ODE trajectories become abnormal curves in Carnot groups.
problem Understanding abnormal curves in Carnot groups.
method Explicit construction of covectors for abnormal curves.
result Polynomial ODE trajectories lift to abnormal curves in Carnot groups.
We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension n≥4 whose group of holomorphic automorphisms has dimension either n2−4, or n2−5, or n2−6. This paper continues a series of articles that achieve classifications for automorphism group dimension n2−3 and greater.
Proves triviality of inertia groups in high-dimensional manifolds.
problem Classifying manifolds in the metastable range.
method Understanding the second extended power functor in synthetic spectra.
result Inertia groups of high-dimensional manifolds are trivial.
Joint sparsity regularization in multi-task learning has attracted much attention in recent years. The traditional convex formulation employs the group Lasso relaxation to achieve joint sparsity across tasks. Although this approach leads to a simple convex formulation, it suffers from several issues due to the loosenes…
Maps in Carnot groups are equivalent to solutions of a PDE system.
problem Understanding maps in Carnot groups of step 2.
method Equivalence between intrinsic Lipschitz maps and solutions to a PDE system.
result Intrinsic Lipschitz maps are equivalent to weak solutions of a PDE system.
In this article we prove that the codimension of the abnormal set of the endpoint map for certain classes of Carnot groups of step 2 is at least three. Our result applies to all step 2 Carnot groups of dimension up to 7 and is a generalisation of a previous analogous result for step 2 free nilpotent groups.
Finite 2-step nilpotent groups studied by Kim and Manturov.
problem Understanding the structure of groups defined by Kim and Manturov.
method Analyzing the groups Γn4 for all n≥6. result The groups are finite and 2-step nilpotent.
In this paper we study contact structure on 2-step nilpotent, Heisenberg type Lie groups. We decompose this Lie groups to center and orthogonal complement, then investigate properties of both orthogonal Lie subgroups. Finally, we provide a connection between matchings in groups and field extensions and 2-step nilpotent…
The H-type deviation measures how close step two Carnot groups are to H-type groups.
problem Quantifying how close step two Carnot groups are to H-type groups.
method Defined and analyzed the H-type deviation for step two Carnot groups.
result Explicitly computed H-type deviation for product of Heisenberg groups and verified the conjectural upper bound.
Researchers describe Casimir functions for 3- and 4-step nilpotent Lie groups.
problem Understanding Casimir functions for free nilpotent Lie groups of steps 3 and 4.
method Construction of Casimir functions for free nilpotent Lie groups of steps 3 and 4.
result For 3-step groups, coadjoint orbits are fully described as affine subspaces or direct products of quadrics.
Left-invariant metrics force 2-step nilpotent groups, preserving Kähler-like conditions.
problem Existence of left-invariant pluriclosed Hermitian metrics on Lie groups.
method Analyzing left-invariant metrics on unimodular Lie groups with abelian complex structures.
result Pluriclosed flow preserves Strominger Kähler-like conditions on 2-step nilpotent Lie groups.
Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
problem Exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
method Combining Varadhan's formula, Loewner's theorem, and the method of stationary phase.
result Characterization of squared sub-Riemannian distance and cut locus on generalized Heisenberg-type groups and star graphs.
Classifies Killing forms of arbitrary degree on specific nilpotent Lie groups.
problem Classifying Killing forms of arbitrary degree on specific Lie groups.
method Analyzing left-invariant Killing forms on simply connected 2-step nilpotent Lie groups with left-invariant metrics.
result Classified Killing forms when center is at most 2-dimensional.
In this article we show that the only 2-step nilpotent Lie groups which carry a non-degenerate left invariant Killing-Yano 2-form are the complex Lie groups. In the case of 2-step nilpotent complex Lie groups arising from connected graphs, we prove that the space of left invariant Killing-Yano 2-forms is one-dimensiona…
New insights into SGD and SGD-M in high dimensions.
problem Understanding and comparing SGD and SGD-M in high-dimensional settings.
method Developed high-dimensional scaling limits for SGD-M and online SGD, examining their dynamics and performance.
result SGD-M amplifies high-dimensional effects, potentially degrading performance compared to online SGD.
The study characterizes G₂-structures on 2-step nilpotent Lie groups.
problem Characterizing G₂-structures on 2-step nilpotent Lie groups.
method Analyzing left-invariant purely coclosed G₂-structures on 7-dimensional 2-step nilpotent Lie groups.
result Criteria for Riemannian metrics induced by these structures and determination of isomorphism classes.
Study conformal Killing forms on specific nilpotent Lie groups.
problem Characterize conformal Killing forms on 2-step nilpotent Lie groups.
method Analyzing left-invariant forms on simply connected groups, proving properties of forms based on center dimension.
result Only specific forms exist under certain conditions.
Characterizes GM-groups via sub-Riemannian geometry properties.
problem Characterizing step-two Carnot groups via sub-Riemannian geometry.
method Sub-Riemannian geometric properties, including squared distance, cut locus, optimal synthesis.
result Characterization of GM-groups and exact expression of d(g)2 for classical cut locus. Tomova, along with results of Bachman and Schleimer, showed that any high distance knot has a stair-step bridge spectrum. In this paper, we compute the bridge spectra and distance of generalized Montesinos knots. In particular, we produce the first example of a class of knots which attain the stair-step bridge spectra …
Study Sard problem in step 2 and filiform Carnot groups.
problem Understanding the Sard problem in specific types of Carnot groups.
method Analyzing endpoint maps in step 2 and filiform Carnot groups.
result Characterized abnormal set in filiform groups and provided bounds in step 2 Carnot groups.
Study magnetic trajectories on 2-step nilpotent Lie groups.
problem Understanding magnetic trajectories on specific Lie groups.
method Formulated magnetic equation, found solutions for invariant Lorentz forces, computed examples in Heisenberg groups.
result Interesting magnetic trajectories involving elliptic integrals found in Heisenberg groups.
Improves Group Lasso for categorical data by reducing dimensionality and selecting models.
problem Sparse modelling of categorical data is challenging, especially for high dimensions.
method Two-step procedure: first, reduce dimensionality using Group Lasso; second, select final model using an information criterion on clustered levels.
result The method produces a sparse solution and performs better than state-of-the-art algorithms in prediction accuracy and model dimension.
New Lie groups generalize H-type groups with nondegenerate centers.
problem Generalizing H-type groups with nondegenerate centers.
method Defined and investigated 2-step nilpotent Lie groups.
result Geometric properties of new Lie groups investigated.
The paper studies automorphisms of 2-step nilpotent Lie groups, showing continuity up to center and field automorphisms.
problem Investigating the continuity of abstract automorphisms in 2-step nilpotent Lie groups.
method Analyzes various types of 2-step nilpotent Lie groups, using tools from Riemannian geometry.
result Abstract automorphisms are continuous 'up to discontinuity due to the center and field automorphisms of C' for many 2-step nilpotent Lie groups. IMM generates high-quality samples in few steps with stable training.
problem Slow inference and instability in generating high-quality samples using diffusion models and Flow Matching.
method Inductive Moment Matching (IMM) is a new generative model for one- or few-step sampling with a single-stage training procedure.
result IMM achieves state-of-the-art 2-step FID of 1.98 on CIFAR-10 for a model trained from scratch.
Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.
problem Estimating asymptotic metrics in 2-step nilpotent Lie groups.
method Developed a novel technique to perturb rectifiable curves.
result Every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone.
New algorithm robustly estimates sparse models in high dimensions with corrupted data.
problem Estimating latent variable models with arbitrarily corrupted samples in high dimensional space.
method Trimmed (Gradient) Expectation Maximization with trimming gradients and hard thresholding steps.
result The algorithm converges to near optimal statistical rate geometrically under certain conditions.
A Carnot group G admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve γ in G and ε>0, there is a C1 horizontal curve Γ such that Γ=γ and Γ′=γ′ outside a set of measure at most ε. We verify this property for free Carno…
Boosting algorithms predict financial vulnerability of farmers in Chile and Tunisia.
problem Predict financial vulnerability of farmers in Chile and Tunisia using environmental data.
method Interpretable boosting algorithms based on ridge-regularized generalized linear models.
result Interaction effects improve predictive power only when included in two-step boosting.
Study of curvature flow on complex Lie groups, leading to soliton convergence.
problem Characterizing long-time behavior of curvature flow on complex 2-step nilpotent Lie groups.
method Analyzing left-invariant metrics and using Cheeger-Gromov topology.
result Normalized solutions converge to a non-flat algebraic soliton.
Sparse Polyak improves high-dimensional statistical estimation.
problem High-dimensional statistical estimation problems with growing problem dimension.
method Sparse Polyak modifies Polyak's adaptive step size to estimate restricted Lipschitz smoothness.
result Sparse Polyak achieves optimal statistical precision with fewer iterations.
This paper characterizes semigenerated Carnot groups and applies it to rectifiability of perimeter sets.
problem Characterizing semigenerated Carnot groups and their applications to rectifiability.
method Algebraic approach focusing on semigroup generation and Engel-type quotients.
result Complete characterization of semigeneration in Carnot groups of step 3 and sufficient criteria for semigeneration in Carnot groups of arbitrary step.
Study G2-instantons on specific Lie groups, finding conditions and structures.
problem Characterize G2-instantons on 2-step nilpotent Lie groups.
method Analyze connections arising from characteristic connections, use Lie group structure and torsion.
result Establish necessary and sufficient conditions for G2-instantons, define naturally reductive structures.
Single-step samplers generate high-quality samples efficiently.
problem Sampling from unnormalized distributions is computationally expensive.
method Developed consistent diffusion samplers that generate samples in a single step.
result Single-step samplers produce high-fidelity samples with less than 1% of traditional samplers' evaluations.
Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie …
Cost-efficient feature selection for multi-label classification in medicine.
problem Feature selection in multi-label classification with cost constraints.
method Sequential feature selection maximizing conditional mutual information, followed by cost-free feature selection using shadow features.
result The method effectively reduces prediction costs in medical applications.
Sharp bounds found on nonabelian quotients of surface braid groups.
problem Finding the smallest nonabelian quotients of surface braid groups.
method Sharp lower bounds and classification of quotients.
result Quotients of minimum order are either symmetric groups or 2-step nilpotent p-groups.
New approach removes data influence in high dimensions with single step.
problem Efficiently removing data influence in high-dimensional settings with strong convexity and smoothness assumptions.
method Introduces ε-Gaussian certifiability and analyzes Newton method performance.
result Single Newton step followed by Gaussian noise achieves privacy and accuracy.
Satellite-based positioning system such as GPS often suffers from large amount of noise that degrades the positioning accuracy dramatically especially in real-time applications. In this work, we consider a data-mining approach to enhance the GPS signal. We build a large-scale high precision GPS receiver grid system to …
Classifies two-step solvable Lie groups with SKT structures.
problem Classifying Lie groups with SKT structures.
method Shear construction and analysis of SKT shear data on Abelian Lie algebras.
result Large part of the classification for two-step solvable SKT algebras of dimension six.
Study of curvature flow on specific Lie groups, leading to soliton solutions.
problem Curvature flow on 2-step nilpotent Lie groups with complex structures.
method Left-invariant metrics and complex structures on Lie groups, convergence analysis.
result Existence and convergence of flow to soliton solutions.
We focus our attention on the notion of intrinsic Lipschitz graphs, inside a special class of metric spaces i.e. the Carnot groups. More precisely, we provide a characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2 in terms of their intrinsic distributional gradients.
We prove that two-step analytic sub-Riemannian structures on a compact analytic manifold equipped with a smooth measure and Lipschitz Carnot groups satisfy measure contraction properties.
Characterizes complex structures on specific Lie groups.
problem Identifying Lie groups with left-invariant complex structures.
method Analyzing Lie algebras and their corresponding Lie groups, considering different nilpotency levels.
result Conditions for the existence of left-invariant complex structures and pluriclosed metrics on 2-step nilpotent Lie groups.
The paper studies algebraic relations of first integrals on specific Lie groups.
problem Algebraic relations of first integrals on step-two and step-three nilpotent Lie groups.
method Analysis of isometry algebra and invariant first integrals.
result Complete families of first integrals can be constructed with Killing vector fields and symmetric Killing 2-tensor fields in low dimensions.
PROBE algorithm efficiently solves sparse high-dimensional linear regression.
problem Sparse high-dimensional linear regression models with complex parameter spaces.
method Partitioned empirical Bayes ECM algorithm for computationally efficient MAP estimation.
result PROBE algorithm provides robust and efficient coordinate-wise optimization.
Continuous functions on graphs in Carnot groups satisfy a Burgers' type equation.
problem Characterizing CH1-regularity of graphs in Carnot groups of step 2. method Proving equivalence between distributional solutions of Burgers' type equations and CH1-regularity of graphs. result Continuous functions on graphs in Carnot groups of step 2 satisfy a Burgers' type equation in the distributional sense.
In Carnot groups of step 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carn…