Counterexample disproves gluing theorem for MCP metric spaces.
problem Gluing theorems for MCP metric measure spaces are not universally valid.
method Used Grushin half-plane as a counterexample.
result The doubling of Grushin half-plane does not satisfy MCP(0,N) for all N.
Proves rectifiability for specific metric spaces with unique tangents.
problem Rectifiability of C D ( K , N ) \mathsf{CD}(K,N) CD ( K , N ) and M C P ( K , N ) \mathsf{MCP}(K,N) MCP ( K , N ) spaces with unique tangents. method Failure of C D \mathsf{CD} CD condition in sub-Finsler Carnot groups, new result on M C P \mathsf{MCP} MCP spaces, recent breakthrough by Bate. result Proves rectifiability for C D ( K , N ) \mathsf{CD}(K,N) CD ( K , N ) and M C P ( K , N ) \mathsf{MCP}(K,N) MCP ( K , N ) spaces under specific conditions. We give two examples of metric measure spaces satisfying the measure contraction property MCP(K,N) but having different topological dimensions at different regions of the space. The first one satisfies MCP(0,3) and contains a subset isometric to R \mathbb{R} R , but does not topologically split. The second space satisfies…
Measure contraction properties M C P ( K , N ) MCP(K,N) M C P ( K , N ) are synthetic Ricci curvature lower bounds for metric measure spaces which do not necessarily have smooth structures. It is known that if a Riemannian manifold has dimension N N N , then M C P ( K , N ) MCP(K,N) M C P ( K , N ) is equivalent to Ricci curvature bounded below by K K K . On the other hand, it was ob…
Proves metric measure spaces with certain properties are one-dimensional.
problem Characterizing metric measure spaces as one-dimensional.
method Analyzes properties of metric measure spaces and uses optimal transport maps.
result Metric measure spaces with specified properties are one-dimensional.
We give necessary and sufficient conditions that show that both the group of isometries and the group of measure-preserving isometries are Lie groups for a large class of metric measure spaces. In addition we study, among other examples, whether spaces having a generalized lower Ricci curvature bound fulfill these requ…
Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.
problem Understanding the structure of Busemann spaces with measures.
method Analyzing geodesic completeness and non-collapse assumptions.
result Rigidity and structure theorems for Busemann spaces with MCP.
Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.
problem Understanding isoperimetric constants in metric measure spaces with measure contraction property.
method Proves local isoperimetric inequalities on essentially non-branching MCP(K,N) spaces with volume constraints and geometric conditions.
result Establishes bounds on isoperimetric constants in smaller geodesic balls.
New sub-Riemannian structures fail synthetic curvature bounds.
problem Failure of synthetic curvature bounds in sub-Riemannian geometry.
method New stability results for local MCP under quotients, applied to specific sub-Riemannian structures.
result Ideal sub-Riemannian structures can fail the MCP, generically for high dimensions and rank > 3.
Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy…
We prove that any corank 1 Carnot group of dimension k + 1 k+1 k + 1 equipped with a left-invariant measure satisfies the M C P ( K , N ) \mathrm{MCP}(K,N) MCP ( K , N ) if and only if K ≤ 0 K \leq 0 K ≤ 0 and N ≥ k + 3 N \geq k+3 N ≥ k + 3 . This generalizes the well known result by Juillet for the Heisenberg group H k + 1 \mathbb{H}_{k+1} H k + 1 to a larger class of structures, which admit non-t…
Proposes a new metric space example showing non-constant topological dimension.
problem Non-constant topological dimension in metric measure spaces.
method Refines Ketterer and Rajala's example to satisfy CD(0,∞) condition.
result Shows non-constancy of topological dimension for CD spaces.
Proposes a new algorithm for online decision-making with high-dimensional data.
problem Online decision-making with high-dimensional data.
method G-MCP-Bandit algorithm with 2-step weighted Lasso procedure.
result Achieves optimal cumulative regret and convergence rate.
Sharp Poincaré inequality proved for specific metric spaces.
problem Proving a sharp Poincaré inequality for certain metric measure spaces.
method Identifying model densities and using localization arguments, without assuming geodesic convexity.
result Best possible Poincaré constant as a function of parameters.
CD converges linearly for MCP/SCAD penalized least squares.
problem Recovering sparse signals from data.
method Coordinate descent for MCP/SCAD penalized least squares.
result CD converges linearly to solutions of MCP/SCAD penalized least squares.
Study on curvature bounds and geodesic dimension in sub-Finsler Heisenberg groups.
problem Investigate synthetic curvature-dimension bounds in sub-Finsler geometry.
method Examine measure contraction property and geodesic dimension on Heisenberg groups with ℓ p \ell^p ℓ p -sub-Finsler norms. result For p ∈ ( 2 , ∞ ] p \in (2, \infty] p ∈ ( 2 , ∞ ] , ℓ p \ell^p ℓ p -Heisenberg group fails to satisfy any measure contraction property. For p ∈ ( 1 , 2 ) p \in (1, 2) p ∈ ( 1 , 2 ) , it satisfies M C P ( K , N ) \mathsf{MCP}(K, N) MCP ( K , N ) under specific conditions. Study on cones over metric spaces with curvature bounds.
problem Establishing curvature bounds for cones over metric spaces.
method Developed a localization technique to prove synthetic curvature bounds.
result Riemannian and Lorentzian cones over CD-spaces satisfy MCP and vice versa.
Paper explores transport maps and measure rigidity in metric spaces.
problem Existence and uniqueness of transport maps in non-branching metric measure spaces.
method Investigates the relationship between transport maps and essentially non-branching measures.
result Essentially non-branching metric measure spaces have unique transport maps under certain conditions.
Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.
problem Tackles constructing Poisson structures on gauge orbits of Maurer-Cartan elements.
method Constructs Poisson structures on gauge orbits of Maurer-Cartan elements of dgla L, associating a compatible Batalin-Vilkovisky algebra to each MC element.
result MCP structures yield a notion of hamiltonian flow of MC elements and define Lie algebroids on gauge orbits.
Sharp uncertainty principle for nodal sets in singular spaces.
problem Estimating the size of nodal sets in non-smooth spaces.
method Uncertainty principle applied to eigenfunctions in metric measure spaces with synthetic Ricci curvature bounds.
result New lower bounds on nodal set sizes in non-smooth spaces.
Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.
problem Determining the curvature exponent of sub-Finsler Heisenberg groups.
method Analyzing the measure contraction property and constructing sub-Finsler structures.
result Proved that curvature exponent N_min ≥ 5, with equality if sub-Riemannian.
MCP learns reusable skills for complex tasks by combining simple ones.
problem Learning complex tasks with many skills requires impractical amounts of data.
method Factorizes skills into primitives that can be combined multiplicatively.
result MCP can learn and reuse skills for novel tasks from pre-training.
The paper examines curvature-dimension bounds on sub-Finsler Heisenberg groups.
problem Investigating synthetic curvature-dimension bounds in sub-Finsler Heisenberg groups.
method Study of measure contraction property (MCP) and curvature-dimension condition (CD).
result Sub-Finsler Heisenberg groups do not satisfy MCP or CD for any parameters.
The curvature-dimension condition fails in sub-Finsler geometry, extending previous results in sub-Riemannian geometry.
problem The failure of the curvature-dimension condition in sub-Finsler geometry.
method Non-trivial adaptation of Juillet's work, introduction of new tools and ideas.
result The C D ( K , N ) \mathsf{CD}(K,N) CD ( K , N ) condition does not hold in sub-Finsler geometry for various norms and measures. QRAFTI uses multi-agent framework to improve equity factor research.
problem Replicating and developing new equity factors in large financial datasets.
method Integrates a research toolkit with MCP servers for data access and custom coding operations.
result Improves performance and explainability in multi-step empirical tasks.
Penalized regression is an attractive framework for variable selection problems. Often, variables possess a grouping structure, and the relevant selection problem is that of selecting groups, not individual variables. The group lasso has been proposed as a way of extending the ideas of the lasso to the problem of group…
MCP extends conformal prediction to vector-valued score functions without data splitting.
problem Fixed prediction set shapes in scalar score functions limit coverage guarantees.
method MCP uses a single optimization problem for prediction set design and calibration, eliminating data splitting.
result RemMCP and RelMCP achieve target coverage with smaller or comparable prediction set sizes, reducing variance.
The paper proves stability properties for quotients of spaces with synthetic Ricci curvature bounds.
problem Stability properties of quotients of spaces with synthetic Ricci curvature bounds.
method Analyzes quotients of Riemannian manifolds with isometric group actions and metric-measure foliations/submersions.
result Quotients of Riemannian manifolds with synthetic Ricci curvature bounds are also Riemannian manifolds with synthetic Ricci curvature bounds.
New examples show strong Kato limits can be branching and not satisfy known conditions.
problem Exploring the boundaries of strong Kato limits and their properties.
method Constructing specific examples of non-collapsed strong Kato limits.
result Found examples of strong Kato limits that are branching and do not satisfy C D ( K , ∞ ) \mathrm{CD}(K,\infty) CD ( K , ∞ ) or M C P ( K , N ) \mathrm{MCP}(K,N) MCP ( K , N ) conditions. Study improves keyword forecasting in earnings-call prediction markets.
problem Accurately predicting future keyword mentions in earnings calls.
method Experiments on earnings-call mention markets, varying context and market probability, introducing MCP.
result Mixture of market probability and MCP yields the best forecasts.
Paper proposes methods to train RBF networks under faults.
problem Training RBF networks with fault tolerance.
method Two novel ADMM-based algorithms using MCP and l0-norm.
result Both methods globally converge to a unique limit point.
Proposes a new confidence criterion for deep neural networks to predict failures.
problem Predicting failures in deep neural networks.
method Introduces True Class Probability (TCP) as a new confidence criterion and proposes a learning scheme to estimate it.
result The proposed approach consistently outperforms existing methods in failure prediction.
Paper proposes sparse classification method for high-dimensional data.
problem Sparse classification in high-dimensional data with positive-confidence samples.
method Developed a novel sparse-penalization framework using L1, SCAD, and MCP penalties for convex and non-convex shrinkage.
result Proved near minimax-optimal sparse recovery rates under Restricted Strong Convexity condition.
Paper proposes algorithms for robust 1-bit compressive sensing with nonconvex penalties.
problem Recovering sparse signals from one-bit measurements.
method Develops algorithms based on convex and nonconvex penalties, providing analytical solutions.
result Analytical solutions for several nonconvex penalties are found, making the recovery process faster and more efficient.
In high-dimensional data analysis, penalized likelihood estimators are shown to provide superior results in both variable selection and parameter estimation. A new algorithm, APPLE, is proposed for calculating the Approximate Path for Penalized Likelihood Estimators. Both the convex penalty (such as LASSO) and the nonc…
A new method for high-dimensional classification using Bernstein polynomials.
problem Computational difficulties in high-dimensional SVM hinge loss.
method Proposes Bernstein support vector machine (BernSVM) and two efficient algorithms.
result Achieves a prediction accuracy rate of s log ( p ) / n \sqrt{s\log(p)/n} s log ( p ) / n with high probability. Picasso is a new library for sparse learning problems in R and Python.
problem Sparse learning problems in high-dimensional data analysis.
method Unified framework of pathwise coordinate optimization with efficient active set selection strategies.
result picasso can efficiently handle large-scale problems.
In this paper we propose and study a family of sparsity-inducing penalty functions. Since the penalty functions are related to the kinetic energy in special relativity, we call them \emph{kinetic energy plus} (KEP) functions. We construct the KEP function by using the concave conjugate of a χ 2 χ^2 χ 2 -distance function and …
Efficient ADMM algorithm solves nonconvex SVMs with various penalties.
problem Solving nonconvex penalized SVMs due to nondifferentiability, nonsmoothness, and nonconvexity.
method ADMM-based algorithm for a wide range of nonconvex penalties.
result The proposed algorithm outperforms other methods on benchmark datasets.
Two new methods improve block-sparse signal recovery from noisy data.
problem Recovering block-sparse signals with unknown partitions.
method LogLOP-l2/l1 and AdaLOP-l2/l1 methods using log-sum penalty and MCP.
result Our methods outperform existing techniques in estimation accuracy.
New nonconvex penalty smooths at origin for deep learning.
problem Improving variable selection and bias in high-dimensional statistical learning.
method Developed a new nonconvex penalty function smooth at origin.
result Asymptotic bias of new penalty function vanishes exponentially fast.
Proposes GAGA algorithm for automatic hyperparameter learning in signal recovery.
problem Difficulty in selecting hyperparameters in traditional signal recovery methods.
method Global Adaptive Generative Adjustment (GAGA) algorithm for automatic hyperparameter learning and signal estimate.
result Consistency of model selection and signal estimate output.
Researchers prove a property for a specific group class, leading to Wasserstein geodesic continuity.
problem Proving a measure contraction property for generalized H-type Carnot groups.
method Analyzing H-type Carnot groups of rank k k k and dimension n n n to establish the M C P ( K , N ) \mathrm{MCP}(K,N) MCP ( K , N ) condition. result Generalized H-type Carnot groups satisfy M C P ( K , N ) \mathrm{MCP}(K,N) MCP ( K , N ) with K ≤ 0 K\leq 0 K ≤ 0 and N ≥ k + 3 ( n − k ) N \geq k+3(n-k) N ≥ k + 3 ( n − k ) , matching geodesic dimension. A new online learning framework selects features and improves convergence.
problem Online learning's limitations in convergence and feature selection.
method A novel online learning framework based on running averages with feature selection.
result The framework achieves high true support recovery and faster convergence.
Develops a method for learning sparse generalized linear models in high-dimensional data.
problem Feature selection in high-dimensional data with many variables.
method GSDAR method based on KKT conditions for ℓ 0 \ell_0 ℓ 0 -penalized maximum likelihood estimations. result The errors of the proposed estimate decay exponentially to the optimal order under certain conditions.
New method improves signal reconstruction with nonconvex penalties and parameter control.
problem Reconstructing sparse signals with nonconvex penalties and nonconvexity control.
method Introduces nonconvex penalties (SCAD, MCP) with nonconvexity parameters and controls them to guide AMP trajectory.
result Achieves perfect reconstruction for relatively dense signals with small nonconvexity parameters.
We provide novel theoretical results regarding local optima of regularized M M M -estimators, allowing for nonconvexity in both loss and penalty functions. Under restricted strong convexity on the loss and suitable regularity conditions on the penalty, we prove that \emph{any stationary point} of the composite objective f…
Paper introduces a novel method for estimating model confidence in deep neural classifiers.
problem Reliable confidence estimation for deep neural classifiers in safety-critical applications.
method Proposes a novel target criterion (true class probability) and learns it from data with an auxiliary model.
result The proposed method outperforms strong baselines in various tasks and network architectures.