  The continuation method is a popular approach in non-convex optimization and
computer vision. The main idea is to start from a simple function that can be
minimized efficiently, and gradually transform it to the more complicated
original objective function. The solution of the simpler problem is used as the
starting point to solve the original problem. We show a continuation method for
discrete optimization problems. Ideally, we would like the evolved function to
be hill-climbing friendly and to have the same global minima as the original
function. We show that the proposed continuation method is the best affine
approximation of a transformation that is guaranteed to transform the function
to a hill-climbing friendly function and to have the same global minima.
  We show the effectiveness of the proposed technique in the problem of nearest
neighbor classification. Although nearest neighbor methods are often
competitive in terms of sample efficiency, the computational complexity in the
test phase has been a major obstacle in their applicability in big data
problems. Using the proposed continuation method, we show an improved
graph-based nearest neighbor algorithm. The method is readily understood and
easy to implement. We show how the computational complexity of the method in
the test phase scales gracefully with the size of the training set, a property
that is particularly important in big data applications.
