Related papers: A generalization of Kruskal's theorem on tensor de…
Tensor train (TT) decomposition is a powerful representation for high-order tensors, which has been successfully applied to various machine learning tasks in recent years. However, since the tensor product is not commutative, permutation of…
Biquadratic tensors play a central role in many areas of science. Examples include elasticity tensor and Eshelby tensor in solid mechanics, and Riemann curvature tensor in relativity theory. The singular values and spectral norm of a…
The tensor rank decomposition, also known as canonical polyadic(CP) or simply tensor decomposition, has a long history in multilinear algebra. However, computing a rank decomposition becomes particularly challenging when the rank lies…
We show that the slice rank of the direct sum of two tensors is equal to the sum of their slice ranks. The upper bound is trivial, but the lower bound needs more than a one-line proof, for reasons we explain. This result generalizes the…
A W-state is an order d symmetric tensor of the form W_d=x^{d-1}y. We prove that the partially symmetric rank of W_{d_1}\otimes \cdots \otimes W_{d_k} is at most 2^{k-1}(d_1+\cdots +d_k-2k+2). The same bound holds for the tensor rank and it…
We prove the existence of an open set of $n_1\times n_2 \times n_3$ tensors of rank $r$ on which a popular and efficient class of algorithms for computing tensor rank decompositions based on a reduction to a linear matrix pencil, typically…
We show that sufficiently low tensor rank for the balanced tripartitioning tensor $P_d(x,y,z)=\sum_{A,B,C\in\binom{[3d]}{d}:A\cup B\cup C=[3d]}x_Ay_Bz_C$ for a large enough constant $d$ implies uniform arithmetic circuits for the matrix…
The celebrated Stallings' decomposition theorem states that the splitting of a finite index subgroup $H$ of a finitely generated group $G$ as an amalgamated free product or an HNN-extension over a finite group implies the same for $G$. We…
Motivated by the study of decompositions of tensors as Hadamard products (i.e., coefficient-wise products) of low-rank tensors, we introduce the notion of Hadamard rank of a given point with respect to a projective variety: if it exists, it…
The Tucker decomposition expresses a given tensor as the product of a small core tensor and a set of factor matrices. Apart from providing data compression, the construction is useful in performing analysis such as principal component…
We compute the expected value of powers of the geometric condition number of random tensor rank decompositions. It is shown in particular that the expected value of the condition number of $n_1\times n_2 \times 2$ tensors with a random…
Given a tensor $f$ in a Euclidean tensor space, we are interested in the critical points of the distance function from $f$ to the set of tensors of rank at most $k$, which we call the critical rank-at-most-$k$ tensors for $f$. When $f$ is a…
Saxl's conjecture (2012) asserts that for the staircase partition $\rho_k = (k, k-1, \ldots, 1)$, the tensor square of the corresponding irreducible representation of the symmetric group $S_{T_k}$ contains every irreducible representation…
Conjugate partial-symmetric (CPS) tensors are the high-order generalization of Hermitian matrices. As the role played by Hermitian matrices in matrix theory and quadratic optimization, CPS tensors have shown growing interest recently in…
We introduce and extend the outer product and contractive product of tensors and matrices, and present some identities in terms of these products. We offer tensor expressions of derivatives of tensors, focus on the tensor forms of…
In this paper, we consider a subclass of quantum states in the multipartite system, namely, the supersymmetric states. We investigate the problem whether they admit the symmetrically separable decomposition, i.e., each term in this…
Since the seminal works of Strassen and Valiant it has been a central theme in algebraic complexity theory to understand the relative complexity of algebraic problems, that is, to understand which algebraic problems (be it bilinear maps…
For polynomial representations of $GL_n$ of a fixed degree, H. Krause defined a new internal tensor product using the language of strict polynomial functors. We show that over an arbitrary commutative base ring $k$, the Schur functor…
Disentanglement is the process which transforms a state $\rho$ of two subsystems into an unentangled state, while not effecting the reduced density matrices of each of the two subsystems. Recently Terno showed that an arbitrary state cannot…
We introduce a notion of amenable normal extension S of a unital ring R with a finite approximation system F, encompassing the amenable algebras over a field of Gromov and Elek, the twisted crossed product by an amenable group, and the…