Small intrinsic scale reveals network structure.
problem Understanding the scale at which network identity is revealed.
method Defined intrinsic scale as distinguishability of subgraphs in random walks.
result Intrinsic scale is surprisingly small (7-20 vertices) across various networks.
Market activity scales near a constant of 0.632 in intrinsic time.
problem Understanding the stability of market scaling laws.
method Modeling market directional changes as a memoryless exponential hazard process and identifying the intrinsic time scaling constant.
result The intrinsic time scaling constant is 1−1/e=0.632. eDCF estimates intrinsic dimension using local connectivity.
problem Challenges in estimating intrinsic dimension due to scale dependence.
method eDCF: a novel, scalable, and parallelizable method based on Connectivity Factor (CF).
result eDCF consistently matches leading estimators with comparable MAE and higher exact intrinsic dimension match rates.
Analyzes intrinsic time in financial markets, linking it to physical time.
problem Understanding the intrinsic nature of time in financial data.
method Presented an analytic relationship linking intrinsic and physical time, using empirical scaling laws.
result A novel empirical scaling law relating intrinsic time variability to overshoots.
Inference-Time Scaling can be extended to domains prone to systematic failure using intrinsic statistics.
problem Scaling inference time in domains prone to systematic failure
method Intrinsic Selection (iS), Intrinsic Particle Filtering (iPF), and Particle Distillation (dPF)
result Intrinsic Selection improves engineering design selection by 20% and pass@1 by 6.1 points on average.
This work introduces a protocol to automatically select the correct range of scales for meaningful Intrinsic Dimension estimation.
problem The Intrinsic Dimension (ID) varies with scale in real-world datasets, leading to erroneous results.
method The protocol selects the correct range of scales by ensuring constant density of data points.
result The method provides a robust and scale-adaptive approach to estimating meaningful Intrinsic Dimension.
Neural networks' performance scales with data size, explained by data manifold dimensionality.
problem Understanding the scaling of neural network performance with the number of parameters.
method Explained by the intrinsic dimension of the data manifold, confirmed through teacher/student framework and various datasets.
result The scaling exponent α is approximately 4 divided by the intrinsic dimension d of the data manifold.
Three training regimes found for scale-invariant neural networks on the sphere.
problem Training scale-invariant neural networks on the sphere with varying effective learning rate.
method Investigated three regimes of training: convergence, chaotic equilibrium, and divergence.
result Discovered three distinct training regimes with unique characteristics.
Study shows one-dimensional location-scale-shape models are flat in Wasserstein geometry.
problem Investigating curvature in location-scale-shape models under Wasserstein metric.
method Introduced location-scale-shape model and investigated its geometry.
result Location-scale-shape model is intrinsically flat but extrinsically curved in Wasserstein geometry.
New estimators for intrinsic dimension and Wasserstein distance improve OT accuracy.
problem Intrinsic dimension estimation and Wasserstein distance estimation in large-scale OT.
method Introduces novel estimators for intrinsic dimension and Wasserstein distance.
result Simple, tuning-free estimator of OT and fast intrinsic dimension estimator.
Scaling laws found for reinforcement learning performance with model size and compute.
problem Challenges in extending generative modeling scaling laws to reinforcement learning.
method Introduced intrinsic performance as a monotonic function of mean episode return.
result Intrinsic performance scales as a power law in model size and environment interactions.
We introduce an event based framework of directional changes and overshoots to map continuous financial data into the so-called Intrinsic Network - a state based discretisation of intrinsically dissected time series. Defining a method for state contraction of Intrinsic Network, we show that it has a consistent hierarch…
LIDL estimates local intrinsic dimension in high dimensions.
problem Estimating local intrinsic dimension in high-dimensional data.
method Approximate likelihood using parametric neural density estimation.
result LIDL scales to thousands of dimensions and yields competitive results.
Develops a new method to compare data distributions on manifolds.
problem Existing techniques for comparing data distributions are limited and uni-scale.
method Intrinsic and multi-scale method using spectral Gromov-Wasserstein distance.
result Effective at discerning data manifold structure and evaluating generative models.
New 3D protein analysis methods improve accuracy.
problem Lack of suitable learning algorithms for protein data.
method Intrinsic-Extrinsic Convolution and Pooling for 3D protein structures.
result Outperforms state-of-the-art methods on protein analysis tasks.
Designing a covariance function that represents the underlying correlation is a crucial step in modeling complex natural systems, such as climate models. Geospatial datasets at a global scale usually suffer from non-stationarity and non-uniformly smooth spatial boundaries. A Gaussian process regression using a non-stat…
A method uses ITD and XGBoost for precise power transformer fault diagnosis.
problem Fault diagnosis of power transformers using DGA data.
method Ranking DGA parameters by skewness, extracting ITD features, and using an XGBoost classifier.
result The method achieves over 95% accuracy in classification.
The study explains transformer scaling laws using statistical and approximation theories.
problem Understanding why transformer scaling laws exist for large models trained on low-dimensional data.
method Established statistical estimation and mathematical approximation theories for transformers on low-dimensional manifolds.
result Predicted a power law between generalization error and model and data sizes, with power depending on intrinsic data dimension.
The paper explores how neural networks generalize differently from natural and medical images.
problem Discrepancies in generalization error between natural and medical images.
method Established and empirically validated a generalization scaling law with respect to intrinsic dataset properties.
result Higher intrinsic 'label sharpness' of medical images leads to higher adversarial vulnerability.
Detects singularities in complex data to improve machine learning models.
problem Real-world data often contains non-manifold structures (singularities) that can mislead machine learning models.
method Develops a topological framework to quantify local intrinsic dimension and Euclidicity score for multiple scales.
result Identifies singularities and captures local geometric complexity in image data.
Shallow nonlinear networks can separate classes linearly with polynomially scaling width.
problem Understanding the linear separability of deep networks' features.
method Modeling inputs as a union of low-dimensional subspaces and using random weights and quadratic activations.
result Shallow nonlinear networks can achieve linear separation with polynomially scaling width.
Introduces Floer functions and Floerfolds for intrinsic properties.
problem Complex transformation of Hessian under chart transition.
method Introduces Floer functions and Floerfolds to address intrinsic properties.
result Floer functions and Floerfolds provide intrinsic conditions for Hessian.
This paper introduces intrinsic time, a new measure of time for complex systems.
problem Traditional time measures fail to capture the dynamic nature of real-world phenomena.
method Intrinsic time uses an event-based, algorithmic framework to analyze time series data.
result Intrinsic time reveals novel structures and regularities in financial markets.
Reinforcement learning algorithms rely on carefully engineering environment rewards that are extrinsic to the agent. However, annotating each environment with hand-designed, dense rewards is not scalable, motivating the need for developing reward functions that are intrinsic to the agent. Curiosity is a type of intrins…
The paper corrects biases in estimating intrinsic dimension and differential entropy.
problem Systematic bias in estimating intrinsic dimension and differential entropy.
method A bias-corrected estimator for both measures is proposed, highlighting shared steps and useful consequences.
result Simultaneous estimation of differential entropy and intrinsic dimension provides complementary perspectives on underlying manifolds.
Estimates intrinsic dimensionality of biological datasets using Fisher separability.
problem High-dimensional biological datasets with complex structures.
method Fisher separability analysis to estimate intrinsic dimensionality.
result The method performs competitively with state-of-the-art measures and is robust to noise.
When analyzing empirical data, we often find that global linear models overestimate the number of parameters required. In such cases, we may ask whether the data lies on or near a manifold or a set of manifolds (a so-called multi-manifold) of lower dimension than the ambient space. This question can be phrased as a (mu…
The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.
problem Understanding properties of sections in metric spaces.
method Definition and investigation of intrinsically quasi-isometric sections in metric spaces.
result Properties of sections, including Ahlfors-David regularity and convexity, are defined and investigated.
The Dirac field is studied in a Lyra space-time background by means of the classical Schwinger Variational Principle. We obtain the equations of motion, establish the conservation laws, and get a scale relation relating the energy-momentum and spin tensors. Such scale relation is an intrinsic property for matter fields…
We provide the proof that the space of time series data is a Kolmogorov space with T0-separation axiom using the loop space of time series data. In our approach we define a cyclic coordinate of intrinsic time scale of time series data after empirical mode decomposition. A spinor field of time series data comes fro…
SAEs struggle with curved activation manifolds, revealing layer-dependent scaling laws.
problem Sparse autoencoders' reconstruction error varies across layers, not fitting existing scaling laws.
method Cross-layer study of 844 SAE checkpoints, fitting and regressing on manifold geometry.
result Manifold geometry predicts layer-dependent width exponents in SAEs, with transferable coefficients.
Learning goal-directed behavior in environments with sparse feedback is a major challenge for reinforcement learning algorithms. The primary difficulty arises due to insufficient exploration, resulting in an agent being unable to learn robust value functions. Intrinsically motivated agents can explore new behavior for …
The abstract explores connections between reinforcement learning, scaling, and diffusion.
problem Aligning reinforcement learning with human feedback and scaling techniques.
method Clarifying connections between reinforcement learning, scaling, and diffusion.
result Introducing a resampling approach for alignment and reward-directed diffusion models.
Python package for estimating intrinsic dimensionality of datasets.
problem Estimating intrinsic dimensionality for machine learning applications.
method Implementation of various intrinsic dimension estimators in Python.
result Benchmarking of ID estimation methods on real and synthetic data.
This paper analyzes deep federated learning for low-dimensional data, revealing intrinsic dimensionality's role in convergence rates.
problem Insufficient investigation of generalization error in heterogeneous federated learning, especially for low-dimensional data.
method Statistical analysis of deep federated regression in a two-stage sampling model.
result Intrinsic dimensionality, characterized by entropic dimension, determines convergence rates for deep learners.
CrossLID evaluates GAN quality by comparing intrinsic dimensions.
problem Measuring GAN quality and detecting mode collapse.
method CrossLID measures local intrinsic dimensionality of real-world data vs. GAN-generated samples.
result CrossLID correlates with GAN training progress and is robust to noise.
The paper examines Wiener process for LID estimation methods.
problem Estimating local intrinsic dimension in high-dimensional datasets.
method Investigates recent LID estimation methods from a Wiener process perspective.
result Explains how methods behave under non-ideal conditions.
Learning shrinks hard tail, improving inference performance.
problem Improving inference performance in neural networks.
method Latent Instance Difficulty (LID) model analyzing fine-tuning of neural networks.
result Training-dependent inference scaling, with βexteff growing with sample size before saturating. For negatively curved manifolds, a condition is found for intrinsic ultracontractivity of heat semigroups.
problem Investigating intrinsic ultracontractivity for domains in negatively curved manifolds.
method Using volume doubling property, Poincaré inequality, and Li-Yau Gaussian estimate for the Dirichlet heat kernel.
result The reciprocal of the bottom of the spectrum and the supremum of the torsion function are comparable with the square of the capacitary width for small capacitary width.
New research shows DDPM can adapt to data's intrinsic low dimensionality efficiently.
problem Theoretical inefficiency of DDPM in high-dimensional data.
method Investigates how DDPM can exploit intrinsic low dimensionality of data.
result Proves DDPM's iteration complexity scales nearly linearly with intrinsic dimension k. Neuro-inspired recurrent neural network algorithms, such as echo state networks, are computationally lightweight and thereby map well onto untethered devices. The baseline echo state network algorithms are shown to be efficient in solving small-scale spatio-temporal problems. However, they underperform for complex task…
Study of Brown--York mass for four-dimensional asymptotically flat manifolds.
problem Calculating mass for hypersurfaces in four-dimensional asymptotically flat manifolds.
method Intrinsic definition of mean curvature, expansion analysis for large uniformly convex hypersurfaces.
result Shape-dependent correction to ADM mass for nearly round surfaces vanishes under certain conditions.
New method for efficient sketching of gradients and Hessians.
problem Memory constraints in training machine learning models.
method A novel framework for scalable gradient and HVP sketching tailored for modern hardware.
result Theoretical guarantees and practical applications in training data attribution and Hessian spectrum analysis.
W2S FT often outperforms weak teachers due to low intrinsic dimensionality.
problem Understanding why weak-to-strong finetuning outperforms weak models.
method Analyzing W2S in ridgeless regression setting, focusing on variance reduction.
result Weak teacher's variance is inherited by strong student in shared feature subspace, reduced in discrepancy subspace.
The paper analyzes how much data points can be altered to change their rank in nearest neighbor searches.
problem Vulnerability of nearest neighbor search in high-dimensional data.
method Statistical analysis of perturbation needed to change neighbor rank.
result Derived statistical distribution of perturbation needed to modify neighbor rank.
New regularizer improves neural network robustness and generalization.
problem Ineffective weight decay for networks with homogeneous activation functions.
method Proposes an invariant regularizer to penalize intrinsic weight norms.
result Improves generalization and adversarial robustness on various datasets.
In the present work we investigate the multiscale nature of the correlations for high frequency data (1 minute) in different futures markets over a period of two years, starting on the 1st of January 2003 and ending on the 31st of December 2004. In particular, by using the concept of "local" Hurst exponent, we point ou…
New method estimates deep neural network's intrinsic dimension for better generalization.
problem Estimating intrinsic dimension of deep neural networks for generalization.
method Topological data analysis (TDA) and persistent homology (PHD).
result Efficient algorithm to estimate PHD in deep neural networks.