cCorrGAN approximates conditional correlation matrices using GANs.
problem Learning empirical conditional distributions in the elliptope of correlation matrices.
method Conditional Generative Adversarial Networks (GANs) applied to correlation matrices.
result Validated through Monte Carlo simulations in finance.
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.
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.
Universal functions derived for topological correlators in Yang-Mills theory.
problem Expressing correlation functions of topologically twisted Yang-Mills theory.
method Combining Seiberg-Witten geometry, u-plane integral, and blowup formula.
result Closed expressions for universal functions determined.
A simple graphical model for correlated defaults is proposed, with explicit formulas for the loss distribution. Algebraic geometry techniques are employed to show that this model is well posed for default dependence: it represents any given marginal distribution for single firms and pairwise correlation matrix. These t…
The article generalizes Pearson correlation to Riemannian manifolds.
problem Analyzing statistical models on non-linear manifolds.
method Reconstitutes Pearson correlation properties and derives a nonlinear generalization.
result Developed the Riemann-Pearson Correlation for manifold analysis.
TabPFN's internal geometry topology correlates with dataset reliability.
problem Understanding TabPFN's behavior on structurally difficult tabular geometries.
method Using zigzag persistent homology, studying TabPFN's internal representations on synthetic tabular tasks with known topology.
result Topology of TabPFN's internal representation geometry is strongly associated with dataset-level reliability.
Develops log-Euclidean Lie groups for SPD and correlation matrices.
problem Unifies various log-Euclidean constructions for SPD and correlation matrices.
method Theory and explicit isometries linking different log-Euclidean metrics.
result Explicit log-Euclidean metrics on SPD and correlation matrices.
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.
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).
New tensor framework connects Fisher information, hypergraphs, and multi-observable correlations.
problem Missing structure in pairwise Fisher graphs for multi-observable radiation patterns.
method Higher-order Fisher tensors and natural exponential-family coordinates.
result Exact triality of Fisher tensors, cumulants, and hypergraphs.
OMD monitors stock market dynamics through matrix trajectories and reveals crisis patterns.
problem Understanding and predicting stock market crises and sector rotations.
method Applying OMD to S\&P 500 returns over three crises, analyzing distance matrices and their spectra.
result Market dynamics show coherent changes during crises, with distinct sector leadership.
In the first part of this paper we provide a short introduction to the AdS/CFT correspondence and to holographic renormalization. We discuss how QFT correlation functions, Ward identities and anomalies are encoded in the bulk geometry. In the second part we develop a Hamiltonian approach to the method of holographic re…
In this paper, we study the multi-asset Black-Scholes model in terms of the importance that the correlation parameter space (equivalent to an N dimensional hypercube) has in the solution of the pricing problem. We show that inside of this hypercube there is a surface, called the Kummer surface ΣK, where the determ…
We started from computer experiments with simple one-dimensional ergodic dynamical systems called interval exchange transformations. Correlators in these systems decay as a power of time. In the simplest non-trivial case the exponent is equal to 1/3. We found a formula connecting characteristic exponents with explicit …
We review the state of the art of clustering financial time series and the study of their correlations alongside other interaction networks. The aim of this review is to gather in one place the relevant material from different fields, e.g. machine learning, information geometry, econophysics, statistical physics, econo…
The paper uses deep learning to detect financial market regimes from correlation matrices.
problem Detecting financial market regimes from correlation dynamics.
method Representation learning on block hierarchical SPD correlation matrices using SPDNet, SPD-NetBN, and U-SPDNet models.
result Deep learning models overfit in financial market data, misleading performance metrics.
Method preserves correlations in synthetic data.
problem Preserving dependence structure of original data.
method Orthogonal Procrustes problem for restoring Pearson correlation.
result Restores Pearson correlation structure while preserving feature distributions and downstream tasks performance.
We propose a methodology to explore and measure the pairwise correlations that exist between variables in a dataset. The methodology leverages copulas for encoding dependence between two variables, state-of-the-art optimal transport for providing a relevant geometry to the copulas, and clustering for summarizing the ma…
One primary focus in multimodal feature extraction is to find the representations of individual modalities that are maximally correlated. As a well-known measure of dependence, the Hirschfeld-Gebelein-Rényi (HGR) maximal correlation becomes an appealing objective because of its operational meaning and desirable propert…
GRIP2 improves deep learning feature selection robustness in correlated and noisy data.
problem Identifying predictive features in correlated and noisy data.
method Integrates first-layer feature activity over a two-dimensional regularization surface to control sparsity and geometry, using efficient block-stochastic sampling.
result Demonstrates improved robustness and power in high correlation and low signal-to-noise ratio regimes.
This paper proposes a geometry-aware active learning framework for spatiotemporal dynamic systems.
problem Challenges in modeling complex dynamic systems with 3D geometries and time evolution.
method Geometry-aware spatiotemporal Gaussian Process (G-ST-GP) and adaptive active learning strategy.
result The proposed framework outperforms traditional methods in predicting high-dimensional dynamic behaviors.
New metrics defined for full-rank correlation matrices, ensuring unique operations.
problem No suitable problem statement as the abstract does not describe a problem to be solved.
method New Riemannian metrics defined on full-rank correlation matrices, providing unique operations.
result Unique Riemannian logarithm and Fréchet mean defined for full-rank correlation matrices.
Enhances graph modeling with hyperbolic geometry and variational inference.
problem Challenges in modeling relational data with complex dependencies.
method Semi-implicit hierarchical variational Bayes with Poincaré embedding and mutual information regularization.
result Improves graph representation quality and flexibility in edge prediction and node classification.
Study uses topological signatures to quantify financial market complexity.
problem Capturing temporal organization beyond volatility measures.
method Null validated topological approach using L1 norm of persistence landscapes. result Persistence landscape norms reveal dynamical structure during market stress.
New measures link neural representation geometry to decoding ability.
problem Understanding how neural representations relate to decoding ability.
method Showed that popular similarity measures can be interpreted from a decoding perspective.
result Proved that measures like CKA and CCA quantify alignment between optimal linear readouts.
Unified framework for Riemannian deep learning across manifold-valued representations.
problem Deep learning on manifold-valued representations often relies on Euclidean approximations or costly geometric operations.
method Develops reusable neural modules, manifold-specific network architectures, and geometric designs.
result Generalizes batch normalization and multinomial logistic regression to broader classes of manifolds.
Unified framework for Riemannian deep learning across manifold-valued representations.
problem Deep learning on manifold-valued data lacks reusable modules, specific network architectures, and efficient geometric operations.
method Develops reusable neural modules, manifold-specific network architectures, and geometric designs for broad classes of Lie groups and gyrogroups.
result Generalizes batch normalization and multinomial logistic regression to Riemannian manifolds, including SPD and hyperbolic spaces.
We show that gamma distributions provide models for departures from randomness since every neighbourhood of an exponential distribution contains a neighbourhood of gamma distributions, using an information theoretic metric topology. We derive also the information geometry of the 3-manifold of McKay bivariate gamma dist…
Network geometry measures predict market instability.
problem Predicting financial market instability using network geometry.
method Discrete Ricci curvatures to capture network fragility.
result Different geometric measures distinguish normal and crash periods.
Study well-posedness of SPDE on Riemannian manifolds with rough initial conditions.
problem Well-posedness of parabolic Anderson model on Riemannian manifolds with rough initial conditions.
method Construct intrinsic Gaussian noises, explore global geometry, use Feynman-Kac formula.
result Show well-posedness with non-positive curvature and conditions on α. OMD monitors stock market dynamics through matrix trajectories, revealing crisis patterns and sector rotations.
problem Understanding and predicting stock market dynamics during crises.
method Applying OMD to S&P 500 returns over three crises, analyzing distance matrices and their spectra.
result Market dynamics show coherent changes during crises, with sector-specific patterns and volatility clustering.
ProbETA models travel time correlations between trips for better navigation.
problem Trip correlations not captured by existing methods.
method Deep hierarchical joint probabilistic model with learnable link representations.
result ProbETA outperforms state-of-the-art methods with 12.60% MAPE reduction.
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…
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.
In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves. Then we show, how intersection numbers can be expressed via Hurwitz numbers. An…
The paper uses geometry to assess how hard examples are for NLP models.
problem Challenges in NLP datasets and classifiers, especially with shallow features.
method Information geometry to quantify example difficulty, exploring BERT, CNN, and fasttext.
result Deep learning models are vulnerable to word substitutions in difficult examples.
The paper presents methods to improve uncertainty calibration in Bayesian Neural Networks.
problem Uncalibrated Bayesian Neural Networks often lead to overconfidence.
method The paper uses alpha-divergences from Information Geometry for calibration.
result Calibration using alpha-divergences provides better uncertainty estimates and is more efficient.
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.
In this short note, using our geometric method introduced in a previous paper \cite{phl} and initiated by \cite{ave}, we derive an asymptotic swaption implied volatility at the first-order for a general stochastic volatility Libor Market Model. This formula is useful to quickly calibrate a model to a full swaption matr…
We consider the problem of identifying universal low-dimensional features from high-dimensional data for inference tasks in settings involving learning. For such problems, we introduce natural notions of universality and we show a local equivalence among them. Our analysis is naturally expressed via information geometr…
DiffeoCFM efficiently generates realistic brain connectivity matrices using pullback metrics.
problem Generating realistic brain connectivity matrices for population heterogeneity analysis.
method Conditional flow matching on matrix manifolds via pullback metrics induced by global diffeomorphisms.
result DiffeoCFM achieves state-of-the-art performance on large-scale fMRI and EEG datasets.
Canonical correlation analysis (CCA) is a powerful technique for discovering whether or not hidden sources are commonly present in two (or more) datasets. Its well-appreciated merits include dimensionality reduction, clustering, classification, feature selection, and data fusion. The standard CCA however, does not expl…
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.
Diffusion maps are a commonly used kernel-based method for manifold learning, which can reveal intrinsic structures in data and embed them in low dimensions. However, as with most kernel methods, its implementation requires a heavy computational load, reaching up to cubic complexity in the number of data points. This l…
This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We ex…
Deep learning dynamics and NTK evolution studied through diverse measures.
problem Understanding the training dynamics of deep neural networks and their loss landscapes.
method Phenomenological analysis of training dynamics in multiple architectures and datasets.
result Training dynamics exhibit a chaotic initial transient followed by a stable phase, with the NTK evolving to match full network performance.
The paper uses geometry to understand how neural networks learn.
problem Understanding the learning capability of neural networks.
method Statistical and differential geometric analysis of neural networks performing simple regression.
result Neural networks with higher generalization capability have a slower convergence rate.