Adaptive template systems improve feature extraction from persistence diagrams for machine learning.
problem Feature extraction from persistence diagrams for machine learning.
method Adaptive template systems using CDER, GMM, and HDBSCAN algorithms.
result Adaptive template systems yield competitive and often superior results in classification tasks.
Paper introduces template functions for featurizing persistence diagrams.
problem Featurizing persistence diagrams for machine learning.
method Characterizes compactness, constructs dense subsets of continuous functions.
result Template functions enable supervised learning with persistence diagrams.
Persistent homology reveals geometric features of metric spaces, especially geodesic circles.
problem Detecting geometric features in metric spaces using persistent homology.
method Analyzing algebraic elements (footprints) in persistent homology of metric spaces and subspace.
result Higher-dimensional persistent homology captures lower-dimensional geometric features.
Persistence landscapes map diagrams into function spaces for statistical and machine learning applications.
problem Mapping persistence diagrams into function spaces for statistical and machine learning.
method Introducing persistence landscapes, weighted persistence landscapes, and Poisson-weighted persistence landscape kernels.
result Persistence landscapes allow for the application of statistical and machine learning tools, and are stable and invertible.
Develops robust persistence diagrams using kernel methods.
problem Persistence diagrams are sensitive to data perturbations.
method Constructs robust persistence diagrams from superlevel filtrations of robust density estimators using reproducing kernels.
result Robust persistence diagrams are consistent estimators in bottleneck distance.
A new method compares persistent cycles in topological data.
problem Comparing persistent homology representations of two spaces.
method Direct comparison of individual persistent cycles based on persistence intervals and spatial placement.
result Demonstrated the effectiveness of the method in topological inference.
STRAND: A single representation for hypothesis testing and vectorisation of persistence diagrams
problem Comparing persistence diagrams
method Survival topological representation analysis
result Non-parametric two-sample test with calibrated Type I error and high power
Regularizes persistent homology gradients for neural network integration.
problem Ill-posed inverse problem in computing gradients of persistent homology.
method Regularization through a grouping term to define gradients for larger entities.
result Ensures gradients are defined with respect to larger entities, not individual points.
Unified pipeline classifies time series using complex networks and persistent homology.
problem Classifying univariate time series using various graph constructions and metrics.
method Time series to graph, graph to dissimilarity matrix, filtration to persistence diagrams, vectorization to features.
result Persistence-based features are robust to noise and optimal graph type depends on signal structure.
Persistent homology enhances graph classification by capturing long-range graph properties.
problem Lack of formal assessment of persistent homology in graph learning.
method Brief introduction and theoretical discussion of persistent homology in graph context, followed by empirical analysis.
result Persistent homology improves graph classification, especially for data with prominent topological structures.
New stable distance for classifying materials from point cloud data.
problem Classifying materials from noisy and sparse data.
method A new distance on persistence diagrams for matching and comparing topological features.
result Stability of the new distance provides theoretical justification for its use in materials classification.
Persistent homology reveals a topological signature of grokking in neural networks.
problem Understanding how neural networks learn and generalize from modular arithmetic tasks.
method Persistent homology on point clouds derived from embedding matrices of models trained on modular arithmetic.
result A sharp increase in first homology persistence indicates grokking, with a dominant long-lived topological feature and structured secondary features.
Develops 2-categorical methods for multi-parameter persistence.
problem Fundamental limitations of traditional persistence modules.
method 2-categorical structures to capture hierarchical interactions.
result New invariants effectively characterize multidimensional topological features.
Paper proves k-means clustering works on persistence diagrams.
problem Complex geometry of persistence diagram space.
method Proves convergence of k-means on persistence diagram space. result Performance of k-means on persistence diagrams and measures is superior. A method for vectorizing persistence diagrams simplifies topological data analysis.
problem Challenges in integrating persistence diagrams into machine learning pipelines.
method Quantized Persistence and Integral transforms of Diagrams (Qupid) using binning and discrete transforms.
result Qupid preserves highly competitive performances compared to state-of-the-art methods across various classification tasks.
Persistence diagrams are two-dimensional plots that summarize the topological features of functions and are an important part of topological data analysis. A problem that has received much attention is how deal with sets of persistence diagrams. How do we summarize them, average them or cluster them? One approach -- th…
Develops persistent Khovanov homology for tangles.
problem Lack of local topological features in evolutionary Khovanov homology for knots and links.
method Introduces a new mathematical framework using persistent Khovanov homology of tangles, employing functor and planar algebra.
result Provides a new method to characterize local features in curve-type data.
Topology-GS improves 3D GS for better structural and feature integrity.
problem Compromised pixel-level and feature-level integrity in 3D GS.
method Incorporates Local Persistent Voronoi Interpolation (LPVI) and PersLoss based on persistent homology.
result Topology-GS outperforms existing methods in PSNR, SSIM, and LPIPS metrics.
The paper examines how long-memory dynamics, rough-volatility, and persistence affect equity volatility forecasting.
problem The study investigates how long-memory dynamics, rough-volatility, and persistence impact equity volatility forecasting.
method The paper combines semiparametric long-memory estimation, rough-volatility diagnostics, and structured forecasting regressions.
result Persistence measures improve out-of-sample volatility forecasts, particularly during periods of elevated market volatility and in volatility-managed portfolio applications.
Optimizes wavelets for graph classification using spectral wavelet signatures and persistence diagrams.
problem Graph classification with geometric properties encoded in persistence diagrams.
method Optimizes spectral wavelets for graph datasets to capture best-suited features for classification.
result Competitive performance in graph classification problems compared to other persistence-based architectures.
Unreduced PDs can perform similarly to reduced PDs in machine learning tasks.
problem Ignoring much of the information in persistence diagrams in machine learning pipelines.
method Developed methods to generate topological feature vectors from unreduced boundary matrices.
result Unreduced PDs can perform on par with, and sometimes outperform, fully-reduced PDs in machine learning tasks.
New method uses topology to analyze data bandwidth.
problem Analyzing the evolution of topological features in changing data.
method Persistence Flamelets, a multiscale version of Persistence Landscape.
result Persistence Flamelets can provide insights into KDE bandwidth parameters.
PLLay adds topological layers to deep learning models efficiently.
problem Efficiently incorporating topological features into deep learning models.
method Persistence landscapes for differentiable topological features.
result PLLay improves model learnability and robustness.
Study how knots occupy space using topological methods.
problem Understanding how knots occupy a volume of space.
method Statistical analysis of persistent homology features from Vietoris-Rips complexes.
result Existence of correlations between geometric and topological features of knots.
Sparse-TDA combines TDA and sparse sampling for multi-way classification.
problem Reconstructing shapes from high-dimensional data for multi-way classification.
method Sparse-TDA algorithm that selects sparse pixel samples from persistent features using QR pivoting.
result Sparse-TDA demonstrates promising performance on human posture recognition and image texture classification.
MuRiT efficiently computes multi-parameter persistence barcodes.
problem Efficient computation of multi-parameter persistent homology.
method Vietoris-Rips transformation to reduce multi-parameter to single-parameter computation.
result MuRiT computes pathwise persistence barcodes for multi-filtered flag complexes.
Study on embedding persistence diagrams into Hilbert spaces, focusing on metric distortion.
problem Understanding metric properties of persistence diagrams in Hilbert spaces.
method Investigate embedding persistence diagrams into separable Hilbert spaces using bi-Lipschitz maps.
result Impossible to find a bi-Lipschitz embedding into finite-dimensional Hilbert spaces.
Generates random persistence diagrams for data analysis.
problem Generating random persistence diagrams for data analysis.
method Based on pairwise interacting point processes and RJ-MCMC algorithm.
result Demonstrates the efficacy and utility of RPDG in materials science.
New feature map for topological data analysis improves classification performance.
problem Lack of effective feature maps for topological data analysis.
method Realize barcodes as paths in a vector space, compute path signature, resulting in a feature map.
result Achieves state-of-the-art results on classification benchmarks.
A new graph classification method using persistent homology.
problem Graph classification with graph connectivity structure.
method Learnable filter function for persistent homology computation.
result Empirically, the method compares favorably to previous techniques.
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful re…
A new method for stable vector representation of persistence diagrams.
problem Finding a stable vector representation of persistence diagrams for ML tasks.
method Persistence B-spline Grid (PBSG) based on data fitting.
result The PBSG method is stable with respect to the 1-Wasserstein distance metric.
New clustering algorithm uses persistent homology for space-time data.
problem Clustering space-time data without labeled examples.
method Persistent homology for topological data analysis, analyzing data at multiple resolutions.
result The algorithm distinguishes true features from noise based on persistence.
A new method uses persistent homology to assess auto-encoders' latent manifold quality.
problem Chaos in auto-encoders' latent manifold and failure of current distance measures.
method Persistent Homology for Wasserstein Auto-Encoders (PHom-WAE).
result PHom-WAE improves auto-encoders' performance in credit card transaction data.
New method uses topological features for chatter detection in turning processes.
problem Chatter detection in turning processes using complex dynamical systems.
method Embedding time series as point clouds, using persistence diagrams, and applying machine learning classifiers.
result TDA-based features yield high accuracy (97%) for two out of four cutting configurations.
Differentiable topology layer for machine learning models.
problem Regularizing and incorporating topological priors in machine learning models.
method Computes persistent homology based on level set and edge-based filtrations.
result Demonstrates three novel applications of the topology layer in machine learning.
An algorithm preserves topological features in dimensionality reduction.
problem Preserving topological features in dimensionality reduction.
method Simulated annealing for finding a linear projection preserving persistent homology.
result Measures of topological equivalence between filtrations.
PHom-GeM uses topological features to assess generative models.
problem Generative models produce chaotic distributions during training.
method Persistent Homology for Generative Models (PHom-GeM) minimizes an objective function between true and reconstructed distributions.
result PHom-GeM is a topological distance measure for generative models.
New method predicts Alzheimer's risk with individual uncertainty estimates.
problem Predicting conversion from mild cognitive impairment to Alzheimer's disease.
method Persistent homology of clinical trajectories combined with stacking ensemble.
result Pipeline achieves high accuracy and individual-level uncertainty quantification.
Study shows how non-uniform scaling affects persistence diagrams.
problem Stability of persistence diagrams under non-uniform scaling.
method Explicit bounds on bottleneck distance derived for Euclidean scaling.
result Explicit bounds on the stability of persistence diagrams under non-uniform scaling.
We compute persistent homology using an intrinsic metric derived from density.
problem Estimating topological features from high-dimensional data.
method Density-based metric learning for persistent homology.
result Persistent homology converges to intrinsic manifold metric.
ChainNet uses blockchain graph topological features to predict cryptocurrency prices.
problem Capturing network dynamics of blockchain graphs to predict cryptocurrency prices.
method Persistent homology for computing topological features of blockchain graphs.
result Topological features outperform standard graph features in predicting cryptocurrency price dynamics.
PHLP uses persistent homology to interpret graph link prediction.
problem Interpreting why graph neural network models perform well in link prediction.
method Employing persistent homology to analyze graph topology and extract features.
result PHLP outperforms state-of-the-art models on most benchmark datasets.
Introduces TSI, a variance-based measure for persistence barcodes.
problem Capturing structural variability in persistence barcodes.
method Variance-based scalar measure, TSI, and complementary TSigI.
result TSI captures structural variability complementary to entropy.
New models for graph sequences capture link and community persistence.
problem Difficulties in extending block model results to graph sequences.
method Two models for graph sequences capturing link and community persistence, with efficient inference algorithms.
result Validated suitability of proposed models and methods on synthetic and real instances.
The paper introduces a scale law for detecting distribution shift in high-dimensional embeddings.
problem Detecting changes in high-dimensional embedding streams.
method The paper presents a scale law that constrains moment-based statistics for detecting distribution shift. It also introduces a calibration rule for kernel tests.
result The scale law predicts the optimal bandwidth for kernel tests, leading to superior performance in detecting distribution shifts.
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.
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.