Study anti-invariant submersions from holomorphic statistical manifolds.
problem Understanding submersions in statistical manifolds.
method Introduced and analyzed anti-invariant holomorphic statistical submersions.
result Supported results with examples.
This paper studies moduli spaces of statistical structures on Lie groups.
problem Understanding statistical structures on Lie groups.
method Introduced and studied moduli spaces for left-invariant statistical structures on Lie groups.
result Moduli spaces of left-invariant Riemannian metrics are singletons for certain Lie groups.
Study invariant connections on multivariate Gaussian distributions.
problem Understanding statistical connections on multivariate Gaussian distributions.
method Investigate invariant connections on N0n with the Fisher metric. result Explicitly determined invariant connections and their moduli spaces.
The paper introduces a statistical version of contact CR-product for Sasakian statistical manifolds.
problem Characterizing geometric properties of contact CR-submanifolds in Sasakian statistical manifolds.
method Characterization of integrability of invariant and anti-invariant distributions, development of results on specific types of contact CR submanifolds, introduction of statistical contact CR-product.
result Introduction of a statistical version of contact CR-product for Sasakian statistical manifolds.
Study on lightlike geometry in indefinite Sasakian statistical manifolds.
problem Exploring lightlike hypersurfaces and their properties in indefinite Sasakian statistical manifolds.
method Introducing indefinite Sasakian statistical manifolds and analyzing lightlike hypersurfaces with respect to dual connections.
result An invariant lightlike submanifold of an indefinite Sasakian statistical manifold is itself an indefinite Sasakian statistical manifold.
Study topological Iwasawa invariants for 3-sphere links, proving density results.
problem Detect and analyze Iwasawa invariants for 3-sphere links.
method Explicit criteria for detecting Iwasawa invariants, statistical analysis of 2-bridge links.
result Density of 2-bridge links with specific Iwasawa invariants.
New connections found on zero-mean multivariate normal distributions.
problem Characterizing statistical connections on zero-mean multivariate normal distributions.
method Investigating invariant conjugate symmetric statistical connections on the submanifold of zero-mean multivariate normal distributions.
result Invariant connections on zero-mean multivariate normal distributions are not uniquely characterized by invariance under the general linear group action.
This work provides statistical guarantees for GANs that are invariant to certain group symmetries.
problem Learning group-invariant distributions efficiently.
method Study of group-invariant GANs and their performance guarantees.
result Group-invariant GANs require fewer samples and have a reduced discriminator approximation error.
The Wintgen inequality (1979) is a sharp geometric inequality for surfaces in the 4-dimensional Euclidean space involving the Gauss curvature (intrinsic invariant) and the normal curvature and squared mean curvature (extrinsic invariants), respectively. In the present paper we obtain a Wintgen inequality for statistica…
Deep learning uncovers patterns between knot types.
problem Discovering connections between combinatorial and hyperbolic knot invariants.
method Statistical approach using linear regression and deep learning.
result Revealed empirical connections between knot types.
The paper extends hypothesis testing to non-diagonalizable matrices, improving network statistics inference.
problem Testing on non-diagonalizable matrices for network statistics.
method Generalizes Wald and t-tests to non-symmetric matrices, controlling convergence rates.
result Improved inference on network statistics from directed networks.
Study invariant minimizers in convex functions under amenable groups.
problem Finding invariant minimizers in convex functions invariant under amenable groups.
method Analyze smallest closed invariant convex subsets and apply to invariant optimality problem.
result Clarifies relations between equivariant neural networks and statistical theorems.
Chentsov's theorem characterizes the Fisher information metric on statistical models as essentially the only Riemannian metric that is invariant under sufficient statistics. This implies that each statistical model is naturally equipped with a geometry, so Chentsov's theorem explains why many statistical properties can…
This paper mainly contributes to a classification of statistical Einstein manifolds, namely statistical manifolds at the same time are Einstein manifolds. A statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. With the Fisher information metric as a Riemannian metric, infor…
SETrLUSI combines diverse knowledge from multiple domains for faster convergence.
problem Handling diverse knowledge from multiple domains in transfer learning.
method Stochastic Ensemble Multi-Source Transfer Learning Using Statistical Invariant (SETrLUSI).
result SETrLUSI accelerates convergence and outperforms related methods.
Data augmented bootstrap unifies various confidence interval construction methods.
problem Constructing confidence intervals from data transformations.
method Data augmented bootstrap (DAB) framework.
result Establishes theoretical coverage results for DAB methods.
Data science enhances knot theory by analyzing invariant relations.
problem Understanding the complex relations between knot invariants.
method Topological data analysis applied to knot theory.
result New insights into long-standing conjectures about knots.
New models exploit invariance to reduce model complexity.
problem Reducing model complexity for tasks with inherent invariances.
method Invariant random features and kernel methods.
result Exploiting invariance saves a dα factor in model complexity. New research optimizes HSIC estimation rate for translation-invariant kernels.
problem Optimizing the rate of HSIC estimation for translation-invariant kernels.
method Proved minimax optimal rate of O(n−1/2) for HSIC estimation. result Optimality of various HSIC estimators proven.
Study aggregation of statistical evidence under unknown dependence using group-invariance.
problem Aggregating statistical evidence under unknown and complex dependence structures.
method Develops a framework using group-invariance and permutation-based constructions to aggregate evidence across transformed datasets.
result Shows uniform improvement in critical values for single-batch aggregation over deterministic calibrations, adapting to unknown dependence structures.
Diffusion models learn simple statistics before complex ones, revealing a sample complexity exponent.
problem Understanding the learning dynamics of diffusion models.
method Empirical observations and theoretical analysis of diffusion models and denoisers.
result Diffusion models learn simple statistics (pair-wise correlations) at linear sample complexity, while higher-order statistics (e.g., fourth cumulant) require cubic sample complexity.
Group-invariant neural networks improve approximation accuracy for symmetric functions.
problem Improving approximation accuracy for symmetric functions using neural networks.
method Investigates the generalization error of group-invariant neural networks within the Barron framework.
result Group invariance introduces a factor δ that can significantly improve approximation accuracy when it is small.
Backdoors in deep neural networks are undetectable and enable invariance-based adversarial examples.
problem Statistically undetectable backdoors in deep neural networks.
method Adversarial model trainer method to plant backdoors, showing invariance-based adversarial examples.
result Backdoors are statistically undetectable and enable generation of adversarial examples for every input.
The abstract discusses transformations on statistical and semi-Weyl manifolds with torsion.
problem Invariance and structure preservation under conformal-projective transformations.
method Proof of invariance and preservation of structures under conformal-projective transformations.
result Semi-Weyl and statistical structures with torsion are invariant under conformal-projective transformations.
In this note we prove certain necessary and sufficient conditions for the existence of an embedding of statistical manifolds. In particular, we prove that any compact smooth (C1 resp.) statistical manifold can be embedded into the space of probability measures on a finite set. As a result, we get an answer to the La…
Time reversal invariance can be summarized as follows: no difference can be measured if a sequence of events is run forward or backward in time. Because price time series are dominated by a randomness that hides possible structures and orders, the existence of time reversal invariance requires care to be investigated. …
This note describes the construction of c U p-invariant differential operators on statistical manifolds, i.e. of operators canonically associated to a geometry which synthetizes the properties of conformal and projective geometries.
Study finds a method to discover causal relationships that are invariant to marginal distributions.
problem Current causal discovery methods are sensitive to marginal distributions, leading to unreliable results.
method Proposes a non-parametric estimator that marginalizes the marginals to find intrinsic causal relationships.
result The proposed method yields causal estimators competitive with current methodologies and emphasizes uncertainty.
New method prevents classifiers from relying on spurious correlations.
problem Group invariant learning fails to prevent classifiers from depending on spurious correlations.
method Statistical independence tests to construct groups and reweight samples by group label proportion.
result New method significantly outperforms existing group invariant learning methods in generalizing to spurious correlation shifts.
Develops theory of homogeneous statistical manifolds and classifies Lie groups.
problem Understanding statistical manifolds and Lie groups.
method Constructs examples and classifies Lie groups using information geometry.
result Explicit examples of homogeneous statistical manifolds of low dimension constructed.
Unified framework for singular statistical models using observable charts.
problem Non-identifiability and breakdown of classical asymptotic theory in singular models.
method Invariant framework based on observable charts to define local coordinate systems in model space.
result Observable order provides a lower bound on KL divergence vanishing rate in singular models.
We present a novel family of deep neural architectures, named partially exchangeable networks (PENs) that leverage probabilistic symmetries. By design, PENs are invariant to block-switch transformations, which characterize the partial exchangeability properties of conditionally Markovian processes. Moreover, we show th…
A new depth function improves multivariate data analysis by considering variability directions.
problem Developing a depth function that respects quantile properties and is affine-invariant.
method Integrating rank-weighted depth with affine-invariance and covariance matrices.
result The AI-IRW depth function provides accurate quantile estimates and is robust to data variability.
In CJKLS quandle cohomology is used to produce invariants for particular embeddings of codimension two; 2-cocycles give to invariants for (classical) knots and 3-cocycles give rise to invariants for knotted surfaces. This is done by way of a notion of coloring of a diagram. Also, these invariants have the form of state…
We propose a statistical model for natural language that begins by considering language as a monoid, then representing it in complex matrices with a compatible translation invariant probability measure. We interpret the probability measure as arising via the Born rule from a translation invariant matrix product state.
AI needs causal inference to avoid being just a correlation machine.
problem AI's inability to distinguish correlation from causation.
method Develops a unified framework connecting various causal statistical estimators and proves a Statistical Necessity Theorem for causal generalization.
result AI systems without causal grounding are brittle and biased, highlighting the need for causal statistics.
New sampling methods improve statistical efficiency for intractable targets.
problem Sampling from complex, intractable probability distributions.
method Gaussian invariant versions of RWM, MALA, and Hessian MALA.
result Gaussian invariant sampling leads to improved statistical efficiency.
This paper improves the robustness of risk estimation for financial positions.
problem Ensuring robustness of risk measures in the presence of data noise.
method Proposes a quantitative approach using the Fortet-Mourier metric to quantify the variation of true probability measures.
result Derives explicit error bounds for discrepancies between laws of estimators based on true and perturbed data.
Researchers study the conformal geometry of bivariate Gaussian manifolds.
problem Exploring the conformal structure of Fisher-Rao metric on statistical manifolds.
method Determined invariants of the conformal structure of the Fisher-Rao metric on the bivariate Gaussian manifold.
result The conformal holonomy group is SO0(1,6) for generic random variables, but SO0(1,4) for independent ones. Develops tests for conditional symmetry under group actions.
problem Testing conditional symmetry in distributions under group actions.
method Nonparametric randomization tests with kernel methods and asymptotic consistency.
result Tests achieve finite-sample Type I error control and power.
Efficient method for shape modeling invariant to rigid motion.
problem Statistical shape modeling for rigidly moving shapes.
method Non-Euclidean Lie group analysis of metric distortion and curvature.
result Outperforms state-of-the-art classifiers in sparse data.
Unified approach to tensor PCA and related problems using tensor cumulants.
problem Statistical inference on invariant distributions, particularly tensor PCA.
method Definition and analysis of tensor cumulants to unify and extend previous results.
result Unified explanation of hardness and subexponential-time algorithms for tensor PCA.
Paper uses algebraic signatures to identify probabilistic structures in empirical data.
problem Identifying probabilistic structure from observed binomials in empirical probability tensors.
method Treating vanishing binomials as algebraic signatures, matching signatures to identify models without parameter estimation.
result The method successfully identified rank-one structures in real language data, revealing interpretable sets of words.
We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers is that we deal here with an elementary, interesting, new mathematics, an…
Proposes learning invariances in neural networks using a weight-space approach.
problem Learning invariances from data in neural networks remains an open problem.
method Minimizes a lower bound on the marginal likelihood in weight space.
result Results in higher performing models with naturally learned invariances.
Proposes a method to compare noisy high-dimensional datasets with low-dimensional manifolds.
problem Comparing distributions on manifolds in noisy high-dimensional datasets.
method Linking low-rank structure to manifold geometry, developing a scale-invariant distance measure.
result Superior robustness and statistical power compared to existing methods.
ADIGen: Automatic, Debiased, and Invariant Counterfactual Generation
problem Generative models for counterfactual outcomes
method ADIGen combines Riesz regression, causal invariance, and orthogonal statistical learning
result ADIGen controls counterfactual risk under general interventions
We improve generative models for heavy-tailed multivariate data using an invariant statistical loss.
problem Traditional generative models struggle with heavy-tailed and multivariate data, leading to unstable training and mode dropping.
method We extend the invariant statistical loss method to handle heavy-tailed and multivariate data using a Pareto-ISL generator trained with input noise from a generalised Pareto distribution.
result Pareto-ISL accurately models the tails of heavy-tailed distributions while capturing central characteristics.