During state perception of a power system, fragments of harmonic data are inevitably lost owing to the loss of synchronization signals, transmission delays, instrument failures, or other factors. A harmonic data recovery method is proposed based on multivariate norm matrix in this paper. The proposed method involves dynamic time warping for correlation analysis of harmonic data, normalized cuts for correlation clustering of power-quality monitoring devices, and adaptive alternating direction method of multipliers for multivariable norm joint optimization. Compared with existing data recovery methods, our proposed method maintains excellent recovery accuracy without requiring prior information or optimization of the power-quality monitoring device. Simulation results on the IEEE 39-bus and IEEE 118-bus test systems demonstrate the low computational complexity of the proposed method and its robustness against noise. In addition, the application of the proposed method to field data from a real-world system provides consistent results with those obtained from simulations.