An Adaptive Test of Independence with Analytic Kernel Embeddings

B. Redundant Test Locations Here, we provide a simple illustration to show that two locations t1 = (v1,w1) and t2 = (v2,w2) which are too close to each other will reduce the optimization objective. We consider the Sinusoid problem described in Section 3.1 with ω = 1, and use J = 2 test locations. In Figure 6, t1 is fixed at the red star, while t2 is varied along the horizontal line. The objective value λ̂n as a function of t2 is shown in the bottom figure. It can be seen that λ̂n decreases sharply when t2 is in the neighborhood of t1. This property implies that two locations which are too close will not maximize the objective function (i.e., the second feature contains no additional information when it matches the first). For J > 2, the objective sharply decreases if any two locations are in the same neighborhood.