Toshio Sakata

A note on testing two-dimensional normal mean

SummaryFor the problem of testing a composite hypothesis with one-sided alternatives of the mean vector of a two-dimensional normal distribution, a characterization of similar tests is presented and an unbiased test dominating the likelihood ratio test is proposed. A sufficient condition for admissibility is given, which implies the result given by Cohen et al. (1983,Studies in Econometrics, Time Series and Multivariate Statistics, Academic Press): the admissibility of the likelihood ratio test.

Rank of 3-tensors with 2 slices and Kronecker canonical forms

Tensor type data are becoming important recently in various application fields. We determine the rank of a 3-tensor with 2 slices in comparison with its Kronecker canonical form over the complex and real number field.

Algebraic and Computational Aspects of Real Tensor Ranks

Basics of Tensor Rank.- 3-Tensors.- Simple Evaluation Methods of Tensor Rank.- Absolutely Nonsingular Tensors and Determinantal Polynomials.- Maximal Ranks.- Typical Ranks.- Global Theory of Tensor Ranks.- 2 x 2 x * * * x 2 Tensors.

Typical ranks of certain 3-tensors and absolutely full column rank tensors

In this paper, we study typical ranks of 3-tensors and show that there are plural typical ranks for tensors over in the following cases: (1) and , where is the Hurwitz–Radon function, (2) , and , (3) , , and , (4) , , and . (5) , and .

Typical ranks of m × n × (m − 1)n tensors with 3 ≤ m ≤ n over the real number field

Let . We study typical ranks of tensors over the real number field. Let be the Hurwitz–Radon function defined as for nonnegative integers such that and . If , then the set of tensors has two typical ranks . In this paper, we show that the converse is also true: if , then the set of tensors has only one typical rank .

The Deutsch-Jozsa algorithm and the bulk ensemble NMR quantum computer

In a bulk quantum computing it is shown by Collins that the iteration number of the Grovers algorithm will be decreased as compared with that number in case of a single quantum computer system. We discuss some of the implications on the Deutsch-Jozsa algorithm in a bulk quantum computing, and have a plan to prove them experimentally.

A new series of rotation invariant moments by Lie transformation group theory

In automatic word recognition, location-scale-rotation invariant features are important. We consider the rotation invariant moments of 2-D imagery. Two new series of invariant moments are derived by Lie theory of the orthogonal transformation group. The infinite series are expressed in terms of regular moments.

\(2\times 2\times \cdots \times 2\) Tensors

We study an upper bound of the ranks of n-tensors with format \((2,2,\ldots ,2)\) over the complex and real number fields. We consider Cayley’s hyperdeterminant and give the necessary and sufficient condition for a tensor of format (2, 2, 2) to have maximal rank 3. Brylinski showed that the maximal rank of complex tensors with format (2, 2, 2, 2) is 4, and Kong and Jiang showed that the maximal rank of real tensors with format (2, 2, 2, 2) is less than or equal to 5. Catalisano, Geramita, and Gimigliano showed that the generic rank of complex n-tensors with format \((2,2,\ldots ,2)\) is equal to \(\lceil 2^n/(n+1)\rceil \), and then, the maximal rank of real n-tensors with format \((2,2,\ldots ,2)\) is greater than or equal to \(\lceil 2^n/(n+1)\rceil \). The maximal rank of real n-tensors with format \((2,2,\ldots ,2)\) is less than or equal to \(2^{n-1}+1\). Bleckherman and Teitler showed that the maximal rank is less than or equal to twice the generic rank, which gives a more strict upper bound of the maximal rank of n-tensors with format \((2,2,\ldots ,2)\) over the real number field.

Global Theory of Tensor Ranks

The generic rank is considered under the complex number field, and it corresponds with the dimension of the secant variety. The dimension is studied in the area of algebraic geometry. In this chapter, we introduce known results and discuss the typical rank from the point of view of the Jacobian matrix. The generic rank attains the minimal typical rank. The typical rank of the set of tensors with format (m, n, p) for \(3\le m\le n\) is equal to \(\min (p,mn)\) if \(p>(m-1)n\). Note that the typical rank is sometimes not unique and the set of typical ranks consists of integers between some integers a and b. For a positive integer r, we consider the image of the summation map, which gives the set of tensors with rank less than or equal to r. Strassen and Tickteig introduced the idea of computing the dimension of the secant variety via the Jacobi criterion and the splitting technique. The generic rank of the set of tensors with format (m, n, p) is \(\lceil mnp/(m+n+p-2)\rceil \) if m, n, and p are sufficiently large integers, although the generic rank of the set of tensors with format (n, n, 3) is equal to \((3n+1)/2\) which is greater than \(\lceil 3n^2/(2n+1)\rceil \) if n is odd.

Simple Evaluation Methods of Tensor Rank

In this chapter, we illustrate simple evaluations of the rank of 3-mode tensors, which might facilitate readers’ understanding of tensor rank.