arXiv subjects
Publications and source records attributed to Jeong-Hoon Ju.
We show that the border rank of the $4 \times 4$ determinant tensor is at least $12$ over $\mathbb{C}$, using the fixed ideal theorem introduced by Buczyńska-Buczyński and the method by Conner-Harper-Landsberg. Together with the known upper bound, this implies that the border rank is exactly $12$.
Every (resp. a generic) complex $n \times n$ matrix can be expressed as a product of $2n+5$ (resp. $\lfloor n/2 \rfloor +1$) Toeplitz matrices. Motivated by this result, it is natural to ask the following question: what is the minimum number of Toeplitz matrices required to factor a given matrix? We generalize this question from Toeplitz structure to more general structures. In this paper, we introduce the notion of structured matrix factorization length when the set of matrices with a given structure is an affine variety $X \subseteq \mathbb{C}^{n \times n}$. Then we introduce the $r$-th $X$-factorization variety, defined as the Zariski closure of the set of products of $r$ matrices in $X$, and use it to define the border structured matrix factorization length. In particular, we study the cases in which $X$ is the affine variety of Toeplitz, Hankel, bidiagonal, tridiagonal, skew-symmetric or companion matrices. We calculate the dimension of the $X$-factorization varieties for all these cases, and discuss how numerical algebraic geometry can be used to obtain computational evidence for the degrees of $X$-factorization varieties with an example. In addition, we propose methods for deriving lower and upper bounds for (border) structured matrix factorization length. For lower bounds, we develop a method based on displacement rank, which can also be used to obtain some defining equations of the $r$-th $X$-factorization variety; for upper bounds, we suggest an approach using alternating minimization.
In this paper we study the computation of both algebraic and non-algebraic tensor functions under the tensor-tensor multiplication with linear maps. In the case of algebraic tensor functions, we prove that the asymptotic exponent of both the tensor-tensor multiplication and the tensor polynomial evaluation problem under this multiplication is the same as that of the matrix multiplication, unless the linear map is injective. As for non-algebraic functions, we define the tensor geometric mean and the tensor Wasserstein mean for pseudo-positive-definite tensors under the tensor-tensor multiplication with invertible linear maps, and we show that the tensor geometric mean can be calculated by solving a specific Riccati tensor equation. Furthermore, we show that the tensor geometric mean does not satisfy the resultantal (determinantal) identity in general, which the matrix geometric mean always satisfies. Then we define a pseudo-SVD for the injective linear map case and we apply it on image data compression.
We investigate new lower bounds on the tensor rank of the determinant and the permanent tensors via recursive usage of the Koszul flattening method introduced by Landsberg-Ottaviani and Hauenstein-Oeding-Ottaviani-Sommese. Our lower bounds on $\mathbf{R} (\det_n)$ completely separate the determinant and the permanent tensors by their tensor ranks. Furthermore, we determine the exact tensor ranks $\mathbf{R} (\det_4) = 12$ and $\mathbf{R} (\operatorname{perm}_4) = 8$ over arbitrary field of characteristic $\neq 2$.
In this paper, we generalize the geometric mean of two positive definite matrices to that of third-order tensors using the notion of T-product. Specifically, we define the geometric mean of two T-positive definite tensors and verify several properties that "mean" should satisfy including the idempotence and the commutative property, and so on. Moreover, it is shown that the geometric mean is a unique T-positive definite solution of an algebraic Riccati tensor equation and can be expressed as solutions of algebraic Riccati matrix equations. In addition, we investigate the Riemannian manifold associated with the geometric mean for T-positive definite tensors, considering it as a totally geodesic embedded submanifold of the Riemannian manifold associated with the case of matrices. It is particularly shown that the geometric mean of two T-positive definite tensors is the midpoint of a unique geodesic joining the tensors, and the manifold is a Cartan-Hadamard-Riemannian manifold.
In this paper, we present a new formula for the determinant of a $4 \times 4$ matrix. We approach via the sparse optimization problem and derive the formula through the Least Absolute Shrinkage and Selection Operator (LASSO). Our formula has the potential to advance understanding of the algebraic structure of determinants, such as an upper bound of the tensor rank, various notions to measure complexity, and effective computational tools in exterior algebras. We also address several numerical experiments which compare our formula with built-in functions in a computer-algebra system.
In this paper, we suggest a new method for a given tensor to find CP decompositions using a less number of rank $1$ tensors. The main ingredient is the Least Absolute Shrinkage and Selection Operator (LASSO) by considering the decomposition problem as a sparse optimization problem. As applications, we design experiments to find some CP decompositions of the matrix multiplication and determinant tensors. In particular, we find a new formula for the $4 \times 4$ determinant tensor as a sum of $12$ rank $1$ tensors.
In this paper, we present a new formula of the determinant tensor $det_n$ for $n \times n$ matrices. In \cite{kim2023newdet4}, Kim, Ju, and Kim found a new formula of $4 \times 4$ determinant tensor $det_4$ which is available when the base field is not of characteristic $2$. Considering some symmetries in that formula, we found a new formula so that \begin{equation*} \operatorname{Crank}(det_n) \leq \operatorname{rank}(det_n) \leq \frac{n!}{2^{\lfloor(n-2)/2 \rfloor}} \end{equation*} when the base field is not of characteristic $2$.