Exploring Tensor Rank
نویسندگان
چکیده
We consider the problem of tensor rank. We define tensor rank, discuss the motivations behind exploring the topic, and give some examples of the difficulties we face when trying to compute tensor rank. Some simpler lower and upper bounds for tensor rank are proven, and two techniques for giving lower bounds are explored. Finally we give one explicit example of a construction of an n×n×n tensors of rank 2nk−O(nk−1). As a corollary we obtain an [n] shaped tensor with rank 2nbr/2c −O(nbr/2c−1) when r is odd, an improvement from the previously best-known construction of nbr/2c.
منابع مشابه
Tensor rank is not multiplicative under the tensor product
The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an `-tensor. The tensor product of s and t is a (k + `)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general...
متن کاملNew Ranks for Even-Order Tensors and Their Applications in Low-Rank Tensor Optimization
In this paper, we propose three new tensor decompositions for even-order tensors corresponding respectively to the rank-one decompositions of some unfolded matrices. Consequently such new decompositions lead to three new notions of (even-order) tensor ranks, to be called the M-rank, the symmetric M-rank, and the strongly symmetric M-rank in this paper. We discuss the bounds between these new te...
متن کاملBeyond Low Rank: A Data-Adaptive Tensor Completion Method
Low rank tensor representation underpins much of recent progress in tensor completion. In real applications, however, this approach is confronted with two challenging problems, namely (1) tensor rank determination; (2) handling real tensor data which only approximately fulfils the low-rank requirement. To address these two issues, we develop a data-adaptive tensor completion model which explici...
متن کاملA note on the gap between rank and border rank
We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a counterexample to a conjecture of Rhodes. At the same time we obtain a new lower bound on the tensor rank of tensor powers of the generalised W-state tensor....
متن کاملun 2 00 9 Subtracting a best rank - 1 approximation may increase tensor rank
It has been shown that a best rank-R approximation of an order-k tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximati...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011