# Questions tagged [tensor-decomposition]

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46 questions
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### 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 ...
1answer
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### 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 $(k,d)$,...
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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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### 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
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### 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 ...
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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 ...
0answers
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### Bounds on tensor rank

What are the results on maximal and expected rank of tensors in $F^{n}\otimes ...\otimes F^{n}$ $k$-times?
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### 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
112 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 ...
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112 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 $S_k \times GL(V)$ ...
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### 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 ...
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### 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 $p \in M$, $\omega_p=e^1_p \wedge e^2_p \wedge \dots \wedge e^k_p$ ...
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### Objective function for CP tensor decomposition

What is the objective function for CP (CANDECOMP/PARAFAC) tensor decomposition? The decomposition tries to decompose a tensor $Z$ to $Z \approx \sum_{l=1}^L \lambda_l a_l \circ b_l \circ c_l$, where ...