The paper models market crashes as phase transitions, finding dynamic transitions offer better predictions.
problem Understanding and predicting extreme financial events like market crashes.
method Employing phase transition theory, focusing on endogenous crashes, and comparing DPT, CPT, and SPT.
result Dynamic phase transitions provide more accurate predictions of market crashes compared to critical and stochastic models.
Generative diffusion models exhibit phase transitions in statistical mechanics, impacting their performance.
problem Understanding the performance and capabilities of generative diffusion models.
method Reformulating generative diffusion models using statistical mechanics, focusing on phase transitions and symmetry breaking.
result Generative diffusion models undergo second-order phase transitions with mean-field universality, critical instability, and mean-field critical exponents.
New model shows natural language exhibits phase transition similar to physics.
problem Understanding critical properties in natural language models.
method Created a context-sensitive random language model.
result Demonstrated a Berezinskii--Kosterlitz--Thouless phase transition.
A neural network model predicts the critical point of the Ising phase transition.
problem Predicting the critical point of the Ising phase transition using supervised learning.
method Proposed a minimal one-free-parameter neural network model to describe the supervised learning problem for the Ising model.
result Just one free parameter is enough to describe the universal finite-size-scaling function in the network output.
Machine learning predicts critical points for directed percolation models.
problem Determining critical points for directed percolation models.
method Supervised and unsupervised machine learning algorithms (CNN and DBSCAN) were used.
result Machine learning accurately predicts critical points for both models.
We show that for three dimensional gravity with higher genus boundary conditions, if the theory possesses a sufficiently light scalar, there is a second order phase transition where the scalar field condenses. This three dimensional version of the holographic superconducting phase transition occurs even though the pure…
Continuous phase transitions identified in Doi-Onsager, noisy transformer, and Hegselmann-Krause models.
problem Phase transitions in multimodal models and their properties.
method Sharp coercivity estimate and constrained Lebedev--Milin inequality.
result Continuous phase transitions at critical coupling strengths for Doi-Onsager, noisy transformer, and Hegselmann-Krause models.
Modeling financial markets as gas molecules, the paper predicts phase transitions similar to water and steam.
problem Understanding the dynamics of financial markets through phase transitions.
method Developed a lattice gas model equivalent to the Ising model on a social network, analyzing critical exponents and auto-correlations.
result Financial market dynamics exhibit phase transition-like behavior, with critical exponents analogous to water and steam.
In this paper, we study the sensitivity of the spectral clustering based community detection algorithm subject to a Erdos-Renyi type random noise model. We prove phase transitions in community detectability as a function of the external edge connection probability and the noisy edge presence probability under a general…
We derive the exact solution of a one-dimensional Markov functional model with log-normally distributed interest rates in discrete time. The model is shown to have two distinct limiting states, corresponding to small and asymptotically large volatilities, respectively. These volatility regimes are separated by a phase …
Catapult phase in neural nets shows exponential loss growth before quick decrease.
problem Understanding phase transitions in neural networks during training.
method Analyzing weight norm and loss behavior for super-critical learning rates.
result Proven existence of catapult phase in quadratic models and two-layer nets.
We fill a void in merging empirical and phenomenological characterisation of the dynamical phase transitions in complex systems by identifying three of them on real-life financial markets. We extract and interpret the empirical, numerical, and semi-analytical evidences for the existence of these phase transitions, by c…
A simple model explains phase transition in large language models.
problem Understanding the emergence of abilities in large language models.
method Modeling LLM as a sequence-to-sequence random function and using a list decoder.
result A critical threshold exists where the expected number of erroneous sequences grows exponentially.
The stability of money value is an important requisite for a functioning economy, yet it critically depends on the actions of participants in the market themselves. Here we model the value of money as a dynamical variable that results from trading between agents. The basic trading scenario can be recast into an Ising t…
High-dimensional models become unstable when sample size falls below a critical level, leading to a phase transition.
problem Instability in high-dimensional learning models when sample size is insufficient.
method Proved the necessity of a Fisher eigenvalue threshold for stability, introduced Fisher floor for verification.
result A sharp phase transition between reliable concentration and inevitable failure in high-dimensional learning.
Tyler's M-estimator's phase transition at DS-SNR = 1 is resolved.
problem Robust Subspace Recovery
method Tyler's M-estimator
result TME converges exactly to the true subspace for DS-SNR >= 1 under a new stability condition.
Study on complexity of random polynomials with deterministic spikes, identifying phase transitions.
problem Complexity of random Gaussian polynomials with deterministic spikes on a sphere.
method Variational formulas, Kac-Rice formula, determinant asymptotics of finite-rank perturbation of Gaussian Wigner matrices.
result Identification of a topological phase transition in the complexity function.
Predicting labels of nodes in a network, such as community memberships or demographic variables, is an important problem with applications in social and biological networks. A recently-discovered phase transition puts fundamental limits on the accuracy of these predictions if we have access only to the network topology…
We study the feasibility and noise sensitivity of portfolio optimization under some downside risk measures (Value-at-Risk, Expected Shortfall, and semivariance) when they are estimated by fitting a parametric distribution on a finite sample of asset returns. We find that the existence of the optimum is a probabilistic …
Study on a pinning model with random walk increments, showing convergence to a critical disordered pinning measure.
problem Understanding the critical behavior of a disordered pinning model.
method Analyzing a disordered pinning model induced by a random walk with specific moment conditions, showing convergence to a limiting measure.
result Convergence of point-to-point partition functions to the critical disordered pinning measure in the critical window.
Deep neural networks near edge of chaos show universal scaling laws.
problem Understanding the behavior of deep neural networks near critical points.
method Analogy to absorbing phase transitions in statistical mechanics, deterministic propagation dynamics, mean-field and directed percolation universality classes.
result Deep neural networks exhibit universal scaling laws near the edge of chaos.
Persistent entropy detects phase transitions in complex systems.
problem Detecting phase transitions in complex systems.
method Established a general theorem for persistent entropy to reliably detect phase transitions, introduced operational framework for finite-time computations.
result Persistent entropy exhibits an asymptotically non-vanishing gap across phases, robust numerical signatures across experiments.
Gradient flow in phase retrieval escapes spurious minima with high probability.
problem Understanding gradient-based optimization in high-dimensional non-convex functions.
method Analytical and numerical study of gradient dynamics in phase retrieval.
result Gradient flow avoids spurious minima by drifting along unstable directions.
Study phase transitions in shuffled regression problems.
problem Phase transitions in shuffled regression problems.
method Transformed permutation recovery into probabilistic graphical model, used message passing (MP) algorithm and branching random walk process.
result Characterized impact of signal-to-noise-ratio ($\snr$) on permutation recovery, proposed Gaussian approximation method.
Optimal spectral initializers impact phase retrieval phase transitions.
problem Understanding the limits of phase retrieval algorithms.
method Developed Random duality theory (RDT) to characterize optimal spectral initializers.
result Optimal spectral initializers can fall into flat regions of the phase retrieval manifold, making phase retrieval difficult.
Quantum phase diagrams for Chern topological insulators show jumps at critical loci.
problem Understanding phase transitions in Chern topological insulators.
method Mathematical formulation and explicit families of physical systems.
result Synthetic design of arbitrary Chern jumps in topological phases.
Study on phase transitions on surfaces using Allen-Cahn equation.
problem Existence of critical points with specific nodal sets on surfaces.
method Analysis of Allen-Cahn functional on compact surfaces, focusing on nodal sets and energy.
result Existence of countable families of critical points with nodal sets converging to geodesics.
PLS-SVD struggles with missing data in multimodal datasets, showing a phase transition in performance.
problem Missing data in PLS-SVD for multimodal datasets.
method Replica-symmetric analysis of spiked rectangular random matrices with missing entries.
result PLS-SVD performance transitions from uninformative to informative singular vectors at a critical signal-to-noise threshold.
Study finds a phase transition in flash crashes involving large and liquid stocks.
problem Systemic risk and propagation of shocks in high frequency trading.
method In-depth investigation of co-crashes in high frequency trading.
result Large co-crashes involve mostly illiquid stocks, while small crashes involve a mix of liquid and illiquid stocks.
We employ unsupervised machine learning techniques to learn latent parameters which best describe states of the two-dimensional Ising model and the three-dimensional XY model. These methods range from principal component analysis to artificial neural network based variational autoencoders. The states are sampled using …
The paper studies phase transitions in random matrices and tensor unfolding for detecting signals.
problem Phase transitions in singular values and vectors of large random matrices.
method Analysis of singular values and vectors of long rectangular random matrices, and tensor unfolding algorithm for asymmetric rank-one spiked tensor models.
result An exact threshold for tensor unfolding to detect signals, independent of unfolding procedure.
Study phase transitions in noisy transformer dynamics on spheres.
problem Understanding phase transitions in noisy transformer dynamics on spheres.
method Sharp Beckner--Onofri/logarithmic HLS inequality, Funk--Hecke/Bessel coefficients, degree-two quartic obstruction.
result Sharp global-minimizer dichotomy and phase transitions in noisy transformer dynamics in arbitrary dimension.
Multilayer graphs are commonly used for representing different relations between entities and handling heterogeneous data processing tasks. New challenges arise in multilayer graph clustering for assigning clusters to a common multilayer node set and for combining information from each layer. This paper presents a theo…
We analyze the landscape of empirical risk minimization for high-dimensional models, predicting phase transitions and critical point properties.
problem Understanding the complexity and structure of high-dimensional empirical risk landscapes.
method Using the Kac-Rice formula, we analyze the expected number of critical points and their spectral properties, providing detailed predictions.
result We derive complete topological phase diagrams for the phase retrieval problem, predicting BBP-type transitions and critical point stability.
Statistical mechanics reveals phase transitions in ε-SVR error.
problem Understanding task precision in neural representations with variability.
method Statistical mechanics applied to ε-SVR. result Double-descent phenomenon in generalization error due to ε. Machine learning approximates phase transitions using Fisher information.
problem Understanding phase transitions from data using machine learning.
method Information geometry and Fisher information.
result Machine learning indicators approximate the square root of Fisher information.
New framework analyzes SGD dynamics in large samples and dimensions.
problem Analyzing stochastic gradient descent in large-scale settings.
method Inspired by random matrix theory, new framework for fixed stepsize and finite sum settings.
result SGD dynamics become deterministic in the large sample and dimensional limit, governed by a Volterra integral equation.
Study the singular limit of a boundary reaction equation, showing energy concentration and varifold support.
problem Analyzing the singular limit of a boundary reaction equation.
method Investigates the critical points of the boundary reaction equation \((-Δ)^{\frac{1}{2}}u = \frac{1}{\varepsilon}(u-u^3)\) in \(U \subset \mathbb{R}^n\).
result Shows existence of an (n−1)-rectifiable energy concentration set and associates limit energy measures to a stationary varifold. New estimator for tensor weights with improved bias.
problem Estimating tensor weights from noisy data.
method Random matrix theory and KKT conditions.
result Asymptotically unbiased estimator for tensor rank.
If a given behavior of a multi-agent system restricts the phase variable to a invariant manifold, then we define a phase transition as change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct…
The paper studies dynamical systems with evolving geometric structure using numerical methods.
problem Qualitative behavior of ODEs with varying geometric structure.
method Fourth-order Runge-Kutta scheme for numerical analysis.
result Qualitative transitions in system dynamics as rotation parameter varies.
Recently, it was shown that there is a phase transition in the community detection problem. This transition was first computed using the cavity method, and has been proved rigorously in the case of q=2 groups. However, analytic calculations using the cavity method are challenging since they require us to understand p…
Diffusion maps help learn complex quantum phase transitions from data.
problem Learning quantum phase transitions from experimental data is challenging.
method Diffusion maps for nonlinear dimensionality reduction and spectral clustering.
result Diffusion maps can learn complex phase transitions unsupervised.
Double-well transitions are stiffer than minimal surfaces.
problem Rigidity of double-well phase transitions compared to minimal hypersurfaces.
method Comparison of rigidity properties between double-well phase transitions and minimal hypersurfaces.
result Double-well phase transitions exhibit more rigidity than minimal hypersurfaces.
New algorithms handle phase retrieval with rank d measurements, revealing phase transitions.
problem Phase retrieval with rank d measurements.
method Random duality theory (RDT) and descending phase retrieval algorithms (dPR).
result Minimal sample complexity ratio for dPR's success exhibits phase transitions.
Characterizing the phase transitions of convex optimizations in recovering structured signals or data is of central importance in compressed sensing, machine learning and statistics. The phase transitions of many convex optimization signal recovery methods such as ℓ1 minimization and nuclear norm minimization are…
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.
Study phase transitions with prescribed mean curvature in Riemannian manifolds.
problem Understanding phase transitions with prescribed mean curvature in geometric settings.
method Analyzing solutions to inhomogeneous semilinear elliptic PDEs, establishing bounds and asymptotics.
result Established upper and lower bounds for eigenvalues of phase transition problems.