arXiv research

Counterexample disproves gluing theorem for MCP metric spaces.

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 $\mathbb{R}$, but does not topologically split. The second space satisfies…

Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.

Proposes a new algorithm for online decision-making with high-dimensional data.

Proves rectifiability for specific metric spaces with unique tangents.

Measure contraction properties $MCP(K,N)$ 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 $N$, then $MCP(K,N)$ is equivalent to Ricci curvature bounded below by $K$. On the other hand, it was ob…

CD converges linearly for MCP/SCAD penalized least squares.

New sub-Riemannian structures fail synthetic curvature bounds.

Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.

MCP learns reusable skills for complex tasks by combining simple ones.

Proves metric measure spaces with certain properties are one-dimensional.

The paper examines curvature-dimension bounds on sub-Finsler Heisenberg groups.

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…

QRAFTI uses multi-agent framework to improve equity factor research.

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.

Study improves keyword forecasting in earnings-call prediction markets.

Paper proposes methods to train RBF networks under faults.

We prove that any corank 1 Carnot group of dimension $k+1$ equipped with a left-invariant measure satisfies the $\mathrm{MCP}(K,N)$ if and only if $K \leq 0$ and $N \geq k+3$. This generalizes the well known result by Juillet for the Heisenberg group $\mathbb{H}_{k+1}$ to a larger class of structures, which admit non-t…

Proposes a new confidence criterion for deep neural networks to predict failures.

One-bit measurements widely exist in the real world, and they can be used to recover sparse signals. This task is known as the problem of learning halfspaces in learning theory and one-bit compressive sensing (1bit-CS) in signal processing. In this paper, we propose novel algorithms based on both convex and nonconvex s…

Study on curvature bounds and geodesic dimension in sub-Finsler Heisenberg groups.

Paper proposes sparse classification method for high-dimensional data.

Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.

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…

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.

The curvature-dimension condition fails in sub-Finsler geometry, extending previous results in sub-Riemannian geometry.

New examples show strong Kato limits can be branching and not satisfy known conditions.

Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.

Picasso is a new library for sparse learning problems in R and Python.

Proposes a new metric space example showing non-constant topological dimension.

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 $χ^2$-distance function and …

Sharp Poincaré inequality proved for specific metric spaces.

We prove that H-type Carnot groups of rank $k$ and dimension $n$ satisfy the $\mathrm{MCP}(K,N)$ if and only if $K\leq 0$ and $N \geq k+3(n-k)$. The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in…

Efficient ADMM algorithm solves nonconvex SVMs with various penalties.

Two new methods improve block-sparse signal recovery from noisy data.

New nonconvex penalty smooths at origin for deep learning.

Proposes GAGA algorithm for automatic hyperparameter learning in signal recovery.

A new online learning framework selects features and improves convergence.

Develops a method for learning sparse generalized linear models in high-dimensional data.

In this paper we investigate the relationship between a general existence of transport maps of optimal couplings with absolutely continuous first marginal and the property of the background measure called essentially non-branching introduced by Rajala-Sturm (Calc.Var.PDE 2014). In particular, it is shown that the quali…

New method improves signal reconstruction with nonconvex penalties and parameter control.

We provide novel theoretical results regarding local optima of regularized $M$-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.

Novel framework for systemic risk analysis in financial markets.

Sharp uncertainty principle for nodal sets in singular spaces.

We study a family of regularized score-based estimators for learning the structure of a directed acyclic graph (DAG) for a multivariate normal distribution from high-dimensional data with $p\gg n$. Our main results establish support recovery guarantees and deviation bounds for a family of penalized least-squares estima…