NGD improves multivariate Gaussian inference by optimizing Fisher information.
problem Efficiently optimizing multivariate Gaussian models.
method Natural Gradient Descent applied to multivariate Gaussian parameters.
result NGD updates are more efficient for symmetric covariance matrices.
The Fisher-Rao geometry is applied to elliptical distributions for optimization and classification.
problem Optimizing and classifying covariance matrices using geometric tools.
method Riemannian optimization and intrinsic Cramér-Rao bounds.
result Geometric tools enhance covariance matrix estimation and classification.
This paper solves the intractability barrier in non-parametric information geometry by introducing a novel framework.
problem The intractability barrier in non-parametric information geometry due to the Fisher-Rao metric being a functional.
method Introducing an Orthogonal Decomposition of the Tangent Space and deriving the Covariate Fisher Information Matrix (cFIM).
result Established a rigorous foundation for the G-entropy and provided fundamental limits of variance for semi-parametric estimators.
Optimal preconditioning improves Langevin sampling efficiency.
problem Improving sampling efficiency in high-dimensional target distributions.
method Optimal preconditioning using Fisher information, applied to MALA.
result Adaptive MCMC scheme significantly outperforms other methods.
Paper presents a rank-1 approximation method for natural policy gradients in deep RL.
problem Computing natural gradients requires inverting the Fisher Information Matrix, which is computationally expensive.
method Develops a rank-1 approximation to the inverse Fisher Information Matrix for efficient natural policy optimization.
result The rank-1 approximation converges faster and has similar sample complexity to stochastic policy gradient methods.
A deep neural network is a hierarchical nonlinear model transforming input signals to output signals. Its input-output relation is considered to be stochastic, being described for a given input by a parameterized conditional probability distribution of outputs. The space of parameters consisting of weights and biases i…
A new measure of model complexity based on Fisher Information.
problem Model complexity measurement in statistical models.
method Effective dimension defined by the number of cubes needed to cover the model space.
result The effective dimension is scale-dependent and measures model complexity.
Two Fisher information matrix estimators are analyzed for neural networks, focusing on their variances and trade-offs.
problem Estimating the Fisher information matrix in neural networks due to its high computational cost.
method Examined two popular diagonal Fisher information matrix estimators and their variances in neural networks for regression and classification.
result The variances of the estimators depend on the non-linearity with respect to different parameter groups and should not be neglected.
FIRE method improves model performance in federated learning by penalizing fragmentation-induced covariate shifts.
problem Performance degradation in federated learning due to data fragmentation and covariate shift.
method FIRE method accumulates fragmentation-induced covariate shift divergences via approximate Fisher information and uses it as a per-fragment loss penalty.
result FIRE outperforms importance weighting and federated learning benchmarks by up to 5.3% on shifted validation sets.
This work tackles sequential data learning challenges by improving neural network robustness to non-iid distribution shifts.
problem Sequential data learning challenges, particularly non-iid distribution shifts across batches.
method Cramér-Rao-based regularization using Fisher Information Matrix to adapt to sequential covariate shifts.
result Achieves 19% accuracy improvement over state-of-the-art methods.
New tensor framework connects Fisher information, hypergraphs, and multi-observable correlations.
problem Missing structure in pairwise Fisher graphs for multi-observable radiation patterns.
method Higher-order Fisher tensors and natural exponential-family coordinates.
result Exact triality of Fisher tensors, cumulants, and hypergraphs.
Paper improves matrix-valued data classification using nonparametric LDA.
problem Classification of matrix-valued data in neuroimaging and signal processing.
method Nonparametric LDA based on NPMLE for vectorized and scaled matrices.
result Improves classification performance across various data structures.
We propose a modified χβ-divergence, give some of its properties, and show that this leads to the definition of a generalized Fisher information. We give generalized Cramér-Rao inequalities, involving this Fisher information, an extension of the Fisher information matrix, and arbitrary norms and power of the estimat…
The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry---the differential geometric approach to statistics. The Rieman…
Natural gradient descent, which preconditions a gradient descent update with the Fisher information matrix of the underlying statistical model, is a way to capture partial second-order information. Several highly visible works have advocated an approximation known as the empirical Fisher, drawing connections between ap…
A new metric tensor improves Riemann manifold Monte Carlo for Bayesian models.
problem Improving sampling efficiency in Bayesian hierarchical models.
method Metric tensor derived from log-density gradient covariance matrices.
result Metric tensors enhance sampling for complex Bayesian models.
The paper refines classical covariance asymptotics using geometric information geometry.
problem Deviation of finite-sample behavior from classical predictions in curved models.
method Develops a curvature-aware refinement by viewing parametric families as Riemannian manifolds with Fisher-Rao metric.
result Derives an \(n^{-2}\) correction to the leading \(n^{-1}I(θ)^{-1}\) covariance term for score-root estimators.
Adaptive classifier optimizes high-dimensional data with spiked covariance structure.
problem Classification of high-dimensional data with spiked covariance structure.
method Adaptive classifier that whitens data, screens features, and applies Fisher linear discriminant.
result The classifier is Bayes optimal under certain conditions and performs well on real and synthetic data.
A new QDA classifier for high-dimensional data with spiked covariance.
problem Classifying high-dimensional data with distinct covariance matrices.
method Proposes a novel quadratic classification technique with parameters chosen to maximize the fisher-discriminant ratio.
result The proposed classifier outperforms classical R-QDA and requires lower computational complexity.
Paper identifies key function spaces for ReLU networks based on Fisher information.
problem Understanding the structure of Fisher information matrices in ReLU networks.
method Spectral decomposition of Fisher information matrices, focusing on the first three eigenspaces.
result The first three eigenspaces account for 97.7% of the trace of the Fisher information matrix, corresponding to spherical harmonic functions of order ≤2.
We introduce RSE to measure robustness in estimation problems.
problem Estimating statistical models from observed data.
method Developed theory for spectral functions of measures to compute RSE.
result RSE reveals a reciprocal relationship with problem complexity.
A family of probability distributions parametrized by an open domain Λ in Rn defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry the standard assumption has been that the Fisher information matrix tensor is positive definite defining in this way a Riemannia…
FADE adapts machine learning models to evolving data efficiently.
problem Sequential covariate shift in dynamic environments.
method FADE uses Fisher information geometry for robust learning under SCS.
result FADE achieves up to 19% higher accuracy under severe shifts.
Proofs Fisher-Rao distance on Gaussian covariance manifold.
problem Proving Fisher-Rao distance on Gaussian covariance manifold.
method Basic Riemannian geometry.
result Proof of Fisher-Rao distance on covariance cone.
Study mini-batch SGD noise and its limits, proving complexity guarantees.
problem Analyzing the noise in mini-batch SGD and its impact on optimization.
method Examined the conditional covariance and diffusion limits of SGD under different sampling designs.
result Proved mean-square upper bounds and Fisher van Trees lower bounds for SGD, linking them to effective dimension and condition number.
Study non-asymptotic estimation bounds for LTI models with Gaussian noise.
problem Estimating parameters of LTI models with non-asymptotic error bounds.
method Sharp non-asymptotic lower bounds using Cramér-Rao and van Trees inequalities, concentration results, and differential geometric constructions.
result Sharp and rate-optimal lower bounds for mean square estimation risk.
It is well known that in a supervised classification setting when the number of features is smaller than the number of observations, Fisher's linear discriminant rule is asymptotically Bayes. However, there are numerous modern applications where classification is needed in the high-dimensional setting. Naive implementa…
Enhances power of covariance matrix tests for high-dimensional data.
problem Testing large covariance matrices in high-dimensional data.
method Proposes a new Fisher's combined probability test for quadratic form and maximum form statistics.
result Boosts power against more general alternatives.
A new method improves uncertainty estimation in deep learning, especially for hard-to-label samples.
problem Improving uncertainty estimation for hard-to-label samples in deep learning.
method Introduces Fisher Information Matrix (FIM) to dynamically reweight objective loss terms.
result Consistently outperforms traditional evidential neural networks in uncertainty estimation tasks.
Early training phase affects deep neural network optimization and generalization.
problem The choice of learning rate influences generalization in deep learning models.
method Showed that SGD implicitly penalizes the trace of the Fisher Information Matrix (FIM) from the start of training, and explicitly penalizing the trace of FIM improves generalization.
result Catastrophic Fisher explosion (large trace of FIM early in training) is linked to poor generalization.
We analyze the variance of Fisher information estimators in deep learning models.
problem Understanding the variance of Fisher information in deep learning models.
method Investigated two unbiased and consistent estimators of Fisher information matrix.
result The variance of estimators is influenced by the model's parametric structure.
Many pattern recognition methods rely on statistical information from centered data, with the eigenanalysis of an empirical central moment, such as the covariance matrix in principal component analysis (PCA), as well as partial least squares regression, canonical-correlation analysis and Fisher discriminant analysis. R…
Study on Gaussian-width complexity on statistical manifolds and its applications in learning and recovery.
problem Understanding the geometry of statistical manifolds and its implications for learning and recovery.
method Analysis of Fisher width and inverse-Fisher width, proving their complementary roles and establishing a relation between them.
result Established a sharp relation between Fisher width and inverse-Fisher width, showing they cannot reduce relative to Euclidean scale.
We study two types of preconditioners and preconditioned stochastic gradient descent (SGD) methods in a unified framework. We call the first one the Newton type due to its close relationship to the Newton method, and the second one the Fisher type as its preconditioner is closely related to the inverse of Fisher inform…
Estimates metric tensor on neuromanifolds using Fisher information and random methods.
problem Computing the metric tensor on high-dimensional neuromanifolds efficiently and accurately.
method Deterministic bounds and unbiased random estimators based on Hutchinson's trace method.
result An efficient random estimator with bounded standard deviation.
Kernel networks' stability edge linked to Fisher Information singularity.
problem Understanding the stability edge in high-capacity kernel Hopfield networks.
method Statistical manifold analysis and Riemannian geometry.
result The Ridge of Optimization corresponds to the Edge of Stability, revealing a dual equilibrium.
Paper defines Fisher co-metric on cotangent bundle and clarifies its relation to variance.
problem Defines Fisher co-metric on cotangent bundle and clarifies its relation to variance.
method Defines Fisher co-metric directly from Fisher metric without going through tangent bundle, using a natural correspondence between cotangent vectors and random variables.
result Clarifies the relation between Fisher co-metric and variance/covariance, trivializing the Cramér-Rao inequality.
Develops new Markov processes with switching rates and past dependence.
problem Modeling processes with dynamic switching rates and path dependence.
method Introduces a new class of Markov jump processes with regime switching and path dependence. Derives distributional properties and maximum likelihood estimates.
result Maximum likelihood estimates of the process parameters are derived in closed form and have asymptotic normality.
Develops information geometry for Lévy processes in finance.
problem Understanding the statistical properties of Lévy processes for financial modeling.
method Deriving α-divergences from Lévy triplets, identifying Fisher information matrix and α-connection. result Identifies statistical implications and differential-geometric structures of Lévy processes.
We derive a single pass algorithm for computing the gradient and Fisher information of Vecchia's Gaussian process loglikelihood approximation, which provides a computationally efficient means for applying the Fisher scoring algorithm for maximizing the loglikelihood. The advantages of the optimization techniques are de…
Natural gradient descent is an optimization method traditionally motivated from the perspective of information geometry, and works well for many applications as an alternative to stochastic gradient descent. In this paper we critically analyze this method and its properties, and show how it can be viewed as a type of 2…
Model analyzes cooccurrence data for recommender systems and item relevance.
problem High-dimensional cooccurrence data from online platforms.
method Shared parameter Alternating Tweedie (SA-Tweedie) model with Fisher scoring and learning rate adjustment.
result SA-Tweedie model outperforms other methods in optimizing parameters.
One way to avoid overfitting in machine learning is to use model parameters distributed according to a Bayesian posterior given the data, rather than the maximum likelihood estimator. Stochastic gradient Langevin dynamics (SGLD) is one algorithm to approximate such Bayesian posteriors for large models and datasets. SGL…
This paper analyzes Barlow Twins' representation efficiency using information-geometric methods.
problem Understanding and comparing the efficiency of self-supervised learning methods.
method Introduces an information-geometric framework to quantify representation efficiency and applies it to Barlow Twins.
result Proves that Barlow Twins achieves optimal representation efficiency (η=1).
A new iterative K-FAC algorithm reduces training time and memory usage.
problem Training deep learning models efficiently.
method Uses conjugate gradient to approximate Fisher information matrix without generating the matrix or factors.
result Time and memory complexity of iterative CG-FAC is less than standard K-FAC.
New method estimates covariance matrices without restrictive assumptions.
problem Estimating high-dimensional covariance matrices under restrictive assumptions.
method Distributionally robust covariance estimation problems with mild conditions.
result Robust estimators are efficient, consistent, and perform well.
Study connects covariance cleaning theory to information theory for heavy-tailed distributions.
problem Optimizing covariance matrices for heavy-tailed distributions using information theory.
method Minimizing Frobenius norm and information loss between true and estimated covariance matrices.
result Asymptotic regime of large matrices minimizes information loss for Student's t distributions.
New methods improve Fisher Matrix approximations for neural networks at low cost.
problem High cost of solving Fisher Information Matrix (FIM) in neural networks.
method Direct minimization via Kronecker product singular value decomposition.
result Improved approximations to FIM provide more accurate and faster optimization.