arXiv research

New method solves complex curvature equations.

In mixed multi-view data, multiple sets of diverse features are measured on the same set of samples. By integrating all available data sources, we seek to discover common group structure among the samples that may be hidden in individualistic cluster analyses of a single data-view. While several techniques for such int…

New weighted surface area measures for convex bodies with applications.

Frame flows on certain symmetric spaces mix exponentially.

Solves Christoffel problem for disk area measures on spheres.

Study proves radial symmetry in convex cones using subharmonic functions.

Characterizes continuity of monotone functionals in mixed topology.

Proposes a convex model for mixed logit to handle individual heterogeneity.

Blind Source Separation (BSS) has proven to be a powerful tool for the analysis of composite patterns in engineering and science. We introduce Convex Analysis of Mixtures (CAM) for separating non-negative well-grounded sources, which learns the mixing matrix by identifying the lateral edges of the convex data scatter p…

Geodesic flow mixing on convex projective manifolds proven.

Study integrates reliability constraints into generation planning models.

Solves a long-standing convex geometry problem about mixed volumes.

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are prove…

Unified Minkowski problem discussed for (p,q)-mixed quermassintegrals.

Solves Christoffel-Minkowski problem for axially symmetric bodies.

Unique entropy measure found for convex projective manifolds.

Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.

The paper develops mixed-integer formulations for neural networks using partitioning.

The paper proves exponential mixing for hyperbolic manifolds, with applications to geodesic holonomy.

Novel network model estimates mixed-membership structure with covariate information.

Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.

Local invertibility of ray transforms on convex manifolds.

New method adapts to unknown mixing time in stochastic optimization.

In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its e…

Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.

New $L_p$-Steiner quermassintegrals defined from Steiner formula.

This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $β(\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, incl…

Local invertibility of higher order tensor transforms on compact manifolds.

We study manifolds endowed with mixed metric 3--contact structures, proving that the distribution spanned by the Reeb vector fields is integrable, with totally geodesic integral manifolds, of constant sectional curvature $k=\pm1$. We also prove a result of projectability of such structures onto paraquaternionic Kähleri…

The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.

This paper tackles convex-submodular minimax problems in mixed continuous-discrete domains.

The paper proves a unique conformal measure for Anosov groups and shows local mixing.

MixMin finds optimal data mixtures for better model performance.

We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which are strictly convex by horospheres, we show the long time existence of mixed volu…

The paper extends geometric results from negatively-curved spaces to strictly convex Hilbert geometry.

We apply conformal flows of metrics restricted to the orthogonal distribution $D$ of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of $D$, and for the case …

At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes…

This paper resolves the Langevin Algorithm's mixing time for log-concave distributions.

Optimizes routing in decentralized exchanges with gas fees.

Let $Γ$ be a Zariski dense convex cocompact subgroup contained in an arithmetic lattice of $\operatorname{SO}(n, 1)^{\circ}$. We prove uniform exponential mixing of the geodesic flow for congruence covers of the hyperbolic manifold $Γ\backslash \mathbb{H}^n$ avoiding finitely many prime ideals. This extends the work of…

We consider the mixed regression problem with two components, under adversarial and stochastic noise. We give a convex optimization formulation that provably recovers the true solution, and provide upper bounds on the recovery errors for both arbitrary noise and stochastic noise settings. We also give matching minimax …

We consider the problem of estimating the phases of K mixed complex signals from a multichannel observation, when the mixing matrix and signal magnitudes are known. This problem can be cast as a non-convex quadratically constrained quadratic program which is known to be NP-hard in general. We propose three approaches t…

New bounds for mixing time and privacy in projected Langevin algorithm and noisy SGD.

Let $X$ be an $n$-dimensional manifold and $V_1,\ldots,V_n\subset C^\infty(X,\mathbb R)$ finite-dimensional vector spaces. For systems of equations $\{f_i = a_i\colon\: f_i\in V_i,\:a_i \in\mathbb R,\:i=1,\ldots,n\}$ we discover a relationship between the average number of their solutions and mixed volumes of convex bo…

The paper studies constant $k$th-mixed curvature on Hermitian manifolds and finds self-duality and Kähler conditions.

The paper explores mixed curvature for Hermitian manifolds and its implications.

PAPAL algorithm finds mixed Nash equilibria in continuous games.

Method improves regression models using unlabeled data.