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.
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.
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…
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…
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.
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.
Unsupervised learning is a discipline of machine learning which aims at discovering patterns in big data sets or classifying the data into several categories without being trained explicitly. We show that unsupervised learning techniques can be readily used to identify phases and phases transitions of many body systems…
Descending phase retrieval algorithms show a phase transition with increasing sample complexity.
problem Theoretical limits of descending phase retrieval algorithms.
method Utilizing Random duality theory (RDT), the study develops a generic program to characterize algorithm performance.
result As sample complexity increases, the parametric manifold transitions from multi to single funneling points, leading to a phase transition in algorithm success.
Study finds phase transition in context-sensitive language model with short-range interactions.
problem Understanding phase transitions in language models with short-range interactions.
method Constructed a random language model with short-range interactions and investigated its statistical properties.
result Phase transition occurs in context-sensitive language models with constant context length.
In the Information Bottleneck (IB), when tuning the relative strength between compression and prediction terms, how do the two terms behave, and what's their relationship with the dataset and the learned representation? In this paper, we set out to answer these questions by studying multiple phase transitions in the IB…
Proves spectra equivalence for Riemannian manifolds.
problem Equivalence of Almgren-Pitts and phase-transition half-volume spectra.
method Proof of spectra equivalence for Riemannian manifolds.
result Confirms conjecture about spectra equivalence.
Deep networks learn features suddenly, akin to a phase transition.
problem Understanding sudden feature learning in deep networks.
method Adaptive kernel approach applied to two teacher-student models.
result Feature learning and Grokking are analogous to a first-order phase transition.
In this paper, we perform statistical segmentation and clustering analysis of the Dow Jones Industrial Average time series between January 1997 and August 2008. Modeling the index movements and log-index movements as stationary Gaussian processes, we find a total of 116 and 119 statistically stationary segments respect…
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.
High-dimensional random geometry shows phase transitions in various problems.
problem Phase transitions in high-dimensional random geometry.
method Analysis of various financial, optimization, and ecological problems.
result Links between seemingly distant fields and further ramifications.
In this paper we study the phenomenon of phase transitions for the geodesic flow on some geometrically finite negatively curved manifolds. We define a class of potentials going slowly to zero through the cusps of M for which the pressure map exhibits a phase transition. By a careful choice of the metric at the cusp w…
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 …
The classification of phase transitions is a central and challenging task in condensed matter physics. Typically, it relies on the identification of order parameters and the analysis of singularities in the free energy and its derivatives. Here, we propose an alternative framework to identify quantum phase transitions,…
Improved simulation of phase transitions using hierarchical autoregressive networks.
problem Simulating phase transitions in complex systems.
method Hierarchical Autoregressive Neural (HAN) network sampling algorithm.
result Significant improvement in statistical uncertainty compared to the Wolff cluster algorithm.
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.
Study shows strong min-max principle for phase transitions.
problem Understanding nodal sets near minimal hypersurfaces.
method Analogous to White's principle, applies to Allen-Cahn energy.
result Strong min-max principle for phase transitions.
Framework for multi-scale clustering using phase transitions.
problem Clustering datasets with multi-scale structures.
method Cascade of phase transitions in simulated annealing of Expectation-Maximisation algorithm with weighted local covariance.
result Approximation of the number and size of clusters at different scales.
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.
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…
The study uses the Merton model to estimate PD and finds a phase transition affecting convergence speed.
problem Estimating the probability of default (PD) using limited historical data.
method Adopted the Merton model and analyzed phase transitions in default correlation.
result PD estimation converges slowly when temporal correlation decays by power law less than one.
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.
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 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…
Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated…
CNN detects phase transitions in Potts models without prior knowledge.
problem Detecting phase transitions in q-state Potts models using deep learning. method Trained a deep CNN on Ising model spin configurations and temperatures, then tested on Potts model images.
result Deep CNN accurately detects phase transitions in Potts models, including high- and low-temperature regions.
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…
Study on estimating signals from shifted and noisy copies in high dimensions, revealing a phase transition.
problem Estimating a signal in high-dimensional space from its circularly-shifted and noisy copies.
method Analysis of sample complexity in the high-dimensional regime, focusing on the parameter α.
result A phase transition phenomenon governed by α, with different sample complexities based on α values.
We find numerical and empirical evidence for dynamical, structural and topological phase transitions on the (German) Frankfurt Stock Exchange (FSE) in the temporal vicinity of the worldwide financial crash. Using the Minimal Spanning Tree (MST) technique, a particularly useful canonical tool of the graph theory, two tr…
Bayesian theory explains abrupt emergence of copy subcircuit in attention.
problem Understanding the abrupt emergence of the copy subcircuit in attention during training.
method Deriving a closed-form posterior over the attention matrix and reducing it to a low-dimensional order parameter space.
result Derive a phase transition in the amount of training data.
We generalize the observable diameter and the separation distance for metric measure spaces to those for pyramids, and prove some limit formulas for these invariants for a convergent sequence of pyramids. We obtain various applications of our limit formulas as follows. We have a criterion of the phase transition proper…
Study uncovers new phase transitions in asymmetric causal inference scenarios.
problem Understanding typical phase transitions in asymmetric causal inference.
method Combining Causal inference (C-inf) and Low-rank recovery (LRR) with Random duality - Free probability theory (RDT-FPT).
result Discovering a doubling low-rankness phenomenon in asymmetric scenarios.
Study reveals phase transition in neural networks near interpolation.
problem Understanding generalization and learning transitions in neural networks.
method Effective theory for approximating Bayes-optimal generalisation error.
result Unveils a discontinuous phase transition between universal and specialisation phases.
Characterizes RFF regression in large n,p,N setting, providing precise learning phases and double descent curve.
problem Characterizes RFF regression in large n,p,N setting. method Characterizes the exact asymptotics of random Fourier feature (RFF) regression in the realistic setting of large n,p,N. result Characterizes two qualitatively different phases of learning and the corresponding double descent test error curve.
The paper analyzes phase transitions in transfer learning for perceptrons.
problem Understanding when transfer learning from a source task to a target task is beneficial.
method Theoretical analysis of a pair of related perceptron learning tasks.
result Reveals a phase transition from negative to positive transfer as task similarity changes.
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.
Study phase transition in liquid crystal droplets using mathematical analysis.
problem Mathematical analysis of phase transition between isotropic and nematic states of liquid crystals.
method Rigorous mathematical analysis using the Ericksen model and Γ-convergence theory.
result Γ-limit provides geometric description and anchoring conditions for liquid crystal orientations.
New findings confirm parallels to De Giorgi's conjecture for phase transitions in higher dimensions.
problem Understanding phase transitions with bounded index in higher-dimensional spaces.
method Establishing parallels to De Giorgi's conjecture for general solutions of bounded Morse index.
result Finite index solutions to the Allen--Cahn equation in R4 are one-dimensional, and this holds for all 4≤n≤7. Enhances neural network dynamics to boost computational capacity.
problem Improving computational capacity of neural networks.
method Introducing Phase Transition Adaptation to drive system dynamics towards edge of stability.
result Consistently achieves enhancement in computational capacity over multiple datasets.
In Maslov (2003), a two level model of the occurrence of financial pyramid (bubbles) has been considered. We also considered the mathematical analogy of this model to Bose condensation. In the present paper, we explain why Ponzi schemes and bubbles result in a crisis in real economics. In Maslov (2005), the law of incr…
Study quantizes energy distribution in inhomogeneous phase transitions.
problem Quantifying energy distribution in inhomogeneous Allen-Cahn phase transitions.
method Analysis of varifolds and convergence of integer rectifiable varifolds.
result Equidistribution of energy between Dirichlet and Potential energy in phase field limit.