Study homogeneous Einstein metrics on specific non-Kähler C-spaces.
problem Classify and analyze homogeneous Einstein metrics on non-Kähler C-spaces.
method Use painted Dynkin diagrams and mapping degree theory to classify and find Einstein metrics.
result Existence and classification of invariant Einstein metrics on specific spaces.
New method generates adversarial images under various non-smooth metrics.
problem Adversarial perturbations misclassify deep neural networks.
method Proposes an attack methodology for non-ℓp adversarial dissimilarity metrics. result ProxLogBarrier outperforms existing methods and reveals new perturbation types.
The study finds invariant Einstein metrics on complex Stiefel manifolds and special unitary groups.
problem Existence of invariant Einstein metrics on complex Stiefel manifolds and special unitary groups.
method Decomposing Lie algebras and tangent spaces, parametrizing scalar products, and computing Ricci tensors for invariant metrics.
result Existence of invariant Einstein metrics on specific special unitary groups and complex Stiefel manifolds.
For an infinite cardinal κ let ℓ2(κ) be the linear hull of the standard othonormal base of the Hilbert space ℓ2(κ) of density κ. We prove that a non-separable convex subset X of density κ in a locally convex linear metric space if homeomorphic to the space (i) ℓ2f(κ) if and only if X can be…
We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible $\Ad(K)$-submodules. To this end we apply a new technique which is based on a fibration of a flag manifold over another flag m…
Let σ be the scattering relation on a compact Riemannian manifold M with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let ℓ be the length of that geodesic ray. We study the que…
Researchers analyze geodesic complexity in robot paths on tree graphs.
problem Understanding optimal paths for robots on tree graphs.
method Examined geodesic complexity in ordered and unordered configuration spaces of graphs in ℓ1 and ℓ2 metrics, finding explicit geodesics and families. result Geodesic complexity matches topological complexity in all cases studied.
The paper constructs stable minimal hypersurfaces with specific singularities.
problem Creating minimal hypersurfaces with controlled singularities.
method Constructing hypersurfaces with a given singular set in a modified Euclidean space.
result Embedded minimal hypersurfaces with stable properties and specified singularities.
Igarashi studies (α,β)-metrics in Cartan spaces and finds invariants.
problem Investigating geometric properties of (α,β)-metrics in Cartan spaces. method Introduced (α,β)-metric in Cartan space ℓn and determined invariants. result Determined invariants for two cases of deformed infinite series metric.
New explicit Calabi-Yau metrics and Kähler-Ricci solitons found on complex n-space.
problem Constructing new explicit Calabi-Yau metrics and Kähler-Ricci solitons.
method Continuous (ℓ−1)-parameter family of explicit complete gradient steady Kähler-Ricci solitons on Cn with Hamiltonian 2-forms. result Construction of new complete gradient steady Kähler-Ricci solitons with positive sectional curvature.
The article explores metrics on buildings and symmetric spaces, proving injectivity and proper actions.
problem Injectivity of metrics on buildings and symmetric spaces.
method Analyzing norms, Helly property, and group actions.
result Most classical buildings and symmetric spaces can be endowed with injective metrics.
We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the 2-dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means…
The paper improves ALO for ℓ1-regularized models.
problem Estimating out-of-sample error for ℓ1-regularized models. method Developed a novel theory for ℓ1-regularized problems, bounding ALO error. result For ℓ1-regularized problems, ALO error goes to zero as p goes to infinity. Improved approximation for socially fair clustering with ℓp-objective.
problem Finding a set of centers minimizing the maximum distance to all points in each group.
method Introduced a strengthened LP relaxation with an integrality gap of Θ(loglogℓlogℓ). result Improved approximation algorithm with (eO(p)loglogℓlogℓ)-approximation. The Schwarzian derivative helps classify minimal surfaces by their degree.
problem Classifying minimal surfaces based on their geometric properties.
method Using the Schwarzian derivative, constructing sequences of meromorphic differentials.
result Minimal surfaces can be approximated by sequences of increasing degree.
In this paper, we study the Lévy-Milman concentration phenomenon of 1-Lipschitz maps into infinite dimensional metric spaces. Our main theorem asserts that the concentration to an infinite dimensional ℓp-ball with the ℓq-distance function for 1≤p<q≤+∞ is equivalent to the concentration to the…
For a Riemannian metric g on the two-sphere, let ℓmin(g) be the length of the shortest closed geodesic and ℓmax(g) be the length of the longest simple closed geodesic. We prove that if the curvature of g is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g…
ALℓ0CORE tensor decomposition reduces computational cost for sparse count data.
problem Efficiently decompose sparse count data matrices.
method Probabilistic Tucker decomposition with ℓ0-norm constraint. result ALℓ0CORE achieves similar results to full Tucker decomposition at a fraction of the cost. Suppose M is a non-compact connected smooth n-manifold. Let D(M) denote the group of diffeomorphisms of M endowed with the compact-open C^\infty-topology and D^c(M) denote the subgroup consisting of diffeomorphisms of M with compact support. Let D(M)_0 and D^c(M)_0 be the connected components of id_M in D(M) and D^c(M)…
Solves a special case of the Hurwitz problem for Riemann surfaces.
problem Finding branched covers with specific branch data.
method Analyzes partitions and uses branched cover theory.
result Proves existence of branched covers for compact Riemann surfaces.
Introduces a new geometric framework for probability distributions.
problem Developing a geometric framework for probability distributions.
method Introduces ℓp-information geometry and defines the ℓ2-probability simplex via the q-root transform. result Defines a noncanonical differentiable structure and q-root map as an isometry. Study non-vanishing ℓ2-Betti numbers for specific groups.
problem Calculating non-vanishing ℓ2-Betti numbers for certain groups. method Using Euler characteristics, higher Kazhdan projections, and Baum-Connes assembly map.
result Non-vanishing calculations for delocalised ℓ2-Betti numbers. The paper proves that certain spaces are injective and Helly graphs.
problem Understanding the structure of certain geometric and algebraic spaces.
method Building Helly graphs and injective metric spaces from lattices.
result The natural piecewise ℓ∞ metric on Euclidean buildings and Deligne complexes is injective. New algorithms for differentially private optimization in convex and non-convex settings with near-optimal rates.
problem Differentially private optimization in convex and non-convex settings.
method Developed algorithms for convex and non-convex settings with near-optimal excess population risk.
result Achieved near-optimal rates in near-linear time for convex settings and nearly dimension independent rates for non-convex settings.
Curvature on Kähler toric manifolds
problem Extending curvature formulas to Kähler toric manifolds
method Using Guillemin--Abreu formalism
result Positive holomorphic sectional curvature on certain manifolds
Improved estimation of concentration using half-spaces for adversarial vulnerability.
problem Understanding the concentration of measure phenomenon and its impact on adversarial vulnerability.
method Extending Gaussian Isoperimetric Inequality to non-spherical Gaussian measures and arbitrary ℓ_p-norms, using half-spaces to estimate concentration.
result Proposed method finds tighter intrinsic robustness bounds, providing evidence against concentration as a cause of adversarial vulnerability.
Improved two-sample testing using L1 geometry for analytic kernels.
problem Detecting differences between distributions.
method Use L1 distance between kernel-based distribution representatives to improve testing power. result Better detection of differences between distributions using L1 norm. Ricci-positive manifolds span the kernel of the A^-genus in rational Spin bordism.
problem Rational Spin bordism classes of Ricci-positive manifolds
method Smooth complete intersections of quadrics
result Ricci-positive manifolds span the kernel of the A^-genus New algorithm solves ℓ0-norm constrained multilinear logistic regression for tensor data.
problem Non-convex and nonsmooth ℓ0-norm constraints in multilinear logistic regression. method APALM+ method for globally convergent optimization. result APALM+ ensures convergence to a first-order critical point. The paper provides generalization bounds for metric learning using neural network embeddings.
problem Generalization guarantees for metric learning with neural network embeddings.
method Uniform generalization bounds for two regimes: sparse and bounded amplification.
result Dimension-free generalization bounds can be achieved even without sparsity in solutions.
New definition of Rumin complex for nilpotent Lie groups.
problem No new problem introduced.
method Alternative definition of Rumin complex on nilpotent Lie groups.
result Direct application of ℓq,p cohomology results to all nilpotent Lie groups. In general, the clustering problem is NP-hard, and global optimality cannot be established for non-trivial instances. For high-dimensional data, distance-based methods for clustering or classification face an additional difficulty, the unreliability of distances in very high-dimensional spaces. We propose a distance-ba…
The vanishing of reduced ℓ2-cohomology for amenable groups can be traced to the work of Cheeger & Gromov. The subject matter here is reduced ℓp-cohomology for p∈]1,∞[, particularly its vanishing. Results showing its triviality are obtained, for example: when p∈]1,2] and G is amenable; whe…
Advances robust principal component analysis with transformed ℓ1 regularization.
problem Recovering low-rank structures from noisy, partially observed data corrupted by sparse outliers.
method Proposes transformed ℓ1 (TL1) regularization to improve approximations of rank and ℓ0 functional.
result Achieves higher accuracy in estimating low-rank and sparse components compared to classical convex models, especially under non-uniform sampling schemes.
Researchers solved the even Lp-Minkowski problem under curvature pinching.
problem Solving the even Lp-Minkowski problem under curvature pinching. method Anisotropic Riemannian metric comparison and anisotropic curvature analysis.
result The even Lp-Minkowski inequality and uniqueness are proven for all p≥pγ. New framework improves adversarial robustness certification for various perturbations.
problem Certifying robustness against adversarial attacks in deep learning models.
method Unified functional optimization approach with non-Gaussian smoothing noise for multiple types of attacks.
result Achieves better certification results and identifies key trade-offs between accuracy and robustness.
This paper assesses Gaussian and Exponential mechanisms for certifying adversarial robustness.
problem Certifying adversarial robustness using randomized smoothing mechanisms.
method Proposes a generic framework to assess the appropriateness of randomized smoothing mechanisms.
result Gaussian mechanism is an appropriate option for certifying both ℓ2-norm and ℓ∞-norm robustness. The paper analyzes ℓ1-LinR for Ising model selection using statistical mechanics.
problem Model selection consistency of ℓ1-LinR for Ising models. method Replica method from statistical mechanics, ℓ1-regularized linear regression (ℓ1-LinR). result Model selection consistency with sample complexity $M=\mathcal{O}\left(\log N
ight)$.
Decomposes string links in a surface into prime components.
problem Decomposing string links in a surface into prime components.
method Proves a prime decomposition theorem for string links in a thickened surface.
result Any non-braid string link can be uniquely decomposed into prime string links up to braid equivalence.
New MPNNs match 2-WL, faster distinguishing graphs.
problem Improving graph neural network expressiveness.
method Introducing ℓ-walk MPNNs and second-order GNNs. result Walk MPNNs match 2-WL and can distinguish graphs faster.
We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which, if M is compact, is defined for w≥0 and T>0 by $$N_{M,w}(T) = \sum\…
Proves connection between ℓ2-Betti numbers and BNSR invariants.
problem Relationship between ℓ2-Betti numbers and BNSR invariants. method Analyzes ℓ2-Betti numbers and BNSR invariants of groups. result If the nth ℓ2-Betti number is non-zero, then the nth BNSR invariant over Q is empty. Multi-task feature learning aims to identity the shared features among tasks to improve generalization. It has been shown that by minimizing non-convex learning models, a better solution than the convex alternatives can be obtained. Therefore, a non-convex model based on the capped-ℓ1,ℓ1 regularization wa…
Generatability in metric spaces studied with novel novelty parameters.
problem Understanding generatability in metric spaces with asymmetric novelty parameters.
method Introducing (ε,ε′)-closure dimension to characterize uniform and non-uniform generatability. result Generatability is stable across novelty scales in doubling spaces but can be highly scale-sensitive in general metric spaces.
Formulae for special almost-complex structures on Vogan diagrams.
problem Existence of special almost-complex structures on almost-Kähler manifolds.
method Combinatorics of Vogan diagrams for classical semisimple Lie groups.
result Explicit formulae for special almost-complex structures.
We construct an infinite family of topologically slice 2--component boundary links ℓi, none of which is smoothly concordant to a split link, such that g4(ℓi)=i.
Introduces injective category number for continuous maps, linking classical and contemporary research.
problem Understanding conditions for a continuous map to be injective.
method Defines injective category number and examines its behavior under various operations.
result Provides a cohomological lower bound and expressions for injective category numbers in specific cases.
Paper optimizes sparse feature selection for cancer detection using GSVP and SVM.
problem Sparse feature selection for cancer detection.
method Regularized GSVP with proximal gradient descent, feature selection via SVM.
result Near-perfect balanced accuracy with few selected features.