1 research outputs found
Robust Testing of Low Dimensional Functions
A natural problem in high-dimensional inference is to decide if a classifier
depends on a small number of linear
directions of its input data. Call a function , a linear -junta if it is completely determined by some
-dimensional subspace of the input space. A recent work of the authors
showed that linear -juntas are testable. Thus there exists an algorithm to
distinguish between: 1. which is a
linear -junta with surface area , 2. is -far from any
linear -junta with surface area , where the query complexity
of the algorithm is independent of the ambient dimension .
Following the surge of interest in noise-tolerant property testing, in this
paper we prove a noise-tolerant (or robust) version of this result. Namely, we
give an algorithm which given any , , distinguishes between 1.
has correlation at least with some
linear -junta with surface area . 2. has correlation at most
with any linear -junta with surface area at most . The query
complexity of our tester is .
Using our techniques, we also obtain a fully noise tolerant tester with the
same query complexity for any class of linear -juntas with
surface area bounded by . As a consequence, we obtain a fully noise tolerant
tester with query complexity for the
class of intersection of -halfspaces (for constant ) over the Gaussian
space. Our query complexity is independent of the ambient dimension .
Previously, no non-trivial noise tolerant testers were known even for a single
halfspace.Comment: We significantly strengthen the results of the previous version. This
includes the first fully noise tolerant testers for linear juntas as well as
subclasses such as any function of constantly many halfspace