Characterizes concept classes for optimistic online learning.
problem Understanding minimal assumptions for online learnability.
method Investigates two questions about concept classes' learnability.
result Characterizes all concept classes for optimistically universal online learnability.
Theory extends optimal learning rates without realizability assumption.
problem Agnostic binary classification without realizability assumption.
method Identifies tetrachotomy of optimal rates and combinatorial structures.
result Optimal universal rates for binary classification in agnostic setting.
Improved multi-group learning with group-realizable concepts.
problem Enhancing multi-group learning efficiency.
method Empirical risk minimization over group-realizable concepts.
result Improved sample complexity in group-realizable settings.
New algorithm learns efficiently with a simple 'yes/no' oracle.
problem Can efficient learning be achieved with a simpler oracle than ERM?
method Developed an oracle that returns 'yes' or 'no' for realizable datasets.
result Learnability is possible with a polynomial price in VC dimension.
Study expands multiclass classification models with new rates and partial concept classes.
problem Multiclass classification with a bounded number of labels under various conditions.
method Extends traditional PAC model to distribution-dependent and data-dependent learning rates, characterizes optimal rates for universal and partial concept classes.
result Characterizes three types of learning rates (exponential, linear, arbitrarily slow) for fixed distributions and complexity measures for partial concept classes.
Improved private agnostic learning with near-optimal sample complexity.
problem Private agnostic learning with arbitrary privacy parameters.
method Near-optimal sample complexity construction.
result Near-optimal extra sample complexity of \(\widetilde{O}(\mathrm{VC}(\mathcal{C})/α^2)\) for any \(\varepsilon \leq 1\).
This paper studies universal rates of ERM for binary classification under agnostic learning.
problem The challenge of achieving universal rates of ERM for binary classification under agnostic learning.
method The paper explores the agnostic universal rates of ERM for binary classification, revealing three possible rates: e−n, o(n−1/2), or arbitrarily slow. result The paper provides a complete characterization of which concept classes fall into each of the three categories of agnostic universal rates.
To date, the tightest upper and lower-bounds for the active learning of general concept classes have been in terms of a parameter of the learning problem called the splitting index. We provide, for the first time, an efficient algorithm that is able to realize this upper bound, and we empirically demonstrate its good p…
Contradiction graphs reveal VC dimension threshold.
problem Determining VC dimension of concept classes.
method Study contradiction graphs of binary concept classes.
result Single contradiction graph Gm(H) determines VC dimension. Paper explores universal rates of ERM in machine learning.
problem Understanding universal learning rates for ERM.
method Analyzes realizable concept classes and ERM principles.
result Four possible universal learning rates by ERM.
Study on tradeoffs between mistakes and ERM oracle calls in online and transductive learning.
problem Analyzing online and transductive learning with limited ERM and weak consistency oracle access.
method Proves lower bounds and upper bounds on mistakes and oracle calls, considering realizable and agnostic cases.
result Achieves optimal mistake bounds with weak consistency queries for certain concept classes.
While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Ma…
Incorrect parity-based descriptions of realizable Gauss diagrams found, but bipartite graphs provide a valid approach.
problem Incorrect descriptions of realizable Gauss diagrams using parity conditions.
method Used bipartite graphs to describe realizable Gauss diagrams.
result Realizable Gauss diagrams can be accurately described using bipartite graphs.
New findings on null measurability in symmetrization interface of VC learning.
problem Null measurability issues in symmetrization interface of VC learning.
method Formalized in Lean 4, using Choquet capacitability and patching properties.
result Null-measurable bad event not Borel measurable, separating regularity levels.
For a smooth (locally trivial) principal bundle in Ehresmann's sense, the relation between the commuting vertical and horizontal actions of the structural Lie group and the structural Lie groupoid (isomorphisms between vertical fibers) is regarded as a special case of a symmetrical concept of conjugation between "princ…
We give a characterization of conformal classes realizing a compact manifold's Yamabe invariant. This characterization is the analogue of an observation of Nadirashvili for metrics realizing the maximal first eigenvalue, and of Fraser and Schoen for metrics realizing the maximal first Steklov eigenvalue.
Study shows certain mapping class groups cannot be realized as subgroup of homeomorphisms.
problem Proving non-realizability of specific mapping class groups.
method Analyzing compactly supported and full mapping class groups of surfaces with genus 3 or order 6 symmetries.
result Proven non-realizability of mapping class groups for surfaces with genus 3 or order 6 symmetries.
Pricing of high-dimensional options is a deep problem of the Theoretical Financial Mathematics. In this article we present a new class of Lévy driven models of stock markets. In our opinion, any market model should be based on a transparent and intuitively easily acceptable concept. In our case this is a linear system …
Study algebraic obstructions to knot-like complex realizability.
problem Algebraic obstructions to knot-like complex realizability.
method Classification of local equivalence classes over F[U,V]. result Classification answers a question about knot-like complexes.
We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the fundamental class of a manifold. Our main result is an explicit purely combinat…
Extends Nielsen realization to infinite-type surfaces, classifying torsion elements and topological groups.
problem Realizing finite subgroups of mapping class groups on infinite-type surfaces.
method Extending Kerckhoff's result to infinite-type surfaces, using hyperbolic metrics and topological group properties.
result Compact subgroups of mapping class groups are finite, and locally compact subgroups are discrete.
Fitting models for non-Poisson point processes is complicated by the lack of tractable models for much of the data. By using large samples of independent and identically distributed realizations and statistical learning, it is possible to identify absence of fit through finding a classification rule that can efficientl…
New bandit algorithm works without realizability assumption.
problem Contextual bandit problems without realizability assumption.
method Computes a constrained regression problem in every epoch, ensuring similar regret guarantees as realizability-based algorithms.
result Ensures similar regret guarantees as realizability-based algorithms, up to a misspecification term.
In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The paper discuss…
In classical differential geometry, a central question has been whether abstract surfaces with given geometric features can be realized as surfaces in Euclidean space. Inspired by the rich theory of embedded triply periodic minimal surfaces, we seek examples of triply periodic polyhedral surfaces that have an identifia…
Solves Nielsen realization problem for hyper-Kähler manifolds.
problem Realization problem for hyper-Kähler manifolds.
method Uses same invariant as for K3 surfaces and determines representation of mapping class group.
result Representation of mapping class group admits a section on its image for some deformation types.
Paper explores how unsupervised learning can be understood through linear algebra concepts.
problem Understanding unsupervised learning through linear algebra concepts.
method Introducing the concept of linearly independent populations and using them to solve for prevalence values.
result Unsupervised learning can be realized as a generalization of supervised learning.
New active learning framework for multiclass classification beyond realizability assumption.
problem Active learning in non-realizable settings with convex model classes.
method Surrogate risk minimization, epoch-based fitting, aggregation of models.
result Achieves label and sample complexity comparable to prior work in non-realizable settings.
N-discount optimality was introduced as a hierarchical form of policy- and value-function optimality, with Blackwell optimality lying at the top level of the hierarchy Veinott (1969); Blackwell (1962). We formalize notions of myopic discount factors, value functions and policies in terms of Blackwell optimality in MDPs…
Researchers solved a complex problem for a specific type of 4-manifolds.
problem Realizing mapping classes of finite order on del Pezzo surfaces.
method Synthesized results from reflection group theory and 4-manifold topology.
result Provided both positive and negative examples of realizability.
Semi-supervised anomaly detection is an approach to identify anomalies by learning the distribution of normal data. Backpropagation neural networks (i.e., BP-NNs) based approaches have recently drawn attention because of their good generalization capability. In a typical situation, BP-NN-based models are iteratively op…
Study shows realizable learnability doesn't imply agnostic learnability for distributions.
problem Learnability and robustness of distribution classes.
method Analyzes the relationship between learnability and robustness for distribution learning.
result Realizable learnability does not imply agnostic learnability for distributions.
ECBMs unify concept-based interpretations in deep learning models.
problem Suboptimal final accuracy and lack of concept interaction and conditional dependencies.
method ECBMs use a set of neural networks to define joint energy, enabling concept correction and conditional dependency quantification.
result ECBMs achieve higher accuracy and richer concept interpretations compared to state-of-the-art methods.
Study shows non-spin 4-manifolds where smooth Nielsen realization fails.
problem Existence of non-spin 4-manifolds with non-realizable mapping class groups.
method Investigation of multi-twists, projective twists, and multi-reflections.
result Examples of non-spin 4-manifolds with non-realizable mapping class groups.
There is an intrinsic notion of what it means for a contact manifold to be the smooth boundary of a Stein manifold. The same concept has another more extrinsic formulation, which is often used as a convenient working hypothesis. We give a simple proof that the two are equivalent. Moreover it is shown that, even though …
The study of which mapping class group elements can be realized as affine automorphisms of dilation surfaces.
problem Which elements of the mapping class group can be realized as affine automorphisms of dilation surfaces?
method Investigation into the affine automorphism groups of dilation surfaces, including the construction of dilation surfaces from multicurves.
result Only certain types of mapping class group elements can arise as affine automorphisms of dilation surfaces.
Proposes new methods for interpreting document classification models.
problem Interpretation fragility of attention-based neural networks.
method Corpus-level and concept-based explanation methods using attention weights.
result Extracts semantically meaningful keywords and concepts for model predictions.
Study on realizing subgroup twists in 3-manifolds.
problem Realizing subgroups of twist groups in 3-manifolds.
method Analyzing Nielsen realization problem for Twist(M) subgroups, applying to Burnside problem.
result Nontrivial subgroups of Twist(M) are realized by diffeomorphisms if and only if they are cyclic and M is a connected sum of lens spaces.
Directly applies Kazdan--Warner results to prescribe scalar curvature on bundles.
problem Prescribing scalar curvature functions on bundles.
method Direct application of Kazdan--Warner results and variational methods.
result Determines which functions are realizable as scalar curvature functions on bundles.
The Schouten bracket (or antibracket) plays a central role in the Poisson formalism and the Batalin-Vilkovisky quantization of gauge systems. There are several (in)equivalent ways to realize this concept on jet spaces. In this paper, we compare the definitions, examining in what ways they agree or disagree and how they…
The paper introduces a new learning model that explains practical aspects of machine learning.
problem Understanding how quickly a concept class can be learned from examples in practical scenarios.
method Introducing a new learning model that considers fixed data sources and varying number of training examples.
result There are only three possible rates of universal learning: exponential, linear, or arbitrarily slow.
The study identifies latent concepts from diverse observations without assuming specific models.
problem Lack of general theoretical support for concept learning.
method Develops a nonparametric framework for identifying latent concepts from multiple classes of observations.
result Correctness guarantees for concept identification without parametric assumptions.
Let Mod(Sg) denote the mapping class group of the closed orientable surface Sg of genus g≥2, and let f∈Mod(Sg) be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on Sg that realizes f as an isometry. In other words, this procedure yield…
The paper optimizes RV estimation by efficient sampling in time-changed diffusion models.
problem Improving realized variance (RV) estimation in time-changed diffusion models.
method Theoretical analysis and simulations of hitting time and realized business time sampling schemes.
result Realized business time sampling is empirically most efficient for high noise levels.
We consider few-body bound state systems and provide precise definitions of Borromean and Brunnian systems. The initial concepts are more than a hundred years old and originated in mathematical knot-theory as purely geometric considerations. About thirty years ago they were generalized and applied to the binding of sys…
Proofs non-realizability of mapping class group via homeomorphisms, resolves Thurston's conjecture.
problem Non-realizability of mapping class group via homeomorphisms
method Short and elementary proof, rigidity results for actions on Euclidean spaces
result Proof of non-realizability of mapping class group via homeomorphisms
New bounds show simple predictors can learn complex concepts online.
problem When can simple predictors learn complex concepts in online learning?
method Characterized optimal mistake bounds for online learning with simple predictors.
result Achieved nearly optimal mistake bounds for online learning using sparse majority-vote of proper predictors.
Study finds roughness in volatility despite diffusive instantaneous volatility.
problem Determining the roughness of volatility in financial assets.
method Non-parametric method based on normalized p-th variation for estimating roughness of sample paths. result Realized volatility exhibits rough behavior with a significantly smaller Hurst exponent than instantaneous volatility.