PERCEPT detects changes in high-dimensional data streams using topological data analysis.
problem Detecting changes in high-dimensional data streams, especially when embedded in a low-dimensional space.
method Leverages topological data analysis to learn embedded topology as a point cloud via persistence diagrams, then applies non-parametric monitoring for detecting changes.
result Demonstrates efficient detection of online changes from high-dimensional data streams.
Topological manifolds can be embedded flatly in high-dimensional Euclidean space and are locally retracts.
problem Embedding and retraction of topological manifolds in Euclidean spaces.
method Locally flat embedding and retraction of manifolds in high-dimensional Euclidean space.
result Every topological n-manifold can be embedded locally flatly in R2n+1 and is a retract of some neighborhood in R2n+1. A novel method extracts topological features from word embeddings for text classification.
problem High dimensional and noisy text representations in natural language processing.
method Persistent homology for topological data analysis on word embeddings.
result Topological features outperform conventional text mining features on long textual documents.
Ellipsoids approach Gaussian distribution in high dimensions.
problem Understanding convergence of high-dimensional ellipsoids to Gaussian spaces.
method Proof of convergence in Gromov's concentration topology.
result Solid ellipsoids converge to Gaussian space in high dimensions.
IVFS simplifies feature selection for high-dimensional data preservation.
problem Maintaining structure and pairwise distances in high-dimensional data.
method IVFS framework based on persistent diagrams from computational topology.
result IVFS well preserves pairwise distances and topological patterns of full data.
Reproduces IVFS for high-dimensional data structure preservation.
problem Preserving high-dimensional data structure in unsupervised feature selection.
method Inspired by random subset method, IVFS maintains data similarity through topological structure.
result IVFS outperforms SPEC and MCFS on most datasets.
Paper proposes a method to improve circular coordinate representation for detecting changes in high-dimensional datasets.
problem Detecting changes in high-dimensional datasets with preserved topological structures.
method Adapt circular coordinate framework using a generalized penalty function instead of an L2 penalty.
result Circular coordinates with generalized penalty can detect changes in high-dimensional datasets under different sampling schemes.
Topological Pontryagin classes are algebraically independent in high-dimensional spaces.
problem Algebraic independence of topological Pontryagin classes.
method Analyzing rationalised cohomology of BTop(d).
result Topological Pontryagin classes are algebraically independent.
The existence theorem for mapping cylinder neighborhoods is discussed as a prototypical example of controlled topology and its applications. The first of a projected series developed from lectures at the Summer School on High-Dimensional Topology, Trieste Italy 2001
Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expand…
New tool helps analyze complex financial data.
problem Difficulty in comprehending high-dimensional financial data.
method Topological Data Analysis Ball Mapper algorithm.
result Shows new way to see detail in financial data.
This is an introductory article on high dimensional knots for the beginners. High dimensional knot theory is an exciting field. It is a field of knot theory, which is one of topology and is connected with many ones. In this article we use few literal expressions, equations, functions, etc. We barely suppose that the re…
Mapper and Ball Mapper tools for complex data analysis.
problem Exploring and visualizing high-dimensional data and scalar functions.
method Combining Mapper and Ball Mapper, adding new features for encoding structure and symmetries.
result A new hybrid algorithm, Mapper on Ball Mapper, for comparing high-dimensional data descriptors.
Geometric framework detects outliers in high-dimensional data.
problem Detecting outliers in high-dimensional data.
method Geometric framework exploiting manifold structure.
result Significant improvement in outlier detection in high-dimensional data.
The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.
problem The Nirenberg problem on high-dimensional spheres with residual mass.
method Analysis of subcritical approximations and blowing up solutions.
result Comprehensive description of blowing up solutions, including blow-up points and rates.
We analyze the landscape of empirical risk minimization for high-dimensional models, predicting phase transitions and critical point properties.
problem Understanding the complexity and structure of high-dimensional empirical risk landscapes.
method Using the Kac-Rice formula, we analyze the expected number of critical points and their spectral properties, providing detailed predictions.
result We derive complete topological phase diagrams for the phase retrieval problem, predicting BBP-type transitions and critical point stability.
ShapeVis visualizes high-dimensional data efficiently.
problem Scalability issues with Mapper in high-dimensional data.
method Landmarks, weighted witness-graph, induced maps, modularity-based pruning.
result ShapeVis scales to millions of data points while preserving visualization quality.
High-dimensional handlebodies are shown to be products of simpler shapes.
problem Understanding the structure of high-dimensional handlebodies.
method Introducing Kirby diagrams to simplify the structure of k-handlebodies. result High-dimensional handlebodies are products of simpler shapes.
An introduction to the applications of algebraic surgery to the structure theory of high-dimensional topological manifolds.
Some generalizations and variations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are cyclic, then these surfaces are topologically unknotted. Using a twist-spinning construction from high-dimensional knot theory, we constr…
The paper analyzes diffusion condensation for data geometry and topology.
problem Understanding the geometry and topology of high-dimensional data.
method Time-inhomogeneous diffusion process with geometric, spectral, and topological analysis.
result The condensation process defines intrinsic condensation homology and ambient persistent homology.
Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to connect persistent homology with machine learning techniques. We contribute a kernel…
Paper explores applying TDA to text classification, improving model performance.
problem Applying TDA to text classification is challenging due to the complexity of text geometry.
method Used word embeddings and TF-IDF vectors to extract topological features from text.
result Topological features improve classification results, especially in ensemble models.
With the rapid adoption of machine learning techniques for large-scale applications in science and engineering comes the convergence of two grand challenges in visualization. First, the utilization of black box models (e.g., deep neural networks) calls for advanced techniques in exploring and interpreting model behavio…
Proposes using continuum percolation to analyze data manifolds and improve generative models.
problem Disentangling geometric support from probability distributions in high-dimensional data.
method Establishes a correspondence between topological phase transitions of random geometric graphs and data manifolds, using Percolation Shift metric.
result Demonstrates that Percolation Shift metric captures structural pathologies like mode collapse and guides training to prevent manifold shrinkage and improve fidelity.
Survey of embedding methods for high-dimensional and network data.
problem Embedding high-dimensional and nonlinear data structures in a lower-dimensional space.
method Survey of various embedding methods including principal curves, multidimensional scaling, graph-based methods, and topological embeddings.
result Discussion of the pros and cons of algorithmic machine learning and statistical modeling approaches.
New Einstein metrics found on manifolds with opposite curvature signs.
problem Finding Einstein metrics with opposite curvature signs on manifolds.
method Reviewing and extending previous work on high-dimensional smooth closed manifolds.
result Proved various related results, including new Einstein metrics.
Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every d there are unbounded degree simplicial co…
In the machine learning field, dimensionality reduction is an important task. It mitigates the undesired properties of high-dimensional spaces to facilitate classification, compression, and visualization of high-dimensional data. During the last decade, researchers proposed many new (non-linear) techniques for dimensio…
New families of embeddings in 4-manifolds, topologically trivial but smoothly non-trivial.
problem Constructing non-trivial smooth embeddings of 3-manifolds in 4-manifolds.
method Parameterized families of embeddings, using high-dimensional spheres.
result Embeddings of homology spheres and any 3-manifold in blown-up K3 surfaces.
We provide a way to infer about existence of topological circularity in high-dimensional data sets in Rd from its projection in R2 obtained through a fast manifold learning map as a function of the high-dimensional dataset X and a particular choice of a positive real σ known as band…
Study on stable Hamiltonian topology finds non-density of certain structures.
problem Non-density of stable hypersurfaces and Hamiltonian structures.
method Proving non-density results for stable hypersurfaces and Hamiltonian structures in various dimensions.
result Non-density of stable hypersurfaces and Hamiltonian structures in specific isotopy and homotopy classes.
We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.
In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer …
We use purely topological methods to prove the semicontinuity of the mod 2 spectrum of local isolated hypersurface singularities in Cn+1, using Seifert forms of high-dimensional non-spherical links, the Levine--Tristram signatures and the generalized Murasugi--Kawauchi inequality obtained in earlier work …
We introduce the notion of an EZ-structure on a group. Delta-hyperbolic groups and CAT(0)-groups have EZ-structures. We show torsion-free groups having an EZ-structure automatically have an action by homeomorphisms on a closed (high-dimensional) ball, which is well-behaved away from a "bad limit set" in the boundary of…
New method integrates topological knowledge into data embeddings.
problem Lack of general tools to incorporate prior topological knowledge into embeddings.
method Introduces new topological losses to topologically regularize data embeddings.
result Natural representation of simple models like clusters and flares.
New methods explain NE embeddings by identifying key variables.
problem Lack of interpretability in NE techniques.
method Combining PCA, Q-residuals, Hotelling's T2, and visualization.
result Identifies discriminatory features not seen in standard approaches.
Regularization preserves topological data structure in autoencoders.
problem Ensuring topological data structure preservation in autoencoders.
method Regularization using Legendre nodes to preserve manifold embedding.
result Regularized autoencoders ensure one-to-one embedding of data manifolds.
New methods improve analysis of single cell RNA sequencing data.
problem High dimensionality and complexity of scRNA-seq data.
method Topological Nonnegative Matrix Factorization (TNMF) and Robust Topological NMF (rTNMF).
result TNMF and rTNMF significantly outperform other NMF-based methods.
A new method for state estimation on complex networks.
problem Reconstructing latent dynamics from multivariate time-series on topological cell complexes.
method Topology-aware state space framework derived from stochastic partial differential equations, with state evolution following heat-like topological diffusion.
result The proposed method successfully recovers latent states and topological structures in real-world networks.
Framework for analyzing dynamic topological changes in point clouds using persistent homology and dynamic optimal transport.
problem Analyzing transient structural reorganizations during dynamic phase transitions in time-evolutionary point clouds.
method Hierarchical dynamic evaluation framework driven by topological and hypergraph reconstruction strategy.
result Combining transport-based alignment with multi-scale entropy diagnostics for dynamic topological analysis.
When doing representation learning on data that lives on a known non-trivial manifold embedded in high dimensional space, it is natural to desire the encoder to be homeomorphic when restricted to the manifold, so that it is bijective and continuous with a continuous inverse. Using topological arguments, we show that wh…
Constructing VAE Latent Spaces with Prescribed Topology
problem Resolving topological mismatch in VAEs for non-Euclidean data
method A constructive framework for product covering spaces
result Topology-aware latent representations with closed-form KL divergences
JORC-UMAP improves UMAP by incorporating geometric and topological priors.
problem UMAP's local Euclidean distance assumption fails to capture intrinsic manifold geometry, leading to topological tearing and structural collapse.
method JORC-UMAP introduces Ollivier-Ricci curvature as a geometric prior and Jaccard similarity as a topological prior to reinforce edges and reduce redundant links.
result JORC-UMAP reduces tearing and collapse more effectively than standard UMAP and other DR methods, as measured by SVM accuracy and triplet preservation scores.
Ensemble decoders to capture latent space topology in deep generative models.
problem Topological mismatch between latent space geometry and data manifolds.
method Using ensembles of decoders to compute geodesics on the expected manifold.
result Ensemble approach provides a simple and reliable way to capture model uncertainty in latent space.
A novel topological method analyzes fMRI data over time.
problem Analyzing time-varying fMRI data due to noise and person-to-person variation.
method Encoding each time point as a persistence diagram of topological features.
result Time-varying persistence diagrams can cluster participants and study brain state trajectories.
Grid security and open markets are two major smart grid goals. Transparency of market data facilitates a competitive and efficient energy environment, yet it may also reveal critical physical system information. Recovering the grid topology based solely on publicly available market data is explored here. Real-time ener…