We investigate serial correlation, periodic, aperiodic and scaling behaviour of eigenmodes, i.e. daily price fluctuation time-series derived from eigenvectors, of correlation matrices of shares listed on the Johannesburg Stock Exchange (JSE) from January 1993 to December 2002. Periodic, or calendar, components are dete…
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Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace-Beltrami operator. Using the direct bou…
We present a simple algorithm for finding eigenmodes of the Laplacian for arbitrary compact hyperbolic 3-manifolds. We apply our algorithm to a sample of twelve manifolds and generate a list of the lowest eigenvalues. We also display a selection of eigenmodes taken from the Weeks space.
Cosmologists are taking a renewed interest in multiconnected spherical 3-manifolds (spherical spaceforms) as possible models for the physical universe. To understand the formation of large scale structures in such a universe, cosmologists express physical quantities, such as density fluctuations in the primordial plasm…
Identifying Relevant Eigenimages - a Random Matrix Approachphysics.data-an
Dimensional reduction of high dimensional data can be achieved by keeping only the relevant eigenmodes after principal component analysis. However, differentiating relevant eigenmodes from the random noise eigenmodes is problematic. A new method based on the random matrix theory and a statistical goodness-of-fit test i…
Observational data hints at a finite universe, with spherical manifolds such as the Poincare dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes…
New method aligns brain data across individuals for better brain decoding.
problem Inter-individual variability in brain response patterns limits decoder generalization.
method SpectralOT method that embeds cortical geometry into Laplace-Beltrami eigenmodes.
result SpectralOT strikes balance between aligning functional features and preserving anatomical structure.
Bootstrap bounds on Einstein manifolds using semidefinite programming.
problem Bounding geometric data of closed Einstein manifolds.
method Semidefinite programming applied to consistency conditions of geometric data.
result Bootstrap bounds translate to constraints on Kaluza-Klein modes.
Emergence of frustration signals systemic riskphysics.soc-ph
Complex systems' systemic risk linked to frustration in network structure.
problem Understanding systemic risk in complex systems.
method Analysis of fluctuation correlations and network structure evolution.
result Emergence of frustration signals systemic risk in complex systems.
The paper analyzes Nordic stock markets' correlation structures and regime shifts.
problem Understanding and exploiting regime shifts in Nordic stock markets.
method Examined two decades of daily data for OMXS30, OMXC20, and OMXH25 universes; proposed an adaptive portfolio allocation framework.
result Documented pronounced regime dependence in rolling correlation matrices; proposed an adaptive portfolio allocation framework.
Spectral denoising recovers meaningful network structure from noisy financial correlations.
problem Noise in empirical correlation matrices from financial returns obscures genuine interactions.
method Spectral decomposition to separate structured and random components.
result Structured networks derived from 10-16 eigenmodes exhibit stronger core-periphery organization and scale-free degree distributions.
With the random matrix theory, we study the spatial structure of the Chinese stock market, American stock market and global market indices. After taking into account the signs of the components in the eigenvectors of the cross-correlation matrix, we detect the subsector structure of the financial systems. The positive …
Commissioned by MIT's in-house artist Jane Philbrick, we evolve an abstract 2D surface (resembling Marta Pan's 1961 "Sculpture Flottante I") under mean curvature, all the while calculating the eigenmodes and eigenvalues of the Laplace-Beltrami operator on the resulting shapes. These are then synthesized into a sound-wa…
Proposes a lattice formulation of APS index using η invariant.
problem Calculating APS index in lattice gauge theory.
method Uses η invariant of domain-wall Dirac operator.
result Index is always an integer with finite lattice spacing.
Method reduces categorical data to lower dimensions using density matrices.
problem Dimensionality reduction for categorical data.
method Density-matrix construction from class-conditional frequencies; spectral embedding.
result Low-dimensional spectral embeddings with controlled rank.
Four-dimensional quaternion-Kahler metrics, or equivalently self-dual Einstein spaces M, are known to be encoded locally into one real function h subject to Przanowski's Heavenly equation. We elucidate the relation between this description and the usual twistor description for quaternion-Kahler spaces. In particular, w…
The salient properties of large empirical covariance and correlation matrices are studied for three datasets of size 54, 55 and 330. The covariance is defined as a simple cross product of the returns, with weights that decay logarithmically slowly. The key general properties of the covariance matrices are the following…
Study on identifying and inferring nonlinear dynamics on unknown networks.
problem Identifying network structure in nonlinear dynamic systems with unknown interactions.
method Showed network structure is not generically identified, requiring sufficient spectral heterogeneity. Developed necessary and sufficient conditions for identification and proposed a semiparametric estimator.
result Necessary and sufficient conditions for identification of network structure in nonlinear dynamic systems.
The paper studies cylindrical singularities in mean curvature flow and proves their local regularity.
problem Understanding the structure and regularity of cylindrical singular sets in mean curvature flow.
method Introduced a new -distance non-concentration property to prove the local regularity of singular sets.
result Locally, cylindrical singular sets are contained in a -dimensional -submanifold.
Perturbation theory improves nonparametric instrumental variable estimation accuracy.
problem Improving nonparametric instrumental variable estimation accuracy in high-dimensional settings.
method Perturbative approach based on physics perturbation theory, extending kernel ridge methods with higher-order corrections.
result First-order perturbative corrections reduce prediction error by up to 99% in high-dimensional ill-defined cases.
Calculates Laplacian spectra on Calabi-Yau hypersurfaces.
problem Computing the spectrum of the Laplacian on complex manifolds.
method Numerical computation of eigenvalues and eigenmodes for line bundles.
result Agreement with exact results for and a torus, first numerical results for Fermat quintic.
Many contemporary statistical learning methods assume a Euclidean feature space. This paper presents a method for defining similarity based on hyperspherical geometry and shows that it often improves the performance of support vector machine compared to other competing similarity measures. Specifically, the idea of usi…
In this study, the fluctuation-dissipation theory is invoked to shed light on input-output interindustrial relations at a macroscopic level by its application to IIP (indices of industrial production) data for Japan. Statistical noise arising from finiteness of the time series data is carefully removed by making use of…
The paper introduces isotropy as a regularizer to enhance portfolio stability.
problem Model uncertainty and estimation errors in diversification strategies.
method Integrates isotropy as a geometric regularizer into mean-variance optimization.
result Isotropy constraint systematically induces negative average-signal exposure, providing a robust crash hedge.
New analysis of Muon and SignSGD on matrix-valued least squares problems.
problem Understanding the behavior of Muon and SignSGD on matrix-valued least squares problems.
method Derive explicit deterministic dynamics to study learning behavior of Muon and SignSGD.
result Muon and SignSGD exhibit different optimal learning rates and convergence characteristics based on batch size and data covariance.
A method to explain disease transformation using biomarker covariance matrices.
problem Understanding disease transformation from a healthy baseline.
method Modeling healthy and disease states of biomarker covariance matrices to characterize perturbations.
result Disease perturbs the biomarker covariance structure, allowing for mechanistic explanations and individual patient prognosis.
Diffusion models' consistency across splits explained by random matrix theory.
problem Consistency of diffusion models trained on non-overlapping subsets.
method Random matrix theory framework to quantify dataset effects on denoiser and sampling map.
result The theory explains and predicts cross-split disagreement in diffusion models.
New defence against data-poisoning attacks in neural networks.
problem Data-poisoning attacks can evade existing defences and increase model efficacy.
method Proved geometric mechanism and identified near clone regime in input space.
result Regularisation and data augmentation reduce data fitting capacity and prevent poisoning.
The limits of interpretability in multiple linear regressioncond-mat.dis-nn
Interpretable machine-learning models can be unstable under multicollinearity, leading to oscillatory weights that do not reflect meaningful contributions.
problem Interpretable machine-learning models can be unstable under multicollinearity.
method Theoretical analysis of eigenmodes of the feature correlation matrix.
result Small-eigenvalue modes associated with multicollinearity amplify fluctuations in the weights and generate oscillatory patterns that do not necessarily reflect meaningful contributions.
New algorithms extract Koopman invariant subspaces from large-scale data.
problem Difficulty in discerning the Koopman invariant subspace from many Koopman eigenmodes.
method Multi-task feature learning and pruning procedure to remove spurious modes.
result Effective in approximating Koopman operator for complex flows.