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The continuation method is a popular approach in non-convex optimization andcomputer vision. The main idea is to start from a simple function that can beminimized efficiently, and gradually transform it to the more complicatedoriginal objective function. The solution of the simpler problem is used as thestarting point to solve the original problem. We show a continuation method fordiscrete optimization problems. Ideally, we would like the evolved function tobe hill-climbing friendly and to have the same global minima as the originalfunction. We show that the proposed continuation method is the best affineapproximation of a transformation that is guaranteed to transform the functionto a hill-climbing friendly function and to have the same global minima.  We show the effectiveness of the proposed technique in the problem of nearestneighbor classification. Although nearest neighbor methods are oftencompetitive in terms of sample efficiency, the computational complexity in thetest phase has been a major obstacle in their applicability in big dataproblems. Using the proposed continuation method, we show an improvedgraph-based nearest neighbor algorithm. The method is readily understood andeasy to implement. We show how the computational complexity of the method inthe test phase scales gracefully with the size of the training set, a propertythat is particularly important in big data applications.

A Continuation Method for Discrete Optimization and its Application to Nearest Neighbor Classification