We propose a neural superstatistics method to estimate dynamic cognitive models from time series data.
problem Memoryless cognitive models ignore parameter fluctuations, leading to inaccurate predictions.
method Developed a simulation-based deep learning method for Bayesian inference of superstatistical models.
result Deep learning method efficiently recovers time-varying and time-invariant parameters.
Share price returns on different time scales can be well modelled by a superstatistical dynamics. Here we provide an investigation which type of superstatistics is most suitable to properly describe share price dynamics on various time scales. It is shown that while chi-square superstatistics works well on a time scale…
We report a general technique to study a given experimental time series with superstatistics. Crucial for the applicability of the superstatistics concept is the existence of a parameter β that fluctuates on a large time scale as compared to the other time scales of the complex system under consideration. The propose…
Superstatistics is a widely employed tool of non-equilibrium statistical physics which plays an important role in analysis of hierarchical complex dynamical systems. Yet, its "canonical" formulation in terms of a single nuisance parameter is often too restrictive when applied to complex empirical data. Here we show tha…
Bayesian inference and superstatistics model financial volatility dynamics across different timescales.
problem Modeling correlated volatility in financial time series with heavy tails and long memory.
method Superstatistical dynamics, Bayesian Inference, Metropolis-Hasting sampling.
result The log-Normal model is reliable for short timescales, while inverse-Gamma is preferred for long timescales.
Financial time series have been investigated to follow fat-tailed distributions. Further, an empirical probability distribution sometimes shows cut-off shapes on its tails. To describe this stylized fact, we incorporate the cut-off effect in superstatistics. Then we confirm that the presented stochastic model is capabl…
We discuss superstatistics theory of labour productivity. Productivity distribution across workers, firms and industrial sectors are studied empirically and found to obey power-distributions, in sharp contrast to the equilibrium theories of mainstream economics. The Pareto index is found to decrease with the level of a…
Labour productivity distribution (dispersion) is studied both theoretically and empirically. Superstatistics is presented as a natural theoretical framework for productivity. The demand index κ is proposed within this framework as a new business index. Japanese productivity data covering small-to-medium to large firm…
We use a continuous-time random walk (CTRW) to model market fluctuation data from times when traders experience excessive losses or excessive profits. We analytically derive "superstatistics" that accurately model empirical market activity data (supplied by Bogachev, Ludescher, Tsallis, and Bunde)that exhibit transitio…
Study characterizes learning from heavy-tailed data in high dimensions using superstatistical methods.
problem Characterizing learning from heavy-tailed data in high-dimensional settings.
method Empirical risk minimization with double-stochastic processes and superstatistical analysis.
result Analytical characterization of separability transition and generalization performance.
We consider an ideal closed stock market, in which 100 traders have economic activities. The assets of the traders change through buying and selling stocks. We simulate the assets under conservation of both total currency and total number of stocks. If the traders are identical, then the assets are distributed as a sta…
The study analyzes river water quality using statistical and machine learning methods.
problem Analyzing spatio-temporal dynamics of dissolved oxygen in the River Thames.
method Superstatistical methods and machine learning (e.g., Light Gradient Boosting Machine, Informer model).
result The Informer model outperforms others in long-term dissolved oxygen concentration forecasting.
Osborne's paradox explained via Bayesian inference and superstatistics.
problem Osborne's paradox regarding stock price changes.
method Bayesian inference and superstatistics.
result Stock market returns are locally described by equilibrium statistical mechanics, globally by superstatistics.
This work models financial market returns with asymmetric Tsallis distributions, improving fit over symmetric q-Gaussians.
problem Non-symmetric behavior of stock market returns over time scales.
method Linear combination of two independent normalized half q-Gaussians with different parameters.
result Asymmetric distributions provide better fits to stock market returns than symmetric q-Gaussians, especially over longer time scales.
We study productivity dispersions across workers, firms and industrial sectors. Empirical study of the Japanese data shows that they all obey the Pareto law, and also that the Pareto index decreases with the level of aggregation. In order to explain these two stylized facts, we propose a theoretical framework built upo…
We consider Feller mean-reverting square-root diffusion, which has been applied to model a wide variety of processes with linearly state-dependent diffusion, such as stochastic volatility and interest rates in finance, and neuronal and populations dynamics in natural sciences. We focus on the statistical mixing (or sup…
Study on price fluctuations and persistence in European electricity spot markets.
problem Analyzing variability and persistence of electricity prices in European spot markets.
method Analysis of hourly, intraday, and 15-min intraday market prices; quantification of fluctuations, correlations, and extreme events; classification into circulation weather types.
result Different time scales in market dynamics; multifractal behavior below 12 hours; anti-correlation and mean reversion above 12 hours; long-term behavior influenced by four-day weather patterns; q-Gaussian distributions as best fit. Path integral techniques for the pricing of financial options are mostly based on models that can be recast in terms of a Fokker-Planck differential equation and that, consequently, neglect jumps and only describe drift and diffusion. We present a method to adapt formulas for both the path-integral propagators and the …
Social, technological and economic time series are divided by events which are usually assumed to be random albeit with some hierarchical structure. It is well known that the interevent statistics observed in these contexts differs from the Poissonian profile by being long-tailed distributed with resting and active per…
Graphs of neural networks are represented to preserve symmetry, improving performance across various tasks.
problem Lack of equivariance in neural network representations of other neural networks.
method Represent neural networks as computational graphs and use graph neural networks to preserve permutation symmetry.
result Single model encodes diverse neural architectures, outperforming state-of-the-art methods.
Convolutional Neural Processes improve data efficiency in neural processes.
problem Improving data efficiency in neural processes for small datasets.
method Convolutional Neural Processes (ConvNPs) improve data efficiency by leveraging translation equivariance and convolutional neural networks.
result ConvNPs enhance the performance of neural processes in small-data problems.
Neural Ordinary Differential Equation (Neural ODE) has been proposed as a continuous approximation to the ResNet architecture. Some commonly used regularization mechanisms in discrete neural networks (e.g. dropout, Gaussian noise) are missing in current Neural ODE networks. In this paper, we propose a new continuous ne…
Investigates how neural network graph structure impacts predictive performance.
problem Lack of understanding between neural network graph structure and predictive performance.
method Developed relational graph representation to analyze neural networks, identifying a 'sweet spot' for improved performance.
result Identified a 'sweet spot' in relational graph structure that significantly improves neural network predictive performance.
Novel framework explains generalization in deep neural networks.
problem Understanding and improving generalization in deep neural networks.
method Topological Quantum Neural Networks as the semi-classical limit of Deep Neural Networks.
result Demonstrates that the perceptron, viewed as the semi-classical limit, achieves similar results to standard neural networks without training.
Neural networks can approximate functions uniformly across various measures.
problem Universal approximation of functions across different probability measures.
method Proving neural networks are dense in Orlicz spaces, extending classical theorems.
result Neural networks uniformly approximate functions for weakly compact families of measures.
Investigates neural codes and their embeddings, proving conjectures and introducing new code types.
problem Analyzing neural codes and their embedding dimensions.
method Combinatorial, topological, and algebraic analysis; proving conjectures; introducing new neural code types.
result Proves conjectures about neural codes and their embeddings, introduces new code types.
Neural dynamical systems are dynamical systems that are described at least in part by neural networks. The class of continuous-time neural dynamical systems must, however, be numerically integrated for simulation and learning. Here, we present a compact neural circuit for two common numerical integrators: the explicit …
Quadratic models explain neural network behavior during training.
problem Understanding neural network dynamics during training with large learning rates.
method Developed and tested Neural Quadratic Models.
result Neural Quadratic Models exhibit the 'catapult phase' similar to neural networks.
Graph Metanetworks process diverse neural architectures efficiently.
problem Processing diverse neural architectures efficiently.
method Builds metanetworks using graph neural networks to process graphs representing input neural networks.
result Proves GMNs are expressive and equivariant to parameter permutation symmetries.
Optimal rates for shallow ReLU networks in nonparametric regression.
problem Approximating smooth and non-smooth functions with shallow ReLU networks.
method Analysis of shallow ReLUk neural networks, using variation norms and deep learning theory. result Optimal approximation rates for shallow ReLU networks in nonparametric regression.
New metric compares noisy neural trajectories using optimal transport.
problem Existing metrics fail to capture differences in noisy, dynamic neural responses.
method Proposed an optimal transport distance metric for Gaussian processes.
result Metric effectively compares neural dynamics in different systems.
The neural tangent kernel equivalence theorem fails in practice.
problem Does the neural tangent kernel (NTK) equivalence theorem hold in practical neural network training?
method Rigorously derived NTK and conducted numerical experiments to evaluate the equivalence theorem.
result Adding a layer to a neural network and the corresponding updated NTK do not yield matching changes in predictor error.
Equivariant neural networks use symmetry to interpret complex data.
problem Interpreting and understanding the behavior of equivariant neural networks.
method Decompose layers into simple representations and analyze nonlinear activation functions.
result Equivariant neural networks can be interpreted using a filtration generalizing Fourier series.
Two new criteria help understand the advantage of deep neural networks.
problem Understanding the advantage of deepening neural networks.
method Proposed two new criteria to evaluate the expressivity of functions computable by deep neural networks.
result Increasing layers is more effective than increasing units in improving the expressivity of deep neural networks.
Use simplified layerwise linear models to understand neural dynamics.
problem Complex neural network dynamics are hard to grasp.
method Apply simplified layerwise linear models to explain neural phenomena.
result Simplified models explain neural collapse, emergence, etc.
Analysis of over-parameterized neural networks has drawn significant attention in recentyears. It was shown that such systems behave like convex systems under various restrictedsettings, such as for two-level neural networks, and when learning is only restricted locally inthe so-called neural tangent kernel space aroun…
The paper proves consistency of neural networks with regularization.
problem Overfitting in neural networks with large scale data.
method Theoretical framework of neural networks with regularization, sieves method, and minimal neural networks theory.
result The estimated neural network converges to the true underlying function as sample size increases.
This study shows why training Neural ODEs is hard and proposes a new method.
problem Training Neural ODEs is challenging, especially in practice.
method Proposed a new stabilization method and provided an analytical convergence analysis.
result Insights and techniques for researchers starting work on Neural ODEs.
We introduce Graph Neural Processes (GNP), inspired by the recent work in conditional and latent neural processes. A Graph Neural Process is defined as a Conditional Neural Process that operates on arbitrary graph data. It takes features of sparsely observed context points as input, and outputs a distribution over targ…
Paper benchmarks quantum neural networks against classical ones for binary classification tasks.
problem Comparing quantum neural networks with classical ones for binary classification.
method Evaluated with two toy examples, focusing on model complexity and training data size.
result EQNN and QNN outperform ENN and DNN for smaller parameter sets and training data samples.
Affine spiking neural networks learn efficiently and generalize well.
problem Learning with spiking neural networks, especially with positive weights.
method Affine encoders and decoders, continuous parameter dependence, gradient-based training.
result Affine spiking neural networks can approximate shallow ReLU networks and generalize well.
Explains equivariant neural networks for machine learning.
problem Understanding equivariance in neural networks.
method Simple mathematical treatment of neural network concepts.
result Clarifies the mathematical basis of equivariant neural networks.
Proposes deep graph persistence to address neural persistence issues in deep learning.
problem Variance of weights and lack of spatial structure in deep neural networks impact neural persistence.
method Extends neural persistence to the whole network, considering interactions between layers.
result Deep graph persistence alleviates variance-related issues and captures persistent paths through the network.
Study of infinitely deep but narrow neural networks using NTK theory.
problem Analyzing the role of depth in deep learning with overparameterized networks.
method Infinite-depth limit analysis of MLP and CNN using Neural Tangent Kernel (NTK) theory.
result Established trainability guarantee for infinitely deep but narrow neural networks.
Q-learning with neural network function approximation (neural Q-learning for short) is among the most prevalent deep reinforcement learning algorithms. Despite its empirical success, the non-asymptotic convergence rate of neural Q-learning remains virtually unknown. In this paper, we present a finite-time analysis of a…
Paper introduces a noise-robust classification method using hypergraph neural networks.
problem Noisy label learning problem in image datasets.
method PCA for dimensionality reduction, then applies graph-based semi-supervised learning methods including hypergraph neural network.
result Our proposed hypergraph neural network achieves the best performance when noise level increases.
Paper introduces new neural network models and theories.
problem Understanding neural networks beyond over-parameterized regime.
method Develops two exact models and a novel representor theory.
result Provides insights into neural network training and kernel evolution.
Researchers analyze neural process architectures and their representational capacities.
problem Understanding what functions can be represented by different neural process architectures.
method Analyzing four types of neural process architectures: CNPs, ANPs, TNPs, and their latent variants.
result Prove these architectures form a strict hierarchy and characterize their representational capabilities.