Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.
problem Understanding infinite volume asymptotically hyperbolic manifolds.
method Survey of geometry, spectral theory, dynamics, and quantum/classical mechanics.
result Recent results, ideas, and conjectures discussed.
Proposes neural dynamic mode decomposition for end-to-end modeling of nonlinear dynamics.
problem Understanding and modeling nonlinear dynamical systems.
method Trains neural networks to minimize forecast error based on spectral decomposition in the lifted space.
result Demonstrates effectiveness in eigenvalue estimation and forecast performance.
New method controls linear systems with partial info and disturbances.
problem Controlling linear dynamical systems under partial observation and adversarial disturbances.
method Double Spectral Control (DSC) using two-level spectral approximation strategy.
result Matches best known regret guarantees with exponential runtime improvement.
Spectral portfolio theory links neural networks to wealth dynamics via SGD weight matrices.
problem Understanding wealth dynamics from neural network training.
method Direct identification of weight matrices as portfolio allocation matrices, linking SGD forces to portfolio dynamics.
result Spectral properties of SGD weight matrices transition between additive and multiplicative regimes, influencing wealth dynamics.
Algorithm learns dynamics from past observations.
problem Learning a nonlinear dynamical system.
method Spectral filtering, online convex optimization.
result Vanishing prediction error for marginally stable systems.
Novel model captures high-dimensional copulas with spectral dynamics and regularization.
problem Modeling time-varying, asymmetric, tail-dependent copulas in high dimensions.
method Score-driven dynamics for eigenvalues, non-linear shrinkage for biases, parsimonious and scalable.
result Model outperforms recent alternatives in capturing co-movements and diversification potential.
DOODL learns shared spectral dynamics across related dynamical systems.
problem Learning independent dynamical operators for each system limits discovery of shared structure.
method DOODL learns a dictionary of characteristic spectral dynamics on a manifold of related systems.
result DOODL achieves errors one to two orders of magnitude lower than independent operator estimation methods.
In this paper we propose the notion of continuous-time dynamic spectral risk-measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk-measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may …
Study of spectral invariants on CR contact manifolds with circle action.
problem Analytic torsion and eta-like invariants on CR contact manifolds.
method Interpret spectral series topologically and dynamically using Reeb flow.
result Spectral series can be interpreted both topologically and dynamically.
Spectral clustering is a widely studied problem, yet its complexity is prohibitive for dynamic graphs of even modest size. We claim that it is possible to reuse information of past cluster assignments to expedite computation. Our approach builds on a recent idea of sidestepping the main bottleneck of spectral clusterin…
A new metric compares dynamical systems using operator eigenvalues.
problem Comparing and interpolating nonlinear dynamical systems from trajectory data.
method Representing systems as distributions of operator eigenvalues and projectors, defining a spectral-Grassmann Wasserstein metric.
result The proposed metric outperforms standard operator-based distances in machine learning applications.
This research introduces dynamic portfolio cuts using a spectral approach for graph-theoretic diversification.
problem Traditional methods for estimating asset-return covariance assume statistical time-invariance, failing to capture the nonstationary nature of asset price movements.
method Introduces graph spectral estimators that account for nonstationarity, partitioning the market graph into time-evolving clusters for dynamic portfolio cuts.
result Demonstrates the advantages of the proposed framework over traditional methods through numerical case studies using real-world price data.
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.
We give a polynomial-time algorithm for learning latent-state linear dynamical systems without system identification, and without assumptions on the spectral radius of the system's transition matrix. The algorithm extends the recently introduced technique of spectral filtering, previously applied only to systems with a…
Sketching reduces data size for accurate spectral estimation.
problem Estimating spectral density from large simulation datasets.
method Sketching for dimensionality reduction and data compression.
result Sketching provides 90% accurate spectral density estimate with 10% data.
Spectral method speeds fitting of binary time series models.
problem Modeling binary time series data with latent linear dynamical systems.
method Spectral learning of probit-Bernoulli latent linear dynamical systems.
result Spectral method provides robust, fixed-cost estimator.
Muon dynamics study uses spectral Wasserstein flow for optimization stability.
problem Optimizing deep learning models with gradient normalization.
method Introduces Spectral Wasserstein distances for matrix flows, proving equivalence with Benamou--Brenier formulation.
result Gradient-flow interpretation of mean-field normalized training dynamics.
Proposes a new framework for risk-sensitive RL using deep nets.
problem Risk-sensitive reinforcement learning problems.
method Conditional elicitability, scoring functions, deep neural networks.
result Dynamic spectral risk measures can be approximated by deep nets.
The paper is an informal report on joint work with Stefan Haller on Dynamics in relation with Topology and Spectral Geometry. By dynamics one means a smooth vector field on a closed smooth manifold; the elements of dynamics of concern are the rest points, instantons and closed trajectories. One discusses their counting…
SPEDER extracts state-action abstraction from dynamics for reinforcement learning.
problem Curse of dimensionality and limited applicability of spectral methods.
method Spectral Decomposition Representation (SPEDER) that extracts state-action abstraction from dynamics without policy dependence.
result Theoretical analysis establishes sample efficiency in online and offline settings.
Survey on strong closing lemmas in Hamiltonian dynamics.
problem Understanding dynamics in Hamiltonian systems.
method Use spectral invariants in symplectic geometry.
result Proofs of strong closing lemmas in various dimensions.
ULES embeds dynamic networks with stability guarantees.
problem Stability of time-varying node embeddings in evolving networks.
method Unfolded Laplacian Spectral Embedding (ULSE) using normalized Laplacian operators.
result ULES satisfies cross-sectional and longitudinal stability under dynamic stochastic block model.
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.
A tutorial on dynamic Laplacian for time-evolving data clusters.
problem Handling data with a time component and lying on a time-evolving manifold.
method Dynamic generalization of the Laplacian, SEBA algorithm for feature separation.
result Clusters correspond to long-lived coherent collections in time-evolving data.
New method embeds dynamic networks with stability for node behavior.
problem Embed time-evolving node representations with stability.
method Unfolded adjacency spectral embedding for dynamic networks.
result Method satisfies cross-sectional and longitudinal stability.
Spectral decomposition of the Koopman operator is attracting attention as a tool for the analysis of nonlinear dynamical systems. Dynamic mode decomposition is a popular numerical algorithm for Koopman spectral analysis; however, we often need to prepare nonlinear observables manually according to the underlying dynami…
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.
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.
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.
Formula derived for zeta functions of 3D foliated systems.
problem Analyzing zeta functions of 3D Riemannian foliated dynamical systems.
method Relating dynamical spectral ξ-functions to zeta functions using the distributional dynamical Lefschetz trace formula. result Proved a regularized determinant formula for zeta functions.
New spectral invariants recover Calabi invariant for surface dynamics.
problem Understanding Hamiltonian homeomorphisms and spectral invariants.
method Defining new spectral invariants for Lagrangian links in surfaces.
result Our invariants recover the Calabi invariant and resolve open questions.
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 an oblique projection technique to approximate a foliation for non-normal dynamics.
problem Modeling dynamics far from a primary Spectral Submanifold (SSM) in non-normal systems.
method Oblique projection technique based on experimental data.
result Approximates a stable invariant foliation for non-normal dynamics efficiently.
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.
Analyzes how diffusion models learn, revealing a spectral bias in structure mastery.
problem Understanding the learning dynamics and bias in diffusion models.
method Developed an analytical framework using a Gaussian-equivalence principle to solve gradient-flow dynamics and integrate probability-flow ODEs.
result Exposes a universal inverse-variance spectral law: high-variance structure is mastered faster than low-variance detail.
Paper explores SNN for learning spectral geometric info from data.
problem Challenges in applying traditional eigensolvers to big data.
method Introduces Spectral Neural Networks (SNN) as an alternative.
result Investigates tradeoffs and optimization landscape of SNN.
This paper improves parameter estimation for autonomous systems with unmodeled dynamics.
problem Accurate parameter estimation for risk-aware autonomous systems with unmodeled dynamics.
method Spectral lines-based approach for estimating parameters of dynamic models, allowing deterministic unmodeled dynamics.
result The proposed method leads to non-asymptotic bounds on parameter estimation error, robust to unmodeled dynamics, and matches existing literature in ideal conditions.
Spectral methods predict long-term signals from linear and nonlinear systems.
problem Forecasting temporal signals from linear and nonlinear systems with arbitrary sampling.
method Introduces a spectral algorithm for linear signals and extends it to nonlinear systems using Koopman theory.
result The spectral methods achieve high accuracy in forecasting and uncertainty quantification.
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.
DynMSA detects market clusters for better portfolio allocation.
problem Identifying stable market clusters for effective portfolio management.
method Combining Random Matrix Theory with modularity optimization and spectral clustering.
result DynMSA outperforms baseline models in intra- and inter-cluster correlation differences.
Develops a dynamical method to prove the sharp Berezin-Li-Yau inequality.
problem Proving the sharp Berezin-Li-Yau inequality for convex domains.
method Volume-preserving mean curvature flow and a new monotonicity principle.
result Shows the sharp Berezin-Li-Yau bound for every smooth convex domain.
Model tracks structural changes in Brownian particle configurations on a sphere.
problem Tracking structural changes in Brownian particle configurations on a sphere.
method Introduces Frustrated Distance Matrix (FDM) model for dynamic distance matrices on S^2.
result Preserves static BBS template with dynamics as redistributed spectral mass.
Introduces new spectral triples for parabolic geometry.
problem Anisotropies and varying orders in parabolic geometry.
method Tangled spectral triples incorporating directional Dirac operators.
result Higher order spectral triples for hypoelliptic complexes and nilpotent group algebras.
New method controls linear systems with adversarial disturbances.
problem Controlling linear dynamical systems under adversarial conditions.
method Novel convex relaxation using spectral filters from Hankel matrix eigenvectors.
result Polylogarithmic running time improvement over prior methods.
New model learns graph spectra accurately, outperforming existing methods.
problem Graph diffusion models struggle to distinguish certain graph families and their spectra.
method Leveraged random matrix theory to analytically extract spectral properties, introducing Dyson Diffusion Model.
result Dyson Diffusion Model learns graph spectra accurately and outperforms existing models.
Interpolates mean shift and spectral clustering on graphs.
problem Data clustering algorithms.
method Fokker-Planck equations on data graphs.
result New theoretical insights on diffusion maps and mean shift dynamics.
New framework tracks communities in dynamic networks.
problem Discovering and tracking communities in evolving networks.
method Spectral framework on Grassmann manifold for subspace tracking.
result Improved dynamic community detection results across various network types.
AUASE embeds dynamic networks with stability guarantees for node comparison.
problem Stability in dynamic network embeddings for comparing nodes across time.
method Attributed unfolded adjacency spectral embedding (AUASE) for stable unsupervised learning.
result AUASE provides significant improvements in link prediction and node classification.