A new way to describe correlation matrices makes modeling easier.
problem Describing correlation matrices in a flexible and positive-definite way.
method Introduces a novel parametrization that allows unrestricted vectors for correlation matrices.
result The new parametrization ensures positive definiteness without additional constraints.
New methods for Z-transform inversion and Wiener-Hopf factorization.
problem Efficient numerical inversion of Z-transforms and factorization of functions. method Sinh-deformations of contours, variable changes, and simplified trapezoid rule.
result High precision and speed in evaluating moments and constructing filters.
New efficient method for inverse Z-transform reduces complexity significantly.
problem Efficient numerical realization of inverse Z-transform for large n.
method Derives sufficient conditions for new scheme, applies to option pricing.
result Significant reduction in complexity for large n, especially for European options.
The paper derives formulas for option pricing and random walk expectations.
problem Calculating the price of barrier and lookback options.
method Inverse Z-transform, Fourier/Laplace inversion, Wiener-Hopf factorization, and numerical methods.
result Efficient numerical methods for option pricing are developed.
Paper introduces a modified Allen-Cahn equation for better energy equipartition.
problem Improving energy equipartition in mean curvature flow.
method Developed a modified Allen-Cahn equation (MAC) by time-dependent parameter and Z-transform.
result Improved estimate on discrepancy energy for better equipartition.
New fast estimation methods stemming from control theory lead to a fresh look at time series, which bears some resemblance to "technical analysis". The results are applied to a typical object of financial engineering, namely the forecast of foreign exchange rates, via a "model-free" setting, i.e., via repeated identifi…
Based on the stochastic model proposed by Patriarca-Kaski-Chakraborti that describes the exchange of wealth between n economic agents, we analyze the evolution of the corresponding economies under the assumption of a Gaussian background, modeling the exchange parameter ε. We demonstrate, that within Gaussian noise,…
Cai, Song and Kou (2015) [Cai, N., Y. Song, S. Kou (2015) A general framework for pricing Asian options under Markov processes. Oper. Res. 63(3): 540-554] made a breakthrough by proposing a general framework for pricing both discretely and continuously monitored Asian options under one-dimensional Markov processes. In …
The z-transform technique is used to investigate the model for distribution of high-tax payers, which is proposed by two of the authors (K. Y and S. M) and others. Our analysis shows an asymptotic power-law of this model with the exponent -5/2 when a total ``mass'' has a certain critical value. Below the critical value…
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.
Fisher auto-encoders use Fisher divergence for more robust generative modeling.
problem Model uncertainty in generative models.
method Minimizing Fisher divergence between true and modeled joint distributions.
result Fisher auto-encoders can more accurately quantify model uncertainty.
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…
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.
In this communication, we describe some interrelations between generalized q-entropies and a generalized version of Fisher information. In information theory, the de Bruijn identity links the Fisher information and the derivative of the entropy. We show that this identity can be extended to generalized versions of en…
Market strategies minimize Fisher information to minimize risk.
problem Applying minimum Fisher information principle to market dynamics.
method Analytical extension to quantum harmonic oscillator eigenstates and Gibbs distribution.
result Minimizing Fisher information reduces information and risk.
Generative Adversarial Networks (GANs) are powerful models for learning complex distributions. Stable training of GANs has been addressed in many recent works which explore different metrics between distributions. In this paper we introduce Fisher GAN which fits within the Integral Probability Metrics (IPM) framework f…
The study examines Fisher-Riemann geodesics for nonparametric probability densities.
problem Understanding nonparametric probability densities using Fisher-Riemann geometry.
method Obtaining Fisher-Riemann geodesics as a limit of parametric cases with increasing parameters.
result The weak limit approach for nonparametric probability densities.
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…
Blog post discusses various implementations of Fisher Information for EWC in continual learning.
problem Improving Elastic Weight Consolidation (EWC) results by optimizing Fisher Information computation.
method Empirically compares different implementations of Fisher Information for EWC.
result Many reported EWC results can be improved by changing Fisher Information computation methods.
Fisher width is a geometric measure of complexity on statistical manifolds.
problem Complexity measures on statistical manifolds
method Introducing Fisher width as a Fisher-geometric analogue of Gaussian width
result Fisher width retains key structural features of Gaussian width while capturing anisotropic geometric effects
The paper studies metrics on Lie groups SO(2) and SO(3) and their integrability.
problem Integrability of gradient systems on Lie groups via Fisher metrics.
method Analysis of Souriau-Fisher metrics and 2-cocycles on Lie groups SO(2) and SO(3).
result Cocycles can locally modify Fisher metrics on Lie group orbits.
This paper shows any Kähler metric can be a Fisher information metric.
problem Establishing a new characterization of Kähler and coKähler manifolds.
method Statistical approach using Fisher information and exponential families.
result Any Kähler metric is a Fisher information metric.
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.
Survey on closed-form Fisher-Rao distance expressions.
problem Finding closed-form expressions for Fisher-Rao distance.
method Collect and present examples of closed-form expressions for Fisher-Rao distance of discrete and continuous distributions.
result Presentation of closed-form expressions for Fisher-Rao distance of various distributions.
New statistics are introduced that maintain the Fisher metric structure closely, akin to sufficient statistics.
problem Maintaining the Fisher metric structure in statistical models.
method Characterizing statistics that maintain the Fisher metric structure bi-Lipschitz equivalently.
result Characterized statistics that preserve the Fisher metric structure closely.
We consider three different approaches to define natural Riemannian metrics on polytopes of stochastic matrices. First, we define a natural class of stochastic maps between these polytopes and give a metric characterization of Chentsov type in terms of invariance with respect to these maps. Second, we consider the Fish…
Fisher score is one of the most widely used supervised feature selection methods. However, it selects each feature independently according to their scores under the Fisher criterion, which leads to a suboptimal subset of features. In this paper, we present a generalized Fisher score to jointly select features. It aims …
We introduce Fisher consistency in the sense of unbiasedness as a desirable property for estimators of class prior probabilities. Lack of Fisher consistency could be used as a criterion to dismiss estimators that are unlikely to deliver precise estimates in test datasets under prior probability and more general dataset…
GANs can be used to extract Fisher vectors for unsupervised feature learning.
problem Unsupervised feature extraction for classification and similarity tasks.
method Derive Fisher Information and Fisher Vectors from GANs' density model.
result GAN-induced Fisher Vectors perform competitively in unsupervised feature extraction.
In information theory, Fisher information and Shannon information (entropy) are respectively used to quantify the uncertainty associated with the distribution modeling and the uncertainty in specifying the outcome of given variables. These two quantities are complementary and are jointly applied to information behavior…
Study on geometry of Dirichlet distributions using Fisher-Rao metric.
problem Understanding the geometry of Dirichlet distributions.
method Analysis of Fisher-Rao metric on Dirichlet distribution parameter space.
result Geodesic completeness and negative sectional curvature of the space.
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…
Paper formulates particle flow using variational inference and Fisher-Rao gradient flow.
problem Estimating posterior densities in probabilistic models.
method Variational formulation of particle flow, Fisher-Rao gradient flow, Gaussian and Gaussian mixture approximations.
result Gaussian and Gaussian mixture approximations of Fisher-Rao particle flow reduce to Exact Daum and Huang particle flow under linear Gaussian assumptions.
Paper discusses the Fisher metric and differentiability in statistical models.
problem Understanding the relationship between Fisher metric and differentiability in statistical models.
method Comparison of different concepts and models in Information Geometry, mathematical statistics, and measure theory.
result Discussion of various models and their differentiability properties.
TopoFisher learns topological summaries by maximizing Fisher information, improving parameter efficiency and inference quality.
problem Simulation-based inference misses key information in low-order statistics, especially for non-Gaussian fields.
method TopoFisher uses a differentiable persistent-homology pipeline that learns topological summaries by maximizing local Gaussian Fisher information.
result TopoFisher recovers much of the available information and outperforms fixed topological vectorizations in weak gravitational lensing.
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.
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.
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.
Proposes a topological framework to study modular invariants and related concepts.
problem Exploring modular invariants and related concepts in topological quantum field theory.
method Topological paradigm in alterfold topological quantum field theory.
result Establishes a novel integral identity for modular invariance across multiple Morita contexts.
The study examines Fisher information matrices and neural tangent kernels for simple ReLU networks with random weights.
problem Understanding the relationship between Fisher information matrices and neural tangent kernels for 2-layer ReLU networks.
method Analyzes Fisher information matrices and neural tangent kernels for 2-layer ReLU networks with random hidden weights, focusing on spectral decomposition and eigenfunctions.
result Obtained an approximation formula for functions represented by 2-layer neural networks.
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.
New method for natural policy gradients converges linearly.
problem Improving natural policy gradient methods for better convergence.
method Fisher-Rao gradient flow applied to state-action distributions.
result Linear convergence rate with geometry-dependent factor.
Global gradient estimates for Fisher-KPP equation on Finsler metric measure spaces.
problem Establishing gradient estimates for the Finslerian Fisher-KPP equation.
method Global gradient estimates on compact and noncompact Finsler metric measure spaces using the traditional CD(K,N) condition and new comparison theorems. result Global gradient estimates for positive solutions of the Finslerian Fisher-KPP equation.
Paper explores Fisher-Rao gradient flows and their kernel approximations.
problem Understanding and analyzing approximations of Fisher-Rao gradient flows.
method Rigorous investigation of Fisher-Rao and Wasserstein type gradient flows, focusing on kernel approximations.
result Proves evolutionary Γ-convergence for kernel-approximated Fisher-Rao flows, providing theoretical guarantees.
Study improves sampling from non-log-concave distributions using Fisher information.
problem Sampling from non-log-concave distributions with high Fisher information guarantees.
method Proximal sampler with RGO implementation, leveraging log-concave sampling results.
result Improved complexity guarantee in relative Fisher information for non-log-concave sampling.
We present a novel synthesis of Fisher information and asset pricing theory that yields a practical method for reconstructing the probability density implicit in security prices. The Fisher information approach to these inverse problems transforms the search for a probability density into the solution of a differential…
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.
Fisher loss improves deep domain adaptation by learning discriminative within-class compact and between-class separable representations.
problem Improving deep domain adaptation performance by learning discriminative representations.
method Proposes a Fisher loss to learn discriminative representations that are within-class compact and between-class separable.
result Noticeable improvements in deep domain adaptation performance, e.g., 6.67% absolute improvement in mean accuracy on the Office-Home dataset.