Maximum Dimension of Subspaces with No Product Basis

• URL: http://arxiv.org/abs/2010.16293v1
• Date: Fri, 30 Oct 2020 14:39:58 GMT
• Title: Maximum Dimension of Subspaces with No Product Basis
• Authors: Yuuya Yoshida
• Abstract summary: We show that the maximum dimension of subspaces of $mathcalFd_otimescdotsotimesmathcalFd_n$ with no product basis is equal to $d_n-2$ if either (i) $n=2$ or (ii) $nge3$ and $#mathcalF>maxd_i. When $mathcalF=bbC$, this result is related to the maximum number of simultaneously distinguishable states in general probabilistic theories
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Let $n\ge2$ and $d_1,\ldots,d_n\ge2$ be integers, and $\mathcal{F}$ be a field. A vector $u\in\mathcal{F}^{d_1}\otimes\cdots\otimes\mathcal{F}^{d_n}$ is called a product vector if $u=u^{[1]}\otimes\cdots\otimes u^{[n]}$ for some $u^{[1]}\in\mathcal{F}^{d_1},\ldots,u^{[n]}\in\mathcal{F}^{d_n}$. A basis composed of product vectors is called a product basis. In this paper, we show that the maximum dimension of subspaces of $\mathcal{F}^{d_1}\otimes\cdots\otimes\mathcal{F}^{d_n}$ with no product basis is equal to $d_1d_2\cdots d_n-2$ if either (i) $n=2$ or (ii) $n\ge3$ and $\#\mathcal{F}>\max\{d_i : i\not=n_1,n_2\}$ for some $n_1$ and $n_2$. When $\mathcal{F}=\mathbb{C}$, this result is related to the maximum number of simultaneously distinguishable states in general probabilistic theories (GPTs).

Related papers

• Weight distribution of a class of $p$-ary codes [0.0]
  We prove all possible weights of codewords in $mathcalC_alpha,beta,beta$, demonstrating that it has at most $p+1$ distinct nonzero weights. We also prove that the dual code $mathcalC_alpha,beta$ is optimal with respect to the sphere packing bound.
  arXiv  Detail & Related papers  (2025-03-24T20:53:04Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• A class of ternary codes with few weights [0.0]
  In this paper, we investigate a ternary code $mathcalC$ of length $n$, defined by $mathcalC$ := (textTr) := (textTr(dx), dots, dots, d_n$. Using recent results on explicit evaluations of exponential sums, we determine the Weil bound, and techniques, we show that the dual code of $mathcalC$ is optimal with respect to the Hamming bound.
  arXiv  Detail & Related papers  (2024-10-05T16:15:50Z)
• The Differential and Boomerang Properties of a Class of Binomials [28.489574654566677]
  We study the differential and boomerang properties of the function $F_2,u(x)=x2big (1+ueta(x)big)$ over $mathbbF_q$. We disproving a conjecture proposed in citebudaghyan2024arithmetization which states that there exist infinitely many $q$ and $u$ such that $F_2,u$ is an APN function.
  arXiv  Detail & Related papers  (2024-09-21T23:33:00Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Efficient Continual Finite-Sum Minimization [52.5238287567572]
  We propose a key twist into the finite-sum minimization, dubbed as continual finite-sum minimization. Our approach significantly improves upon the $mathcalO(n/epsilon)$ FOs that $mathrmStochasticGradientDescent$ requires. We also prove that there is no natural first-order method with $mathcalOleft(n/epsilonalpharight)$ complexity gradient for $alpha 1/4$, establishing that the first-order complexity of our method is nearly tight.
  arXiv  Detail & Related papers  (2024-06-07T08:26:31Z)
• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• Dimension-free Remez Inequalities and norm designs [48.5897526636987]
  A class of domains $X$ and test sets $Y$ -- termed emphnorm -- enjoy dimension-free Remez-type estimates. We show that the supremum of $f$ does not increase by more than $mathcalO(log K)2d$ when $f$ is extended to the polytorus.
  arXiv  Detail & Related papers  (2023-10-11T22:46:09Z)
• On character table of Clifford groups [0.0]
  We construct the character table of the Clifford group $mathcalC_n$ for $n=1,2,3$. As an application, we can efficiently decompose the (higher power of) tensor product of the matrix representation. As a byproduct, we give a presentation of the finite symplectic group $Sp(2n,2)$ in terms of generators and relations.
  arXiv  Detail & Related papers  (2023-09-26T11:29:35Z)
• A Fast Optimization View: Reformulating Single Layer Attention in LLM Based on Tensor and SVM Trick, and Solving It in Matrix Multiplication Time [7.613259578185218]
  We focus on giving a provable guarantee for the one-layer attention network objective function $L(X,Y). In a multi-layer LLM network, the matrix $B in mathbbRn times d2$ can be viewed as the output of a layer. We provide an iterative algorithm to train loss function $L(X,Y)$ up $epsilon$ that runs in $widetildeO( (cal T_mathrmmat(n,d) + d
  arXiv  Detail & Related papers  (2023-09-14T04:23:40Z)
• Increasing subsequences, matrix loci, and Viennot shadows [0.0]
  We show that the quotient $mathbbF[mathbfx_n times n]/I_n$ admits a standard monomial basis. We also calculate the structure of $mathbbF[mathbfx_n times n]/I_n$ as a graded $mathfrakS_n times mathfrakS_n$-module.
  arXiv  Detail & Related papers  (2023-06-14T19:48:01Z)
• Learning low-degree functions from a logarithmic number of random queries [77.34726150561087]
  We prove that for any integer $ninmathbbN$, $din1,ldots,n$ and any $varepsilon,deltain(0,1)$, a bounded function $f:-1,1nto[-1,1]$ of degree at most $d$ can be learned.
  arXiv  Detail & Related papers  (2021-09-21T13:19:04Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.