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@@ -96,6 +96,10 @@ $k$. I think it would make more sense from the learning perspective to consider
 uniformly random sets. In this case, the above approach allows us to conclude
 directly.
 
+More generally, I think the “right” way to think about this is to look at $A$
+as the adjacency matrix of a random $k$-regular graph. There are tons of
+results about the probability of the adjacency matrix to be non singular.
+
 \section{Passive Optimization of Additive Functions}
 
 Let $\Omega$ be a universe of elements of size $n$. For $k \in [n]$, let $D_k$ be the uniform probability distribution over all sets of size $k$, and let $D_p$ be the product probability distribution over sets such that $\forall j \in \Omega, \mathbb{P}_{S \sim D_p}(j \in S) = p$. Note that $D_{k}$ and $D_{p}$ for $p \equiv \frac{k}{n}$ are very similar, and will be considered almost interchangeably throughout the following notes.