Analyzing aging-induced delay degradations of ring oscillators (ROs) is an effective way to detect recycled field-programmable gate arrays (FPGAs). However, it requires a large number of RO measurements for all FPGAs before shipping, which increases the measurement costs. We propose a cost-efficient recycled FPGA detection method using a statistical performance characterization technique called virtual probe (VP) based on compressed sensing. The VP technique enables the accurate prediction of the spatial process variation of RO frequencies on a die by using a very small number of sample RO measurements. Using the predicted frequency variation as a supervisor, the machine-learning model classifies target FPGAs as either recycled or fresh. Through experiments conducted using 50 commercial FPGAs, we demonstrate that the proposed method achieves 90% cost reduction for RO measurements while preserving the detection accuracy. Furthermore, a one-class support vector machine algorithm was used to classify target FPGAs with around 94% detection accuracy.