New method clusters directed graphs using Koopman operators.
problem Challenges in clustering directed graphs, especially complex eigenvalues and lack of cluster definition.
method Relate graph Laplacians to transfer operators and metastable sets in stochastic systems, derive clustering algorithms for directed and time-evolving graphs.
result Clusters can be interpreted as coherent sets, useful for analyzing transport and mixing processes.
The paper proposes a method to learn low-dimensional state embeddings from time series data.
problem Finding compact state embeddings from high-dimensional Markov state trajectories.
method The paper introduces a method based on diffusion maps to learn a low-dimensional state embedding and captures the dynamics of the process.
result The method reveals metastable structures in state clustering, providing sharp statistical error bounds and misclassification rates.
IDM learns clustered representations for complex FELs.
problem Learning meaningful representations for complex free energy landscapes.
method Information Distilling of Metastability (IDM) method.
result IDM achieves physically meaningful representations of FELs.
The paper studies how search and distillation improve reasoning in large language models.
problem Improving reasoning capabilities of large language models.
method Viewing chain-of-thought generation as a metastable Markov process, proving benefits of search and distillation.
result Search protocol rewards sparse edges, reducing the expected number of steps to reach different clusters.
New method detects metastable basins in high dimensions using trajectory sampling.
problem Identifying distinct basins in high-dimensional Markov processes.
method Discriminative approach based on marginal trajectory distribution comparison.
result Bayes-optimal classifier achieves high accuracy distinguishing between basins.
Learn true model from metastable samples of discrete distributions.
problem Learning true model from samples that get stuck in metastable states.
method Conditional likelihood estimator based on strong metastability condition.
result True model can be recovered from metastable samples of discrete distributions.
New method makes machine learning approximations unbiased and efficient.
problem Efficient sampling of complex probability distributions.
method Uses autoregressive neural networks with cluster updates and physical symmetries.
result Shows unbiased and low-variance approximations for phase transitions.
This paper addresses metaconsistency in Bayesian inference for metastable systems.
problem Inference for metastable systems may not be consistent, but can be metaconsistent over large but finite time intervals.
method Introduces metaconsistency in a Bayesian framework, discusses its relation to spectral properties of model dynamics.
result Metaconsistency can be exploited to infer sub-systems efficiently from larger systems.
New methods use machine learning to simulate rare transitions in molecular systems.
problem Simulating rare transitions between metastable states in molecular dynamics.
method Generative models and reinforcement learning for importance sampling.
result Efficiently generated transition paths linking metastable states.
Proves triviality of inertia groups in high-dimensional manifolds.
problem Classifying manifolds in the metastable range.
method Understanding the second extended power functor in synthetic spectra.
result Inertia groups of high-dimensional manifolds are trivial.
Lipid necks, large curvature bridges, are shown to be metastable.
problem Understanding the energetically prohibitive yet ubiquitous lipid necks in cell membranes.
method Geometric triality approach to demonstrate metastability.
result Lipid necks can exist for finite but potentially long times without stabilizing mechanisms.
Paper uses deep learning to compute committor functions for rare events in complex systems.
problem Computing committor functions for low-temperature, high-dimensional systems is challenging.
method Combines deep learning, data sampling, and feature engineering.
result Achieves good performance on complex benchmark problems with rough energy landscapes.
Study finds the number of modes in Gaussian kernel density estimators scales with sqrt(β log β).
problem Determining the number of clusters in Transformers.
method Used Kac-Rice formula and Edgeworth expansion to prove scaling.
result The expected number of modes scales as Θ(√(β log β)).
This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings N→Sm. We study the specific case of knotted tori, i. e. the embeddings Sp×Sq→Sm. The classification of knotted tori up to isotopy in the metastable dimension ran…
We propose a deep generative Markov State Model (DeepGenMSM) learning framework for inference of metastable dynamical systems and prediction of trajectories. After unsupervised training on time series data, the model contains (i) a probabilistic encoder that maps from high-dimensional configuration space to a small-siz…
Study shows social reinforcement learning can lead to persistent but metastable polarization.
problem Understanding the persistence and dynamics of opinion polarization.
method Simulation and mathematical analysis of a reinforcement learning model of opinion dynamics.
result Polarization observed in the model is metastable and will eventually lead to consensus.
ISOKANN learns collective variables and effective dynamics for metastable transitions.
problem Understanding metastable transitions in complex molecular systems.
method Integrates Koopman operators with neural networks to extract CVs and effective dynamics.
result Reconstructs coarse-grained kinetics and reproduces transition times across barriers.
An algorithmically hard phase was described in a range of inference problems: even if the signal can be reconstructed with a small error from an information theoretic point of view, known algorithms fail unless the noise-to-signal ratio is sufficiently small. This hard phase is typically understood as a metastable bran…
GMVAE improves clustering in molecular simulations data.
problem Clustering metastable states in multi-basin free-energy landscapes.
method Gaussian mixture variational autoencoder (GMVAE) for dimensionality reduction and clustering.
result Enhanced clustering of metastable states compared to standard VAEs.
Embeddability of simplicial complex [3]∗K linked to K's embeddability.
problem Relating embeddability of [3]∗K to K's embeddability. method Comparing embeddability conditions for K and [3]∗K. result Embeddability of [3]∗K equivalent to vanishing van Kampen obstruction for K. Analyzes SGD's behavior under heavy-tailed noise, deriving step-size conditions for metastability.
problem Analyzing SGD's performance under heavy-tailed gradient noise.
method Modeling SGD as a discretized SDE driven by Lévy motion, deriving step-size conditions.
result Identifies small step-sizes for discrete system to inherit continuous-time system's metastability behavior.
Study of phase separation and geometry on a closed elastic curve, including dynamics and free energy minimization.
problem Free energy and dynamics of a closed elastic filament coupled to a scalar concentration field.
method Analytical and numerical simulations of coupled Willmore flow and Cahn--Hilliard gradient flow on differential geometry.
result Qualitative changes in free energy landscape due to closure constraint, leading to metastable and stable multi-domain morphologies.
We analyze wealth condensation for a wide class of stochastic economy models on the basis of the economic analog of thermodynamic potentials, termed transfer potentials. The economy model is based on three common transfers modes of wealth: random transfer, profit proportional to wealth and motivation of poor agents to …
Develops methods to simulate rare transitions in molecular systems.
problem Rare transitions between metastable states in molecular systems are difficult to study due to limited data.
method Two novel methods: chain-based and midpoint-based approaches.
result Demonstrates effectiveness of methods in both data-rich and data-scarce scenarios.
We study the detailed path-wise behavior of the discrete-time Langevin algorithm for non-convex Empirical Risk Minimization (ERM) through the lens of metastability, adopting some techniques from Berglund and Gentz (2003. For a particular local optimum of the empirical risk, with an arbitrary initialization, we show tha…
A new framework uses stochastic optimal control to estimate rare events more accurately.
problem Estimating rare events like chemical reactions in biomolecules is computationally challenging.
method The approach casts committor estimation as a stochastic optimal control problem, developing direct and off-policy Value Matching losses.
result The framework yields more accurate committor estimates, reaction rates, and equilibrium constants.
Discovering quasipotential equations from data using machine learning.
problem Understanding escape mechanisms from metastable states in nonlinear systems.
method Combining neural networks and sparse regression to symbolically reconstruct quasipotential equations.
result Model-unbiased analytical forms of quasipotential discovered directly from data.
Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-multiplicity intersections. We focus on conditions for the existence of almost r-embe…
Given Poincare spaces M and X, we study the possibility of compressing embeddings of M x I in X x I down to embeddings of M in X. This results in a new approach to embedding in the metastable range both in the smooth and Poincare duality categories.
We introduce a machine learning approach for extracting fine-grained representations of protein evolution from molecular dynamics datasets. Metastable switching linear dynamical systems extend standard switching models with a physically-inspired stability constraint. This constraint enables the learning of nuanced repr…
An image pattern can be represented by a probability distribution whose density is concentrated on different low-dimensional subspaces in the high-dimensional image space. Such probability densities have an astronomical number of local modes corresponding to typical pattern appearances. Related groups of modes can join…
The purpose of this note is to attract attention to the following conjecture (metastable r-fold Whitney trick) by clarifying its status as not having a complete proof, in the sense described in the paper. Assume that D=D1⊔…⊔Dr is disjoint union of r disks of dimension s, f:D→Bd a proper …
RC flow learns molecular kinetics in low dimensions.
problem Discovering interpretable low-dimensional models of molecular kinetics.
method Normalizing flow for coordinate transformation and Brownian dynamics for kinetics approximation.
result Tractable and trainable model of reduced kinetics in continuous time and space.
Study shows bifurcating price dynamics in ASME with traders.
problem Understanding price dynamics in artificial stock markets.
method Agent-based model of endogenous traders interacting through a LOB.
result Bistability in price equilibria: zero-price and persistent positive-price states.
Let EMBED(k,d) be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R^d? Known results easily imply polynomiality of EMBED(k,2) (k=1,2; the case k=1, d=2 is graph planarity) and of EMBED(k,2k) for all k>2 (even if k i…
Markov state models (MSMs) and Master equation models are popular approaches to approximate molecular kinetics, equilibria, metastable states, and reaction coordinates in terms of a state space discretization usually obtained by clustering. Recently, a powerful generalization of MSMs has been introduced, the variationa…
We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection-diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or h…
We develop a new approach to the classical problem on isotopy classification of embeddings of manifolds into Euclidean spaces. This approach involves studying of a new embedding invariant, of almost-embeddings and of smoothing, as well as explicit constructions of embeddings. Using this approach we obtain complete conc…
Efficiently simulates slow dynamics of high-dimensional stochastic systems.
problem Simulating high-dimensional stochastic systems with slow dynamics and fast modes.
method Designs an algorithm to estimate an invariant manifold and its dynamics, averaging out fast modes.
result Efficient simulator of effective dynamics on low-dimensional invariant manifold.
This paper extends some results of Hatcher and Quinn beyond the metastable range. We give a bordism theoretic obstruction to deforming a map between manifolds simultaneously off of a collection of pairwise disjoint submanifolds under the assumption that it can be deformed off of any proper subcollection in a homotopy c…
Paper tackles sampling from non-log-concave distributions using denoising diffusion.
problem Sampling from non-log-concave distributions efficiently.
method DDMC framework, Zeroth-Order Diffusion Monte Carlo (ZOD-MC) algorithm.
result ZOD-MC achieves inverse polynomial dependence on sampling accuracy, efficient for low dimensions.
Enhanced VMC methods improve neural wavefunction training.
problem Efficiently training neural wavefunctions in VMC to converge to energy minimum.
method Rayleigh-Gauss-Newton (RGN) optimization and parallel tempering sampling.
result RGN method achieves superlinear convergence with reduced computational cost.
Timewarp accelerates molecular dynamics by learning to simulate long timescales.
problem Efficiently simulating long timescales in molecular dynamics.
method Uses a normalizing flow to learn large time steps in Markov chain Monte Carlo.
result Generalizes to unseen small peptides, accelerating sampling.
A clear understanding of topology of higher-dimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higher-dimensional topology in a way which makes clear the visual and intuitive part of the constructions and the arguments. In particu…
Although standard economics textbooks are seldom interested in production networks, modern economies are more and more based upon suppliers/customers interactions. One can consider entire sectors of the economy as generalised supply chains. We will take this view in the present paper and study under which conditions lo…
This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings Sp×Sq→Sm, which we call knotted tori. The set of knotted tori in the the space of sufficiently high dimension, namely in the metastable range m≥p+3q/2+2, p≤q, which is a nat…
Measures mode separation in high-dimensional densities via a reversible diffusion process.
problem Quantifying how sharply a distribution fragments into barrier-separated clusters in high dimensions.
method A unique reversible diffusion process with f as stationary distribution, extracting SSA and DA from its autocovariance matrix.
result Empirical autocovariance spectrum and readouts (SSA, DA) quantify mode separation using only samples and pretrained score-based models.
Study improves policy search in continuous control by using heavy-tailed distributions.
problem Challenges in continuous space policy search due to non-convexity and myopic-farsighted incentives.
method Introduced heavy-tailed policy parameterizations and analyzed convergence rates and stability.
result Convergence rate to stationarity depends on policy's tail index and exploration tolerance.