The abstract proves the existence of self-Cheeger sets on Riemannian manifolds.
problem Existence of self-Cheeger sets on Riemannian manifolds.
method Perturbations of geodesic balls centered at points on the manifold.
result Existence of a family of self-Cheeger sets on Riemannian manifolds.
Self-Cheeger sets exist on Riemannian manifolds, proven through perturbations of geodesic balls.
problem Existence of self-Cheeger sets on Riemannian manifolds.
method Proving existence through perturbations of geodesic balls centered at points on the manifold.
result Existence of a family of self-Cheeger sets on compact Riemannian manifolds.
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.
New deep learning model for matching sets of items, preserving exchangeability.
problem Matching two different sets of items while preserving exchangeability.
method Exchangeable deep neural networks architecture and efficient training framework.
result Significant improvements in fashion set recommendation and group re-identification.
New set type with no uniformly perfect subsets.
problem Understanding compact sets without uniformly perfect subsets.
method Introduced hereditarily non uniformly perfect sets and compared them with other types of sets.
result Example of a compact set with Hausdorff dimension 2 and positive logarithmic capacity is hereditarily non uniformly perfect.
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.
Maps sets to probability distributions to minimize information loss.
problem Learning to map sets to probability distributions to preserve information.
method Relates set operations to probability distribution interpolations and demonstrates a preliminary solution.
result Experimental results show the effectiveness of the set embedding approach.
Unified framework for generating set-valued outputs.
problem Handling unordered set outputs with varying sizes.
method Sequential Set Generation (SSG) framework.
result SSG outperforms baseline methods in experiments.
New model predicts sets from feature vectors without discontinuity issues.
problem Discontinuity issues in predicting sets from feature vectors.
method General model that respects set structure, auto-encodes point sets, predicts bounding boxes, and attributes.
result Model successfully predicts sets from a single feature vector without discontinuity.
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.
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.
Bayesian optimization for set inputs using approximate set kernels.
problem Permutation-invariant optimization over sets with black-box functions.
method Developed a Bayesian optimization method with set kernel, efficient approximate set kernel, and constrained acquisition function.
result Our method outperforms other methods in numerical experiments.
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.
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.
Model learns set representations through optimized permutations.
problem Challenges in learning set representations due to permutation-invariance.
method Proposes a Permutation-Optimisation module to learn set permutations.
result Achieves state-of-the-art results on various set learning tasks.
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.
MAGIC generates image collages from set templates using attention and set representations.
problem Generating image collages from set templates is challenging for classical models.
method Memory Attentive Generation of Image Collages (MAGIC) using Set-Transformer layers and set-pooling.
result MAGIC can generate image collages from set templates in one forward pass.
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.
The paper explores the equivalence of self-affine sets using a new pseudo-norm.
problem Tackles the equivalence of self-affine sets using a novel approach.
method Constructs a hyperbolic graph and uses a pseudo-norm to prove equivalence.
result Two totally disconnected integral self-affine sets are Lipschitz equivalent if and only if they have the same w-Hausdorff dimension. New methods for signal reconstruction using guiding sets and frame-less pathways.
problem Signal reconstruction in Hilbert spaces with specified properties.
method Axiomatic approach involving sample consistent and guiding sets, with reconstruction set defined as a shortest pathway.
result Existence and uniqueness of reconstruction set in Hilbert space, with derived stability and error bounds.
DMPS learns set data by connecting graph and set learning.
problem Lack of relational learning for set data.
method Deep Message Passing on Sets (DMPS) that connects graph and set learning.
result DMPS achieves competitive or superior results on synthetic and real-world datasets.
We adapted the Covertype data set for unsupervised learning.
problem Lack of suitable unsupervised learning data sets.
method Transformed the Covertype data set into the Wilderness Area data set.
result The Wilderness Area data set is more suitable for unsupervised learning.
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.
Study fixed points in digital images, introducing new invariants.
problem Understanding properties of digital images through fixed points.
method Introduce new invariants and freezing/cold sets to analyze fixed point sets.
result Existence of fixed point sets restricts maps on their complements.
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.
New findings show mapping degree sets can be realized as sums of arithmetic progressions.
problem Realizing mapping degree sets as sums of arithmetic progressions.
method Using multiplication by finite subsets of Z and sets from additive submonoids of Z.
result Every set of the described type occurs as a mapping degree set.
The paper studies Lipschitz equivalence of Cantor sets via polynomial irreducibility.
problem Lipschitz equivalence of Cantor sets.
method Analyzing the irreducibility of polynomials.
result Two Cantor sets are Lipschitz equivalent if their contraction vectors are equivalent.
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.
Set Flow models sets of data, learns dependencies, and achieves state-of-the-art likelihoods.
problem Modeling and sampling from finite, potentially high-dimensional, non-i.i.d. sets of data.
method Extends RealNVPs to handle finite sets, maintaining invertibility and exact log-likelihood evaluation.
result Achieves state-of-the-art likelihoods on 3D point clouds.
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.
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…
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.
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.
Examines manifold cardinality and compares it to structure set.
problem Cardinality of manifold sets and structure sets.
method Study of homotopy equivalence and comparison of cardinalities.
result Cardinality of manifold sets is compared to structure sets.
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…