# Questions tagged [tensor-decomposition]

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47 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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### 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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### 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 ...
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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 ...
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### CP-ALS Tensor Decomposition

I'm trying to implement CP-ALS (alternating least squares algorithm for canonical polyadic decomposition) tensor rank decomposition, but I cannot find any references for good guesses for the matrix ...
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### 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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### 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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### 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 $A_k\otimes B_k$, ...
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### 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 ...
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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 ...