Robust Tensor Completion via Reflective Convolution Nuclear Norm Minimization

Robust tensor completion recovers multidimensional data from partial observations corrupted by sparse gross errors. Existing convolutional low-rank models typically construct translated copies using circular continuation, which introduces artificial wrap-around neighborhoods for finite nonperiodic data. We propose reflective convolution nuclear norm minimization (RCNNM), which replaces circular shifts with endpoint-nonrepeating reflection. The resulting lifting has nonuniform entry multiplicities and satisfies a weighted Gram identity that supports both the recovery analysis and the optimization method. Under random sampling and sparse corruption, we establish high-probability exact recovery of the underlying tensor and sparse errors, together with stability under bounded dense perturbations. We further develop a two-block ADMM with a closed-form entrywise tensor update, while singular-value thresholding is implemented through the smaller right Gram matrix. Experiments on synthetic tensors, BSDS color images, and CAVE multispectral images show that RCNNM consistently improves over its circular-lifting counterpart, with the clearest gains near image boundaries. In particular, average boundary-PSNR improvements reach 3.26 dB while global reconstruction quality remains competitive.