Related papers: New lower bounds for matrix multiplication and the…
We present new lower bounds for the symmetric border rank of the n x n determinant for all n. Further lower bounds are given for the 3 x 3 permanent.
An $(m,n,a,b)$-tensor code consists of $m\times n$ matrices whose columns satisfy `$a$' parity checks and rows satisfy `$b$' parity checks (i.e., a tensor code is the tensor product of a column code and row code). Tensor codes are useful in…
In a paper published in 1981, Sch\"onhage showed that large total matrix multiplications can be reduced to powers of partial matrix multiplication tensors, which correspond to the bilinear computation task of multiplying matrices with some…
We prove that in any $\mathbb{Z}^n$-periodic triangulation of $\mathbb{R}^n$ the number of $\mathbb{Z}^n$-orbits of $n$-dimensional simplices is at least the tensor rank of the $n$th determinant tensor. The latter is known to be at least…
We determine defining equations for the set of concise tensors of minimal border rank in $C^m\otimes C^m\otimes C^m$ when $m=5$ and the set of concise minimal border rank $1_*$-generic tensors when $m=5,6$. We solve this classical problem…
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…
In numerical analysis it is often necessary to estimate the condition number $CN(T)=||T||_{} \cdot||T^{-1}||_{}$ and the norm of the resolvent $||(\zeta-T)^{-1}||_{}$ of a given $n\times n$ matrix $T$. We derive new spectral estimates for…
A tight $\Omega((n/\sqrt{M})^{\log_2 7}M)$ lower bound is derived on the \io complexity of Strassen's algorithm to multiply two $n \times n$ matrices, in a two-level storage hierarchy with $M$ words of fast memory. A proof technique is…
It is proved that there is no an algorithm for multiplication of $3\times3$ matrices of multiplicative length $\leq23$ that is invariant under a certain group isomorphic to $S_4\times S_3$. The proof makes use of description of the orbits…
We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in…
Matrix multiplication is a fundamental kernel in high performance computing. Many algorithms for fast matrix multiplication can only be applied to enormous matrices ($n>10^{100}$) and thus cannot be used in practice. Of all algorithms…
We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various…
Let $m,n\geq 3$, $(m-1)(n-1)+2\leq p\leq mn$, and $u=mn-p$. The set $\mathbb{R}^{u\times n\times m}$ of all real tensors with size $u\times n\times m$ is one to one corresponding to the set of bilinear maps $\mathbb{R}^m\times…
We give constructions of n^k x n^k x n tensors of rank at least 2n^k - O(n^(k-1)). As a corollary we obtain an [n]^r shaped tensor with rank at least 2n^(r/2) - O(n^(r/2)-1) when r is odd. The tensors are constructed from a simple recursive…
There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher, that parallels the Eckart-Young theorem for matrices. In this paper, we argue that the naive approach to this…
Matrix-vector multiplication is one of the most fundamental computing primitives. Given a matrix $A\in\mathbb{F}^{N\times N}$ and a vector $b$, it is known that in the worst case $\Theta(N^2)$ operations over $\mathbb{F}$ are needed to…
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 Buczy\'nska-Buczy\'nski and the method by Conner-Harper-Landsberg. Together with the known…
We provide a new lower bound on the number of $(\leq k)$-edges of a set of $n$ points in the plane in general position. We show that for $0 \leq k \leq\lfloor\frac{n-2}{2}\rfloor$ the number of $(\leq k)$-edges is at least $$ E_k(S) \geq…
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…
An integer-valued matrix $\mathbf{A}$ is $\Delta$-modular if each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix has determinant at most $\Delta$ in absolute value. The column number problem is to determine the maximum…