Hamilton–Jacobi equations for finite-rank matrix inference

Mourrat, Jean-Christophe

doi:10.1214/19-aap1556

Cited by 25 publications

(33 citation statements)

References 14 publications

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“…We have thus obtained a formal relation between the Hamilton-Jacobi equation in (1.2) and that in (5.3), which itself can be rigorously connected with the Hopf-Lax formula (5.1)-see [21] and [22] on how to handle the boundary condition on ∂(R k + ) in the contexts of viscosity solutions and weak solutions respectively.…”

Section: The Parisi Formula Is a Hamilton-jacobi Equation In Wasserstein Spacementioning

confidence: 99%

The Parisi formula is a Hamilton–Jacobi equation in Wasserstein space

Mourrat

2021

Can. J. Math.-J. Can. Math.

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Section: The Parisi Formula Is a Hamilton-jacobi Equation In Wasserstein Spacementioning

confidence: 99%

The Parisi formula is a Hamilton–Jacobi equation in Wasserstein space

Mourrat

2021

Can. J. Math.-J. Can. Math.

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“…Later, Panchenko proved the Parisi formula for general mixed p-spin models [58] by showing that the Ghirlanda-Guerra identities imply ultrametricity for replica overlaps [56]. Recently Mourrat [48] has reinterpreted these Parisi formulas as the solution to a Hamilton-Jacobi equation in the Wasserstein space of probability measures on the positive half-line; see [50,49,21,23,22] for finite-dimensional analogues, and [52] for a generalized result.…”

Section: 21)mentioning

confidence: 99%

Free energy in multi-species mixed $p$-spin spherical models

Bates,

Sohn

2021

Preprint

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We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed p-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model's covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constraint set is a compact metric space. This space is not related to the model's spherical structure and can be carried over to other multi-species settings.

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“…Considerations of using Hamilton-Jacobi equations to study the free energy of mean-field disordered models first appeared in physics literature [30,31,6,5]. The approach was mathematically initiated in [38], and used subsequently in [37,12,15,13,16] to treat statistical inference models. There, the equations also take the form (1.1) but are defined on finite-dimensional cones.…”

Section: Introductionmentioning

confidence: 99%

Hamilton-Jacobi equations from mean-field spin glasses

Chen¹,

Xia²

2022

Preprint

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We establish the well-posedness of Hamilton-Jacobi equations arising from meanfield spin glass models in the viscosity sense. Originally defined on the set of monotone probability measures, these equation can be interpreted, via an isometry, to be defined on an infinite-dimensional closed convex cone with empty interior in a Hilbert space. We prove the comparison principle, and the convergence of finite-dimensional approximations furnishing the existence of solutions. Under additional convexity conditions, we show that the solution can be represented by a version of the Hopf-Lax formula, or the Hopf formula on cones. As the first step, we show the well-posedness of equations on finite-dimensional cones, which is self-contained and, we believe, is of independent interest. The surprising observation key to our work is that a monotonicity assumption on the nonlinearity allows us to prescribe that the solution satisfies the equation in the viscosity sense at any point, without having to distinguish between interior and boundary points.Previously, two notions of solutions were considered, one defined directly as the Hopf-Lax formula, and another as limits of finite-dimensional approximations. They have been proven to describe the limit free energy in a wide class of mean-field spin glass models. This work shows that these two kinds of solutions are viscosity solutions.

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Hamilton–Jacobi equations for finite-rank matrix inference

Cited by 25 publications

References 14 publications

The Parisi formula is a Hamilton–Jacobi equation in Wasserstein space

The Parisi formula is a Hamilton–Jacobi equation in Wasserstein space

Free energy in multi-species mixed $p$-spin spherical models

Hamilton-Jacobi equations from mean-field spin glasses

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