The tensor–tensor product-induced tensor nuclear norm (t-TNN) (Lu et al. , 2020) minimization for low-tubal-rank tensor recovery attracts broad attention recently. However, minimizing the t-TNN faces some drawbacks. For example, the obtained solution could be suboptimal to the original problem due to its loose approximation. In this article, we extract a unified nonconvex surrogate of the tensor tubal rank as a tighter regularizer, which involves many popular nonconvex penalty functions. An iterative reweighted t-TNN algorithm is proposed to solve the resulting generalized nonconvex tubal rank minimization for tensor recovery. It converges to a critical point globally with rigorous proofs based on the Kurdyka–Łojasiwicz property. Furthermore, we provide the theoretical guarantees for exact and robust recovery by developing the tensor null space property. Extensive experiments demonstrate that our approach markedly enhances recovery performance compared with several state-of-the-art convex and nonconvex methods.

One-bit compressed sensing aims to recover unknown sparse signals from extremely quantized linear measurements which just capture their signs. In this paper, we propose a nonconvex ℓ _p (0 < p < 1) minimization model for one-bit compressed sensing problem and define the set of ℓ _p effectively s -sparse signals which contains genuinely s -sparse signals. Utilizing properties of covering number, we show that our method can recover the direction of ℓ _p effectively s -sparse signals with error order O ( ( s / m log ( m n / s ) ) 2 − p 2 + p ) . We also employ thresholded one-bit measurements to estimate the magnitude of signals and prove that any ℓ _p effectively s -sparse bounded signal x can be estimated using augmented ℓ _p minimization model and empirical distribution function method respectively. Especially, to recover ℓ _p effectively s -sparse signals in practice, we introduce an adaptive binary iterative thresholding algorithm which can be utilized without knowing the sparsity of underlying signals. Numerical experiments on both synthetic and real-world data sets are conducted to demonstrate the superiority of our algorithm.