A new geometric framework embeds correlation matrices into Euclidean space for scalable brain network analysis.
problem Inefficient and unstable analysis of functional brain networks in high-dimensional contexts.
method Diffeomorphic transformations to embed correlation matrices into Euclidean space, preserving manifold properties.
result Improved computational speed and enhanced accuracy compared to conventional manifold-based approaches.
Study uses geometric algebra to analyze credit cycles, revealing dangerous feedback loops.
problem Understanding and predicting dangerous feedback loops in credit cycles.
method Represent economic states as multi-vectors in Clifford algebra, focusing on bivector elements for rotational coupling.
result Geometric relationship between unemployment and credit contraction shifts from simple correlation to dangerous rotational dynamics during crises.
Geometric QHD tests improve hub detection in correlated data.
problem Detecting hubs in correlated data with evolving correlations.
method Geometric QHD tests combining QCD and QHD, clustering.
result Improved hub detection in correlated data.
Complex network analysis reveals dominant stocks in financial stock returns correlations.
problem Inferring financial stock returns correlations from complex network analysis.
method Simulated geometric Brownian motion for stocks, complex network analysis, eigenvector centrality, clustering.
result Returns correlation matrix is dominated by stocks with high eigenvector centrality and clustering.
The evolution of prices on ideal market is given by geometrical Brownian motion, where Gaussian white noise describes fluctuations. We study the effect of correlations introduced by a color noise.
We study velocity correlations induced by diffusion and dissipation in a simple dissipative dynamical system. We observe that diffusion, as a result of time reversible microscopic processes, leads to correlations with different spatial parity from those caused by dissipation, consisting of time irreversible microscopic…
Study shows how correlations between neural activity affect classification capacity.
problem Understanding how correlations between neural activity impact classification performance.
method Calculated the capacity of neural activity on spherical manifolds with and without correlations between centroids and axes.
result Introducing correlations between neural activity centroids pushes spheres closer together, while correlations between axes shrink their radii, revealing a duality between correlations and geometry in classification.
Study models illiquid stock prices and finds low correlation due to constant prices.
problem Modeling illiquid stock prices and measuring correlation accurately.
method Combined Markov model with Ornstein Uhlenbeck and geometric Brownian motion.
result Low correlation in USE stocks due to constant prices and illiquidity.
The paper introduces a method to decorrelate circular coordinates using lattice reduction.
problem Geometric correlation between circle-valued maps when multiple cohomology classes are used.
method Systematic procedure using the Lenstra--Lenstra--Lovász algorithm for constructing low energy torus-valued maps.
result A method to obtain less correlated maps from cohomology classes using integer linear combinations.
This note improves correlation stress tests using geodesic distance.
problem Improving financial risk management through better covariance stress tests.
method Proposes a new geometrically invariant definition of correlation stress tests.
result Demonstrates a submanifold approach to stress testing covariance matrices.
A novel circuit motif uses sister cells for inference with correlated priors.
problem Structured priors in neural systems pose architectural challenges.
method Proposes a novel circuit motif using sister cells to implement correlated priors without direct interactions.
result Demonstrates the efficacy of correlated priors for inference in noisy environments.
This paper tackles matching two complete graphs with correlated edge weights in geometric models.
problem Matching two complete graphs with edge weights correlated through latent geometries.
method Derives an approximate maximum likelihood estimator for recovering hidden vertex correspondence.
result The estimator provably achieves perfect recovery under certain noise conditions.
Study topological quantum mechanics on orbifolds with geometric interpretation.
problem Quantum mechanical models on symplectic orbifolds.
method Explicit orbifold version of quantum HKR map and exact semi-classical approximation.
result Geometric and quantum field theoretic interpretation of orbifold algebraic index.
A new method for approximating softmax and Gaussian kernels with reduced error.
problem Approximating softmax and Gaussian kernels with low error.
method Simplex Random Features (SimRFs) and SimRFs+.
result SimRFs provide the smallest MSE among weight-independent geometrically-coupled PRF mechanisms.
Geometrically constructs twist-field correlation functions in CFT.
problem Understanding entanglement entropy in quantum systems.
method Using Cauchy-Hadamard renormalization of Polyakov anomaly integral on surfaces with conical singularities.
result Provides a purely mathematical interpretation of entanglement entropy results.
Improved sample complexity for Gaussian Mixture Models using Pair Correlation Factor.
problem Understanding the sample complexity of Gaussian Mixture Models.
method Introducing Pair Correlation Factor (PCF) to measure clustering of component means and improving sample complexity bounds.
result The Pair Correlation Factor (PCF) more accurately determines the difficulty of parameter recovery in Gaussian Mixture Models.
This paper develops geometric tools for causal inference using information flow concepts.
problem Developing a geometric interpretation of causal inference from probabilistic measures.
method Introducing a new measure, GeoC_{y
ightarrow x}, based on fractal correlation dimension.
result Avoids boundedness issues in transfer entropy, providing a more robust measure of causal inference.
A new method treats all variables equally in fitting data.
problem Fitting relationships to data with multiple variables, especially when dependent and independent variables are not clearly defined.
method A general method treating all variables impartially, using geometric mean functional relationships and correlation.
result The method provides coefficients that are easily calculated from covariances or correlations, making it scale-invariant and applicable to various units.
A new model for multiview data analysis using graph autoencoders.
problem Nonlinear multiview canonical correlation analysis for large datasets.
method Variational approach with graph convolutional neural networks.
result Competitive performance on classification, clustering, and recommendation tasks.
Develops methods to analyze feature-outcome associations in subpopulations.
problem Challenges in understanding feature-outcome associations in high-dimensional data.
method Geometric decomposition framework using gradient flow and co-monotonicity decomposition.
result Identifies context-dependent patterns and improves statistical power and interpretability.
Based on the classical Plücker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in CP3. Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the e…
The study of record statistics of correlated series is gaining momentum. In this work, we study the records statistics of the time series of select stock market data and the geometric random walk, primarily through simulations. We show that the distribution of the age of records is a power law with the exponent α lyi…
Study reveals LLM personas have two distinct components: frame-robust aggregated traits and frame-dependent geometric features.
problem Evaluation of LLM personas via psychometric questionnaires discards within-instance correlation structure.
method Constructed within-instance correlation matrices from IPIP-50 responses and analyzed geometry on SPD manifolds under manipulated question orderings.
result Persona expression comprises two dissociable components: aggregated features (Big Five scores) and geometric features (SPD manifold).
Geometric stability measures neural network robustness, distinguishing from similarity metrics.
problem Lack of robustness in neural network representations.
method Introduces geometric stability, quantified by Shesha metric measuring self-consistency.
result Stability and similarity are uncorrelated, revealing distinct properties of neural network robustness.
We derive a closed-form solution for the price of an average price as well as an average strike geometric Asian option, by making use of the path integral formulation. Our results are compared to a numerical Monte Carlo simulation. We also develop a pricing formula for an Asian option with a barrier on a control proces…
Study geodesics on flat tori, focusing on orthogonal lengths and their distribution.
problem Properties of geodesics on flat tori and their lengths.
method Fine study of dynamical correlation function and anisotropic Sobolev spaces.
result Properties of the length distribution and singularities of its Fourier transform.
This paper develops a two-dimensional structural framework for valuing credit default swaps and corporate bonds in the presence of default contagion. Modelling the values of related firms as correlated geometric Brownian motions with exponential default barriers, analytical formulae are obtained for both credit default…
We propose an artificial market model based on deterministic agents. The agents modify their ask/bid price depending on past price changes. The temporal development of market price fluctuations is calculated numerically. A probability density function of market price changes has power law tails. Autocorrelation coeffic…
We study the problem of rank aggregation: given a set of ranked lists, we want to form a consensus ranking. Furthermore, we consider the case of extreme lists: i.e., only the rank of the best or worst elements are known. We impute missing ranks by the average value and generalise Spearman's ρto extreme ranks. Our main …
We develop a ``canonical Wick rotation-rescaling theory in 3-dimensional gravity''. This includes: (a) A simultaneous classification that shows how generic maximal globally hyperbolic spacetimes of constant curvature, which admit a complete Cauchy surface (in particular a compact one), as well as complex projective str…
Study quantizes ropelength and writhe of 12-crossing knots.
problem Quantifying geometric and topological properties of knots.
method Computation of ropelength and space writhe for 12-crossing knots.
result Non-alternating knots show evidence of writhe quantization, while alternating knots do.
Researchers identify critical protein residues using advanced graph theory.
problem Identifying essential residues in proteins for function.
method Learning Random Geometric Graphs (RGG) with Cramer's V correlation and organic thresholding.
result Advanced RGG methods accurately identify critical residues compared to existing techniques.
Let U(r),r∈Ω⊂R2 be a harmonic function that solves an exterior Dirichlet problem. If all the level sets of U(r),r∈Ω are smooth Jordan curves, then there are several geometric inequalities that correlate the curvature κ(r) with the ma…
Study non-asymptotic bounds on correlation in high-dimensional linear systems, revealing invariant subspaces and bottlenecks.
problem Understanding correlation and mixing in high-dimensional linear systems with Gaussian noise.
method Sampling from sub-trajectories, using Talagrand's inequality, and analyzing invariant subspaces.
result Large discrepancy between algebraic and geometric multiplicity leads to bottlenecks between invariant subspaces.
Unified framework for nonconvex matrix completion with linearly parameterized factors.
problem Matrix completion with improved accuracy using linearly parameterized factors.
method Unified nonconvex optimization framework with Correlated Parametric Factorization condition.
result Uniform upper bounds for low-rank estimation at any local minimum.
s-OTDD compares datasets efficiently without training, robust to class variations.
problem Efficiently compare datasets without training or class variations.
method Moment Transform Projection (MTP) and sliced optimal transport.
result s-OTDD correlates with optimal transport and transfer learning performance.
Manifold Markov chain Monte Carlo algorithms have been introduced to sample more effectively from challenging target densities exhibiting multiple modes or strong correlations. Such algorithms exploit the local geometry of the parameter space, thus enabling chains to achieve a faster convergence rate when measured in n…
This paper analyzes Barlow Twins' representation efficiency using information-geometric methods.
problem Understanding and comparing the efficiency of self-supervised learning methods.
method Introduces an information-geometric framework to quantify representation efficiency and applies it to Barlow Twins.
result Proves that Barlow Twins achieves optimal representation efficiency (η=1).
Study identifies failure modes of machine learning models in out-of-distribution settings.
problem Machine learning models fail to generalize well to new, unseen data.
method Theoretical study of gradient-descent-trained linear classifiers on easy-to-learn tasks, followed by experiments on modern neural networks.
result Two failure modes of spurious correlations are uncovered: geometric and statistical.
Graph Ricci flow reveals hidden hierarchies in stock market correlations.
problem Detecting hidden structures in the complex stock market graph.
method Using graph Ricci curvature and flow techniques to analyze the NASDAQ 100 index.
result Algorithm detects hidden hierarchies, community behavior, and clustering in financial markets.
Analytical pricing formulas and Greeks are obtained for European and American basket put options using Mellin transforms. We assume assets are driven by geometric Brownian motion which exhibit correlation and pay a continuous dividend rate. A novel approach to numerical Mellin inversion is achieved via the fast Fourier…
PyTorch Geometric Signed Directed fills the gap for GNNs on signed and directed graphs.
problem Lack of unified software packages for GNNs on signed and directed networks.
method Developed a software package with GNN models, synthetic and real-world data, and evaluation metrics.
result Demonstrates the effectiveness of the implemented methods through experiments.
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.
The volume of a credal set correlates with epistemic uncertainty in binary classification but not in multi-class.
problem Representing and quantifying epistemic uncertainty in machine learning.
method Examined the geometric representation of credal sets as d-dimensional polytopes and their volume as a measure of uncertainty. result The volume of a credal set is a meaningful measure of epistemic uncertainty in binary classification but not in multi-class.
We determine the minimal entropy martingale measure for a general class of stochastic volatility models where both price process and volatility process contain jump terms which are correlated. This generalizes previous studies which have treated either the geometric Lévy case or continuous price processes with an ortho…
New algorithm learns HMM parameters on Riemannian manifolds.
problem Learning hidden Markov models on non-Euclidean spaces.
method Geometric method of moments algorithm for Riemannian manifolds.
result Significantly improved speed and accuracy compared to existing methods.
We give geometric explanations and proofs of various mirror symmetry conjectures for Tn-invariant Calabi-Yau manifolds when instanton corrections are absent. This uses fiberwise Fourier transformation together with base Legendre transformation. We discuss mirror transformations of (i) moduli spaces of complex stru…
GCPNet improves molecular graph learning for protein structure and binding.
problem Learning from 3D molecular graphs for protein structure and binding.
method SE(3)-equivariant graph neural network for 3D molecular graphs.
result GCPNet achieves state-of-the-art performance in multiple molecular tasks.