Mapper instability quantified for robust graph construction.
problem Variability in Mapper output due to parameter selection.
method Defined an intrinsic instability measure for Mapper outputs.
result Identified local minima of instability for reliable outputs.
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.
Optimizes cover parameter in Mapper algorithm for better visualization.
problem Tuning the cover parameter in Mapper algorithm to generate a ``nice'' graph.
method Optimizes cover by repeatedly splitting using statistical tests and Gaussian mixture model.
result Algorithm generates covers that retain dataset essence while being faster.
The construction of Mapper has emerged in the last decade as a powerful and effective topological data analysis tool that approximates and generalizes other topological summaries, such as the Reeb graph, the contour tree, split, and joint trees. In this paper, we study the parallel analysis of the construction of Mappe…
Mapper classifier improves robustness over CNNs.
problem Improving robustness of classifiers against attacks.
method Mapper algorithm applied to PCA or autoencoders projected data.
result Mapper classifier is immune to gradient-based attacks.
Mapper-GIN simplifies 3D point cloud classification with lightweight structure.
problem Robust 3D point cloud classification under corruption.
method Mapper algorithm for structural decomposition, GIN for graph classification.
result Mapper-GIN achieves competitive accuracy with minimal parameters.
Improved Mapper algorithm for datasets with varying density.
problem Difficulty in tuning resolution for datasets with varying density.
method Generalized cover type and incorporated lens-space density into the cover.
result Graph produced by Mapper converges to Reeb graph of Rips complex.
Bi-filtration stabilizes TDA mapper results under noise.
problem Stability issues in TDA mapper results under data perturbation.
method Introduced bi-filtration approach to stabilize mapper graphs.
result Persistent homology of perturbed data set is 2δ-interleaved with original.
A new Mapper algorithm optimizes data visualization through automatic parameter tuning.
problem Manual parameter tuning and fixed intervals limit the performance of the standard Mapper algorithm.
method Introduces a soft Mapper framework based on Gaussian mixture models for automatic interval construction and optimization via stochastic gradient descent.
result Demonstrates effectiveness in capturing underlying topological structures and identifying distinct subgroups.
New statistical method for estimating Reeb spaces from multivariate data.
problem Stability and convergence of Mapper to Reeb spaces in multivariate and metric spaces.
method Modified Mapper construction with risk bounds for Gromov-Hausdorff distance.
result Risk bounds for estimating Reeb spaces using the modified Mapper.
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.
Mapper merges GNNs with TDA for graph visualisation.
problem Visualising complex graphs for understanding model structure.
method Combining Mapper, GNNs, and PageRank for graph pooling.
result Hierarchical visualisations reveal graph structure and model application.
Node-link diagrams are a popular method for representing graphs that capture relationships between individuals, businesses, proteins, and telecommunication endpoints. However, node-link diagrams may fail to convey insights regarding graph structures, even for moderately sized data of a few hundred nodes, due to visual …
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.
AuToMATo clusters data without tuning parameters, outperforming others.
problem Clustering data efficiently and without manual tuning.
method Combines ToMATo with bootstrapping for density estimation.
result Performs well across various clustering algorithms and applications.
This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.
problem Improving topological inference and visualization of large-scale geometric datasets.
method Proposes a method for learning topologically-faithful covers of geometric datasets using optimization.
result Simplicial complexes obtained from learned covers outperform standard methods in terms of size and representation of large-scale topology.
We improve current instability-based methods for the selection of the number of clusters k in cluster analysis by developing a normalized cluster instability measure that corrects for the distribution of cluster sizes, a previously unaccounted driver of cluster instability. We show that our normalized instability mea…
Modeling HFT interactions reveals market instability.
problem Market instability caused by HFT dynamic coupling.
method Developed a recurrence relations framework to model HFT interactions.
result Unexpected latency and feedback can trigger market instability.
New proof of instability for certain Einstein metrics.
problem Einstein metrics on specific 4-manifolds.
method Proving instability of conformally Kähler, Einstein metrics.
result Proven instability of certain Einstein metrics.
Study stability and instability of Ricci-flat metrics under generalized Ricci flow.
problem Stability and instability of Ricci-flat metrics under Ricci flow.
method Analysis of generalized Ricci flow for Ricci-flat metrics and vanishing 3-forms.
result Dynamical stability and instability results for Ricci-flat metrics and vanishing 3-forms.
Some exotic compact objects possess evanescent ergosurfaces: timelike submanifolds on which a Killing vector field, which is timelike everywhere else, becomes null. We show that any manifold possessing an evanescent ergosurface but no event horizon exhibits a linear instability of a peculiar kind: either there are solu…
Interval Neural Networks detect instabilities in image reconstructions.
problem Detecting instabilities in deep learning image reconstructions.
method Employed uncertainty quantification methods with Interval Neural Networks.
result Interval Neural Networks effectively reveal image reconstruction instabilities.
Note on instabilities in super-time-stepping methods for Heston model.
problem Instabilities in super-time-stepping methods applied to Heston model.
method Exploration of explicit super-time-stepping schemes (RK-Chebyshev, RK-Legendre) for Heston model.
result Relevance of stability remarks beyond super-time-stepping schemes.
The paper explores how word embeddings affect the stability of downstream NLP models.
problem Small changes in training data can cause significant changes in model predictions.
method Empirical and theoretical analysis of embedding instability, including the introduction of eigenspace instability measure.
result Increasing embedding memory can reduce the disagreement in predictions by 5% to 37%.
Binary perceptron's instability linked to replica symmetry breaking.
problem Understanding the relationship between algorithmic instability and replica symmetry breaking in binary perceptron learning.
method Established the connection between algorithmic instability and replica symmetry breaking by comparing the instability condition around the fixed point to the instability for breaking the replica symmetric solution of the free energy function.
result The instability condition around the algorithmic fixed point is identical to the instability for breaking the replica symmetric saddle point solution of the free energy function.
A central area of research in nonlinear science is the study of instabilities that drive the emergence of extreme events. Unfortunately, experimental techniques for measuring such phenomena often provide only partial characterization. For example, real-time studies of instabilities in nonlinear fibre optics frequently …
Develops a geometric framework for soap film instabilities.
problem Soap film instabilities on minimal surfaces.
method Björling representation of minimal surfaces.
result Jacobi fields are maximally localized with lowest eigenfunction amplitude.
The study examines stability and instability of Poincaré-Einstein metrics using Ricci flow.
problem Stability and instability of Poincaré-Einstein metrics.
method Variant of expander entropy for asymptotically hyperbolic manifolds, local positive mass theorem, volume comparison.
result Characterization of stability and instability in terms of local positive mass theorem and volume comparison.
New algorithm explains neural network extrema for better model generalization.
problem Understanding and improving neural network generalization.
method Dynamic extremum mapping and solving linear equations.
result Number of extrema correlates with model parameters, and new algorithm simplifies gradient issues.
Study shows instability of naked singularities in perfect fluid models.
problem Instability of naked singularities in Einstein equations coupled with isothermal perfect fluid.
method Investigated spherically symmetric self-similar naked singularities under C1,α perturbations of an external massless scalar field. result Spherically symmetric self-similar naked singularities are unstable to trapped surface formation.
Study on spectral stability of Riemannian coverings.
problem Stability of eigenvalues in Riemannian coverings.
method Analysis of Laplacian eigenvalues under finite coverings.
result Necessary conditions for spectral stability or instability.
Clinical models can be unstable, leading to unreliable predictions.
problem Stability of clinical prediction models developed using statistical or machine learning methods.
method Simulation and case studies of statistical and machine learning approaches to show instability in model predictions.
result Model instability often leads to miscalibration of predictions in new data.
A data-driven approach predicts morphological development under structural instability.
problem Understanding and predicting spatiotemporal complexities of morphogenesis under structural instability.
method Machine-learning framework based on physical modeling of morphogenesis.
result Identification of key bifurcation characteristics and prediction of history-dependent development.
Study shows instability of naked singularities in scalar field models.
problem Stability of naked singularities in spherically symmetric Einstein-Scalar field systems.
method Analysis of a family of incoming null cones becoming increasingly singular.
result Naked singularities are unstable to black hole formation under certain perturbations.
Study shows instability of certain MOTSs with continuous symmetry.
problem Stability of MOTSs with continuous symmetry.
method Analysis of initial data sets with continuous symmetry and non-preserved MOTSs.
result Exotic MOTSs are unstable except in exceptional cases.
The paper studies stability and instability of minimal submanifolds in complex Einstein spaces.
problem Stability and instability of minimal submanifolds in complex Einstein spaces.
method Computation of index and nullity, investigation of stability, and algorithm for higher eigenvalues.
result Criterion for instability of minimal submanifolds in some cases.
Paper extends Simons theorem to F-Yang-Mills connections for instability.
problem Tackles instability of F-Yang-Mills connections. method Extends Simons theorem to F-Yang-Mills connections using Kobayashi-Ohnita-Takeuchi's method. result Derives a sufficient condition for instability of non-flat F-Yang-Mills connections. The study examines the stability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.
problem Linear instability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.
method Analysis of the second and third Betti numbers for Sasaki Einstein and nearly parallel G2 manifolds.
result Positive second and third Betti numbers lead to linear instability for the respective manifolds.
In this short article, we improve the dynamical stability and instability results for Ricci-flat metrics under Ricci flow proved by Sesum and Haslhofer, getting rid of the integrability assumption.
Proposes a new measure to evaluate stability of statistical parameters under distributional shifts.
problem Difficulty in transferring knowledge across data sets due to distributional changes.
method Introduces a measure of instability quantifying sensitivity of statistical parameters to Kullback-Leibler divergence and directional shifts.
result The proposed measure can elucidate the type of shifts a parameter is sensitive to and improve estimation accuracy under shifted distributions.
The paper analyzes numerical instability in variational flows and proposes a diagnostic method.
problem Numerical instability in variational flows affects sampling, density evaluation, and ELBO estimation.
method Treated variational flows as dynamical systems, used shadowing theory for theoretical guarantees, and developed a diagnostic procedure.
result Despite numerical instability, results from variational flows can be accurate enough for practical applications.
Proposes a continuous flow model to understand and control instability in gradient descent for deep learning.
problem Understanding and controlling the instability of gradient descent in deep learning.
method Introduces the Principal Flow (PF), a continuous time flow that approximates gradient descent dynamics.
result The PF captures divergent and oscillatory behaviors of gradient descent, including escaping local minima and saddle points.
In this paper, we study stability for harmonic foliations on locally conformal Kähler manifolds with complex leaves. We also discuss instability for harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces.
Study on instability of extreme Reissner-Nordström spacetime perturbations.
problem Linear stability of gravitational and electromagnetic perturbations in extreme Reissner-Nordström spacetime.
method Extends Giorgi's framework to prove instability results for a set of gauge invariant quantities along the event horizon.
result Proves decay, non-decay, and polynomial blow-up estimates for certain quantities along the event horizon, depending on the number of derivatives.
Dual-objective GANs reduce training instabilities with tunable α-loss parameters.
problem Training instabilities in Generative Adversarial Networks (GANs).
method Introduce (αD,αG)-GANs with dual objectives modeled using α-loss. result Upper bounds on estimation error show improved performance under certain conditions.
Study uses DNM theory to detect early warning signals of market instability.
problem Detecting early warning signals of financial market instability.
method Applying Dynamical Network Marker (DNM) theory to trading data from the Tokyo Stock Exchange.
result Early warning signals of large price movements can be detected on a daily time scale.
We prove the instability of some families of Riemannian manifolds with non-trivial real Killing spinors. These include the invariant Einstein metrics on the Aloff-Wallach spaces Nk,l=SU(3)/ik,l(S1) (which are all nearly G2 except N1,0), and Sasaki Einstein circle bundles over certain ir…
Following the financial crisis of 2007-2008, a deep analogy between the origins of instability in financial systems and complex ecosystems has been pointed out: in both cases, topological features of network structures influence how easily distress can spread within the system. However, in financial network models, the…