Eigen-stratified models reduce model size and improve performance.
problem Large model size in Laplacian-regularized stratified models.
method Formulate eigen-stratified models with linear combinations of bottom eigenvectors of the graph Laplacian.
result Significant reduction in model size with eigen-stratified models.
The paper addresses bias in stratified classification models and proposes a calibration method.
problem Stratified sampling introduces bias in classification models, affecting ranking performance.
method Developed an analytical solution to optimize ROC curve for stratified sampling bias.
result The proposed ranking algorithm effectively addresses stratified sampling bias in classification models.
The paper studies control data and equivariant properties of stratified spaces.
problem Equivariant properties of stratified spaces with group actions.
method Equivariant submersion theorem and control data construction.
result Inclusion of G-stratified subspaces is a G-equivariant cofibration. Efficiently fits stratified models with Laplacian regularization for categorical features.
problem Traditional stratified models fit separate models for each categorical value, which can be inefficient and lack flexibility.
method Proposes a distributed ADMM method for fitting stratified models with Laplacian regularization.
result Improves model performance and allows fitting for categorical values not in training data.
The paper introduces knot invariants using stratified homotopy groups.
problem Defining knot invariants in stratified spaces.
method Introducing stratified homotopy groups and proving their properties.
result Stratified Whitehead's theorem holds for these groups.
Paper classifies pseudomanifolds over stratified spaces.
problem Classifying pseudomanifolds over stratified spaces.
method Introducing locally standard T-pseudomanifolds and using characteristic data. result Locally standard T-pseudomanifolds over topological stratified pseudomanifolds are classified by their characteristic data. Control data constructed for smooth weak deformation retraction of stratified spaces.
problem Construct control data for smooth weak deformation retraction of stratified spaces.
method Show smooth local triviality with conical fibers, construct control data, use fiber-wise scalar multiplications.
result Obtain neighbourhood smooth weak deformation retraction of stratified spaces.
StratPPI improves prediction-powered inference with stratified sampling.
problem Improving statistical estimates with limited human-labeled data.
method Combining small human-labeled data with large automatic-labeled data, stratifying data for tighter confidence intervals.
result StratPPI provides substantially tighter confidence intervals than unstratified approaches.
Study of hyperkahler spaces from Lie algebras.
problem Understanding hyperkahler quotients of cotangent bundles.
method Analyzing stratified spaces of cotangent bundles of complex semisimple Lie groups.
result Explicit description of partial order on strata using Lie theoretic data.
Paper introduces stratified vector bundles and their properties.
problem Understanding singular spaces and their vector bundles.
method Characterization via monoid actions and examples from various fields.
result Functorial properties extended to the stratified case.
Extends h-principle to stratified spaces using sheaf and jet theories.
problem Applying h-principle to stratified spaces.
method Developed new sheaf and bundle theories for stratified spaces, and proved the h-principle.
result Stratified continuous sheaves and homotopy fiber sheaves lead to the parametric h-principle.
Symplectic embedding extended to stratified spaces.
problem Symplectic embedding theorem for stratified spaces.
method Defined symplectic structure on stratified spaces, demonstrated embedding in complex projective space.
result Symplectic embedding theorem extended to stratified spaces.
We derive spectral sequences for the intersection homology of stratified fibrations and approximate tubular neighborhoods in manifold stratified spaces. These neighborhoods include regular neighborhoods in PL stratified spaces.
Study minimax estimation of stratified structure from i.i.d. samples.
problem Estimating stratified structure from i.i.d. samples of stratified mixtures of immersed manifolds.
method Ascending hierarchical co-detection of points belonging to different layers, identifying number of layers and their dimensions, assigning points to layers accurately, estimating tangent spaces optimally.
result Achieves optimal estimation of mixture components at their optimal dimension-specific rates adaptively.
Two algorithms reduce label complexity in machine learning using stratified sampling.
problem Reducing the number of true labels needed for machine learning evaluation.
method Proposes two algorithms that estimate strata properties and optimize evaluation accuracy.
result Demonstrates algorithms are rate optimal and reduce label complexity.
Develops deep generative models for stratified learning.
problem Challenges in learning distributions on stratified spaces with varying dimensions and singularities.
method Two generative frameworks: sieve maximum likelihood and diffusion-based.
result Establishes convergence rates and consistency for estimating intrinsic dimensions and number of strata.
Paper proves Whitney stratified spaces can be given a conically smooth structure.
problem Proving Whitney stratified spaces can be given a conically smooth structure.
method Introduced conically smooth structure by Ayala, Francis, and Tanaka. Proved conjecture that any Whitney stratified space admits a canonical conically smooth structure.
result Established a connection between Whitney stratified spaces and conically smooth spaces.
A new method reduces data valuation variance for more trustworthy data trading.
problem Data valuation and trustworthy data trading in algorithmic prediction.
method Variance reduced Shapley value estimation using stratified sampling.
result VRDS method reduces estimation variance and improves data marketplace development.
New groups contactomorphic to stratified ones found.
problem Understanding contactomorphic relationships between polarised and stratified Lie groups.
method Constructing modifications of stratified groups and proving contactomorphic relationships.
result Polarised groups are contactomorphic to stratified groups under specific conditions.
We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples…
The paper proves symplectic neighbourhood theorems for stratified subspaces.
problem Finding symplectic neighbourhoods of stratified subspaces.
method Analogy with Weinstein's neighbourhood theorem, strong version of Moser's trick, and tubular neighbourhood theorem.
result Generalization of existing constructions for exotic Lagrangians.
The paper establishes conditions for q-parabolicity in stratified pseudomanifolds and related singular spaces.
problem Conditions for q-parabolicity in stratified pseudomanifolds and related singular spaces. method Investigation of sufficient conditions for q-parabolicity in compact Thom-Mather stratified pseudomanifolds with c^-iterated edge metrics. result Established sufficient conditions for q-parabolicity in stratified pseudomanifolds and related singular spaces. In this paper we introduce the notion of a smooth structure on a stratified space, the notion of a Poisson smooth structure and the notion of a weakly symplectic smooth structure on a stratified symplectic space, refining the concept of a stratified symplectic Poisson algebra introduced by Sjamaar and Lerman. We show t…
Sphere theorems extended to RCD spaces and improved for Einstein stratified spaces.
problem Generalizing sphere theorems to new types of spaces.
method Proved sphere theorems for RCD(n-1, n) spaces and Einstein stratified spaces.
result Extended sphere theorems to RCD spaces and improved results for Einstein stratified spaces.
Skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations (under dimension and compactness assumptions). In general, the stratified approximate fibrations cannot be replaced by bundles, and the teardrops cannot be replaced by ma…
A new method for finding significant patterns in stratified data.
problem Statistical challenges in finding enriched itemsets in stratified data.
method Proposes a strategy and efficient algorithm for significant pattern mining in the presence of categorical covariates.
result Efficiently corrects for multiple testing in pattern mining while retaining statistical power.
Jet spaces on Carnot groups have a canonical Lie group structure.
problem Understanding jet spaces on Carnot groups.
method Constructing jet spaces over stratified Lie groups and showing they are stratified Lie groups.
result Every stratified Lie group of step s+1 can be embedded in a jet space over a stratified Lie group of step s. Paper proves collapsing result for orbifolds without curvature bounds.
problem Proving collapsing result for orbifolds without curvature bounds.
method Introduces weak submersions and stratified Riemannian metrics.
result Allows Gromov-Hausdorff limits of orbifolds with strictly lower dimension.
We show that conically smooth stratified spaces embed fully faithfully into ∞-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each ∞-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions…
Gauss-Green theorem proven for vector fields in stratified groups.
problem Establishing the Gauss-Green theorem for vector fields in noncommutative stratified Lie groups.
method Developed a new family of function spaces for divergence-measure fields and proved the Gauss-Green theorem.
result Gauss-Green theorem achieved for vector fields of low regularity on sets of finite perimeter in stratified groups.
We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the cla…
Characterizes hypergenerated stratified groups with flat boundaries.
problem Characterizing stratified groups with flat boundaries.
method Algebraic characterization and embedding analysis.
result Hypergenerated groups have locally bi-Lipschitz embeddings of non-characteristic hypersurfaces.
Survey of Hodge theory on stratified spaces and related compactifications.
problem Analyzing Hodge theory on complex stratified spaces.
method Resolution of Thom-Mather stratified spaces to manifolds with corners, introduction of mezzoperversity, definition of Cheeger spaces.
result Novikov conjecture verified for Cheeger spaces.
Method estimates multiple related Gaussian distributions using Laplacian regularization.
problem Jointly estimate multiple related zero-mean Gaussian distributions.
method Laplacian regularized stratified model fitting with hyper-parameters to encourage covariance closeness.
result The method performs well, especially in low data regimes, as demonstrated in finance, radar, and weather.
Studied L2−invariants on stratified spaces, proving their stability.
problem Stability of L2−invariants on stratified spaces. method Defined and analyzed L2-Betti numbers and Novikov-Shubin invariants for compact smoothly stratified pseudo-manifolds with a wedge metric, extending results to these pseudo-manifolds. result Invariance of L2-Betti numbers and Novikov-Shubin invariants under smoothly stratified, strongly stratum preserving homotopy equivalence. SBSS uses similarity to split data for better classifier training.
problem Training better classifiers with realistic performance estimation.
method SBSS uses both input and output space information to split data using similarity functions.
result SBSS outperformed ordinary stratified 10-fold cross-validation in 75% of scenarios.
Analyzes solutions of stratified Lie systems and their geometric structures.
problem Nonautonomous systems of differential equations on manifolds.
method Analyzes particular solutions and properties of stratified Lie systems.
result Generalizes properties of Lie systems to stratified Lie systems.
Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the loc…
The study examines filling functions and divergence functions for stratified nilpotent Lie groups.
problem Analyzing the behavior of isoperimetric inequalities in stratified nilpotent Lie groups.
method Developed algebraic conditions to prove growth rates of filling functions and divergence functions.
result For stratified nilpotent Lie groups, filling functions grow as fast as Euclidean space in low dimensions and slower in high dimensions.
Differentiable groupoids and their inertia spaces are studied with a de Rham theorem.
problem Understanding the de Rham cohomology of inertia spaces.
method Introducing differentiable stratified groupoids and proving a de Rham theorem.
result Inertia groupoids of proper Lie groupoids are locally contractible differentiable stratified groupoids.
A new algorithm SSAG reduces gradient variance to achieve linear convergence in large-scale optimization.
problem Achieving linear convergence rate in large-scale optimization problems due to gradient variance.
method Introduces SSAG, a novel algorithm that combines stratified sampling and averaging over iterations to reduce gradient variance.
result SSAG achieves linear convergence rate of O((1-μ/(8CL))^k) with smaller storage and iterative costs, depending mainly on class variance.
R package stagedtrees learns staged tree structures from data.
problem Learning the structure of staged trees from data.
method Score-based and clustering-based algorithms implemented.
result Illustrated capabilities using two datasets.
We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are us…
A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theor…
Proves Verdier duality for sheaves on stratified spaces.
problem Verdier duality for constructible sheaves on stratified spaces.
method Uses conically smooth stratified spaces and Lurie's Verdier duality.
result Shows equivalence between constructible sheaves and cosheaves.
The study lists low-dimensional stratified groups and their properties.
problem Understanding the algebraic structure of stratified groups.
method Explicitly provided a list of low-dimensional stratified groups and their properties.
result All stratified groups in dimensions up to 7 and some free-nilpotent groups in dimensions up to 14 were studied.
In a previous article the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPh…
The paper proves a rigidity theorem for stratified spaces with positive Ricci curvature.
problem Proving a rigidity theorem for stratified spaces with positive Ricci curvature.
method Analyzing stratified spaces with positive Ricci lower bounds and no cone angles larger than 2π.
result The first non-zero eigenvalue of the Laplacian is equal to the dimension if and only if the stratified space is isometric to a spherical suspension.