Model separates overall uncertainty into aleatoric and epistemic components for active learning.
problem Active learning with uncertainty quantification.
method Non-stationary Heteroscedastic Gaussian process model.
result Model separates overall uncertainty into aleatoric and epistemic components.
Cooperative model disentangles data uncertainties.
problem Disentangling aleatoric and epistemic uncertainties in real-world data.
method Cooperatively trains a variance estimation network with a Bayesian neural network.
result Improves mean estimation and disentangles uncertainties.
Proposes a new principle for active learning based on epistemic uncertainty.
problem Active learning and uncertainty quantification in machine learning.
method Distinction between epistemic and aleatoric uncertainty; proposes epistemic uncertainty sampling.
result Epistemic uncertainty sampling shows promising performance in experimental studies.
Estimating how uncertain an AI system is in its predictions is important to improve the safety of such systems. Uncertainty in predictive can result from uncertainty in model parameters, irreducible data uncertainty and uncertainty due to distributional mismatch between the test and training data distributions. Differe…
The standard taxonomy of predictive uncertainty is inconsistent with standard measures.
problem Uncertainty taxonomy and measure inconsistency
method Proof of inconsistency
result Uncertainty is not reducible to data collection
Deep ensembles effectively capture epistemic uncertainty through training stochasticity, providing a frequentist perspective.
problem Understanding and quantifying epistemic uncertainty in machine learning models.
method Bootstrap-based estimator and decomposition of deep ensembles into data variability and training stochasticity.
result Deep ensembles primarily capture training stochasticity, explaining their effectiveness in quantifying epistemic uncertainty.
The paper decomposes probabilistic scores into reliability, uncertainty, and information loss.
problem Understanding the reliability and uncertainty of probabilistic predictions.
method Developed decomposition identities for proper losses, quantifying reliability, residual uncertainty, and information gain.
result A three-term identity for classification scores, revealing miscalibration, grouping term, and feature-level uncertainty.
New research shows calibration error is flawed when dealing with model uncertainty.
problem Current model evaluation techniques conflate model uncertainty with aleatoric uncertainty.
method Posterior predictive checks to evaluate deep learning models.
result Calibration error and variants are incorrect when model uncertainty is present.
Improves Bayesian optimization by focusing on well-behaved structure in objectives.
problem Bayesian optimization struggles with real-world objectives that are often poorly behaved.
method Proposes surrogate models that focus on well-behaved structure, absorbing challenging structures as irreducible uncertainty.
result Surrogate models with appropriate noise distributions improve reliability and performance in challenging objective functions.
Benchmark assesses fairness in algorithmic uncertainty, revealing consistent and calibrated estimates improve fairness.
problem Challenges in managing uncertainty in fairness evaluations for predictive algorithms.
method Introduces FairlyUncertain, an axiomatic benchmark for evaluating uncertainty in fairness.
result Consistent and calibrated uncertainty estimates improve fairness without explicit fairness interventions.
OOD detection methods often misidentify OOD points, leading to ineffective safety improvements.
problem Improving model safety through OOD detection methods often leads to incorrect identification of out-of-distribution points.
method Re-examine popular OOD detection procedures based on predictive uncertainty or features of supervised models trained on in-distribution data.
result Popular OOD detection methods incorrectly conflate high uncertainty and far feature-space distance with being out-of-distribution.
Framework combines machine learning and inverse methods to quantify uncertainties in model parameters.
problem Combining aleatoric and epistemic uncertainties in engineered systems modeling.
method Develops a robust filtering step in LUQ to learn useful QoI maps from noisy datasets, iterates over time, and uses sufficiency tests.
result Transforms datasets into distributions for DC-based inversion, improving parameter quantification.
Paper defines numerical criteria to test handlebody link irreducibility.
problem Determining the irreducibility of handlebody links.
method A set of numerical criteria to test handlebody links for irreducibility.
result Effective method to recognize irreducibility of handlebody knots and most handlebody links.
Study bounds growth rate of irreducible meanders, showing proportion vanishes.
problem Understanding growth rate of irreducible meanders.
method Provided upper and lower bounds for growth rate.
result Proportion of irreducible meanders among prime meanders approaches 0 as n increases.
Examines how irreducibility and rigidity affect digital images.
problem Understanding interactions between irreducibility and rigidity in digital images.
method Analyzes Cartesian products, wedges, and cold and freezing sets.
result Interactions between irreducibility and rigidity in digital images.
Decoupled PFNs improve sequential decision-making by separating epistemic and aleatoric uncertainties.
problem Sequential decision-making requires distinguishing between epistemic uncertainty about latent signals and irreducible aleatoric observation noise.
method Developed a decoupled PFN architecture that uses query-level labels to train separate heads for latent signal and aleatoric noise.
result Empirically, decoupled PFNs mitigate the failure mode of total-variance exploration in noisy and heteroscedastic settings.
Paper finds a counterexample showing rectangle condition doesn't detect strong irreducibility.
problem Detecting strong irreducibility from the rectangle condition.
method Constructing a double branched cover of a knot in S^3.
result Found a genus 2 Heegaard splitting that is strongly irreducible but fails rectangle condition.
Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.
The study classifies strongly irreducible Heegaard splittings in hyperbolic 3-manifolds.
problem Understanding the structure of Heegaard splittings in hyperbolic 3-manifolds.
method Effective version of Li's theorem applied to hyperbolic 3-manifolds, focusing on irreducible and strongly irreducible surfaces.
result Haken hyperbolic 3-manifolds have a finite collection of strongly irreducible Heegaard surfaces and incompressible surfaces, which classify all strongly irreducible Heegaard splittings.
An irreducible open 3-manifold W is {\bf R}2-irreducible if every proper plane in W splits off a halfspace. In this paper it is shown that if such a W is the universal cover of a connected, {\bf P}2-irreducible open 3-manifold M with finitely generated fundamental group, then either W is homeomorphic to…
Advocates for a new posterior that predicts better than classical and generalised Bayes.
problem Combining parameter inference and density estimation for better predictive models.
method Predictively Oriented (PrO) posterior using mean field Langevin dynamics.
result PrO posteriors converge to the predictively optimal model average, adapting to model misspecification.
Shows uniqueness of irreducible generating tuples for Fuchsian groups.
problem Identifying irreducible generating tuples in Fuchsian groups.
method Variation of ideas from \cite{W2} to show uniqueness of almost orbifold covers with rigid generating tuples.
result Irreducible generating tuples are unique up to equivalence and are irreducible.
A graph embedded in the 3-sphere is called irreducible if it is non-splittable and for any 2-sphere embedded in the 3-sphere that intersects the graph at one point the graph is contained in one of the 3-balls bounded by the 2-sphere. We show that irreducibility is preserved under certain deformations of embedded graphs…
Study on Yang-Mills fields on S4 proving self-duality constraints.
problem Proving self-duality of weakly stable Yang-Mills fields on S4. method Investigating irreducible connections under self-duality assumption.
result Irreducible Yang-Mills fields on S4 are either self-dual or anti-self-dual. Study investigates lattices fibring over the circle, focusing on BNSR invariants.
problem Investigating BNSR invariants of irreducible uniform lattices.
method Examines BNSR invariants and Bestvina-Brady groups to understand lattice properties.
result Irreducibility of lattices is linked to the vanishing of BNSR invariants for all finite-index subgroups.
Paper proposes a new method for estimating mixture proportions without irreducibility assumption.
problem Estimating mixture proportions when component distributions are not irreducible.
method Developed a resampling-based meta-algorithm that adapts existing MPE algorithms to non-irreducible settings.
result Empirical results show improved estimation performance compared to baseline methods and regrouping-based algorithms.
A surface automorphism is strongly irreducible if every essential simple closed curve in the surface has nontrivial geometric intersection with its image. We show that a three-manifold admits only finitely many inequivalent surface bundle structures with strongly irreducible monodromy.
The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreduci…
Connected weakly irreducible not irreducible subgroups of Sp(1,n+1)⊂SO(4,4n+4) that satisfy a certain additional condition are classified. This will be used to classify connected holonomy groups of pseudo-hyper-Kählerian manifolds of index 4.
We give necessary and sufficient topological conditions for the existence of an irreducible SO(3)-structure on a 5-manifold. Using these conditions we provide some new examples of 5-manifolds with an irreducible SO(3)-structure.
A complex disproves a curvature property.
problem Proving non-equivalence of curvature properties.
method Analyzing a specific 2-complex presentation.
result The contracting non-positive immersions property is not equivalent to positive maximal irreducible curvature.
Study the smallest Laplace eigenvalue in special geometric spaces.
problem Finding the smallest positive eigenvalue of Laplace-Beltrami operator in strongly isotropy irreducible spaces.
method Explicit expression for simply connected cases, proving Einstein manifold properties and eigenvalue bounds.
result Proved E<λ1≤16E for all strongly isotropy irreducible spaces. A complete list of irreducible triangulations is identified on the Möbius band.
We prove for any positive integer n there exist boundary-sum irreducible Zn-corks with Stein structure. Here `boundary-sum irreducible' means the manifold is indecomposable with respect to boundary-sum. We also verify that some of the finite order corks admit hyperbolic boundary by HIKMOT.
Possible irreducible holonomy algebras $\g\subset\sp(2m,\Real)$ of odd Riemannian supermanifolds and irreducible subalgebras $\g\subset\gl(n,\Real)$ with non-trivial first skew-symmetric prolongations are classified. An approach to the classification of some classes of the holonomy algebras of Riemannian supermanifolds…
Irreducible skew-Berger algebras $\g\subset\gl(n,\Co)$, i.e. algebras spanned by the images of the linear maps $R:\odot^2\Co^n\to\g$ satisfying the Bianchi identity, are classified. These Lie algebras can be interpreted as irreducible complex Berger superalgebras contained in $\gl(0|n,\Co)$.
New techniques create irreducible 4-manifolds with specific properties.
problem Creating irreducible 4-manifolds with specific topological and geometric properties.
method Performing various operations on irreducible simply-connected 4-manifolds, including torus surgeries, symplectic fiber sums, rational blow-downs, and Lefschetz fibrations.
result For most (e,σ) coordinates, irreducible smooth structures can be found on 4-manifolds with order two fundamental group. A polynomial f(t) with rational coefficients is strongly irreducible if f(t^k) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(t^k) and g(t^l) are relatively prime for all positive integers k and l. We provide some sufficient conditions for strong irreducibility a…
Train track automata for fully irreducible elements in Out(F_r).
problem Understanding fully irreducible elements in Out(F_r).
method Describing train track automata and geodesics in Outer Space.
result Geodesics in Culler-Vogtmann Outer Space for fully irreducible elements.
New train tracks for complex homeomorphisms found.
problem Existence of irreducible train tracks for pseudo-Anosov homeomorphisms.
method Starting from a veering triangulation, identify and modify branches to bypass obstructions.
result Construction of invariant train tracks with irreducible transition matrix.
A triangulation of a punctured or pinched surface is irreducible if no edge can be shrunk without producing multiple edges or changing the topological type of the surface. The finiteness of the set of (non-isomorphic) irreducible triangulations of any punctured surface is established. Complete lists of irreducible tria…
Automorphisms of handlebodies arise naturally in the a classification of automorphisms of three-manifolds. Among automorphisms of handlebodies, there are certain automorphisms called irreducible (or generic), which are analogues of pseudo-Anosov automorphisms of surfaces. We show that irreducible automorphisms of handl…
For product manifolds, cohomologically calibrated affine connections are geometrically irreducible.
problem Establishing geometric irreducibility of cohomologically calibrated affine connections on product manifolds.
method Proof relies on Hodge theory and integral arguments showing non-cancellation of off-diagonal components in the Riemann curvature tensor.
result Cohomologically calibrated affine connections on product manifolds are holonomically irreducible.
Example found of subgroup not a lattice in product of Lie groups
problem Finding irreducible discrete subgroups that are not lattices
method Produced an example in SL(2,R)imesSL(2,R) result Example of subgroup not a lattice in product of Lie groups
Geodesic orbit metrics proven on specific homogeneous spaces.
problem Geodesic orbit metrics on homogeneous spaces.
method Using strongly isotropy irreducible spaces and proving natural reductivity.
result Geodesic orbit metrics are naturally reductive on constructed homogeneous spaces.
New proof of Alesker's Irreducibility Theorem using localization techniques.
problem Representing polynomial valuations on convex bodies.
method Introducing a localization technique for polynomial valuations and reducing to a representation problem for differential forms.
result Smooth and translation invariant valuations are representable by integration with the normal cycle.
Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equ…
Possible irreducible holonomy algebras $\g\subset\osp(p,q|2m)$ of Riemannian supermanifolds under the assumption that $\g$ is a direct sum of simple Lie superalgebras of classical type and possibly of a one-dimensional center are classified. This generalizes the classical result of Marcel Berger about the classificatio…