The spectral p-norm and nuclear p-norm of matrices and tensors appear in various applications, albeit both are NP-hard to compute. The former sets a foundation of [Formula: see text]-sphere-constrained polynomial optimization problems, and the latter has been found in many rank minimization problems in machine learning. We study approximation algorithms of the tensor nuclear p-norm with an aim to establish the approximation bound matching the best one of its dual norm, the tensor spectral p-norm. Driven by the application of sphere covering to approximate both tensor spectral and nuclear norms ([Formula: see text]), we propose several types of hitting sets that approximately represent the [Formula: see text]-sphere with adjustable parameters for different levels of approximations and cardinalities, providing an independent toolbox for decision making on [Formula: see text]-spheres. Using the idea in robust optimization and second-order cone programming, we obtain the first polynomial-time algorithm with an [Formula: see text]-approximation bound for the computation of the matrix nuclear p-norm when [Formula: see text] is a rational, paving a way for applications in modeling with the matrix nuclear p-norm. These two new results enable us to propose various polynomial-time approximation algorithms for the computation of the tensor nuclear p-norm using tensor partitions, convex optimization, and duality theory, attaining the same approximation bound to the best one of the tensor spectral p-norms. Effective performance of the proposed algorithms for the tensor nuclear p-norm is shown by numerical implementations. We believe the ideas of [Formula: see text]-sphere covering with its applications in approximating nuclear p-norm would be useful for tackling optimization problems on other sets, such as the binary hypercube with its applications in graph theory and neural networks and the nonnegative sphere with its applications in copositive programming and nonnegative matrix factorization. Funding: This work was supported by the National Natural Science Foundation of China [Grants 71825003, 72394360, 72394364, 72171141, 72192830, and 72192832], and the program for Innovative Research Team of Shanghai University of Finance and Economics.
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