Study shows how metric space fundamental groups can be approximated by discrete groups.
problem Understanding fundamental groups of metric spaces through discrete approximations.
method Investigates the existence of isomorphisms between metric space fundamental groups and their discrete limits.
result Mild conditions are sufficient to establish an isomorphism between metric space fundamental groups and their discrete limits.
Study shows different fundamental groups for manifolds with same limit.
problem Understanding fundamental groups of manifolds with non-negative Ricci curvature.
method Constructed sequences of manifolds with specific properties.
result Found manifolds with same limit but different fundamental groups.
Estimating data limits easier than building algorithms to achieve them.
problem Achieving the fundamental limits of data processing.
method Case studies on binary classification, data compression, and prediction.
result Estimators of limits can be constructed with fewer samples than explicit algorithms to achieve limits.
The paper sets limits on prediction accuracy and generalization.
problem Fundamental limits of prediction accuracy and generalization.
method Combining entropic analysis and innovations approach.
result Conditions for achieving prediction error bounds.
Study shows a central limit theorem for random coverings of manifolds with nilpotent groups.
problem Understanding the distribution of connected components in random coverings of manifolds with nilpotent fundamental groups.
method Used sampling homomorphisms from the fundamental group into the symmetric group and subgroup growth zeta functions of nilpotent groups.
result Proved a central limit theorem for the number of connected components of these random coverings.
This work characterizes the fundamental limit of network pruning using statistical dimension and convex geometry.
problem The fundamental limit of network pruning is still lacking, especially for deep neural networks.
method Directly imposing sparsity constraint on the loss function and using statistical dimension in convex geometry.
result Characterizes the sharp phase transition point as the fundamental limit of pruning ratio.
The paper uses information theory to find limits of feedback control systems.
problem Fundamental performance limitations of feedback control systems.
method Utilizes information theory to derive Lp bounds on control error. result Bounds are characterized by the conditional entropy of the disturbance.
The paper sets limits for sequential prediction and recursive algorithms using entropy analysis.
problem Fundamental limitations in sequential prediction and recursive algorithms.
method Entropic analysis to investigate underlying relationships of data and noises.
result Derives Lp bounds quantifiable in conditional entropy. Study on matching nodes between graphs to preserve edges, focusing on limits and algorithms.
problem Matching nodes between graphs to preserve most edges, especially in random graphs.
method Investigates fundamental limits and designs algorithms to recover alignments in planted graphs.
result High probability guarantees on the success or failure of graph alignment algorithms.
Paper explores limits of exact inference in structured prediction models.
problem Exact recovery of true labels in graph-based structured prediction models.
method Analyzes necessary and sufficient conditions for exact recovery using maximum likelihood estimation.
result Derives tight conditions for exact recovery, revealing a gap with computationally tractable methods.
Study on the limit set of spherical CR uniformization of cusped hyperbolic manifolds.
problem Understanding the limit set of spherical CR uniformization of cusped hyperbolic manifolds.
method Analyzes the limit set as a closure of a countable union of R-circles, proving properties and structure. result Proves the limit set is connected and contains a Hopf link with three components, and the fundamental group of its complement is not finitely generated.
Characterizes fundamental groups of disjointly tree-graded spaces.
problem Understanding fundamental groups of complex geometric structures.
method Defines and analyzes disjointly tree-graded spaces, characterizing their fundamental groups.
result Fundamental groups of disjointly tree-graded spaces embed into inverse limits of free products of fundamental groups of pieces.
Study on the limits of projective special real manifolds and their symmetries.
problem Understanding the limits of projective special real manifolds.
method Evolution of defining polynomial and centro-affine fundamental form along curves.
result Found a list of possible limit geometries and a lower bound for symmetry groups.
A study shows a limit on the dimension of certain 4-manifolds.
problem Understanding the dimension of specific 4-manifolds with non-spin property.
method Analyzing 4-manifolds with residually finite fundamental groups and non-spin universal coverings.
result Proves a dimension limit for these 4-manifolds.
Proves fundamental theorem for singular surfaces with limiting tangent planes.
problem Extending classical differential geometry to singular surfaces.
method Characterizes singular surfaces and their fundamental forms, introduces new curvatures.
result Characterizes wave fronts and introduces new types of curvatures.
We study the limits and methods of training two-layer autoencoders.
problem Understanding the limits and methods of training two-layer autoencoders.
method Focus on non-linear two-layer autoencoders trained in the proportional regime, using gradient methods.
result Gradient methods achieve the minimizers of the population risk and reveal the structure of the features.
Study of limits of Cantor and \sier sets, showing homeomorphic spaces.
problem Understanding topological properties of certain geometric structures.
method Analyzing direct limits of embedded Cantor sets and \sier curves, showing homeomorphism.
result Morse boundaries of specific groups are homeomorphic to limits of Cantor and \sier sets.
This paper sets fundamental limits for rank-one matrix estimation with varying noise levels.
problem Estimating a rank-one matrix from Gaussian observations with different noise levels across blocks.
method Novel reduction from heterogeneous noise to homogeneous noise, proving asymptotic error bounds.
result Asymptotically exact formulas for minimum mean-squared error in estimating rank-one matrix and factors.
Belief Propagation optimally infers true labels from crowdsourced data.
problem Inferring true labels from crowdsourced data with errors.
method Introduced tighter lower bound and proved BP matches it.
result Belief Propagation optimally infers true labels.
In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a regular domain G of R^3. We prove that the second fundamental form of the boundary surface of G is nonnegatively defined at every point of the limit set of such immersions.
Paper proposes MMC to avoid high-density bias in clustering.
problem High-density bias in density-based clustering.
method Introduces mass distribution as a better foundation for clustering, proposing mass-maximization clustering (MMC).
result MMC avoids high-density bias and discovers clusters of arbitrary shapes, sizes, and densities.
New insights into high-dimensional inference limits and optimal algorithms.
problem Fundamental limits on parameter inference in high-dimensional datasets.
method Statistical physics of quenched disorder to analyze high-dimensional inference.
result Optimal algorithms achieve fundamental limits in high dimensions, simpler than MAP and ML.
The study finds surface subgroups in specific types of groups.
problem Finding surface subgroups in certain groups.
method Analyzing graph pairs and using properties of fundamental groups and limit groups.
result Surface subgroups found in graph pairs and limit groups.
When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give…
Study on limits of detecting a rank-one perturbation in Wigner matrices.
problem Detecting an additive rank-one perturbation in Wigner matrices.
method Gaussian interpolation methods and rigorous incarnation of the cavity method.
result Established the maximal region of contiguity between planted and null models, marking a phase transition for both estimation and detection.
A new method combines simple binary classifiers to build complex multiclass classifiers, achieving performance limits in a Gaussian setting.
problem Building a sophisticated multiclass classifier from simple binary decisions.
method Combining O(logK) simple binary classifiers to form a K-class classifier. result Explicit performance bounds across various decoding and dimensional regimes for a stylized Gaussian setting.
New research limits how well attackers can guess if data points were in a model's training set.
problem Revealing membership of data points in machine learning models.
method Theoretical analysis of statistical limits for membership inference attacks.
result The effectiveness of membership inference attacks is limited by a constant that quantifies data distribution diversity.
The fundamental group of a hyperbolic manifold acts on the limit set, giving rise to a cross-product C^* algebra. We construct nontrivial K-cycles for the cross-product algebra, thereby extending some results of Connes and Sullivan to higher dimensions. We also show how the Patterson-Sullivan measure on the limit set c…
Minimal surfaces in spherical space forms have limited genus.
problem Understanding minimal surfaces in spherical space forms.
method Analyzing orientable index one minimal surfaces with large fundamental groups.
result Surfaces have genus at most two, confirming a conjecture.
In this note we establish several versions of a compactness theorem for submanifolds. In particular we require only bounds on the second fundamental form and do not assume volume or diameter bounds. As an application we prove a compactness theorem for mean curvature flows and use it to construct smooth blow-up limits a…
The paper studies random covers of torus knot complements and their statistical properties.
problem Understanding the statistical behavior of finite covers of torus knot complements.
method Asymptotic subgroup growth analysis and Benjamini-Schramm limit theorems.
result Determination of the linear growth rate of Betti numbers for random covers of torus knot complements.
Turnover-adjusted IR is always lower than classic IR, suggesting managers can improve performance by limiting turnover.
problem The classic relationship between IR and its determinants does not account for turnover costs.
method Mathematical derivations and simulations considering volatility of information coefficient and portfolio turnover.
result Turnover-adjusted IR is lower and managers can improve performance by limiting turnover.
Shannon's theory sets limits on information transmission.
problem Limits of information transmission in systems.
method Mathematical theory of communication.
result Defines fundamental limits on information transmission.
Survey of network analysis limits and optimal methods.
problem Graphon estimation, community detection, and hypothesis testing.
method Review of minimax optimal rates and procedures.
result Optimal algorithms for network analysis.
Esnault asked whether every smooth complex projective variety with infinite fundamental group has a nonzero symmetric differential (a section of a symmetric power of the cotangent bundle). In a sense, this would mean that every variety with infinite fundamental group has some nonpositive curvature. We show that the ans…
The paper sets fundamental limits for ERM in high dimensions.
problem Understanding statistical accuracy of ERM in high-dimensional settings.
method Sharp performance characterizations and tight lower bounds derived for generalized linear models.
result Optimal tuning of loss function and regularization parameter.
We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's pr…
Endowed with natural topologies, the fundamental group of the Hawaiian earring continuously injects into the inverse limit of free groups. This note shows the injection fails to have a continuous inverse. Such a phenomenon was unexpected and appears to contradict results of another author.
Paper shows adversarial classification algorithms are inherently more sensitive to data manipulation.
problem Inadequate provable guarantees for machine learning performance, especially in unreliable data environments.
method Formal analysis of binary classification algorithms' sensitivity to adversarial manipulation.
result Fundamental tradeoff curve between accuracy and sensitivity is determined by data statistics, not algorithm tuning.
This research sets limits on robust learning from data with errors.
problem Learning from data with malicious errors and outliers.
method Lower bounds on communication and space complexities for robust learning.
result Gaining robustness in learning usually increases complexity.
The paper characterizes the efficiency of transferring knowledge from a teacher to a student classifier over finite domains.
problem Characterizing the statistical efficiency of knowledge transfer over finite domains.
method Three progressive levels of privileged information: hard labels, teacher probabilities, and soft labels. Novel empirical loss functions used to achieve the fundamental limits.
result Achieving the fundamental limits of knowledge transfer through specific levels of privileged information and novel loss functions.
By using Thurston's bending construction we obtain a sequence of faithful discrete representations ρ_n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic space H^4. The algebraic limit of ρ_n contains a finitely generated subgroup F whos…
This work examines fundamental limits in model falsification without assuming specific distributions.
problem Establishing lower bounds on model class risk in distribution-free settings.
method Model-agnostic fundamental hardness result for constructing lower bounds on test error.
result No positive lower bound on model class risk is possible in certain settings.
In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in Bk×(0,T) to the minimal fundamental solut…
Study finds exact limits for sparse regression with fewer observations than usual.
problem Understanding sparse linear regression with sublinear sparsity.
method Adaptive interpolation method and modified AMP algorithm.
result Exact asymptotic expressions for mutual information and MMSE in sublinear sparsity.
Paper models and compresses wideband CSI feedback in FDD MIMO systems.
problem Fundamental limits of channel state information (CSI) feedback in FDD massive MIMO systems.
method Modeling CSI as a Gaussian-mixture source with latent geometry states, proposing Gaussian-mixture transform coding (GMTC).
result Near-optimal CSI compression achieved through state-adaptive transform coding without large neural encoders.
Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.
problem Proving a fundamental gap lower bound for horoconvex domains in hyperbolic space.
method Reduces the problem to a radial-height problem, compares Dirichlet forms with angular operators, and uses Green estimates.
result Establishes a polynomial \(D^{-3}\) scale fundamental gap lower bound.
We prove that any complete hyperbolic 3--manifold with finitely generated fundamental group, with a single topological end, and which embeds into $\BS^3$ is the geometric limit of a sequence of hyperbolic knot complements in $\BS^3$. In particular, we derive the existence of hyperbolic knot complements which contain ba…