From d73d3cdc03337fde998db900ed3232151e75f729 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 25 Oct 2024 19:00:16 +0200
Subject: Clarified the definitions of marginal minima and pseudogap in the
 opening paragraphs

---
 marginal.tex | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index acd3d4a..d9e1d47 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -61,7 +61,10 @@ dynamics would get stuck at a specific energy level, called the threshold
 energy. The threshold energy is the energy level at which level sets of the
 landscape transition from containing mostly saddle points to containing mostly
 minima. The level set associated with this threshold energy contains mostly \emph{marginal
-minima}, or minima that have a pseudogap in the spectrum of their Hessian.
+minima}, or minima whose Hessian matrix has a continuous spectral density over
+all sufficiently small positive eigenvalues. In most circumstances the spectrum
+is \emph{pseudogapped}, which means that the spectral density smoothly
+approaches zero as zero eigenvalue is approached from above.
 
 However, recent work found that the threshold energy is not important even for
 simple gradient descent dynamics \cite{Folena_2020_Rethinking, Folena_2023_On, ElAlaoui_2020_Algorithmic}.
-- 
cgit v1.2.3-70-g09d2

