Questions tagged [tensor-decomposition]

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46 questions
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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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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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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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 ...
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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. ...
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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 ...