We develop polynomial time algorithms for dual volume sampling.
problem Lack of polynomial time algorithms for dual volume sampling.
method Developed exact and derandomized polynomial time sampling algorithms.
result Dual volume sampling satisfies the Strong Rayleigh property, enabling fast mixing Markov chains.
A simpler edge-based discretization method without dual volumes.
problem Efficiently computing edge-based discretization vectors without forming dual volumes.
method Directly compute edge-midpoint vectors and reduce dual volume formation.
result Significant reduction in computing time for tetrahedral grids.
New general volume concept solves Minkowski problem for star bodies.
problem Minkowski problem for star bodies
method General volume concept, new curvature measure, variational formulas, Minkowski-type inequality
result Solution to the Minkowski problem for the new general dual Orlicz curvature measure
Dual volume bound linked to Weil-Petersson distance in quasi-Fuchsian manifolds.
problem Bounding dual volume in quasi-Fuchsian manifolds.
method Using dual Bonahon-Schläfli formula, proving bound on dual volume.
result Explicit constant bound on dual volume related to Weil-Petersson distance.
Study shows infimum of dual volume equals convex core volume for hyperbolic 3-manifolds.
problem Infimum of dual volume of convex co-compact hyperbolic 3-manifolds.
method Varying geometry by quasi-isometric deformations to deduce infimum.
result Linear lower bound on quasi-Fuchsian manifold volume based on bending lamination length.
This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz φ-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz φ-radial addition of two star bodies, we derive a f…
A new method for SSMF improves upon existing algorithms.
problem Identify identifiable solutions in simplex-structured matrix factorization.
method Dual simplex volume maximization approach.
result The proposed method outperforms state-of-the-art SSMF algorithms.
A new proof of dual Bonahon-Schläfli formula for 3-manifolds.
problem Deriving the dual Bonahon-Schläfli formula for 3-manifolds.
method Using differential Schläfli formula and properties of dual volume.
result A self-contained proof of the dual Bonahon-Schläfli formula.
Study on volumes of quasifuchsian manifolds, focusing on similarities and proximities.
problem Understanding the relationship between renormalized volume and dual volume of quasifuchsian manifolds.
method Analyzing similarities and proximities between renormalized volume and dual volume, using variational formulas and Weil-Petersson distance.
result Renormalized volume and dual volume are closely related, with bounded distance between related objects.
Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.
problem Understanding the group of volume preserving diffeomorphisms through symplectic geometry.
method Using cotangent bundles of spaces of smooth embeddings, symplectic reduction, and nonlinear Grassmannians of augmented submanifolds.
result Descriptions of coadjoint orbits of the group of volume preserving diffeomorphisms in terms of submanifolds of augmented spaces.
The Mahler volume of a centrally symmetric convex body K is defined as M(K)= (Vol K)(Vol K^dual). Mahler conjectured that this volume is minimized when K is a cube. We introduce the bottleneck conjecture, which stipulates that a certain convex body K^diamond subset K X K^dual has least volume when K is an ellipsoid. If…
Extends dual volume and curvature measures to broader functions and sets, solving Minkowski problems.
problem Characterize measures for which there exists a convex body with a given dual Orlicz curvature measure.
method Extends dual volume and curvature measures to broader functions and sets, proving existence and existence of solutions for Minkowski problems.
result Existence of convex polytopes and solutions for Minkowski problems when measures are discrete or even.
Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable real-valued functions on n-dimensional Euclidean space. This generalizes intrinsic vol…
New algorithm beats traditional methods for seismic data interpolation.
problem Efficiently filling in missing seismic data volumes.
method Primal-dual alternating approach using matrix factors and block-coordinate algorithm.
result Successfully interpolated a large 5D seismic data volume from a 3D model.
We derive an analytic formula for the dual Jacobian matrix of a generalised hyperbolic tetrahedron. Two cases are considered: a mildly truncated and a prism truncated tetrahedron. The Jacobian for the latter arises as an analytic continuation of the former, that falls in line with a similar behaviour of the correspondi…
The main results in this paper provide upper bounds of the second order Dehn functions for three-dimensional groups Nil and Sol. These upper bounds are obtained by using the Varopoulos transport argument on dual graphs. The first step is to start with reduced handlebody diagrams of the three-dimensional balls either im…
Unified description of tetrahedra in various spacetimes.
problem Characterizing tetrahedra with lightlike faces in different spacetimes.
method Using a generalized cross-ratio and edge lengths/dihedral angles to describe tetrahedra and their duals.
result Generalized ideal tetrahedra are the duals of tetrahedra with lightlike faces.
This research develops a dual-level reinforcement learning strategy to track daily VWAP accurately.
problem Inaccurate tracking of daily VWAP due to short trading horizons.
method Dual-level architecture using Transformer and LSTM models.
result Improves accuracy in approximating daily VWAP compared to previous models.
This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz-Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is…
Meta-learning predicts stock trading volumes by learning from each stock's unique patterns.
problem Predicting trading volumes for different stocks using a universal model.
method Dual-process meta-learning framework that learns common patterns with a meta-learner and specific patterns with stock-dependent parameters.
result Improves performance of various baseline models in volume predictions.
We show that the Reeb vector, and hence in particular the volume, of a Sasaki-Einstein metric on the base of a toric Calabi-Yau cone of complex dimension n may be computed by minimising a function Z on R^n which depends only on the toric data that defines the singularity. In this way one can extract certain geometric i…
A neural network predicts coarse-scale basis functions for efficient uncertainty quantification.
problem Efficiently estimating coarse-scale basis functions for multiscale methods.
method Data-driven approach using neural networks fitted to solution samples.
result Significant computational savings for uncertainty quantification tasks.
A convex projective surface is the quotient of a properly convex open Ω of P(R) by a discret subgroup Γ of SL3(R). We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if Ω is not a triangle then …
New method speeds up optimization with non-uniform sampling.
problem Optimizing complex stochastic problems efficiently.
method Stochastic Primal Dual Coordinate Method with Optimality Violation-based Sampling.
result The proposed method and variants outperform other methods in speed.
Improved pathwidth bound for hyperbolic 3-manifolds.
problem Finding a more efficient triangulation for hyperbolic 3-manifolds.
method Synthesis of tools from 3-manifold theory: Heegaard splittings, amalgamations, and thick-thin decomposition.
result Volume provides a linear upper bound on pathwidth of triangulations.
In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cashflows which are subject to volume constraints modeled by integer valued adapted processes and refraction period…
Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of resu…
This work tackles multi-objective optimization using constrained formulations for internet applications.
problem Handling user and item level constraints, scalability, and variance of dual estimates.
method Constrained optimization formulations, Lagrangian duality, user and item level constraints handling, scalability improvements, variance reduction.
result Significant advancement in state-of-the-art multi-objective optimization for internet applications.
APDO optimizes CMDPs with off-policy dual updates for faster convergence.
problem Learning policies that maximize long-term reward while satisfying safety constraints.
method Accelerated Primal-Dual Optimization (APDO) incorporating off-policy dual updates.
result APDO achieves better sample efficiency and faster convergence than existing methods.
Study relates Gromov norm to harmonic norm on non-positively curved manifolds.
problem Relating norms on homology classes to cohomology.
method Relates Gromov norm to harmonic norm, using volume and geometric quantities.
result Obtains double-sided bounds on norms.
Estimates the dual Thurston norm for foliations on negative curvature 3-manifolds.
problem Bounding the dual Thurston norm of foliations on 3-manifolds of negative curvature.
method Uses constants like injectivity radius, volume, curvature, and mean curvature of foliation leaves to estimate the dual Thurston norm.
result Provides an upper bound estimate on the dual Thurston norm of the Euler class of a foliation.
New decompositions of link complements into bipyramids.
problem Understanding the geometry of link complements.
method Presenting dual bipyramid decompositions for multi-crossing links.
result New upper bounds on hyperbolic link volumes from projections.
This paper proves the following: A volume preserving vector field on a compact 3-manifold whose dual 2-form is exact can not generate uniquely ergodic dynamics unless its asymptotic linking number is zero.
Study on volumes of random inscribed polytopes in projective geometries.
problem Estimating volumes of random inscribed polytopes in projective geometries.
method Central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
result Established central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
We develop an efficient algorithm to find confidence ellipsoids with volume guarantees in high dimensions.
problem Finding robust confidence ellipsoids in high-dimensional data.
method Polynomial time algorithm using primal-dual structure and geometric Brascamp-Lieb inequality.
result Algorithm finds ellipsoids within a O(β)γd volume factor of best β-conditioned ellipsoid. Let X be a closed, four-dimensional, oriented, smooth manifold with a Riemannian metric, g, let G be a compact Lie group, and P be a principal G bundle over X. D. Groisser and T. Parker (1987, 1989) and S. K. Donaldson (1990) conjectured that the moduli space of g-anti-self-dual connections on P, endowe…
Simply connected 4-manifolds with specific Weyl tensor are geodesic balls in space forms.
problem Characterizing 4-manifolds with harmonic anti-self dual Weyl tensor.
method Proving isometry to geodesic balls in space forms.
result Simply connected critical metrics are geodesic balls in space forms.
This work generalizes a geometric Laplacian determinant description to higher dimensions.
problem Defining and understanding the Laplacian determinant in higher dimensions with non-Delaunay triangulations.
method Geometric description of the Laplacian determinant in higher dimensions, relating it to volume quantities derived from simplex geometry.
result Generalizes geometric Laplacian determinant description to higher dimensions, showing negative semidefiniteness and kernel of constants.
In this note we introduce a natural Finsler structure on convex surfaces, referred to as the projective Finsler structure, which is dual in a sense to the obvious inclusion of a convex surface in a normed space. It has an associated projective girth, which is similar to the notion of girth defined by Schäffer. We prove…
Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.
problem Finding complete hyperbolic metrics on cusped 3-manifolds.
method Analogue of surface and compact 3-manifold flows, minimizing co-volume, extending through singularities.
result Existence of complete hyperbolic metric is equivalent to flow convergence.
New partition designs reduce star discrepancy in high-dimensional sampling.
problem Improving the expected star discrepancy in high-dimensional sampling.
method Developed non-equal volume partitions to achieve lower expected star discrepancy.
result Explicit upper bounds for expected star discrepancy under non-equal volume partitions.
We generalize the Riesz potential of a compact domain in Rm by introducing a renormalization of the rα−m-potential for α≤0. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renorm…
We investigate the rigidity of hyperbolic cone metrics on 3-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…
Accelerated Gibbs sampling for Gaussian graphical models using dual factor graphs.
problem Improving convergence rate of Gibbs sampling for Gaussian graphical models.
method Dual normal factor graph approach to accelerate convergence.
result Universal convergence rate improvement in dual domain for all homogeneous models.
We show that the complex projective space has maximal degree (volume) among all n-dimensional Kahler-Einstein Fano manifolds admitting a holomorphic C^*-action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart's volume conjecture fo…
Correcting bias in least squares regression with volume-rescaled sampling.
problem Bias in linear least squares solutions without distributional assumptions.
method Volume-rescaled sampling to correct bias in i.i.d. samples.
result Combined sample becomes unbiased with rescaled volume additional sample.
Study of supersymmetric AdS solutions using fibred GK geometries.
problem Understanding flux quantization and supersymmetric Killing vectors in AdS solutions.
method Analysis of GK geometries that fibre over Kähler bases, using master volume and global data.
result Quantum mechanical and SCFT central charges and entropy can be computed without explicit geometry.
The renormalized volume of hyperbolic manifolds is a quantity motivated by the AdS/CFT correspondence of string theory and computed via a certain regularization procedure. The main aim of the present paper is to elucidate its geometrical meaning. We use another regularization procedure based on surfaces equidistant to …