We introduce a new test for detection of power-law cross-correlations among a pair of time series - the rescaled covariance test. The test is based on a power-law divergence of the covariance of the partial sums of the long-range cross-correlated processes. Utilizing a heteroskedasticity and auto-correlation robust est…
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48 results for “power-law covariance”
This work analyzes neural scaling laws using power-law data spectra and derives analytical expressions for generalization error.
problem Understanding how neural network performance scales with key factors like data size and model complexity.
method Statistical mechanics techniques applied to one-pass stochastic gradient descent in a student-teacher framework.
result Derivation of analytical expressions for generalization error under power-law data spectra and identification of conditions for power-law scaling.
Power-law spectrum of random feature model is preserved in neural networks.
problem Preserving power-law spectrum in neural networks through random feature model.
method Characterized eigenvalues of population random-feature covariance using dyadic head-tail decomposition and Wick chaos expansions.
result Power-law exponent is inherited from input covariance, modified by a logarithmic correction.
Paper examines the structure of stochastic gradients in deep learning.
problem Exploring the structure and heavy tails of stochastic gradients in deep learning.
method Conducted formal statistical tests on stochastic gradients and gradient noise.
result Stochastic gradients and gradient noise do not exhibit power-law heavy tails, but their covariance spectra do.
Signatures of universality are detected by comparing individual eigenvalue distributions and level spacings from financial covariance matrices to random matrix predictions. A chopping procedure is devised in order to produce a statistical ensemble of asset-price covariances from a single instance of financial data sets…
Study on KRR with power-law data, showing better sample complexity.
problem High-dimensional kernel ridge regression with anisotropic power-law covariance.
method Explicit characterization of kernel spectrum and asymptotic analysis of excess risk.
result Sample complexity is governed by effective dimension, not ambient dimension.
Hybrid ResNet and RMT improve covariance matrix estimation for cryptocurrency portfolios.
problem Noisy, non-Gaussian financial data leads to unstable covariance matrices.
method Combines RMT regularization and ResNet learning for data-driven corrections.
result Hybrid estimator outperforms traditional methods in portfolio optimization.
Study shows how anisotropic data affects learning dynamics in phase retrieval.
problem Understanding learning dynamics in phase retrieval with anisotropic Gaussian inputs.
method Developed a tractable reduction to reveal a three-phase trajectory and derived scaling laws.
result Found that anisotropy leads to a three-phase trajectory: fast escape, slow convergence, and spectral-tail learning.
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.
Detrended Cross-Correlation Analysis: A New Method for Analyzing Two Non-stationary Time Seriesq-fin.ST
Here we propose a method, based on detrended covariance which we call detrended cross-correlation analysis (DXA), to investigate power-law cross-correlations between different simultaneously-recorded time series in the presence of non-stationarity. We illustrate the method by selected examples from physics, physiology,…
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.
Analyzes SGD dynamics on multi-class problems with exact expressions.
problem Analyzing SGD dynamics on multi-class problems.
method Developed a framework for analyzing training and learning rate dynamics using exact expressions.
result Exact expressions for risk and overlap with true signal in terms of ODEs.
Ridge regression reveals surprising high-dimensional behaviors via random matrix theory.
problem Understanding power-law scalings in high-dimensional regression models.
method Random matrix theory and free probability.
result Analytic formulas for training and generalization errors derived from -transform.
Scaling laws in linear regression explain model performance improvements with size and data.
problem Disagreement between empirical neural scaling laws and conventional wisdom on variance error.
method Infinite dimensional linear regression setup, one-pass SGD, Gaussian prior, power-law spectrum.
result Variance error is dominated by other errors, disappearing from the bound due to SGD's implicit regularization.
Detection of power-law behavior and studies of scaling exponents uncover the characteristics of complexity in many real world phenomena. The complexity of financial markets has always presented challenging issues and provided interesting findings, such as the inverse cubic law in the tails of stock price fluctuation di…
Spectral densities of Wishart-Levy free stable random matrices: Analytical results and Monte Carlo validationcond-mat.stat-mech
Random matrix theory is used to assess the significance of weak correlations and is well established for Gaussian statistics. However, many complex systems, with stock markets as a prominent example, exhibit statistics with power-law tails, that can be modelled with Levy stable distributions. We review comprehensively …
Study of SGD with state-dependent noise, improving escape from local minima.
problem Understanding and improving the dynamics of SGD in non-convex optimization.
method Formal study on SGD with state-dependent noise, proposing power-law dynamic with state-dependent diffusion.
result Power-law dynamic can escape from sharp minima exponentially faster than flat minima.
Comparing Classes of Estimators: When does Gradient Descent Beat Ridge Regression in Linear Models?math.ST
Gradient descent outperforms ridge regression under certain covariance matrix decay conditions.
problem Comparing the performance of gradient descent and ridge regression in linear models.
method Investigated gradient descent and ridge regression for linear regression with random isotropic ground truth.
result Gradient descent outperforms ridge regression under specific covariance matrix decay conditions.
Improved scaling laws in linear regression using data reuse.
problem Sustainability of neural scaling laws when running out of new data.
method Data reuse in multi-pass stochastic gradient descent (multi-pass SGD) for -dimensional linear models trained on data with sketched features.
result Multi-pass SGD achieves a test error of with , improving scaling laws in data-constrained regimes.
Extends QHawkes to MQHawkes for analyzing financial co-jumps.
problem Capturing endogenous co-jumps in financial markets.
method Develops MQHawkes process with quadratic kernels, investigates stationarity, and derives Yule-Walker equations.
result Volatility distribution exhibits power-law behavior with computable exponents.
New SDE model from machine learning optimization with unique stationary distribution.
problem Stationary distribution of machine learning optimization models.
method Proved ergodicity and unique stationary distribution of power-law dynamic SDE.
result Power-law dynamic has a unique stationary distribution and is ergodic.
New analysis of Muon and SignSGD on matrix-valued least squares problems.
problem Understanding the behavior of Muon and SignSGD on matrix-valued least squares problems.
method Derive explicit deterministic dynamics to study learning behavior of Muon and SignSGD.
result Muon and SignSGD exhibit different optimal learning rates and convergence characteristics based on batch size and data covariance.
Superposition accelerates training to a universal power-law exponent.
problem Training dynamics in neural networks.
method Teacher-student framework and analytic theory.
result Superposition leads to a universal power-law exponent of ~1, independent of data and channel statistics.
Market lawsq-fin.ST
Power laws detected in financial data, modeled with random multipliers.
problem Detecting power laws in financial data.
method Investigated data from financial instruments, proposed a model based on sums of Maxwell-Boltzmann distributions with random multipliers.
result Detected power laws with various exponents in financial data, proposed a universal model.
The High Line: Exact Risk and Learning Rate Curves of Stochastic Adaptive Learning Rate Algorithmsmath.OC
Exact risk and learning rate curves derived for adaptive SGD on high-dimensional problems.
problem Analyzing risk and learning rate dynamics in high-dimensional optimization problems.
method Developed a framework to give exact expressions for risk and learning rate curves using ODEs.
result Exact expressions for risk and learning rate curves, with detailed analysis of two adaptive learning rates.
We use data on wealth of the richest persons taken from the "rich lists" provided by business magazines like Forbes to verify if upper tails of wealth distributions follow, as often claimed, a power-law behaviour. The data sets used cover the world's richest persons over 1996-2012, the richest Americans over 1988-2012,…
A mechanism to derive multi-power law functions: an application in the econophysics frameworkphysics.data-an
It is generally recognized that economical systems, and more in general complex systems, are characterized by power law distributions. Sometime, these distributions show a changing of the slope in the tail so that, more appropriately, they show a multi-power law behavior. We present a method to derive analytically a tw…
Study non-integer power-law potentials for Schrödinger operators using Lie-Rinehart algebras.
problem Analyzing Schrödinger operators with non-integer power-law potentials.
method Using Lie-Rinehart algebras and microlocal analysis.
result Microlocal analysis can be applied to Schrödinger operators with non-integer power-law potentials.
The role of kernels is central to machine learning. Motivated by the importance of power-law distributions in statistical modeling, in this paper, we propose the notion of power-law kernels to investigate power-laws in learning problem. We propose two power-law kernels by generalizing Gaussian and Laplacian kernels. Th…
Detrended fluctuation analysis as a regression framework: Estimating dependence at different scalesq-fin.ST
We propose a framework combining detrended fluctuation analysis with standard regression methodology. The method is built on detrended variances and covariances and it is designed to estimate regression parameters at different scales and under potential non-stationarity and power-law correlations. The former feature al…
Limit order placement as an utility maximization problem and the origin of power law distribution of limit order pricesphysics.soc-ph
I consider the problem of the optimal limit order price of a financial asset in the framework of the maximization of the utility function of the investor. The analytical solution of the problem gives insight on the origin of the recently empirically observed power law distribution of limit order prices. In the framewor…
Modelling financial markets by the multiplicative sequence of tradescond-mat.stat-mech
We introduce the stochastic multiplicative point process modelling trading activity of financial markets. Such a model system exhibits power-law spectral density S(f) ~ 1/f**beta, scaled as power of frequency for various values of beta between 0.5 and 2. Furthermore, we analyze the relation between the power-law autoco…
Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processescond-mat.stat-mech
Non-Markovian point process shows power-law scaling, similar to nonlinear Markovian process.
problem Understanding the scaling behavior of non-Markovian point processes.
method Analyzed a confined fractional Brownian motion-driven point process and compared it to a nonlinear Markovian process.
result A nonlinear Markovian process can reproduce the power-law scaling behavior of a non-Markovian point process.
The Power-law Tail Exponent of Income Distributionsphysics.soc-ph
In this paper we tackle the problem of estimating the power-law tail exponent of income distributions by using the Hill's estimator. A subsample semi-parametric bootstrap procedure minimising the mean squared error is used to choose the power-law cutoff value optimally. This technique is applied to personal income data…
On the interplay between short and long term memory in the power-law cross-correlations settingstat.ME
We focus on emergence of the power-law cross-correlations from processes with both short and long term memory properties. In the case of correlated error-terms, the power-law decay of the cross-correlation function comes automatically with the characteristics of separate processes. Bivariate Hurst exponent is then equa…
We define a numerical method that provides a non-parametric estimation of the kernel shape in symmetric multivariate Hawkes processes. This method relies on second order statistical properties of Hawkes processes that relate the covariance matrix of the process to the kernel matrix. The square root of the correlation f…
We present a model for random simple graphs with a degree distribution that obeys a power law (i.e., is heavy-tailed). To attain this behavior, the edge probabilities in the graph are constructed from Bertoin-Fujita-Roynette-Yor (BFRY) random variables, which have been recently utilized in Bayesian statistics for the c…
We introduce preferential behavior into the study on statistical mechanics of money circulation. The computer simulation results show that the preferential behavior can lead to power laws on distributions over both holding time and amount of money held by agents. However, some constraints are needed in generation mecha…
Paper proves SVV model reproduces power-law skew in implied volatilities.
problem Reproducing power-law behavior in implied volatility skew.
method Analytical proof using Malliavin calculus and Volterra kernel selection.
result SVV model reproduces power-law skew under correct kernel choice.
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.
One of the first steps to understand and forecast economic downturns is identifying their frequency distribution, but it remains uncertain. This problem is common in phenomena displaying power-law-like distributions. Power laws play a central role in complex systems theory; therefore, the current limitations in the ide…
On the Origin of Power-Law Fluctuations in Stock Pricescond-mat.dis-nn
We respond to the issues discussed by Farmer and Lillo (FL) related to our proposed approach to understanding the origin of power-law distributions in stock price fluctuations. First, we extend our previous analysis to 1000 US stocks and perform a new estimation of market impact that accounts for splitting of large ord…
We introduce a new statistical tool (the TP-statistic and TE-statistic) designed specifically to compare the behavior of the sample tail of distributions with power-law and exponential tails as a function of the lower threshold u. One important property of these statistics is that they converge to zero for power laws o…
Study uses OT to simulate markets, revealing power-law returns are driven by informational effect.
problem Reproduce power-law returns in financial markets using realistic simulations.
method Constructed artificial markets, used optimal transport (OT) to measure similarity, incrementally introduced behavioral components.
result Informational effect of prices is dominant in reproducing power-law returns, and multiple components interact synergistically.
Auto-regressive conditionally heteroskedastic (ARCH) family models are still used, by practitioners in business and economic policy making, as a conditional volatility forecasting models. Furthermore ARCH models still are attracting an interest of the researchers. In this contribution we consider the well known GARCH(1…
The study finds significant power-law cross correlations in Bitcoin's return-volatility dynamics.
problem Investigating asymmetry in Bitcoin's return-volatility relationships.
method Analysis of daily and high-frequency Bitcoin data to identify cross correlations.
result Power-law cross correlations between returns and future volatilities are observed, indicating long-range dependencies.
Robust CD method for real-world time series with power-law distributions.
problem Challenges in causal discovery due to noise sensitivity.
method Power-law spectral feature extraction for robust CD.
result Consistently outperforms state-of-the-art alternatives on real-world datasets.
Critical volatility triggers log-normal to power-law transitions in interconnected systems.
problem Understanding the transition from log-normal to power-law distributions in interconnected systems.
method Analyzing an infinite option-on-option chain model, deriving a critical volatility threshold.
result A critical volatility threshold of approximately 250.66% for unconditional cases, dropping to 125.3% with selective survival.