# Questions tagged [tensor-decomposition]

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21 questions
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### A characterization of the subgroup of $\text{GL}(\bigwedge^k V)$ which preserves pure tensors?

Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set $$H=\{B\in\text{GL}(\bigwedge^k V) \, | \, B \,\text{ preserves pure tensors }\}$$ (i.e. $B \in H$ if it maps ...
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Suppose we have two canonical forms $A, B \in \mathbb{F}_2^{2 \times 2 \times 2}$ of a 3-dimensional tensor product space over the finite field with two elements, where $A = e_1 \otimes e_2 \otimes ... 1answer 33 views ### Maximal rank of tensors in$F^n\otimes …\otimes F^n$. What is the largest possible rank of a tensor in the space$F^n\otimes ...\otimes F^n$where we have$k$copies of$F^n$? It is quite easy to see that it is at most$n^{k-1}$(I have commented the ... 1answer 71 views ### Maximum Singular Value of$(A \cos\phi+B \sin\phi) \in \mathbb{R}^{2 \times 2}Problem I am wondering if there is a way to efficiently compute the maximum singular value of \begin{align} C(\phi)=A \cos\phi+B \sin\phi, \end{align} where A, B and C are real 2x2 matrices. ... 1answer 70 views ### Singular value decomposition for some matrix Consider unit vectorsu_i\in \mathbb{R}^n,\ m>n$s.t.$A=[u_1\cdots u_m]$has rank$n$. If$c_i>0$and$c=(c_1,\cdots, c_m),\ C={\rm diag}\ (c_1,\cdots, c_m)$s.t. $$Ac=0,$$ then $$ACA^T =\... 0answers 107 views ### principal curvature in high dimensions The principal curvature of a 2D (m=2) manifold in a 3D (n=3) ambient Euclidean space, is given by the eigenvalues of the second fundamental form (or the Hessian matrix) \Pi \in \Re^{m \times m} at ... 0answers 56 views ### Papers on Tensor factorization I started reading a few papers on Tensors. Right now I am reading a paper by Chi and Kolda called “On Tensors, Sparsity, and Nonnegative Factorizations”. I passed linear algebra and calculus courses. ... 0answers 89 views ### Is this a known operator decomposition? Consider a Hermitian operator H_\mathcal{AB} acting on the Hilbert space \mathcal{A}\otimes\mathcal{B}. What is the smallest commuting set of separable operators in the form A_k\otimes B_k, ... 0answers 131 views ### How is Tensor Decomposition (Factorization) related to Topological Data Analysis? I have been researching modern exploratory data analysis techniques, and came across two promising approaches: Topological Data Analysis (TDA) and Tensor Decomposition/Factorization (TF). I am ... 0answers 48 views ### Is every basis for \bigwedge^kV satisfying a “complementary” property a rescaling of a “standard” basis? This question was inspired by this beautiful answer: Let V be a 4-dimensional real vector space. Let \omega_{i_1,i_2} (1 \le i_1 < \ldots < i_2 \le 4) be a basis for \bigwedge^2V, ... 0answers 69 views ### Young operator acting on tensor with symmetries I am performing an irreducible decomposition of a tensor of rank 4, where it is symmetric in the first two indices: T_{abmn} = T_{bamn}. In English notation, the Young tableaux I need to evaluate ... 0answers 65 views ### Can I decompose a tensor in this way? I found several "generalizations" of the singular value decomposition (with some overlap), here is a short list to some references: Tucker, Tensor Train, Tensor Train rank-1, CANDECOMP/PARAFAC (CP), ... 0answers 67 views ### computing dual matrix trace norm and tensor gradient in python I'm trying to write the following function in python:$$ f_\mu(\mathcal X) = f_0(\mathcal X) + \sum_{i = 1}^n \max_{||\mathcal Y_{i(i)}|| \leq1} \alpha_i\langle \mathcal X_{(i)},\mathcal Y_{i(i)} \... 0answers 26 views ### Simultaneous diagonalization and SVD of 4th order tensor Given a 4th order tensor$L_{ijkl}$, how can I find four unit vectors$\mathbf x$,$\mathbf y$,$\mathbf z$and$\mathbf w$such that:$\mathbf{w}$and$\mathbf z$are the left and right singular ... 0answers 153 views ### Higher order singular value decomposition projection matrices Given an$N$-th order tensor$\mathcal{A} \in \mathbb{R}^{I_1\times I_2\times\ldots\times I_N}$we can find an approximation of$\mathcal{A}$,$\hat{\mathcal{A}}$by means of Higher Order Singular ... 0answers 17 views ### Which subspaces of exterior power have decomposable bases? Let$V$be a real$n$-dimensional vector space, and let$1<k<n,r>1$. I wonder: Is there a way to characterise which$r$-dimensional subspaces of the exterior power$\bigwedge^k V$have ... 0answers 11 views ### How to solve the tensor approximation optimization problem? Given a third-order tensor$\mathcal{X}\in\mathbb{R}^{I_1\times I_2\times I_3}$, we want to find an approximation tensor$\hat{\mathcal{X}}$of$\mathcal{X}$with$R$rank-one components, and some ... 0answers 33 views ### Does every subspace of the exterior algebra of dimension$>1$contain a decomposable element? Let$V$be a real$n$-dimensional vector space, and let$W \le \bigwedge^k V$be a subspace . Suppose that$\dim W \ge 2$. Does$W$contain a non-zero decomposable element? If$\dim W=1$, then ... 0answers 12 views ### Bounds on tensor rank What are the results on maximal and expected rank of tensors in$F^{n}\otimes ...\otimes F^{n}k\$-times?
I have a tensor three dimension tensor $$H$$ of size 4x8x128. I divide this tensor into two sub tensors (ill be using matlab language to demonstrate) $$H_1 = H(:,:,1:64)$$ $$H_2= H(:,:, 65:128)$$ ...
Here is a pattern of group embedding, $$SU(4) \supset SU(3) \times U(1)$$ such that the irrep of SU(4) can be decomposed as the sum of tensor product of irrep of SU(3) and irrep of U(1). Two ...