Universal Bayes consistency proved in metric spaces.
problem Proving universal Bayes consistency in metric spaces.
method Extending a multiclass learning algorithm and proving its Bayes-consistency in all metric spaces.
result First learning algorithm universally strongly Bayes-consistent in all metric spaces.
New graph types help identify complex relationships.
problem Understanding complex relationships in data.
method Introducing separable and essentially separable graphs to characterize and identify graphical models.
result Developed algorithms to identify equivalence classes of essentially separable graphs.
Characterizes Bayesian networks up to unconditional equivalence.
problem Characterizing Bayesian networks up to unconditional equivalence.
method Transformational characterization via undirected graphs and specified moves.
result Two DAGs are in the same UEC if and only if one can be transformed into the other via a finite sequence of moves.
Study shows constraints on slopes for knot manifolds with specific tori.
problem Constraints on slopes for knot manifolds containing essential twice-punctured tori.
method Analyzes four cases of essential twice-punctured tori in hyperbolic knot manifolds and determines slopes distances.
result Distance between slopes is ≤ 5 unless the knot is figure eight, with bounds realized on infinitely many manifolds.
Study essential diagrams of knots in SgimesS1 and their relation to virtual knots.
problem Understanding essential diagrams and their relation to virtual knots in SgimesS1. method Analyzing knots with minimal double lines and embedding virtual knot theory.
result Virtual knot theory is embedded in the theory of knots in SgimesS1. We construct a hyperbolic 3-manifold M (with ∂M totally geodesic) which contains no essential closed surfaces, but for any even integer g>0 there are infinitely many separating slopes r on ∂M so that M[r], the 3-manifold obtained by attaching 2-handle to M along r, contains an essential…
We consider perturbed quadharmonic operators, Δ4+V, acting on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential V satisfying a bound from below by a non-positive function depending on the distance from a point. Under a bounded geometry assumption on the Hermitian vecto…
Kernel embeddings separate distinct probability distributions, simplifying testing.
problem Testing equality of non-atomic probability distributions.
method Kernel covariance embeddings and Gaussian measures in reproducing kernel Hilbert spaces.
result Testing for singularity between Gaussian measures is equivalent to testing for equality of non-atomic probability distributions.
We construct a small, hyperbolic 3-manifold M such that, for any integer g≥2, there are infinitely many separating slopes r in ∂M so that M(r), the 3-manifold obtained by attaching a 2-handle to M along r, is hyperbolic and contains an essential separating closed surface of genus g. The resu…
Geometrically convex return risk measures on AM-algebras
problem Quantifying risk in time series analysis
method Extending return risk measures to general ordered vector spaces
result Establishing results on finiteness, continuity, separability, and dual and aggregation-based representations
Gradient methods avoid overfitting on separable data.
problem Overfitting on separable data with gradient methods.
method Gradient methods (flow, descent, SGD) on separable data.
result Gradient methods asymptotically do not overfit on separable data.
Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.
problem Estimating the expected max-sliced Wasserstein distance between a probability measure and its empirical distribution.
method Banach space version and operator norm approach for upper bounds.
result Upper bounds for max-sliced Wasserstein distances are essentially matching and sharp up to a log factor.
Tropical SVM tackles phylogenomics by classifying multi-locus data.
problem Classifying multi-locus data sets for phylogenetic analysis.
method Proposes tropical support vector machines (SVMs) for phylogenomics, formulated as linear programming problems.
result Developed methods for hard and soft margin tropical SVMs, proving necessary and sufficient conditions for separation.
Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.
problem Computing shortest non-separating simple closed curves on non-orientable surfaces.
method Developed tools for computing shortest curves, proving NP-hardness and tractability.
result Proved NP-hardness and fixed-parameter tractability for computing shortest orienting curves, and polynomial-time algorithm for non-orienting curves.
The paper connects decision tree interpretability and robustness through separation.
problem Empirical observation of a connection between robustness and interpretability in decision trees.
method Investigation of the connection through decision trees and l∞-perturbation robustness, proving bounds on tree size. result First algorithm with guarantees on robustness, interpretability, and accuracy for decision trees.
Artificial Intelligence (AI) systems sometimes make errors and will make errors in the future, from time to time. These errors are usually unexpected, and can lead to dramatic consequences. Intensive development of AI and its practical applications makes the problem of errors more important. Total re-engineering of the…
Dimensionality reduction techniques play an essential role in data analytics, signal processing and machine learning. Dimensionality reduction is usually performed in a preprocessing stage that is separate from subsequent data analysis, such as clustering or classification. Finding reduced-dimension representations tha…
Paper estimates GMMs with unknown covariances using sparse regularization.
problem Estimating GMMs with unknown diagonal covariances from samples.
method Employed Beurling-LASSO (BLASSO) for sparse estimation of component means, covariances, and weights.
result Established non-asymptotic recovery guarantees with nearly parametric convergence rates.
This work investigates implicit bias in multiclass separable data using a novel geometry-aware optimizer.
problem Understanding implicit bias in overparameterized models on multiclass separable data.
method Introduces NucGD, a geometry-aware optimizer enforcing low-rank structures through nuclear norm constraints.
result NucGD enables scalable training and characterizes the impact of stochastic optimization dynamics.
We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by a result of Heintze and Liu, and associate to such a submanifold M and a point x in M a canonical homogeneous structure (a certain bilinear map on the tangent space). We prove that the homogeneous…
DISCoVeR learns disentangled representations by separating shared and condition-specific factors.
problem Learning disentangled representations for multi-condition data.
method Dual-latent architecture, parallel reconstructions, max-min objective.
result DISCoVeR achieves improved disentanglement on various datasets.
L-space knots lack essential Conway spheres, proven with Floer theory.
problem Essential Conway spheres in L-space knots.
method Floer theoretic invariant for tangles.
result L-space knots have no essential Conway spheres.
Study on diagonal and separating coordinates for symmetric spaces of rank 1.
problem Existence and nonexistence of diagonal and separating coordinates for symmetric spaces of rank 1.
method Generalization of results by Gauduchon and Moroianu, 2020, and analysis of constant sectional curvature and orthogonal separation of variables.
result Diagonal coordinates exist if and only if the symmetric space has constant sectional curvature.
Constructs orthogonal coordinates in curved spaces.
problem Separating variables in curved spaces.
method Explicit construction of orthogonal coordinates and transformations.
result Explicit formulas for Killing tensors and Stäckel matrices.
Proposes a two-step method for sound source separation.
problem Improving sound source separation performance.
method First, learn a latent space transform. Second, train a separation module in the latent space.
result The proposed method achieves better performance than joint learning approaches.
Improves arc separation result for homogeneous spaces.
problem Separating regions in homogeneous spaces by arcs.
method Using homogeneity instead of strong local homogeneity, and considering arcs with one interior point.
result Regions in homogeneous spaces of dimension ≥ 2 are not separated by arcs.
We establish bounds on the KL divergence between two multivariate Gaussian distributions in terms of the Hamming distance between the edge sets of the corresponding graphical models. We show that the KL divergence is bounded below by a constant when the graphs differ by at least one edge; this is essentially the tighte…
The paper studies essential spectra of submanifolds in Euclidean spaces.
problem Investigating the essential spectrum of submanifolds under geometric conditions.
method Analyzing submanifolds in Euclidean spaces with various geometric constraints.
result The essential spectrum of a complete non-compact submanifold is [0,+∞) if the second fundamental form satisfies certain Lp norms. Extends BV functions and divergence-measure fields to metric spaces.
problem Defining BV functions and divergence-measure fields in metric spaces.
method Employing differential structure developed by N. Gigli, extending BV functions and divergence-measure fields to metric spaces.
result Gauss-Green formulas established for BV functions and divergence-measure fields in metric spaces.
The paper classifies hypersurfaces with constant curvature in Euclidean spaces.
problem Classifying separable hypersurfaces with constant sectional curvature.
method Analytical proof and classification of hypersurfaces in Euclidean spaces.
result Hyperspheres are the only separable hypersurfaces with nonzero constant sectional curvature.
Study Poincaré inequality in metric spaces via separating sets.
problem Geometric characterization of Poincaré inequality in metric spaces.
method Properties of separating sets and various notions of energy.
result Equivalence of conditions for 1-Poincaré inequality.
The aim of this paper is to classify the cohomogeneity one conformal actions on the three-dimensional essential Riemannian spaces, up to orbit equivalence. Among other results, the representations of all connected Lie groups acting with cohomogeneity one or zero within the full conformal group of a given three-dimensio…
Sharp conditions link separators to R-trees for space transformations.
problem Conditions for separators to form R-trees under group actions.
method Sharp conditions linking separators to R-trees for space transformations.
result Sharp conditions linking separators to R-trees for space transformations.
Paper tackles online learning on curved spaces without projections.
problem Online learning on Riemannian manifolds with computational constraints.
method Develops projection-free algorithms for geodesically convex optimization.
result Achieves sub-linear regret guarantees in online geodesically convex optimization.
Proves conditions for separating regions in homogeneous spaces without trivial topology.
problem Separating regions in homogeneous, locally compact spaces without trivial topology.
method Analyzes properties of closed subsets and their boundaries in Čech cohomology.
result Conditions for irreducible separation without trivial topology.
Computes expected number of real intersection points of essential variety with random linear spaces.
problem Computing the expected number of real intersection points of the essential variety with random linear spaces.
method Two probability distributions for linear spaces: invariant under orthogonal group action and one motivated from computer vision. Used Monte Carlo simulation for the latter.
result Expected number of real intersection points lies in the interval (3.95 - 0.05, 3.95 + 0.05) with high probability.
We introduce a new metric to evaluate corruption robustness of ML classifiers.
problem Evaluating corruption robustness of machine learning classifiers.
method We propose a test data augmentation method using minimal class separation distance to derive a robustness distance ε and a metric MSCR.
result The MSCR metric allows interpretable comparison of classifier robustness on different datasets.
New statistical measures assess group separability in low-dimensional geometrical spaces.
problem Lack of statistical measures to evaluate group separability in low-dimensional geometrical spaces.
method Proposed three statistical measures (PSI-ROC, PSI-PR, PSI-P) based on Projection Separability rationale.
result Statistical-based measures outperform traditional cluster validity indices in evaluating group separability.
Study submanifolds in hyperbolic space, focusing on their boundary and Laplace operator.
problem Understanding the geometry and regularity of submanifolds in hyperbolic space.
method Analyzing asymptotic geometry and regularity properties near the ideal boundary, computing essential spectra.
result Computed essential spectra of the Laplace operator on certain submanifolds.
Existing depth separation results for constant-depth networks essentially show that certain radial functions in Rd, which can be easily approximated with depth 3 networks, cannot be approximated by depth 2 networks, even up to constant accuracy, unless their size is exponential in d. However, the func…
We prove that the 8^4_2 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V_8 of the ideal regular octahedron, i.e. 7.32... = 2V_8. The proof relies on Agol's argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also nee…
Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.
problem Topology of collapsed spaces with lower Ricci bounds
method Prove that collapsed spaces are topological surfaces
result Collapsed spaces are topological surfaces
In the present paper, we consider the family of all compact Alexandrov spaces with curvature bound below having a definite upper diameter bound of a fixed dimension. We introduce the notion of essential coverings by contractible metric balls, and provide a uniform bound on the numbers of contractible metric balls formi…
Study on minimal hypersurfaces in a special normed space.
problem Characterizing minimal hypersurfaces in a specific normed space.
method Investigate translation and separable minimal hypersurfaces.
result New insights into the properties of minimal hypersurfaces.
The paper extends optimal transport for linear separability of sheared distributions in supervised learning.
problem Learning on the space of probability measures using shifts and scalings.
method Embedding probability measures into L2 spaces using optimal transport, then applying regular machine learning techniques. result Sheared distributions can be linearly separated under certain conditions, with bounds on transformations.
Classifies zero mean curvature surfaces in Lorentz-Minkowski space.
problem Classifying surfaces with zero mean curvature.
method Using separable surface equations and constructing examples.
result All zero mean curvature surfaces of separable type have been classified.
In this article we prove a generalization of Weyl's criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for its essential spectrum.
Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.
problem Characterize essential surfaces in Seifert fiber spaces with singular surfaces.
method Extends Frohman and Rannard's approach to handle surfaces with singular fibers.
result Characterizes essential surfaces in Seifert fiber spaces with singular surfaces.