New tighter bounds for learning algorithms from Steinke & Zakynthinou's supersample setting.
problem Improving generalization bounds for machine learning algorithms.
method Information-theoretic approach using projected loss and Rademacher sequence.
result The new bounds are tighter than previous information-theoretic bounds.
Develops a new framework for analyzing sequential decision-making problems using information theory.
problem Lack of information-theoretic generalization bounds for sequential decision-making problems.
method Introduces a sequential supersample framework that separates learner filtration from proof-side enlargement, controlling the generalization gap by sequential CMI.
result Establishes a sequential CMI that controls the generalization gap in sequential decision-making problems.
The paper establishes bounds for transductive learning using information theory.
problem Transductive learning generalization gap control.
method Information theory, PAC-Bayes, mutual information, conditional mutual information, different information measures.
result Established transductive information-theoretic and PAC-Bayesian bounds.
New bounds derived using conditional f-information for machine learning models.
problem Improving generalization bounds in machine learning.
method Introducing novel information-theoretic generalization bounds via conditional f-information. result Derives generalization bounds applicable to both bounded and unbounded loss functions.
Meta-learning bound uses conditional mutual information.
problem Bounding generalization performance in meta-learning.
method Extends CMI framework to meta-learning with a meta-supersample.
result Explicit bound involving two CMI terms.
Artificial neural network training with stochastic gradient descent can be destabilized by "bad batches" with high losses. This is often problematic for training with small batch sizes, high order loss functions or unstably high learning rates. To stabilize learning, we have developed adaptive learning rate clipping (A…
This paper analyzes multi-view learning using information theory to improve generalization.
problem Lack of theoretical understanding of multi-view learning's generalization behavior.
method Developed information-theoretic generalization bounds for multi-view learning.
result Capturing both consensus and complementary information maximizes representation disentanglement.
This work analyzes generalization in federated learning using information theory.
problem Generalization performance in federated learning is less explored compared to centralized learning.
method The work applies an information-theoretic analysis via the conditional mutual information (CMI) framework to study federated learning's two-level generalization.
result The work derives multiple CMI-based bounds, including hypothesis-based CMI bounds and fast-rate evaluated CMI bounds, which improve convergence rates for specific model aggregation strategies and structured loss functions.
The study sets limits on how well halfspaces can be learned when labels are corrupted.
problem Learning halfspaces in the presence of Massart noise.
method Statistical query (SQ) lower bounds.
result No SQ algorithm can achieve misclassification error better than the corruption rate η with superpolynomial accuracy or a superpolynomial number of queries. Hierarchical Federated Learning bounds generalize using Wasserstein distance.
problem Bounding generalization error in Federated Learning with hierarchical sampling.
method Introduced a hierarchical sampling framework and derived generalization bounds using Wasserstein distance.
result Recover and strictly imply existing CMI bounds for bounded losses.
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.