The paper extends a spectral evolution model for link prediction in evolving networks.
problem Link prediction in evolving networks.
method Approximated eigenvalue trajectories using Rayleigh quotient and extrapolation.
result Learning algorithms based on approximated trajectories outperform traditional methods.
FNOs learn solution operators of dissipative equations efficiently via spectral methods.
problem Learning and approximation of solution operators for dissipative equations.
method Introducing spectral methods and deriving FNO approximation bounds and sample complexity guarantees.
result Polynomial sample complexity guarantees for FNOs learning solution operators of dissipative equations.
Gradient descent solves rank-one matrix estimation problem with detailed time evolution analysis.
problem Estimating a rank-one symmetric matrix corrupted by noise.
method Gradient descent on a sphere, using local versions of the semi-circle law.
result Explicit formulas for the time evolution of the estimator and cost function, revealing phase transitions.
Study of meromorphic connections and their spectral duals in gl3(C).
problem Exploring ℏ-deformed meromorphic connections and their spectral duals. method Using apparent singularities and their dual partners as Darboux coordinates, the Hamiltonian evolutions and reductions are derived.
result Spectral duality extends to Hamiltonian evolutions, tau-functions, and Hermitian matrix models on both sides.
New method clusters evolving networks using spatio-temporal graph Laplacian.
problem Clustering communities in time-varying graphs.
method Extends spectral clustering to dynamic graphs using CCA and spatio-temporal graph Laplacian.
result The spatio-temporal graph Laplacian clearly interprets cluster evolution over time.
The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-b…
Modeling financial market dynamics with 2D Levy flights.
problem Capturing the complex, scaling laws in financial market dynamics.
method 2D Lévy flight model applied to S\&P 500 index prices.
result Empirical spectral properties match model predictions.
Investigates spectral properties of neural networks, showing invariance under certain conditions.
problem Understanding the spectral evolution and invariance in linear-width neural networks.
method Empirical and theoretical analysis of spectra of weight matrices in high-dimensional settings.
result Spectra of weight matrices are invariant under certain training conditions, with implications for feature learning.
New methods avoid spectral pollution in transfer operators for accurate analysis.
problem Spectral pollution in finite-dimensional approximations of transfer operators.
method Algorithms for computing spectral properties of transfer operators without spectral pollution.
result Accurate spectral estimation across various applications, including protein folding models.
Spectral analysis detects structural changes in financial networks.
problem Detecting structural transitions in financial networks to assess systemic risk.
method Ensemble properties of spectral radius of random graph models calibrated on real-world evolving networks.
result The spectral deviation captures ongoing topological changes in financial networks.
Study cryptocurrency price dynamics using adaptive EMD and spectral analysis.
problem Analyze the time-varying volatility of cryptocurrency prices.
method Adaptive complementary ensemble empirical mode decomposition (ACE-EMD) and Hilbert spectral analysis.
result Reveal the properties of various timescales in cryptocurrency price dynamics.
A framework uses free probability to analyze Transformer models.
problem Understanding the dynamics and complexity of Transformer-based language models.
method Formal operator-theoretic analysis using free probability theory.
result Entropy-based generalization bounds derived under freeness assumptions.
Develops finite-gap solutions for Pohlmeyer--Lund--Regge equation and Lund--Regge curve evolution.
problem Solving the Pohlmeyer--Lund--Regge equation and understanding Lund--Regge curve evolution.
method Finite-gap construction using hyperelliptic spectral data, Baker--Akhiezer function, and SU(2)-frame. result Explicit theta-quotient formula for PLR solutions and criteria for Lund--Regge curve evolution.
A fast spectral algorithm detects community structure in evolving graphs.
problem Detecting community structure in time-evolving sparse graphs.
method Extension of the Bethe-Hessian matrix for spectral community detection.
result The algorithm reaches the optimal detectability threshold and outperforms other methods.
New method extrapolates spectral densities from smaller models to larger ones.
problem Limited practical computations for large machine learning models.
method Algebraic spectral curve theory for free decompression.
result Framework enables extrapolation of spectral densities with multiple or multi-modal bulks.
The paper studies non-integer curvature flows and proves convergence to spheres under specific conditions.
problem Analyzing the convergence of non-integer curvature flows on rotationally symmetric surfaces.
method Spectral theory of singular Sturm-Liouville operators to construct an eigenbasis and prove convergence.
result The flow converges to a round sphere if the focal points coincide at the poles, otherwise to a non-round Hopf sphere.
This paper is devoted to the horizontal (``characteristic'') cohomology of systems of differential equations. Recent results on computing the horizontal cohomology via the compatibility complex are generalized. New results on the Vinogradov C-spectral sequence and the Krasil'shchik C-cohomology are obtained. As an appl…
Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise. If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform simple spectral approaches. Recent work characterizes the asymptotic acc…
This paper proposes a spectral clustering algorithm for hyperbolic spaces, improving efficiency over Euclidean methods.
problem Inefficient clustering in Euclidean spaces for complex data structures.
method Developed a spectral clustering algorithm using hyperbolic similarity matrices.
result The algorithm converges at least as fast as Euclidean spectral clustering and performs better on complex datasets.
We consider the problem of training input-output recurrent neural networks (RNN) for sequence labeling tasks. We propose a novel spectral approach for learning the network parameters. It is based on decomposition of the cross-moment tensor between the output and a non-linear transformation of the input, based on score …
HSSE framework embeds single-cell RNA-seq data at multiple scales.
problem Capturing heterogeneous local structure in single-cell RNA-seq data.
method Hierarchical sheaf spectral embedding (HSSE) framework.
result HSSE achieves competitive or improved performance in single-cell RNA-seq data representation learning.
The paper analyzes diffusion condensation for data geometry and topology.
problem Understanding the geometry and topology of high-dimensional data.
method Time-inhomogeneous diffusion process with geometric, spectral, and topological analysis.
result The condensation process defines intrinsic condensation homology and ambient persistent homology.
Unified theory of θ-expectations derived from chaotic dynamics.
problem Non-convex stochastic control problems outside G-expectations.
method Spectral theory of transfer operators for uniformly hyperbolic flows, viscosity solutions to HJB equations.
result Affine Hessian, non-convex gradient structure of θ-expectation. Study optimal algorithms for recovering signals through inhomogeneous low-rank channels.
problem Recovering signals through an inhomogeneous low-rank matrix channel.
method Derive and analyze an approximate message-passing algorithm (AMP) and a spectral method.
result The AMP iteration matches the conjectured optimal computational phase transition.
A time-varying network reveals community structure in cryptocurrencies.
problem Investing in cryptocurrencies from different communities can diversify risk.
method Dynamic covariate-assisted spectral clustering method.
result Investors can earn 1.08% daily return by diversifying across communities.
Heat flow on lens spaces settles into Morse functions with four critical points.
problem Understanding the behavior of heat flow on lens spaces.
method Analyzing the asymptotic spectral expansion of the heat flow.
result Generic heat evolutions on lens spaces \(L(p,q)\) with \(p\geq2\) and \(1\leq q\leq p/2\) tend to settle into Morse functions with exactly four critical points.
Graph neural networks (GNNs) have become increasingly popular for classification tasks on graph-structured data. Yet, the interplay between graph topology and feature evolution in GNNs is not well understood. In this paper, we focus on node-wise classification, illustrated with community detection on stochastic block m…
This paper concerns the numerical solution of the finite-horizon Optimal Investment problem with transaction costs under Potential Utility. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints. In Finite-Horizon Optimal Investment with Transaction Costs: A Parabolic Double Obst…
New interpretation reconciles country and product complexity.
problem Difficulty in interpreting Economic and Product Complexity Indices.
method Spectral clustering algorithm to separately group similar countries and products.
result Indices identify two co-clusters of similar countries and products.
The study examines spectral dynamics in deep neural networks, predicting how outliers evolve during training.
problem Understanding spectral evolution in deep neural networks during training.
method Developed a two-level dynamical mean-field theory (DMFT) to track spectral dynamics.
result The theory predicts how outliers evolve with training time, width, output scale, and initialization variance.
Higher-order motif structures and multi-vertex interactions are becoming increasingly important in studies that aim to improve our understanding of functionalities and evolution patterns of networks. To elucidate the role of higher-order structures in community detection problems over complex networks, we introduce the…
The paper connects isomonodromic and isospectral deformations for sl2(C) connections.
problem Connecting isomonodromic and isospectral deformations for sl2(C) connections. method Explicitly constructing Lax pairs and Darboux coordinates to bridge isomonodromic and isospectral deformations.
result Explicit change of Darboux coordinates to match spectral invariants, solving an open issue.
We propose an extension of the differential system for constant mean curvature (CMC) surfaces in a three dimensional space form to an associated hierarchy of evolution equations by the higher-order commuting symmetries. The infinite sequence of higher-order conservation laws of CMC surfaces admit the corresponding exte…
We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemann…
Study local geometry of mixture models via spectral theory, revealing transitions in training dynamics.
problem Understanding the local geometry of high-dimensional mixture models.
method Spectral theory of Hessian and information matrices, focusing on i.i.d. Gaussian mixtures.
result Exact formulas for limits of spectral distribution and outlier eigenvalues, connecting training dynamics to effective dynamics.
New findings show neural network training loss follows a power law over time.
problem Understanding the optimization process of neural networks during training.
method Spectral analysis of the integral operator representing the linearized evolution of a large network.
result The loss function in neural network training follows a power law behavior, L(t)∼t−ξ, with exponent ξ determined by network parameters and data characteristics. Partitioning a graph into groups of vertices such that those within each group are more densely connected than vertices assigned to different groups, known as graph clustering, is often used to gain insight into the organisation of large scale networks and for visualisation purposes. Whereas a large number of dedicated…
New insights into why neural networks generalize well.
problem Understanding why neural networks generalize well despite heavy-tailed weight distributions.
method Developed a simple model to analyze the emergence of heavy-tailed empirical spectral densities (ESDs) in two-layer neural networks without gradient noise.
result Learning rates play a crucial role in shaping the ESDs of two-layer neural networks, leading to better generalization.
We consider the heat flow of corotational harmonic maps from R3 to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self…
This paper tackles graph translation challenges by predicting both node and edge attributes simultaneously.
problem Challenges in predicting both node and edge attributes in graph translation, especially in interactive, iterative, and asynchronous processes.
method Developed a novel framework integrating both node and edge translations seamlessly, using spectral graph regularization to maintain consistency.
result Demonstrated the effectiveness of the proposed method on both synthetic and real-world application data.
OLS is a special case of Transformer, revealing its linear nature.
problem Understanding the statistical essence of Transformer architecture.
method Algebraic proof and spectral decomposition of covariance matrix.
result Attention mechanism in Transformers is mathematically equivalent to OLS.
A new model captures forward curve dynamics with stochastic volatility.
problem Modeling continuous-time evolution of forward curves in financial markets.
method Affine stochastic volatility model with modulated dynamics.
result Model allows for maturity-specific risk and volatility clustering.
Cryptocurrencies return cross-predictability and technological similarity yield information on risk propagation and market segmentation. To investigate these effects, we build a time-varying network for cryptocurrencies, based on the evolution of return cross-predictability and technological similarities. We develop a …
We introduce a new elliptic operator on null hypersurfaces of four-dimensional Lorentzian manifolds. This operator depends on the first and second fundamental forms of the sections of a foliation of the null hypersurface and its novelty originates from its covariant transformation under change of foliation. It thus pro…
New method models covariates and responses without parametric assumptions using manifold learning.
problem Losing explanatory power for responses in standard factor models applied to covariates alone.
method Anisotropic diffusion maps for learning low-dimensional embeddings.
result Kalman filtering in diffusion-map coordinates improves joint covariate-response prediction.
Study shows how anisotropic data affects learning dynamics in phase retrieval.
problem Understanding learning dynamics in phase retrieval with anisotropic Gaussian inputs.
method Developed a tractable reduction to reveal a three-phase trajectory and derived scaling laws.
result Found that anisotropy leads to a three-phase trajectory: fast escape, slow convergence, and spectral-tail learning.
KOMET identifies Koopman operators from model parameter trajectories to adapt to evolving data distributions.
problem Adaptation of parametric models to non-stationary environments.
method Data-driven framework using Koopman operator identification and Extended Dynamic Mode Decomposition (EDMD).
result KOMET achieves high autonomous-rollout accuracies of 0.981 to 1.000 over 100 time steps on various drifting datasets.
SPT predicts age and mass of red giants from spectra.
problem Challenges in age and mass estimation of red giants using traditional methods.
method SPT framework with Multi-head Hadamard Self-Attention and Mahalanobis distance-based loss function.
result Remarkable age and mass estimations with low errors and uncertainties.