Abstract
It is known that computing the spectral norm and the nuclear norm of a tensor is NP-hard in general. In this paper, we provide neat bounds for the spectral norm and the nuclear norm of a tensor based on tensor partitions. The spectral norm (respectively, the nuclear norm) can be lower and upper bounded by manipulating the spectral norms (respectively, the nuclear norms) of its subtensors. The bounds are sharp in general. When a tensor is partitioned into its matrix slices, our inequalities provide polynomial-time worst-case approximation bounds for computing the spectral norm and the nuclear norm of the tensor.
Highlights
Given any tensor T that is well partitioned into any set of subtensors {T1, T2, . . . , Tm}, its spectral norm T σ and its nuclear norm T ∗ are bounded as follows:(1) ( T1 σ, T2 σ, . . . , Tm σ) ∞ ≤ T σ ≤ ( T1 σ, T2 σ, . . . , Tm σ) 2, (2)( T1 ∗, T2 ∗, . . . , Tm ∗) 2 ≤ T ∗ ≤ ( T1 ∗, T2 ∗, . . . , Tm ∗) 1, where · p stands for the Lp norm of a vector for 1 ≤ p ≤ ∞
According to our main result, one useful regular partition is to cut a tensor into matrix slices since both the matrix spectral norm and the matrix nuclear norm can be computed in polynomial time
We study the spectral norm and the nuclear norm of a tensor from a new perspective
Summary
Given any tensor T that is well partitioned (a formal definition called a regular partition is given in Definition 2.5) into any set of subtensors {T1, T2, . . . , Tm}, its spectral norm T σ and its nuclear norm T ∗ are bounded as follows:. Friedland and Lim [3] showed that the computational complexity of the tensor nuclear norm is NP-hard They proposed simple lower and upper bounds for the tensor spectral norm and the tensor nuclear norm [3, Lemma 9.1], which are implied by a special case of our main results in this paper. Li, and Zhang [7] essentially applied the matrix flattening to obtain a worst-case approximation bound for the tensor spectral norm. They provide easy computable worst-case approximation bounds for the tensor spectral norm and nuclear norm. The approximation bound for the tensor nuclear norm is currently the best We believe these inequalities will have potential both in theory and in practice. We may define the Lp norm of a tensor ( known as the Holder p-norm) for 1 ≤ p ≤ ∞ by looking at a tensor as a vector, as follows:
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