Decomposes ultrametric spaces into scaled simplices.
problem Understanding the structure of ultrametric spaces.
method Introducing metric resolutions and coarse disjoint union.
result Constructs universal spaces for asymptotic dimension 0.
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48 results for “universal scaling”
A universal learner achieves best rates for all distributions.
problem Improving learning algorithm rates under various settings.
method Simple extension of Levin's universal search.
result Achieves best-possible rates for all distributions.
We present evidence, that if a large enough set of high resolution stock market data is analyzed, certain analogies with physics -- such as scaling and universality -- fail to capture the full complexity of such data. Despite earlier expectations, the mean value per trade, the mean number of trades per minute and the m…
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.
Comment on ``Tests of scaling and universality of the distributions of trade size and share volume: Evidence from three distinct markets" by Plerou and Stanley, Phys. Rev. E 76, 046109 (2007)
Collective phenomena with universal properties have been observed in many complex systems with a large number of components. Here we present a microscopic model of the emergence of scaling behavior in such systems, where the interaction dynamics between individual components is mediated by a global variable making the …
Knot probabilities in equilateral random polygonscond-mat.stat-mech
Study reveals a universal formula for knotting in random equilateral polygons.
problem Probability of knotting in equilateral random polygons.
method Extensive Monte Carlo simulations with improved algorithms and knot invariants.
result A universal scaling formula for knotting probability with number of edges, involving exponential and power law factors.
The investigations of financial markets from a complex network perspective have unveiled many phenomenological properties, in which the majority of these studies map the financial markets into one complex network. In this work, we investigate 30 world stock market indices through their visibility graphs by adopting the…
The common assumption of universal behavior in stock market data can sometimes lead to false conclusions. In statistical physics, the Hurst exponents characterizing long-range correlations are often closely related to universal exponents. We show, that in the case of time series of the traded value, these Hurst exponen…
Field theory for optimal signal propagation in ResNetscond-mat.dis-nn
Field theory explains optimal scaling in ResNets for signal propagation.
problem Understanding optimal scaling parameter for ResNet performance.
method Finite-size field theory for ResNets to study signal propagation and scaling.
result Analytical expressions for optimal scaling parameter, independent of other hyperparameters.
In the framework of Lorentzian warped products, we study the Friedmann-Robertson-Walker cosmological model to investigate non-smooth curvatures associated with multiple discontinuities involved in the evolution of the universe. In particular we analyze non-smooth features of the spatially flat Friedmann-Robertson-Walke…
We investigated the critical dynamics on the daily Taiwan stock exchange index (TSE) from 1971 to 2005, and the 5-min intraday data from 1996 to 2005. A global persistence exponent was defined for non-equilibrium critical phenomena \cite{Janssen,Majumdar}, and describing dynamic behavior in an economic index \c…
Dropout schedules can be optimized to significantly reduce model test loss.
problem Improving model performance in neural networks.
method Developed a mean-field theory of dropout at the edge of chaos, proposing front-loaded dropout schedules.
result Front-loaded dropout schedules reduce test loss by 18-35% over constant dropout.
A grand challenge of the 21st century cosmology is to accurately estimate the cosmological parameters of our Universe. A major approach to estimating the cosmological parameters is to use the large-scale matter distribution of the Universe. Galaxy surveys provide the means to map out cosmic large-scale structure in thr…
This work explains scaling laws as redundancy laws in deep learning.
problem The mathematical origins of scaling laws in deep learning models remain unclear.
method Kernel regression and analysis of data covariance spectra.
result Scaling laws can be explained as redundancy laws, revealing the learning curve's slope depends on data redundancy.
Develops a simple model to understand learning curves for arbitrary power laws.
problem Lack of theoretical understanding of scaling laws in machine learning.
method Analyzes a toy model to determine if learning curves are universal or depend on data distribution.
result Determines that learning curves can exhibit for arbitrary power .
We uncover scaling laws and statistical structure in complex datasets.
problem Understanding universal traits in complex datasets.
method Analogizing data to physical systems, using statistical physics and RMT.
result Real-world datasets and Gaussian data with long-range correlations share the same RMT universality class.
In this paper we explore coarse properties of cusp-decomposable manifolds first defined by Nguyên Phan. We describe the large scale geometry of the universal cover of a cusp-decomposable manifold and of quasi-isometries between two such universal covers. This description will provide us the tools to prove quasi-isometr…
Universal Patterns in the Blockchain: Analysis of EOAs and Smart Contracts in ERC20 Token Networksq-fin.ST
This study reveals statistical patterns in ERC20 token transactions on Ethereum blockchain.
problem Understanding transactional dynamics in decentralized systems.
method Examined over 44 million ERC20 token transfers, categorized by address type (EOA or SC), and analyzed using scaling laws.
result EOA-driven transactions exhibit consistent statistical behavior, while SC-driven activity displays sublinear scaling and bursty activity.
The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spacesmath.MG
Develops a new framework for large-scale geometry.
problem Characterizing large-scale models of metric spaces.
method Categorical framework for metric Rips filtration and universal quasigeodesic cones.
result Establishes universal properties and adjointness of the Rips colimit.
LLMs learn peaked distributions slowly due to power-law losses.
problem Slow convergence of loss in training large language models.
method Systematic analysis of toy models and empirical evaluation of LLMs.
result Power-law time scaling with an exponent of 1/3 for learning peaked distributions.
For many externally driven complex systems neither the noisy driving force, nor the internal dynamics are a priori known. Here we focus on systems for which the time dependent activity of a large number of components can be monitored, allowing us to separate each signal into a component attributed to the external drivi…
The paper explores simple and relatively simple transformation groups and their universal coverings.
problem Understanding the structure of universal coverings of transformation groups.
method Study of relatively simple groups and generalization of Tsuboi's metric space.
result Tsuboi's metric space of is not quasi-isometric to the half line.
Gaussian equivalence fails for simple polynomial embeddings in quadratic scaling RF models.
problem Failure of Gaussian equivalence in polynomial feature embeddings under quadratic scaling.
method Introduced Conditional Gaussian Equivalent (CGE) model to capture non-Gaussian behavior.
result Correct asymptotics derived for training and test errors in CGE model.
Study on scalar curvature bounds and manifold topological complexity.
problem Understanding the topological complexity of manifolds with scalar curvature constraints.
method Introduced a small scale index theorem to establish bounds for Gromov's simplicial norm.
result Upper bound for Gromov's simplicial norm established in terms of scalar curvature, volume, and injectivity radius.
Endogenous and exogenous dynamics in the fluctuations of capital fluxes: An empirical analysis of the Chinese stock marketphysics.soc-ph
A phenomenological investigation of the endogenous and exogenous dynamics in the fluctuations of capital fluxes is investigated on the Chinese stock market using mean-variance analysis, fluctuation analysis and their generalizations to higher orders. Non-universal dynamics have been found not only in exponents diff…
Transfer learning improves chaotic dynamics predictions with less data.
problem Efficiently predicting chaotic dynamics with limited data.
method Transfer learning for nonlinear dynamics, optimizing transfer rate and leveraging small-scale turbulence universality.
result Significantly more accurate inference of chaotic dynamics achieved.
Study free energy in spherical spin glasses, proving universality dichotomy.
problem Analyzing free energy in spherical spin glass models with different tail exponents.
method Introduced a tail-adapted normalization and used universality dichotomy.
result Sharp universality dichotomy for free energy across different tail exponents.
The paper shows instability in Minkowski spacetime for a quantum system.
problem Linear instability of the semiclassical Einstein-Klein-Gordon system in Minkowski spacetime.
method Formulated a forcing problem for metric and state perturbations, used tensor decomposition and quantum Møller operator.
result Metric perturbations grow exponentially, bounded by a universal scale H, indicating quantum backreaction.
Yau's Affine-Normal Descent for Large-Scale Unrestricted Higher-Moment Portfolio Optimizationq-fin.PM
Algorithm tackles large-scale portfolio optimization with higher moments, improving computational efficiency.
problem Optimizing portfolios with higher moments (variance, skewness, kurtosis) for large asset universes is computationally infeasible.
method Developed a structure-exploiting algorithm based on Yau's affine-normal descent, working directly with return matrix.
result Algorithm avoids explicit higher-order tensors and exploits quartic structure for efficient computation.
Develops hierarchical reinforcement learning value function approximators.
problem Estimating long-term returns in reinforcement learning with multiple goals.
method Introduces hierarchical universal value function approximators (H-UVFAs) using the options framework.
result Demonstrates generalization and improved performance of H-UVFAs over UVFAs.
ELF simplifies normalizing flows, making them more efficient and universal.
problem Computational inefficiency of normalizing flows.
method ELF introduces a simple, one-layer network with closed-form Lipschitz constants, combining the ease of residual flows with the performance of autoregressive flows.
result ELF is a provably universal density approximator, more efficient computationally and parameter-wise.
For each , we construct a separable metric space that is universal in the coarse category of separable metric spaces with asymptotic dimension () at most and universal in the uniform category of separable metric spaces with uniform dimension () at most . Thus, $\m…
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, , with exponent determined by network parameters and data characteristics.
We present a relatively detailed analysis of the persistence probability distributions in financial dynamics. Compared with the auto-correlation function, the persistence probability distributions describe dynamic correlations non-local in time. Universal and non-universal behaviors of the German DAX and Shanghai Index…
Emergence of a finite-size-scaling function in the supervised learning of the Ising phase transitioncond-mat.stat-mech
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.
Scaling analysis of multivariate intermittent time seriescond-mat.stat-mech
The scaling properties of the time series of asset prices and trading volumes of stock markets are analysed. It is shown that similarly to the asset prices, the trading volume data obey multi-scaling length-distribution of low-variability periods. In the case of asset prices, such scaling behaviour can be used for risk…
Take N sites distributed randomly and uniformly on a smooth closed surface. We express the expected distance <D_k(N)> from an arbitrary point on the surface to its kth-nearest neighboring site, in terms of the function A(l) giving the area of a disc of radius l about that point. We then find two universalities. First, …
How and why stock prices move is a centuries-old question still not answered conclusively. More recently, attention shifted to higher frequencies, where trades are processed piecewise across different timescales. Here we reveal that price impact has a universal non-linear shape for trades aggregated on any intra-day sc…
The dynamics of traded value revisitedphysics.soc-ph
We conclude from an analysis of high resolution NYSE data that the distribution of the traded value (or volume) has a finite variance for the very large majority of stocks , and the distribution itself is non-universal across stocks. The Hurst exponent of the same time series displays a crossover from we…
The growth of business firms is an example of a system of complex interacting units that resembles complex interacting systems in nature such as earthquakes. Remarkably, work in econophysics has provided evidence that the statistical properties of the growth of business firms follow the same sorts of power laws that ch…
Researchers use quantum chaos and RMT to analyze turbulence, revealing unique scaling laws.
problem Understanding the statistical structure and scaling laws of turbulence.
method Applied tools from quantum chaos and Random Matrix Theory to analyze turbulence datasets.
result Turbulence Gram matrices exhibit power-law scalings distinct from classical chaos and random data.
A Generalized Online Algorithm for Translation and Scale Invariant Prediction with Expert Advicecs.LG
Generalized algorithm for translation and scale-invariant prediction.
problem Sequential prediction with expert advice, focusing on translation and scale invariance.
method Designing a generalized online algorithm using the universal prediction perspective to compete against a generic class of expert selection strategies.
result No preliminary knowledge of loss sequences is required; performance bounds are stable under arbitrary scalings and translations.
Common Scaling Patterns in Intertrade Times of U. S. Stockscond-mat.stat-mech
We analyze the sequence of time intervals between consecutive stock trades of thirty companies representing eight sectors of the U. S. economy over a period of four years. For all companies we find that: (i) the probability density function of intertrade times may be fit by a Weibull distribution; (ii) when appropriate…
Unified framework for comparing classification metrics across different imbalance rates.
problem Differences in scale and sensitivity to class imbalance rates in classification metrics.
method Introduces outperformance standardization (OPS) function to map metrics to a common scale.
result Unified o-value metric provides clear comparison across different imbalance rates.
In the study manifolds of Ricci curvature bounded below, a stumbling obstruction is the lack of links between large-scale geometry and small-scale geometry at a fixed reference point. There have been few links (volume, dimension) when the unit ball at the point is not collapsed, that is, . …
Large Investment Modelq-fin.ST
LIM enhances investment performance and efficiency at scale.
problem Diminishing returns and rising labor/time costs in traditional quantitative investment research.
method End-to-end learning and universal modeling to create a global patterns foundation model.
result Optimized performance for specific tasks through transfer learning of global patterns.
Stochastic gradient descent converges to universal limits in high dimensions.
problem Statistical tasks in high dimensions with specific data projections.
method Stochastic gradient descent applied to mixture distributions, proving universality of limits.
result The ODE limits are universal for mixtures of arbitrary product distributions.