Clarified the definitions of marginal minima and pseudogap in the

opening paragraphs

1 files changed, 4 insertions, 1 deletions

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}.