures. We provide access to the code for those who are interested in other applications and different setups (Code: https://github.com/jeremy-levy/Classification-of-audio-signals-using-spectrogram-surfaces-and-extrinsic-distortion-measures).

A Latent Encoder Coupled Generative Adversarial Network (LE-GAN) for Efficient Hyperspectral Image Super-Resolution

Realistic hyperspectral image (HSI) super-resolution (SR) techniques aim to generate a high-resolution (HR) HSI with higher spectral and spatial fidelity from its low-resolution (LR) counterpart. The generative adversarial network (GAN) has proven to be an effective deep learning framework for image super-resolution. However, the optimisation process of existing GAN-based models frequently suffers from the problem of mode collapse, leading to the limited capacity of spectral-spatial invariant reconstruction. This may cause the spectral-spatial distortion on the generated HSI, especially with a large upscaling factor. To alleviate the problem of mode collapse, this work has proposed a novel GAN model coupled with a latent encoder (LE-GAN), which can map the generated spectral-spatial features from the image space to the latent space and produce a coupling component to regularise the generated samples. Essentially, we treat an HSI as a high-dimensional manifold embedded in a latent space. Thus, the optimisation of GAN models is converted to the problem of learning the distributions of high-resolution HSI samples in the latent space, making the distributions of the generated super-resolution HSIs closer to those of their original high-resolution counterparts. We have conducted experimental evaluations on the model performance of super-resolution and its capability in alleviating mode collapse. The proposed approach has been tested and validated based on two real HSI datasets with different sensors (i.e. AVIRIS and UHD-185) for various upscaling factors and added noise levels, and compared with the state-of-the-art super-resolution models (i.e. HyCoNet, LTTR, BAGAN, SR- GAN, WGAN).

Locally linear embedding vote: A novel filter method for feature selection

Feature selection algorithm based on the locally linear embedding (LLE) cannot effectively detect the changes in the neighborhood of data and cannot take full advantage of the graph-preserving ability. To address these issues, a novel filter method integrated LLE, named LLE vote, is proposed. In the proposed algorithm, we first construct two graphs of each feature through computing the local structure and sparse structure of data in high-dimensional manifold, and then a novel metric criterion is employed to rank the features by measuring the difference between the reconstructed feature and the corresponding original input. Extensive experiments are carried out on benchmark fault data set and the other kind of data from our own laboratory, and the experimental results not only demonstrate the effectiveness of the proposed method, but also indicate that the LLE vote outperforms the existing state-of-art methods.

Fracton physics of spatially extended excitations. II. Polynomial ground state degeneracy of exactly solvable models

Generally, ``fracton'' topological orders are referred to as gapped phases that support \textit{point-like topological excitations} whose mobility is, to some extent, restricted. In our previous work [Phys. Rev. B 101, 245134 (2020)], a large class of exactly solvable models on hypercubic lattices are constructed. In these models, \textit{spatially extended excitations} possess generalized fracton-like properties: not only mobility but also deformability is restricted. As a series work, in this paper, we proceed further to compute ground state degeneracy (GSD) in both isotropic and anisotropic lattices. We decompose and reconstruct ground states through a consistent collection of subsystem ground state sectors, in which mathematical game ``coloring method'' is applied. Finally, we are able to systematically obtain GSD formulas (expressed as $\log_2 GSD$) which exhibit diverse kinds of polynomial dependence on system sizes. For example, the GSD of the model labeled as $[0,1,2,4]$ in four dimensional isotropic hypercubic lattice shows $ 12L^2-12L+4$ dependence on the linear size $L$ of the lattice. Inspired by existing results [Phys. Rev. X 8, 031051 (2018)], we expect that the polynomial formulas encode geometrical and topological fingerprints of higher-dimensional manifolds beyond toric manifolds used in this work. This is left to future investigation.

Fridman Function, Injectivity Radius Function, and Squeezing Function

Very recently, the Fridman function of a complex manifold X has been identified as a dual of the squeezing function of X. In this paper, we prove that the Fridman function for certain hyperbolic complex manifold X is bounded above by the injectivity radius function of X. This result also suggests us to use the Fridman function to extend the definition of uniform thickness to higher dimensional hyperbolic complex manifolds. We also establish an expression for the Fridman function (with respect to the Kobayashi metric) when $$X = \mathbb {D} \diagup \varGamma $$ and $$\varGamma $$ is a torsion-free discrete subgroup of isometries on the standard open unit disk $$\mathbb {D}$$ . Hence explicit formulae of the Fridman functions for the annulus $$A_r$$ and the punctured disk $$\mathbb {D}^*$$ are derived. These are the first explicit non-constant Fridman functions. Finally, we explore the boundary behavior of the Fridman functions (with respect to the Kobayashi metric) and the squeezing functions for regular type hyperbolic Riemann surfaces and planar domains, respectively.

Counterexamples in synchronization: Pathologies of consensus seeking gradient descent flows on surfaces

Certain consensus seeking multi-agent systems can be formulated as gradient descent flows of a disagreement function. We study how known pathologies of gradient descent flows in Euclidean spaces carry over to consensus seeking systems that evolve on nonlinear manifolds. In particular, we show that the norms of agent states can diverge to infinity, but this will not happen if the manifold is the boundary of a convex set. Moreover, the system can be initialized arbitrarily close to consensus without converging to it, but this will not happen if the manifold is analytic. For analytic manifolds, consensus is asymptotically stable. This last result summarizes a number of previous findings in the literature on generalizations of the well-known Kuramoto model to high-dimensional manifolds.

WSN Localization Using Similarity-Based Ricci Flow Embedding

In low-cost Wireless Sensor Networks (WSN), node localization in a noisy environment is a significant problem as additional hardware like GPS is avoided due to high cost. The pairwise distances between sensor nodes are estimated using the Received Signal Strength Indicator (RSSI) because of the low computational complexity and cost-effectiveness. However, the RSSI distance estimates are rough and fluctuate owing to various environmental conditions like noise, interference, and fading. This results in non-Euclidean pairwise distance estimates. To compute the actual inter-node distances, we consider the manifold assumption that the nodes embedded in the low dimensional space apparently lie in an unknown high dimensional manifold. The inter-node distances can be estimated by reducing the dimension through manifold learning. This work proposes a localization technique based on Similarity-Based Ricci Flow Embedding (SBRFE), an unsupervised manifold learning method that reduces the non-Euclidean artefacts of a distance matrix and converts it into a Euclidean one. Each inter-node distance is considered independently, and the corresponding sectional curvature is updated iteratively till all the distances become Euclidean. Experiment results show that the SBRFE algorithm outperforms all the conventional manifold learning algorithms and can localize all the sensor nodes with an increased accuracy up to 94.55%.

Residual finiteness and Abelian subgroup separability of some high dimensional graph manifolds

We generalize $3$-manifolds supporting non-positively curved metric to construct manifolds which have the following properties : (1) They are not locally $\mathrm{CAT}(0)$. (2) Their fundamental groups are residually finite. (3) They have subgroup separability for some abelian subgroups.