  We construct near optimal linear decision trees for a variety of decision
problems in combinatorics and discrete geometry. For example, for any constant
$k$, we construct linear decision trees that solve the $k$-SUM problem on $n$
elements using $O(n \log^2 n)$ linear queries. Moreover, the queries we use are
comparison queries, which compare the sums of two $k$-subsets; when viewed as
linear queries, comparison queries are $2k$-sparse and have only $\{-1,0,1\}$
coefficients. We give similar constructions for sorting sumsets $A+B$ and for
solving the SUBSET-SUM problem, both with optimal number of queries, up to
poly-logarithmic terms.
  Our constructions are based on the notion of "inference dimension", recently
introduced by the authors in the context of active classification with
comparison queries. This can be viewed as another contribution to the fruitful
link between machine learning and discrete geometry, which goes back to the
discovery of the VC dimension.
