• Corpus ID: 250072109

Fenchel-Moreau identities on convex cones

@inproceedings{Chen2020FenchelMoreauIO,
  title={Fenchel-Moreau identities on convex cones},
  author={Hong-Bin Chen and Jiaming Xia},
  year={2020}
}
. A pointed convex cone naturally induces a partial order, and further a notion of nondecreasingness for functions. We consider extended real-valued functions defined on the cone. Monotone conjugates for these functions can be defined in an analogous way to the standard convex conjugate. The only difference is that the supremum is taken over the cone instead of the entire space. We give sufficient conditions for the cone under which the corresponding Fenchel–Moreau biconjugation identity holds for… 

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References

SHOWING 1-10 OF 26 REFERENCES

Free energy upper bound for mean-field vector spin glasses

We consider vector spin glasses whose energy function is a Gaussian random field with covariance given in terms of the matrix of scalar products. For essentially any model in this class, we give an

Hamilton-Jacobi equations for inference of matrix tensor products

  • Hong-Bin ChenJ. Xia
  • Mathematics
    Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
  • 2022
TLDR
The high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products is studied and the limit is identified with the solution of a certain Hamilton-Jacobi equation.

Hamilton–Jacobi equations for nonsymmetric matrix inference

We study the high-dimensional limit of the free energy associated with the inference problem of a rank-one nonsymmetric matrix. The matrix is expressed as the outer product of two vectors, not

Nonconvex interactions in mean-field spin glasses

  • J. Mourrat
  • Mathematics
    Probability and Mathematical Physics
  • 2020
We propose a conjecture for the limit of mean-field spin glasses with a bipartite structure, and show that the conjectured limit is an upper bound. The conjectured limit is described in terms of the

Monotonic Analysis over Cones: I

In this article, we study increasing and positively homogeneous functions defined on convex cones of locally convex spaces. This work is the first part in a series of studies to have a general view

Parisi's formula is a Hamilton-Jacobi equation in Wasserstein space

Parisi's formula is a self-contained description of the infinite-volume limit of the free energy of mean-field spin glass models. We show that this quantity can be recast as the solution of a

Self-dual cones in Hilbert space

Homogeneous and facially homogeneous self-dual cones

Monotonic analysis over cones: II

In this article, we study the class of increasing and convex along rays (ICAR) functions over a cone. Apart from studying its basic properties, we study them from the point of view of Abstract

Faces and duality in convex cones