# Questions tagged [tensor-decomposition]

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### Proof of 3-fold Tensor Uniqueness

In the notes linked below Professor Roughgarden states Theorem 3.1 without proof. http://timroughgarden.org/s17/l/l10.pdf Any ideas how one would go about proving this statement or references to ...
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### Tensor product raised to power $N$.

Can the following quantity be reduced to further $[A \otimes B + (\mathbb{1} - A) \otimes D]^N$, for positive interger $N$? Here, $\otimes$ denotes the Kronecker (tensor) product and the matrices $A$...
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### What are the matricization and vectorization of tensor products?

I'm trying to understand the concepts of matricization (matrix unfolding) and vectorization of tensor products. In the past, I've only dealt with tensor products of infinite-dimensional Banach and ...
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### If a tensor's multilinear rank is $(R,R,R)$, then is its canonical/CP rank also $R$?

Given an order $2$ tensor (i.e. a matrix), one always has that row rank is equal to the column rank, so that its multilinear rank or $n$-ranks is always equal to $(R,R)$ for some $R$. Moreover that $R$...
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Tensor ring decomposition of a given tensor $\boldsymbol{X}\in \mathbb{R}^{n_1\times\cdots\times n_d}$ is to find tensors $G^1,\cdots,G^d\in \mathbb{R}^{r\times n_i\times r}$ which minimize $$\sum_{\... 0answers 33 views ### Missing value imputation by tensor decomposition As a beginner in linear algebra and tensor decomposition techniques, I may be posting a naive question. I have a tensor A that has some missing values. I am trying to predict the missing values using ... 0answers 41 views ### Symmetrizing an arbitrary rank tensor by pairwise permutations Given a rank two tensor T_{ab}, we can symmetrize and antisymmetrize it as T_{(ab)} = \frac{1}{2}(T_{ab}+T_{ba}), \quad T_{[ab]} = \frac{1}{2}(T_{ab}-T_{ba}). In this case it is trivial to see ... 1answer 50 views ### Does the kernel of every alternating form contain a decomposable element? Let V be a real n-dimensional vector space, and let 1 < k < n. Let \alpha \in \bigwedge^k (V^*) \cong (\bigwedge^k V)^*. Thinking of \alpha as a linear functional \bigwedge^k V \to \... 3answers 139 views ### Can every closed differential form be expressed via constant coefficients? Let M be a smooth n dimensional manifold, and let 1 \le k < n. Let \omega \in \Omega^k(M) be a closed k-form on M. Let p \in M. Do there exist coordinates around p, such that \... 2answers 100 views ### Are standard bases for the exterior power essentially unique? Let V be a real d-dimensional vector space, and let 1<k<d be fixed. Let v_i be a basis for V. Consider the induced basis for the k-th exterior power \bigwedge^k V, given by v_{i_1}... 0answers 40 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 ... 1answer 74 views ### Can we “mod out” a common subspace in the Grassmannian inside the exterior algebra? While reading this paper, I have seen the following claim stated without a proof: Let V be an n-dimensional vector space over a field, and let \alpha,\beta \in \bigwedge^k V be decomposable ... 1answer 29 views ### Is “being decomposable” preserved under taking a subspace? Let V be a vector space over some field, and W \le V a vector subspace. Let 1<k<\dim V be an integer. Suppose \omega \in \bigwedge^k W is decomposable as an element in \bigwedge^k V.... 0answers 53 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, ... 1answer 66 views ### Is every decomposable basis for \bigwedge^kV “standard”? This is a curiosity: Let V be a d-dimensional real vector space, and let 1<k<d. Set Let \omega^{i_1,\ldots,i_k} be a basis for \bigwedge^kV, whose elements are all decomposable. Is \... 1answer 54 views ### In which degrees there exist non-decomposable elements in the exterior algebra? I am trying to get a better understanding of the concept "decomposable" element in an exterior algebra. Let V be a d-dimensional real vector space, and let 1<k<d. For which tuples (k,d),... 1answer 111 views ### 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 }\}$$(i.e. B \in H if it maps ... 1answer 56 views ### Convert from one tensor canonical form to another Suppose we have two canonical forms A, B \in \mathbb{F}_2^{2 \times 2 \times 2} of a 3-dimensional tensor product space over the finite field with two elements, where A = e_1 \otimes e_2 \otimes ... 1answer 367 views ### Why is the exterior power \bigwedge^kV an irreducible representation of GL(V)? \newcommand{\GL}{\operatorname{GL}} Let V be a real n-dimensional vector space. For 1<k<n we have a natural representation of \GL(V) via the k exterior power: \rho:\GL(V) \to \GL(\... 0answers 72 views ### Does every subspace of the exterior algebra of dimension >1 contain a decomposable element? Let V be a real n-dimensional vector space, and let W \le \bigwedge^k V be a subspace . Suppose that \dim W \ge 2. Does W contain a non-zero decomposable element? If \dim W=1, then ... 1answer 57 views ### what does it mean to be symmetric for tensors and Kronecker delta symbols and help explain this answer to me i understand how to change 2 tensors into Kronecker delta symbols but unsure how they managed to transform back to just one. If someone could add all the steps to get to the answer that would be ... 1answer 69 views ### Maximal rank of tensors in F^n\otimes …\otimes F^n. What is the largest possible rank of a tensor in the space F^n\otimes ...\otimes F^n where we have k copies of F^n? It is quite easy to see that it is at most n^{k-1} (I have commented the ... 2answers 391 views ### Decomposition of tensor product of two representations in S_3. Consider the group S_3. There are three irreducible representations, the trivial, \varphi^{triv}, the sign representation \varphi^\epsilon (both 1-dimensional), and the two-dimensional one \... 0answers 114 views ### Embed SU(2) into SO(6) or SU(4) Here is a pattern of group embedding,$$ 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 ... 1answer 84 views ### On the decomposition of (2,0)- and (0,2)-tensors So, consider the n-dimensional vector space V, on a field \mathbb{F}, with a chosen basis \{e_i\} and its dual space V^* with the chosen basis \{e^j\} with the property e^j(e_i)=\delta^... 1answer 23 views ### Find real s.p.d. matrix C to which there is no real F with F^{\sf T}F=C Say C is a real, symmetric and positive definite matrix. Then it has a square root F such that F^2=C. However, I wonder if one can construct C in any way such that a real (optional: positive ... 0answers 261 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 ... 1answer 176 views ### Schur-Weyl Duality (Example for k = 3) I am an undergraduate student interested in representation theory. I know that you can decompose the iterated tensor product (C^n)^{\otimes k} into the direct sum of irreducible S_k \times GL(V) ... 1answer 89 views ### How to expand a vector valued multilinear mapping as a tensor. I wrote this in my notes: Let V_1,\ldots,V_k be real vector spaces of dimensions n_1,\ldots,n_k and let V_j have basis (\mathbf{e}^{(j)}_1,\ldots,\mathbf{e}^{(j)}_{n_j}) and corresponding ... 1answer 121 views ### Is a pointwise decomposable differential form smoothly decomposable? Let \omega be a smooth differential form on a smooth manifold M. Suppose \omega is pointwise decomposable, that is for every p \in M, \omega_p=e^1_p \wedge e^2_p \wedge \dots \wedge e^k_p ... 1answer 93 views ### Can we always span a decomposable form via constant coefficients? Let M be a smooth manifold of dimension d. Let 1 < k <d be an integer. Let \omega^i be a local frame of the exterior power bundle \Lambda_{k}(T^*M). Does there exists numbers a_i ... 1answer 3k views ### Actual example of tensor contraction So I'm having trouble to compute tensor contractions with "actual" numbers from the matrix representations of the tensors. I have only seen abstract theoretical examples on the internet so I'm asking ... 1answer 265 views ### Uniqueness of Tensor Decompositions (Aren't Matrix Decompositions a Special Case?) It seems that higher-order tensors (of order 3 or higher) generally have unique decompositions under relatively mild conditions. For example, Kruskal proved that if an order-3 Tensor T can be ... 0answers 57 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. ... 1answer 842 views ### Inertia Tensor of an ellipsoid Given is the following inertia tensor of a certain mass distribution \rho(\vec{r}) :$$ I_{ij} = \int dV \rho(\vec{r}) \left( \vec{r}^2 \cdot \delta_{ij} - r_ir_j \right) $$I should compute the ... 0answers 83 views ### computing dual matrix trace norm and tensor gradient in python I'm trying to write the following function in python:$$ 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)} \...
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 ...