Research classifies L-space knots using Floer simple manifolds.
problem Understanding L-space fillings and their topological significance.
method Topological characterisation of Floer simple manifolds and classification of L-space knots.
result Partial classification of L-space twisted torus knots.
Study finds the order of Dehn twists in various groups.
problem Determining the order of Dehn twists in different topological groups.
method Analyzing the specific groups mentioned in the abstract.
result The order of Dehn twists was calculated in the specified groups.
Survey on categorifying Jones polynomial.
problem Categorification of Jones polynomial.
method Not explicitly stated, likely involves algebraic and geometric categorification techniques.
result Significance and ramifications in geometry, algebra, and topology.
Study enhances neural network predictions for wave height using topological features.
problem Challenges in predicting wave heights due to short-term and long-term factors.
method Hybridization of persistent homology with neural networks for feature engineering.
result Significant improvements in R2 score and reductions in errors for various neural network models. New method detects uncertainty in neural networks for out-of-distribution detection.
problem Detecting out-of-distribution inputs to ensure model reliability.
method Predictive topological uncertainty (pTU) based on persistent homology.
result pTU provides a statistical framework for OOD detection.
Study minimal freezing sets in convex digital disks.
problem Finding minimal freezing sets in convex digital disks.
method Showed how to find minimal freezing sets for convex disks in digital plane.
result Found minimal freezing sets for convex disks in digital plane.
This paper reduces the complexity of deep CNNs by optimizing their topology.
problem Reduces computational complexity and storage in deep CNNs for resource-constrained platforms.
method Analyzes the impact of CNN topology (depth and width) on feature extraction, focusing on scattering networks.
result Designs networks of fixed depth to retain a significant portion of input signal energy in the feature vector.
DeepGDL models create realistic power grids from confidential data.
problem Creating realistic power grids from confidential data.
method Graph distribution learning (GDL) with a deep nonlinear recurrent structure.
result DeepGDL models accurately create synthetic power grids.
Overview of Thurston's work in math.
problem None explicitly stated, focuses on Thurston's contributions.
method Presentation of significant results.
result Impact of Thurston's work on mathematics.
Study evaluates synthetic data augmentation for small datasets, highlighting inconsistencies in traditional metrics.
problem Inconsistent validation of synthetic data generated for small sample sizes.
method Proposes a normalized Bottleneck distance metric to evaluate synthetic tabular data.
result Common metrics like propensity scoring and MMD fail for small datasets, showing instability and high variability.
Study reveals how network topology affects credit risk distribution among firms.
problem Understanding how network topology influences credit risk distribution.
method Investigation of a large dataset of Italian firms' payments and credit risk ratings.
result Significant correlations between local topological properties of firms and their risk profiles.
The paper analyzes the crash of stock and commodity markets during COVID-19 using Topological Data Analysis.
problem Identifying and understanding the dynamics and interdependence of stock and commodity markets during the COVID-19 crash.
method Topological Data Analysis (TDA) and Wasserstein Distance (WD) to identify crashes and compare market dynamics.
result Significant topological differences and interdependence between stock and commodity markets during the crash period.
Paper introduces topological eigenvalue theorems for tensor analysis in multi-modal data.
problem Lack of deep understanding of tensor structures in multi-modal data fusion.
method Introduces topological perspective to tensor eigenvalue analysis, linking eigenvalues to topological features.
result Establishes new theorems that enhance understanding of tensor structures in data fusion.
Paper proves a new criterion for time-like geodesics in flat spacetimes.
problem Existence and nature of time-like geodesics in asymptotically flat spacetimes.
method Generalized topological criterion using the Jordan-Brouwer Separation Theorem and differential geometry.
result Conclusively affirms the presence of time-like geodesics intersecting transversally.
Paper proposes learnable topological features for efficient phylogenetic inference.
problem Finding appropriate topological structures for phylogenetic inference tasks requires significant design effort and domain expertise.
method Combines raw node features with graph neural networks to automatically adapt to different tasks.
result Demonstrates effectiveness and efficiency on simulated and real data phylogenetic inference tasks.
Paper introduces TVaRD, a new topological risk measure for financial portfolios.
problem Traditional risk measures like VaR and CVaR are insufficient for complex market conditions.
method Topological data analysis (TDA) using cohomology groups on financial time series data.
result TVaRD reveals significant changes in financial time series during stress conditions.
The paper proves UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.
problem Proving UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.
method Rigorously proving UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories on Rd′imesCd. result Proves vanishing anomalies for hybrid topological-holomorphic field theories, allowing for the definition of a factorization algebra structure for quantum observables.
New lattice path method for statistical inference of persistent diagrams.
problem Statistical inference on persistent diagrams.
method Lattice path representation and combinatorial enumerations.
result Topological changes observed in spike proteins of COVID-19 virus.
Paper speeds up topological signal identification and cycle matching.
problem Efficiently identifying and matching topological signals across datasets.
method Cohomological approach to persistent homology computation.
result Significantly faster performance on large-scale datasets.
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. We investigate the properties of these mappin…
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.
We introduce a graph-theoretic approach to extract clusters and hierarchies in complex data-sets in an unsupervised and deterministic manner, without the use of any prior information. This is achieved by building topologically embedded networks containing the subset of most significant links and analyzing the network s…
A new topology improves decentralized learning efficiency and accuracy.
problem Finding efficient decentralized learning topologies with fast consensus and low maximum degree.
method Proposed the Base-(k+1) Graph topology for decentralized learning. result The Base-(k+1) Graph enables faster convergence and better communication efficiency than the exponential graph. Develops a braid-theoretic framework to analyze chirality in molecular knots.
problem Analyzing chirality in molecular knots constructed using circuit topology.
method Translated circuit topology approach to knot engineering into braid-theoretic framework, calculating Jones polynomial for binary combinations.
result Jones polynomial provides a powerful tool for analyzing chirality of molecular knots.
Sparse neural networks can improve performance with less memory.
problem Lack of fast memory limits deep neural network performance.
method Experimented with sparse neural network topologies, including pruning-based and RadiX-Nets.
result Sparse networks achieve comparable accuracy to dense networks but suffer instability at extreme sparsity.
This study investigates porosity and topological properties of TPMS using machine learning.
problem Understanding the relationships between porosity and topological properties of TPMS.
method Application of machine learning techniques to analyze porosity and shape factor of TPMS.
result Conjectures suggesting polynomial relationships between porosity and shape factor of TPMS.
Neural networks improve error correction in topological codes.
problem Finding optimal correction of errors in generic stabilizer codes is computationally hard.
method Systematic study of versatile neural-network decoders for topological codes.
result Neural decoders significantly improve error-correction threshold over leading efficient decoders.
TDA-based portfolios show better risk-adjusted returns than classical methods.
problem Traditional portfolio selection methods fail to capture complex asset dynamics.
method Topological Data Analysis (TDA) using persistence landscapes to quantify portfolio risk.
result TDA-based portfolios outperform classical models in excess mean return and financial ratios.
This research quantifies neural networks using magnitude, a topological invariant.
problem Understanding the generalization capabilities of neural networks.
method Using a novel topological invariant called magnitude to study neural network representations.
result Magnitude dimension is theoretically connected to generalisation error and can predict it.
This work uses differential topology to address challenges in DNNs.
problem Challenges in Deep Neural Networks: expressibility, optimisability, and generalisability.
method Modeling the dataset as a smooth manifold and applying differential topology to loss landscape, expressibility, and generalisability.
result A differential topological view offers new insights into DNNs' challenges.
Study topological Iwasawa invariants for 3-sphere links, proving density results.
problem Detect and analyze Iwasawa invariants for 3-sphere links.
method Explicit criteria for detecting Iwasawa invariants, statistical analysis of 2-bridge links.
result Density of 2-bridge links with specific Iwasawa invariants.
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.
A CNN with U-Net improves structural topology optimization efficiency and generalization.
problem Structural topology optimization with reduced computation cost and improved generalization.
method Deep Convolutional Neural Network (CNN) with U-Net architecture, using SIMP-generated dataset.
result Significant reduction in computation cost with little sacrifice on design optimality.
Improved modeling of persistence diagrams for data analysis.
problem Determining significant outliers in persistence diagrams.
method Modification of the RST (Replicating Statistical Topology) model using MCMC Metropolis-Hastings algorithm.
result The modified RST model improves the goodness of fit in persistence diagram analysis.
TensorQuant simulates quantization of DNNs for efficient computation.
problem Optimal quantization method and bit number for DNNs are topology-dependent.
method TensorQuant simulates quantization of existing DNN topologies.
result TensorQuant supports experimental evaluation of quantization impact.
Paper introduces a framework for diagnosing Alzheimer's disease using higher-order topological features from fMRI.
problem Diagnosing Alzheimer's disease using brain network topology.
method Persistent homology to extract higher-order features (cycles, cavities) from fMRI data.
result Framework significantly outperforms existing methods in AD classification.
Persistence norms explain financial uncertainty better than volatility.
problem Capturing financial instability and predictability.
method Applied topological data analysis to financial markets.
result Persistence norms are significant in explaining financial uncertainty, while volatility is less effective.
This paper finds efficient algorithms for approximating Markov networks with k-tree topologies.
problem Efficiently approximating Markov networks with complex topologies.
method Developed O(n^{k+1})-time algorithms for finding maximum spanning k-trees (MSkT) that retain certain subgraphs.
result Optimal approximation of Markov networks with k-tree topology is achieved in polynomial time.
Characterizes winding of braided vector fields in tubular domains.
problem Understanding the topology of braided vector fields in complex domains.
method Defines field line winding as a measure of entanglement, proving its uniqueness in classifying vector field topology.
result Field line winding uniquely classifies the topology of braided vector fields.
This paper formalizes the h-principle and sphere eversion in differential topology.
problem Formalizing the h-principle and sphere eversion in differential topology.
method Lean formalization of the local h-principle for first-order partial differential relations, using convex integration.
result Reproves Smale's sphere eversion theorem and formalizes advanced mathematics.
In this paper we settle Thurston's old question of whether the Weber-Seifert dodecahedral space is non-Haken, a problem that has been a benchmark for progress in computational 3-manifold topology over recent decades. We resolve this question by combining recent significant advances in normal surface enumeration, new he…
Study shows effective resistance distance yields more accurate network barycenter than Hamming distance.
problem Identifying the best metric for computing the Fréchet mean network.
method Compared the effectiveness of Hamming distance and effective resistance distance in capturing network topology.
result Effective resistance distance produces a more accurate Fréchet mean network.
Topology-aware generative models improve manifold-based defenses against adversarial examples.
problem Adversarial examples compromise the reliability of ML models, especially DNNs.
method Investigate if generative models used in manifold-based defenses need to be topology-aware.
result Topology-aware generative models enhance the robustness of manifold-based defenses.
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.
New method reduces clustering time and improves accuracy.
problem High time and space complexity in spectral clustering.
method Approximate spectral clustering using GNG network topology.
result Equal or better clustering performance than traditional SC.
This study uses persistent homology to analyze complex transitional networks from time series data.
problem Lack of effective tools to summarize complex topology in transitional networks.
method Persistent homology from topological data analysis applied to coarse-grained state-space networks (CGSSN).
result CGSSN improves dynamic state detection and noise robustness compared to other methods.
This paper introduces anti-correlation networks to study China's stock market.
problem Previous studies ignored anti-correlation in financial networks.
method Constructed weighted temporal anti-correlation and positive correlation networks.
result Unveiled differences in topological measurements between anti-correlation and positive correlation networks.
We investigated the topological properties of stock networks through a comparison of the original stock network with the estimated stock network from the correlation matrix created by the random matrix theory (RMT). We used individual stocks traded on the market indices of Korea, Japan, Canada, the USA, Italy, and the …