The paper tackles learning from non-irreducible Markov chains, proving learnability and generalization bounds.
problem Learning from temporal dependent data with non-irreducible Markov chains.
method Uniform convergence and generalization bounds for sample error under uniform ergodicity.
result Learnability and generalization bounds for approximate sample error minimization algorithm.
The paper analyzes learning rates for non-irreducible Markov chains.
problem Real-world data often violates i.i.d. assumptions.
method Examines iterated random functions and contractive functions.
result Derives data-distribution dependent learning rates.
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.
We investigate geometric properties of indecomposable but non-irreducible Lorentzian manifolds, which are total spaces of circle bundles. We investigate under which conditions these manifolds are complete and give examples which fulfill the obtained conditions. In particular we investigate the Einstein equation for the…
That announcement gives the structure of totally reducible linear Lie algebras which are the Lie algebra of the holonomy group of (at least) one torsion-free connection. The result uses the (already known) classi cation of the irreducible ones and some previous (unpublished) works by the author giving the classi cation…
The holonomy group of an (n+2)-dimensional simply-connected, indecomposable but non-irreducible Lorentzian manifold (M,h) is contained in the parabolic group (R×SO(n))⋉Rn. The main ingredient of such a holonomy group is the SO(n)--projection G:=prSO(n)(Holp(M,h)) and one may ask…
Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.
problem Understand how algebraic conditions on isotropy group affect the geometry and curvature of Lorentzian homogeneous spaces.
method Prove that a Lorentzian locally homogeneous space is locally isometric to a plane wave if it admits an Ambrose--Singer connection with indecomposable, non-irreducible holonomy.
result Generalize existing results about Lorentzian homogeneous spaces with irreducible isotropy and prove results about Lorentzian connections with parallel torsion and 2-symmetric connections.
We study the geometry and holonomy of semi-Riemannian, time-like metric cones that are indecomposable, i.e., which do not admit a local decomposition into a semi-Riemannian product. This includes irreducible cones, for which the holonomy can be classified, as well as non irreducible cones. The latter admit a parallel d…
Due to a result by Gallot a Riemannian cone over a complete Riemannian manifold is either flat or has an irreducible holonomy representation. This is false in general for indefinite cones but the structures induced on the cone by holonomy invariant subspaces can be used to study the geometry on the base of the cone. Th…
Study shows Julia sets and gasket limit sets are quasiconformally different.
problem Quasiconformal non-equivalence of Julia sets and gasket limit sets.
method Proved quasiconformal non-equivalence of Julia sets and gasket limit sets.
result Julia sets and gasket limit sets are quasiconformally different.
Study shows non-symmetric convex sets have full boundary limits.
problem Understanding boundaries of non-symmetric convex sets.
method Proved using proximal limit set analysis.
result Proximal limit set equals full projective boundary for non-symmetric irreducible divisible convex sets.
The paper analyzes set-to-set matching with neural networks, focusing on theoretical generalization.
problem Theoretical analysis of set-to-set matching with neural networks.
method Generalization error analysis of set-to-set matching with neural networks.
result Theoretical insights into the behavior of set-to-set matching models.
Generative model learns to autoencode and generate sets of images.
problem Learning to represent and generate sets of images with unknown number of sets.
method Set Distribution Networks (SDNs) learn set encoder, discriminator, generator, and prior.
result SDNs can reconstruct and generate sets of images with preserved attributes.
Study on cold and freezing sets in digital images.
problem Properties of cold sets in digital images.
method Analysis of properties and relationships between cold and freezing sets.
result Examined relationships between cold and freezing sets.
Paper solves whether zero sets are mapping degree sets.
problem Whether finite sets containing zero are mapping degree sets.
method Examined oriented closed connected manifolds of the same dimension.
result Affirmative answer given for both integer and rational settings.
Matching two different sets of items, called heterogeneous set-to-set matching problem, has recently received attention as a promising problem. The difficulties are to extract features to match a correct pair of different sets and also preserve two types of exchangeability required for set-to-set matching: the pair of …
New set-valued star-shaped risk measures introduced for better risk assessment.
problem Improving risk assessment in financial contexts.
method Developed new set-valued star-shaped risk measures and proved their representation theorems.
result Set-valued star-shaped risk measures can be represented as unions of set-valued convex risk measures.
We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an exa…
Study dynamics and topology of flows near non-saddle sets or W-sets.
problem Understanding the dynamics and topology of flows near specific invariant sets.
method Cohomological relations and global properties analysis.
result Dynamical classification of surfaces and robustness of non-saddle-sets.
The study explores mapping degree sets and their properties for manifolds.
problem Understanding the structure and properties of mapping degree sets for manifolds.
method Analyzes the properties of mapping degree sets and their relationships with self-mapping degree sets.
result Not every multiplicative set containing 0,1 is a self-mapping degree set.
Current approaches for predicting sets from feature vectors ignore the unordered nature of sets and suffer from discontinuity issues as a result. We propose a general model for predicting sets that properly respects the structure of sets and avoids this problem. With a single feature vector as input, we show that our m…
This paper studies the geometry of minimum-volume confidence sets for multinomial parameters.
problem Determining if minimum-volume confidence sets for multinomial outcomes are disjoint.
method Enumerating and covering the continuous regions of the exact p-value function to study the geometry of minimum-volume confidence sets.
result The geometry of minimum-volume confidence sets for multinomial parameters is studied, providing insights into their structure and properties.
Consider a general machine learning setting where the output is a set of labels or sequences. This output set is unordered and its size varies with the input. Whereas multi-label classification methods seem a natural first resort, they are not readily applicable to set-valued outputs because of the growth rate of the o…
Deep Sets approximates functions on sets with high-dimensional latent space.
problem Modeling functions of sets (permutation-invariant functions).
method Deep Sets, a method known to be a universal approximator for continuous set functions.
result Deep Sets' universal approximation property is only guaranteed with a sufficiently high-dimensional latent space.
Study online learning with set-valued feedback, showing differences between deterministic and randomized approaches.
problem Online learning with set-valued feedback, where labels are sets rather than single labels.
method Introduced new combinatorial dimensions (Set Littlestone and Measure Shattering) to characterize learnability.
result Characterized deterministic and randomized online learnability, and established bounds for various learning settings.
A stability-based method selects the most desirable conformal prediction set.
problem Selecting the most desirable conformal prediction set from multiple valid sets invalidates coverage guarantees.
method A stability-based approach that ensures coverage for the selected prediction set.
result The stability-based approach maintains coverage guarantees for the selected prediction set.
The paper links set cuspidality to function regularity and flatness.
problem Linking set cuspidality to function regularity and flatness.
method Analyzes arc-smooth functions and their properties on various sets.
result Establishes a precise link between set cuspidality and function regularity.
This work establishes properties on diffeological structures for set-valued maps and measures.
problem Establish rigorous properties on diffeological structures for set-valued maps and measures.
method Using diffeologies, the authors link various structures including set-valued maps, relations, gradients, measures, and shape analysis.
result Established rigorous properties on sample diffeologies.
Causal Set Theory's Hauptvermutung is resolved in two ways, one of which is true.
problem Formulating and resolving the Hauptvermutung in Causal Set Theory.
method Two mathematically well-defined formulations of the Hauptvermutung, one of which is true.
result The Hauptvermutung is true when finite sets are replaced by countable sets.
This letter introduces an abstract learning problem called the "set embedding": The objective is to map sets into probability distributions so as to lose less information. We relate set union and intersection operations with corresponding interpolations of probability distributions. We also demonstrate a preliminary so…
Find limiting sets for digital cones and suspensions.
problem Digital topology cone and suspension constructions.
method Identify (m, n)-limiting sets, especially (0, 0)-freezing sets.
result Discover (0, 0)-limiting sets for digital cones and suspensions.
Analytic sets with unique infinite tangent cone are algebraic.
problem Characterizing analytic sets with unique infinite tangent cones.
method Analytic and algebraic set properties, degree of complex algebraic sets.
result Degree of Lipschitz normally embedded sets equals their infinite tangent cone degree.
Study freezing sets for digital images in a 2D grid.
problem Determine minimal freezing sets for digital images.
method Prove methods to obtain freezing sets for digital images (X, c_i) where X is a subset of Z^2.
result Examples show how methods can lead to the determination of minimal freezing sets.
Fuzzy prediction sets generalize binary predictions to include elements at varying confidence levels.
problem Binary prediction sets are limited; fuzzy prediction sets offer richer guarantees.
method Generalize prediction sets to fuzzy sets, showing they are e-values with merging properties.
result Optimal e-values lead to optimal fuzzy prediction sets, including optimal conformal prediction.
Representations of sets are challenging to learn because operations on sets should be permutation-invariant. To this end, we propose a Permutation-Optimisation module that learns how to permute a set end-to-end. The permuted set can be further processed to learn a permutation-invariant representation of that set, avoid…
Develops deep neural network techniques for sets as input and output.
problem Bottlenecks in set representation and discontinuity issues in set prediction.
method Techniques for set representation and prediction, addressing unordered nature and relations.
result Improvements in set prediction and representation across various experiments.
Proves a theorem for Assouad dimension with applications to distance sets and radial projections.
problem Problems related to Assouad dimension and distance sets.
method General nonlinear projection theorem for Assouad dimension.
result Sharp estimates for sets with Assouad dimension less than 1 and exceptional set estimates.
The paper explores connections between perimeter, area, and visual angle of convex sets.
problem Understanding geometric properties of convex sets through visual angle and related measurements.
method Establishing universal formulas and characterizing convex sets of constant width.
result Crofton's formula is the unique universal formula relating visual angle, length, and area.
The paper defines cyclic sets from ribbon string links and connects them to quantum invariants.
problem Defining and relating cyclic sets from ribbon string links.
method Endowing ribbon string links with cyclic and cocyclic structures, relating to coend of a ribbon category via quantum invariants.
result Established a relationship between ribbon string links and quantum invariants.
New tools for constructing fixed point sets in digital topology.
problem Constructing fixed point sets in digital topology.
method Defining excludable points and articulation points, and showing their exclusion from freezing sets.
result Excludable points and articulation points can be excluded from all freezing sets.
Set risk measures extend traditional risk measures to handle sets of positions.
problem Handling sets of positions with a single capital requirement.
method Developed an axiomatic framework for set risk measures, dual representation through topology and measures.
result Characterized worst-case set risk measures and provided examples.
Study contractibility of boundaries in convex sets and limit sets of subgroups.
problem Understanding contractibility of boundaries and wildness of limit sets in geometric structures.
method Use sufficient conditions for contractibility, study coarse upper curvature bounds, and analyze interpolation in geodesic metric spaces.
result Conditions for contractibility of boundaries and properties of limit sets are established.
Set classification problems arise when classification tasks are based on sets of observations as opposed to individual observations. In set classification, a classification rule is trained with N sets of observations, where each set is labeled with class information, and the prediction of a class label is performed a…
Building machine translation (MT) test sets is a relatively expensive task. As MT becomes increasingly desired for more and more language pairs and more and more domains, it becomes necessary to build test sets for each case. In this paper, we investigate using Amazon's Mechanical Turk (MTurk) to make MT test sets chea…
New algorithms approximate Rashomon set for sparse models, aiding expert interaction.
problem Lack of interaction between models and domain experts in classical machine learning.
method Approximate Rashomon set of sparse, generalized additive models using ellipsoids.
result Efficiently approximated Rashomon set facilitates model selection and exploration.
This note fixes a small gap in Kerckhoff's proof that the limit set of the handlebody set has measure zero.
Minimal generating sets of Reidemeister moves identified and classified.
problem Classifying minimal generating sets of Reidemeister moves.
method Determined minimal generating sets, provided classifications, and identified candidates.
result 12 out of 16 candidates for minimal generating sets were proven minimal.
In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance betwee…