  A common challenge in nonparametric inference is its high computational
complexity when data volume is large. In this paper, we develop computationally
efficient nonparametric testing by employing a random projection strategy. In
the specific kernel ridge regression setup, a simple distance-based test
statistic is proposed. Notably, we derive the minimum number of random
projections that is sufficient for achieving testing optimality in terms of the
minimax rate. An adaptive testing procedure is further established without
prior knowledge of regularity. One technical contribution is to establish upper
bounds for a range of tail sums of empirical kernel eigenvalues. Simulations
and real data analysis are conducted to support our theory.
