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Questions tagged [tensor-decomposition]

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8
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1answer
101 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 ...
1
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1answer
23 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 ...
1
vote
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 ...
0
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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. ...
0
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1answer
70 views

Singular value decomposition for some matrix

Consider unit vectors $u_i\in \mathbb{R}^n,\ m>n$ 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 =\...
3
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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 ...
3
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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. ...
3
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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 $A_k\otimes B_k$, ...
3
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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 ...
2
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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 $\bigwedge^2V$, ...
2
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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 ...
2
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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), ...
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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)} \...
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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 ...
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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 ...
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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 ...
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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 ...
0
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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 ...
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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?
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23 views

Canonical polyadic decomposition of tensors that are common in some dimension

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) $$H_1 = H(:,:,1:64)$$ $$H_2= H(:,:, 65:128)$$ ...
0
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0answers
67 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 ...