Left in a comment in Sec 3

hash_const.tex

@@ -1,5 +1,10 @@
 % -*- root: main.tex -*-
 \section{Hash Function Construction}
+
+\AR{Aaron: Please re-write this section more generally. I.e. instead of assuming $h_i$ and $s_i$ are specifically defined as linear functions, define them generally: i.e. keep $h_i:W\to [B]$ and $s_i:W\to \{-1,1\}$ as generic function but abstract out the properties we want from them-- i.e. (1) $h_i$ is pair-wise independent, (2) $s_i$ is $4$-wise independent and (3) given any $\buck\in \{0,1\}^b$, we want to be able to compute the following quantity in $\mathrm{poly}(N)$ time (or an approximation of it):
+\[|\{\mathbf{w}\in W| h_i(\mathbf{w})=\buck, s_i(\mathbf{w})=1\}|-|\{\mathbf{w}\in W| h_i(\mathbf{w})=\buck, s_i(\mathbf{w})=-1\}|.\]
+From my discussion with the folks here at the workshop the requirement (3) seems to be new for $k$-wise independent hash functions and we should highlight this definition too. Once things have been defined this way, you can state the definition of $h_i$ as you have stated below. But in the next section, it would be good to state the algorithm only in terms of these more general properties of the hash functions. Once you have made this change, I can make a more careful pass over this section and the next.}
+
 As with world identification, bucket identification can be viewed as a binary vector.  This vector is of length $\lenB = \log\sketchCols$, where $\buck \in \{0, 1\}^\lenB$.  Similarly, we can define a set of hash vectors $\matrixH$ as a matrix of $\lenB$ precomputed vectors $\hVec$ where each $\hVec \in \{0, 1\}^\numTup$, formally
 \begin{equation*}
 	\begin{pmatrix*}[l]
@@ -30,4 +35,4 @@ Finally, we augment $\matrixH$ to $\matrixH$' by adding $\mathbf{\sketchPolar}$
 		s_{i, 0} &\cdots &s_{i, \numTup}
 		\end{pmatrix*}.
 \end{equation*}  
-Note that this also turns $\buck$ into a $b + 1$ size column vector, with the last element being the polarity of the hashed world vector.
+Note that this also turns $\buck$ into a $b + 1$ size column vector, with the last element being the polarity of the hashed world vector.