Fenchel lifted networks improve neural network training without performance loss.
problem The difficulty and non-convexity of training deep neural networks.
method Introduces Fenchel lifted networks that represent activation functions as biconvex constraints and uses Lagrange Multipliers to create a lower bound of the training problem.
result Fenchel lifted networks match or outperform traditional neural networks in performance.
Unified framework for lifted training and inversion of neural networks.
problem Challenges in gradient-based training of deep neural networks.
method Unified framework encapsulating various lifted training strategies.
result Unified framework improves training landscape and stability.
We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abe…
The SL(2)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group G of M. Using the ergodicity of G on the SU(2)-character variety, we deduce that every G-invariant meromorphic function on X is constant. The trace functions of closed curves on M de…
Paper proposes new loss functions for training energy networks.
problem Challenges in computing gradients for training energy networks.
method Proposes generalized Fenchel-Young losses for efficient gradient computation.
result Demonstrates the calibration of excess risk for linear-concave energies.
The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.
problem Defining Fenchel conjugate and biconjugate on curved spaces.
method Introduced a new definition of Fenchel conjugate and biconjugate on Hadamard manifolds based on the tangent bundle.
result Developed a Fenchel-Moreau Theorem for geodesically convex functions on Hadamard manifolds.
Defines Fenchel-Nielsen coordinates for SL(3,C) representations.
problem No specific problem stated; coordinates defined for a new context.
method Introduced Fenchel-Nielsen coordinates for mSL(3,C) representations. result Relates to classical and generalized Fenchel-Nielsen coordinates.
Generalizes Fenchel conjugation to nonlinear functions on arbitrary sets.
problem Extending Fenchel conjugation to functions on arbitrary sets without structure.
method Replacing linear test functions with nonlinear ones, investigating properties including biconjugation.
result Derived further results on smooth manifolds and Lie groups, relating to convexity.
The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.
problem Finding minimizers of nonlocal curvature energies.
method Combining Fenchel-type theorems with geometric analysis techniques.
result Circles and disks minimize specific energy functionals.
Calculates twist in Teichmüller space using cross ratios.
problem Calculating the Fenchel-Nielsen twist in Teichmüller space.
method Using cross ratio coordinates.
result Compact calculation of twist in Teichmüller space.
Introduces Fitzpatrick losses, tighter than Fenchel-Young losses.
problem Improving loss functions for machine learning.
method Introduces Fitzpatrick losses based on the Fitzpatrick function.
result Fitzpatrick losses are tighter than Fenchel-Young losses.
Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.
problem Developing inequalities for submanifolds in curved spaces.
method Connecting Fenchel-Willmore and logarithmic Sobolev inequalities for mean-convex submanifolds.
result Established extensions of Fenchel-Willmore inequality and derived new Sobolev-type inequalities.
Various Alexandrov-Fenchel type inequalities have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. The purpose of this article is to shed new light on this by c…
Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
problem Relationship between Fenchel-Nielsen coordinates and shear coordinates on Riemann surfaces.
method Explicitly showed the Poisson bracket on shear coordinates induces the Fenchel-Nielsen bracket on gluing parameters.
result Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
Paper introduces Fenchel-Young losses for supervised learning tasks.
problem Choosing the right loss function for supervised learning tasks.
method Introduces Fenchel-Young losses as a generic way to construct convex loss functions.
result Fenchel-Young losses unify and create new loss functions.
Study flows to analyze sphere quermassintegrals.
problem Analyze quermassintegrals on the sphere.
method Use two types of flows to study quermassintegrals.
result Establish Alexandrov-Fenchel inequalities for the sphere.
This paper studies Fenchel-Young losses, a generic way to construct convex loss functions from a regularization function. We analyze their properties in depth, showing that they unify many well-known loss functions and allow to create useful new ones easily. Fenchel-Young losses constructed from a generalized entropy, …
Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.
problem Establishing a Fenchel-Willmore inequality for submanifolds in manifolds with non-negative Ricci curvature.
method Analyzing submanifolds in manifolds with non-negative intermediate Ricci curvature and Euclidean volume growth.
result Sharp Fenchel-Willmore inequality for submanifolds in manifolds with non-negative intermediate Ricci curvature.
Given a real-valued function defined on the Heisenberg group, we provide a definition of abstract convexity and Fenchel transform that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the Euclidean case, a convex function can be characterized via its iterated Fen…
Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.
problem Extending capillary convex body results to anisotropic setting.
method Developed theory for anisotropic capillary convex bodies in half-space and established Alexandrov-Fenchel inequality for mixed volumes.
result Established a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies, weakening and extending previous results.
Gradient descent converges with arbitrary stepsize for separable data under Fenchel-Young losses.
problem Understanding the conditions under which gradient descent converges with arbitrary stepsize.
method Using Fenchel-Young losses and leveraging the classical perceptron argument to derive convergence rates.
result GD converges with arbitrary stepsize for a majority of Fenchel-Young losses, with better rates for specific loss functions.
Paper proves a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
problem Proving a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
method Using a locally constrained nonlinear curvature flow to preserve the n-th quermassintegral and decrease the k-th quermassintegral. result Obtained the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in Bn+1. The paper computes Fenchel-Nielsen coordinates for fixed points of cyclic actions on Teichmüller space.
problem Computing Fenchel-Nielsen coordinates for cyclic actions on Teichmüller space.
method Developed algorithms to describe Fenchel-Nielsen coordinates of fixed points of cyclic subgroups of Mod(S_g) on Teich(S_g).
result Computed Fenchel-Nielsen coordinates for cyclic subgroups of orders 10, 8, and 4 in Mod(S_2).
Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition P and given a base complex structure X on S, there is an associated deformation space of complex structures on S, which we call the Fenchel-Nielsen Teichmüller space associated to the pair $(\…
Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.
problem Proving Wolpert's formula for the Weil-Petersson symplectic form.
method Introducing a cell decomposition and groupoid cocycle on a surface to represent points in Teichmüller space.
result Topological proof of Wolpert's formula for the Weil-Petersson symplectic form.
We extend the classical Aleksandrov-Fenchel inequality for mixed volumes to functionals arising naturally in hermitian integral geometry. As a consequence, we obtain Brunn-Minkowski and isoperimetric inequalities for hermitian quermassintegrals.
Study flow on de Sitter space for convex hypersurfaces.
problem Behavior of locally constrained inverse curvature flow in de Sitter space.
method Analyze flow on de Sitter space with specific initial conditions and inequalities.
result Derive Alexandrov-Fenchel type inequalities.
We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov-Fenchel-type inequality for strictly convex hypersurfaces in the n-dimensional sphere, n≥3.
Paper solves inequalities for capillary hypersurfaces in half-spaces.
problem Finding inequalities for convex capillary hypersurfaces in half-spaces.
method Introduced quermassintegrals and constructed a new locally constrained curvature flow to prove convergence to spherical caps.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
The paper defines a new coordinate system for anti-de Sitter structures.
problem Understanding the geometry of anti-de Sitter spaces.
method Combining recent work on anti-de Sitter structures with classical theory.
result Introduced a coordinate system resembling Fenchel-Nielsen coordinates.
In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space $\H^n$, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov-Fenchel inequalities.
We develop Fenchel-Nielsen coordinates for representations of surface groups into Sp(2n,R) with maximal Toledo invariant. Analogous to classical Fenchel-Nielsen coordinates on the Teichmüller space they consist of a parametrization of representations of the fundamental group of a pair of pants and a careful investigati…
New inequalities for convex hypersurfaces in various spaces.
problem Deriving inequalities for hypersurfaces under convex weight.
method Sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces in Euclidean, spherical, and hyperbolic spaces.
result Incorporates convex, non-decreasing positive functions as weights, yielding a broad family of geometric inequalities.
The article proves inequalities for capillary hypersurfaces in hyperbolic space.
problem Proving inequalities for capillary hypersurfaces in hyperbolic space.
method Constructing a new locally constrained inverse curvature flow.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space.
Study geodesic flow on symmetric surfaces to determine parabolic type.
problem Determine conditions for a Riemann surface to be of parabolic type.
method Analyze Fenchel-Nielsen coordinates and covering group properties.
result Conditions for a surface to be parabolic are equivalent to specific properties of its Fenchel-Nielsen coordinates.
Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.
problem Understanding the evolution of hypersurfaces with capillary boundaries.
method Locally constrained mean curvature flow for star-shaped hypersurfaces in the half-space.
result Established new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary.
Paper extends Fenchel's conjecture to non-Euclidean crystallographic groups.
problem Does every non-Euclidean crystallographic group have a specific subgroup?
method Examined various cases of non-Euclidean crystallographic groups and their orbit spaces.
result Affirmative answer in many cases, open in others.
Proof of wall-crossing formula using spectral networks.
problem Proving the Kontsevich-Soibelman wall-crossing formula.
method Path-lifting rules for spectral networks, convergence justification.
result Definition and justification of path lifting rules for spectral networks.
Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.
problem Deriving upper bounds for the Alexandrov-Fenchel deficit.
method Using weighted Minkowski integral formulas and an integral formula for the deficit in Jensen's inequality.
result Quantitative estimates under weaker convexity assumptions, including a distance term.
The article uses harmonic mean curvature flow to prove new geometric inequalities for convex hypersurfaces.
problem Proving new geometric inequalities for convex hypersurfaces in hyperbolic space.
method Harmonic mean curvature flow, Alexandrov-Fenchel inequalities, inverse mean curvature flow, Heintze-Karcher type inequality.
result New geometric inequalities for convex hypersurfaces in hyperbolic space.
We cut a hyperbolic surface of finite area along some analytic simple closed curves, and glue in cylinders of varying moduli. We prove that as the moduli of the glued cylinders go to infinity, the Fenchel-Nielsen twist coordinates for the resulting surface around those cylinders converge.
The paper proves new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
problem Proving new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
method Locally constrained inverse curvature flows in hyperbolic and spherical spaces.
result Established new Alexandrov-Fenchel and Minkowski inequalities involving general convex weight functions.
The paper proves stability of inequalities for nearly spherical sets in various spaces.
problem Stability of geometric inequalities for nearly spherical sets.
method Deriving a quantitative quermassintegral inequality and applying it to derive stability results.
result Stability of geometric inequalities involving weighted curvature integrals and quermassintegrals for nearly spherical sets in Rn+1 and Hn+1. Researchers develop a new framework to control neural network sensitivity.
problem Understanding and controlling the behavior of neural networks.
method Direct parameterization of bi-Lipschitzness in convex neural networks.
result A clear and tight control of neural network sensitivity achieved.
Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.
problem Geometric inequalities and mixed Hodge-Riemann relations for translation-invariant valuations.
method Proves mixed Hodge-Riemann relations for various convex bodies and their mixed volumes.
result Strengthened geometric inequalities for lower dimensional convex bodies.
Dual-based algorithms optimize distributed convex problems over networks.
problem Optimizing distributed convex problems over network constraints.
method Dual formulation of primal problem, distributed algorithms achieving optimal rates.
result Achieves optimal rates similar to centralized algorithms with additional cost related to network spectral properties.
Researchers define new quantum representations for a Lorentz algebra and study their Clebsch-Gordan decomposition.
problem Quantum representations of a Lorentz algebra and their Clebsch-Gordan decomposition.
method Defined new infinite-dimensional irreducible representations using quantum torus algebra and quantized Chern-Simons theory.
result The Clebsch-Gordan decomposition of tensor product representations reduces to problems in Fenchel-Nielson length operators in quantized Chern-Simons theory.
DCCNNs reduce computational overhead and ambiguity in convolutional neural networks.
problem Reducing computational overhead and ambiguity in convolutional neural networks.
method Introducing a primal learning problem and constructing a dual convex training program, using Fenchel conjugates and Karush-Kuhn-Tucker conditions.
result Eliminates ambiguity and reduces computational overhead in constructing a large kernel matrix.