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| -rw-r--r-- | marginal.tex | 4 |
1 files changed, 2 insertions, 2 deletions
diff --git a/marginal.tex b/marginal.tex index 8414310..508b674 100644 --- a/marginal.tex +++ b/marginal.tex @@ -95,7 +95,7 @@ Hessian are necessary to lead to marginal minima. This strategy is so successful in the spherical spin glasses because it is straightforward to implement. First, the shape of the Hessian's spectrum is independent of energy and even whether one sits at a stationary point or not. This is a property of models -whose energy is a Gaussian random variable \cite{Bray_2007_Statistics}. +whose energy is a Gaussian random variable \cite{Fyodorov_2004_Complexity, Bray_2007_Statistics}. Furthermore, a natural parameter in the analysis of these models linearly shifts the spectrum of the Hessian. Therefore, tuning this parameter to a specific constant value allows one to require that the Hessian spectrum have a @@ -761,7 +761,7 @@ The marginal optima of these models can be studied without the methods introduced in this paper, and have been in the past \cite{Folena_2020_Rethinking, Kent-Dobias_2023_How}. First, these models are Gaussian, so at large $N$ the Hessian is statistically independent of the gradient and energy -\cite{Bray_2007_Statistics}. Therefore, conditioning the Hessian can be done +\cite{Fyodorov_2004_Complexity, Bray_2007_Statistics}. Therefore, conditioning the Hessian can be done mostly independently from the problem of counting stationary points. Second, in these models the Hessian at every point in the landscape belongs to the GOE class with the same width of the spectrum $\mu_\mathrm m=2\sqrt{f''(1)}$. |