منابع مشابه
Exact probabilities for typical ranks of 2 × 2 × 2 and 3 × 3 × 2 tensors
We show that the probability to be of rank 2 for a 2× 2× 2 tensor with elements from a standard normal distribution is π/4, and that the probability to be of rank 3 for a 3× 3× 2 tensor is 1/2. In the proof results on the expected number of real generalized eigenvalues of random matrices are applied. For n × n × 2 tensors with n ≥ 4 we also present some new aspects of their rank.
متن کاملOn the Ranks and Border Ranks of Symmetric Tensors
Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for po...
متن کاملAn Improvement on Ranks of Explicit Tensors
We give constructions of n × n × n tensors of rank at least 2n − O(n). As a corollary we obtain an [n] shaped tensor with rank at least 2n − O(n) when r is odd. The tensors are constructed from a simple recursive pattern, and the lower bounds are proven using a partitioning theorem developed by Brockett and Dobkin. These two bounds are improvements over the previous best-known explicit tensors ...
متن کاملOn Maximum, Typical, and Generic Ranks
We show that for several notions of rank including tensor rank, Waring rank, and generalized rank with respect to a projective variety, the maximum value of rank is at most twice the generic rank. We show that over the real numbers, the maximum value of the real rank is at most twice the smallest typical rank, which is equal to the (complex) generic rank.
متن کاملThe Sign-Real Spectral Radius for Real Tensors
In this paper a new quantity for real tensors, the sign-real spectral radius, is defined and investigated. Various characterizations, bounds and some properties are derived. In certain aspects our quantity shows similar behavior to the spectral radius of a nonnegative tensor. In fact, we generalize the Perron Frobenius theorem for nonnegative tensors to the class of real tensors.
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ژورنال
عنوان ژورنال: Linear and Multilinear Algebra
سال: 2019
ISSN: 0308-1087,1563-5139
DOI: 10.1080/03081087.2019.1637811