Topological Regularity Research Articles

Deep learning of multimodal networks with topological regularization for drug repositioning

MotivationComputational techniques for drug-disease prediction are essential in enhancing drug discovery and repositioning. While many methods utilize multimodal networks from various biological databases, few integrate comprehensive multi-omics data, including transcriptomes, proteomes, and metabolomes. We introduce STRGNN, a novel graph deep learning approach that predicts drug-disease relationships using extensive multimodal networks comprising proteins, RNAs, metabolites, and compounds. We have constructed a detailed dataset incorporating multi-omics data and developed a learning algorithm with topological regularization. This algorithm selectively leverages informative modalities while filtering out redundancies.ResultsSTRGNN demonstrates superior accuracy compared to existing methods and has identified several novel drug effects, corroborating existing literature. STRGNN emerges as a powerful tool for drug prediction and discovery. The source code for STRGNN, along with the dataset for performance evaluation, is available at https://github.com/yuto-ohnuki/STRGNN.git.

AIGO-DTI: Predicting Drug-Target Interactions Based on Improved Drug Properties Combined with Adaptive Iterative Algorithms.

Artificial intelligence-based methods for predicting drug-target interactions (DTIs) aim to explore reliable drug candidate targets rapidly and cost-effectively to accelerate the drug development process. However, current methods are often limited by the topological regularities of drug molecules, making them difficult to generalize to a broader chemical space. Additionally, the use of similarity to measure DTI network links often introduces noise, leading to false DTI relationships and affecting the prediction accuracy. To address these issues, this study proposes an Adaptive Iterative Graph Optimization (AIGO)-DTI prediction framework. This framework integrates atomic cluster information and enhances molecular features through the design of functional group prompts and graph encoders, optimizing the construction of DTI association networks. Furthermore, the optimization of graph structure is transformed into a node similarity learning problem, utilizing multihead similarity metric functions to iteratively update the network structure to improve the quality of DTI information. Experimental results demonstrate the outstanding performance of AIGO-DTI on multiple public data sets and label reversal data sets. Case studies, molecular docking, and existing research validate its effectiveness and reliability. Overall, the method proposed in this study can construct comprehensive and reliable DTI association network information, providing new graphing and optimization strategies for DTI prediction, which contribute to efficient drug development and reduce target discovery costs.

Topological regularization on numerical simulations of the advection equation

Topological regularization on numerical simulations of the advection equation

Incorporating Topological Priors Into Low-Dimensional Visualizations Through Topological Regularization

Incorporating Topological Priors Into Low-Dimensional Visualizations Through Topological Regularization

Topological regularization of networks in temporal lobe epilepsy: a structural MRI study.

Patients with temporal lobe epilepsy (TLE) often exhibit neurocognitive disorders; however, we still know very little about the pathogenesis of cognitive impairment in patients with TLE. Therefore, our aim is to detect changes in the structural connectivity networks (SCN) of patients with TLE. Thirty-five patients with TLE were compared with 47 normal controls (NC) matched according to age, gender, handedness, and education level. All subjects underwent thin-slice T1WI scanning of the brain using a 3.0 T MRI. Then, a large-scale structural covariance network was constructed based on the gray matter volume extracted from the structural MRI. Graph theory was then used to determine the topological changes in the structural covariance network of TLE patients. Although small-world networks were retained, the structural covariance network of TLE patients exhibited topological irregularities in regular architecture as evidenced by an increase in the small world properties (p < 0.001), normalized clustering coefficient (p < 0.001), and a decrease in the transfer coefficient (p < 0.001) compared with the NC group. Locally, TLE patients showed a decrease in nodal betweenness and degree in the left lingual gyrus, right middle occipital gyrus and right thalamus compared with the NC group (p < 0.05, uncorrected). The degree of structural networks in both TLE (Temporal Lobe Epilepsy) and control groups was distributed exponentially in truncated power law. In addition, the stability of random faults in the structural covariance network of TLE patients was stronger (p = 0.01), but its fault tolerance was lower (p = 0.03). The objective of this study is to investigate the potential neurobiological mechanisms associated with temporal lobe epilepsy through graph theoretical analysis, and to examine the topological characteristics and robustness of gray matter structural networks at the network level.

Topological regularity of isoperimetric sets in PI spaces having a deformation property

We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we prove that isoperimetric sets are open, satisfy boundary density estimates and, under a uniform lower bound on the volumes of unit balls, are bounded. Our results apply, in particular, to the class of possibly collapsed $\mathrm {RCD}(K,N)$ spaces. As a consequence, the rigidity in the isoperimetric inequality on possibly collapsed $\mathrm {RCD}(0,N)$ spaces with Euclidean volume growth holds without the additional assumption on the boundedness of isoperimetric sets. Our strategy is of interest even in the Euclidean setting, as it simplifies some classical arguments.

Universal structural patterns in sparse recurrent neural networks

Sparse neural networks can achieve performance comparable to fully connected networks but need less energy and memory, showing great promise for deploying artificial intelligence in resource-limited devices. While significant progress has been made in recent years in developing approaches to sparsify neural networks, artificial neural networks are notorious as black boxes, and it remains an open question whether well-performing neural networks have common structural features. Here, we analyze the evolution of recurrent neural networks (RNNs) trained by different sparsification strategies and for different tasks, and explore the topological regularities of these sparsified networks. We find that the optimized sparse topologies share a universal pattern of signed motifs, RNNs evolve towards structurally balanced configurations during sparsification, and structural balance can improve the performance of sparse RNNs in a variety of tasks. Such structural balance patterns also emerge in other state-of-the-art models, including neural ordinary differential equation networks and continuous-time RNNs. Taken together, our findings not only reveal universal structural features accompanying optimized network sparsification but also offer an avenue for optimal architecture searching.

Topological Regularity for Solutions to the Generalised Hopf Equation

The generalised Hopf equation is the first order nonlinear equation defined on a planar domain Omega subset {mathbb {C}}, with data Phi a holomorphic function and eta ge 1 a positive weight on Omega , hwhw¯¯η(w)=Φ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} h_w\\,\\overline{h_{\\overline{w}}}\\,\\eta (w) = \\Phi . \\end{aligned}$$\\end{document}The Hopf equation is the special case eta (w)={tilde{eta }}(h(w)) and reflects that h is harmonic with respect to the conformal metric sqrt{{tilde{eta }}(z)}|dz|, usually eta is the hyperbolic metric. This article obtains conditions on the data to ensure that a solution is open and discrete. We also prove a strong uniqueness result.

Topological Regularization for Representation Learning via Persistent Homology

Generalization is challenging in small-sample-size regimes with over-parameterized deep neural networks, and a better representation is generally beneficial for generalization. In this paper, we present a novel method for controlling the internal representation of deep neural networks from a topological perspective. Leveraging the power of topology data analysis (TDA), we study the push-forward probability measure induced by the feature extractor, and we formulate a notion of “separation” to characterize a property of this measure in terms of persistent homology for the first time. Moreover, we perform a theoretical analysis of this property and prove that enforcing this property leads to better generalization. To impose this property, we propose a novel weight function to extract topological information, and we introduce a new regularizer including three items to guide the representation learning in a topology-aware manner. Experimental results in the point cloud optimization task show that our method is effective and powerful. Furthermore, results in the image classification task show that our method outperforms the previous methods by a significant margin.

Topology as a didactic tool for the development of students’ mathematical thinking

The article is aimed at revealing main methodical aspects of teaching topology for students of mathematical specialties of pedagogical universities. The article emphasizes the special importance of topology as a didactic tool for the development of the mathematical worldview of students of mathematical specialties at universities. Topology is a section of mathematics studying the properties of figures (or spaces) which they preserve in continuous deformations such as stretching, compression or bending. Continuous deformation takes place when there is no ruptures (that is integrity of a figure is not broken) or patching. Topological problems are solved through methods that are radically different from methods of geometry. The studying of topology plays an important role due to broad theoretic and practical application of topological regularities in such branches of science as physics, astronomy, anthology, cybernetics, geology, architecture, geography, cosmology, genetics etc. Topology is particularly useful in improving students’ intellectual and cognitive abilities and developing their mathematical skills.

Fine-grained, nonlinear registration of live cell movies reveals spatiotemporal organization of diffuse molecular processes.

We present an application of nonlinear image registration to align in microscopy time lapse sequences for every frame the cell outline and interior with the outline and interior of the same cell in a reference frame. The registration relies on a subcellular fiducial marker, a cell motion mask, and a topological regularization that enforces diffeomorphism on the registration without significant loss of granularity. This allows spatiotemporal analysis of extremely noisy and diffuse molecular processes across the entire cell. We validate the registration method for different fiducial markers by measuring the intensity differences between predicted and original time lapse sequences of Actin cytoskeleton images and by uncovering zones of spatially organized GEF- and GTPase signaling dynamics visualized by FRET-based activity biosensors in MDA-MB-231 cells. We then demonstrate applications of the registration method in conjunction with stochastic time-series analysis. We describe distinct zones of locally coherent dynamics of the cytoplasmic protein Profilin in U2OS cells. Further analysis of the Profilin dynamics revealed strong relationships with Actin cytoskeleton reorganization during cell symmetry-breaking and polarization. This study thus provides a framework for extracting information to explore functional interactions between cell morphodynamics, protein distributions, and signaling in cells undergoing continuous shape changes. Matlab code implementing the proposed registration method is available at https://github.com/DanuserLab/Mask-Regularized-Diffeomorphic-Cell-Registration.

Distances and isomorphism between networks: stability and convergence of network invariants

We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional representations of networks, or networks for short, specialize in the finite setting to (possibly asymmetric) adjacency matrices and derived representations such as distance or kernel matrices. Existing literature utilizing these constructions cannot, however, benefit from continuous formulations because the continuum limits of finite networks under this distance are not well-understood. For example, while there are currently numerous persistent homology methods on finite networks, it is unclear if these methods produce well-defined persistence diagrams in the infinite setting. We resolve this situation by introducing the collection of compact networks that arises by taking continuum limits of finite networks and developing sampling results showing that this collection admits well-defined persistence diagrams. Compared to metric spaces, the isomorphism class of the generalized Gromov-Hausdorff distance over networks is rather complex, and contains representatives having different cardinalities and different topologies. We provide an exact characterization of a suitable notion of isomorphism for compact networks as well as alternative, stronger characterizations under additional topological regularity assumptions. Toward data applications, we describe a unified framework for developing quantitatively stable network invariants, provide basic examples, and cast existing results on the stability of persistent homology methods in this extended framework. To illustrate our theoretical results, we introduce a model of directed circles with finite reversibility and characterize their Dowker persistence diagrams.

LEO Satellite Networks: When Do All Shortest Distance Paths Belong to Minimum Hop Path Set?

A low Earth orbit (LEO) satellite constellationnetwork (SCN) has become a promising solution for nonterrestrial networks. In LEO-SCNs, the shortest distance path (SDP) and the minimum hop path (MHP) are two types of important paths. This letter focuses on the proposition that all the SDPs belong to the MHP set and studies the conditions when the proposition holds or not. Based on the topological regularity and link distance variation patterns, this letter proves several simplified equivalent propositions and derives a discriminant function to judge if the proposition holds in an arbitrary constellation. Simulations verify the judging method and find that all the SDPs belong to the MHP set in constellations with small inclinations (less than 68$^\circ$) or large phasing offsets. The propositions can help to simplify the calculation of the SDP.

Topological regularization with information filtering networks

This paper introduces a novel methodology to perform topological regularization in multivariate probabilistic modeling by using sparse, complex, networks which represent the system’s dependency structure and are called information filtering networks (IFN). This methodology can be directly applied to covariance selection problem providing an instrument for sparse probabilistic modeling with both linear and non-linear multivariate probability distributions such as the elliptical and generalized hyperbolic families. It can also be directly implemented for topological regularization of multicollinear regression. In this paper, I describe in detail an application to sparse modeling with multivariate Student-t. A specific expectation–maximization likelihood maximization procedure over a sparse chordal network representation is proposed for this sparse Student-t case. Examples with real data from stock prices log-returns and from artificially generated data demonstrate applicability, performances, robustness and potentials of this methodology.

Asymptotic topological regularity of CAT(0) spaces

We study asymptotic topological regularity of \({\text {CAT}}(0)\) spaces. We prove that if a purely n-dimensional, proper, geodesically complete \({\text {CAT}}(0)\) space has small volume growth, then it is homeomorphic to the n-dimensional Euclidean space. We also discuss asymptotic geometry of proper, geodesically complete \({\text {CAT}}(0)\) spaces of small volume growth.

Topological regularity of spaces with an upper curvature bound

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits of Riemannian manifolds and deduce a sphere theorem.

On the topology and the boundary of N–dimensional RCD(K,N) spaces

We establish topological regularity and stability of N-dimensional RCD(K,N) spaces (up to a small singular set), also called non-collapsed RCD(K,N) in the literature. We also introduce the notion of a boundary of such spaces and study its properties, including its behavior under Gromov-Hausdorff convergence.

Ballistic random walks in random environment as rough paths: convergence and area anomaly

Annealed functional CLT in the rough path topology is proved for the standard class of ballistic random walks in random environment. Moreover, the `area anomaly', i.e. a deterministic linear correction for the second level iterated integral of the rescaled path, is identified in terms of a stochastic area on a regeneration interval. The main theorem is formulated in more general settings, namely for any discrete process with uniformly bounded increments which admits a regeneration structure where the regeneration times have finite moments. Here the largest finite moment translates into the degree of regularity of the rough path topology. In particular, the convergence holds in the $\alpha$-Holder rough path topology for all $\alpha<1/2$ whenever all moments are finite, which is the case for the class of ballistic random walks in random environment. The latter may be compared to a special class of random walks in Dirichlet environments for which the regularity $\alpha<1/2$ is bounded away from $1/2$, explicitly in terms of the corresponding trap parameter.

Behavioral and physiological correlates of kinetically tracking a chaotic target.

Humans can innately track a moving target by anticipating its future position from a brief history of observations. While ballistic trajectories can be readily extrapolated, many natural and artificial systems are governed by more general nonlinear dynamics and, therefore, can produce highly irregular motion. Yet, relatively little is known regarding the behavioral and physiological underpinnings of prediction and tracking in the presence of chaos. Here, we investigated in lab settings whether participants could manually follow the orbit of a paradigmatic chaotic system, the Rössler equations, on the (x,y) plane under different settings of a control parameter, which determined the prominence of transients in the target position. Tracking accuracy was negatively related to the level of unpredictability and folding. Nevertheless, while participants initially reacted to the transients, they gradually learned to anticipate it. This was accompanied by a decrease in muscular co-contraction, alongside enhanced activity in the theta and beta EEG bands for the highest levels of chaoticity. Furthermore, greater phase synchronization of breathing was observed. Taken together, these findings point to the possible ability of the nervous system to implicitly learn topological regularities even in the context of highly irregular motion, reflecting in multiple observables at the physiological level.

Regularity in Topological Modules

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory called strong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, if M is a topological R-module and m∗∈M∗ is a continuous linear functional on M which is open as a map between topological spaces, then m∗−1(int(B)) is regular open and m∗−1(B) is regular closed, for B any closed-unit zero-neighborhood in R.