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Questions tagged [tensor-decomposition]

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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}...
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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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2answers
57 views

Comparing the representation $T \otimes T $ in terms of matrices and $T^2$

Interpret the representation $ T \otimes T$ in terms of matrices, and compare it with $T^2$. Could anyone give me a hint on how to solve this please ? EDIT: enter image description here enter ...
4
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1answer
63 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 ...
0
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1answer
21 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 $\bigwedge^k V$....
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 $\bigwedge^2V$, ...
3
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1answer
50 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 $\...
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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 ...
2
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1answer
34 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 $(k,d)$,...
8
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1answer
100 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
19 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 ...
5
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1answer
154 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(\...
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0answers
32 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 ...
0
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1answer
21 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 ...
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1answer
31 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 ...
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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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2answers
118 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 $\...
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0answers
22 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) $$H_1 = H(:,:,1:64)$$ $$H_2= H(:,:, 65:128)$$ ...
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0answers
61 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 ...
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1answer
33 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^...
0
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1answer
22 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 ...
3
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0answers
83 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 ...
2
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1answer
93 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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1answer
48 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 ...
4
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1answer
65 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 $p \in M$, $\omega_p=e^1_p \wedge e^2_p \wedge \dots \wedge e^k_p$ ...
1
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1answer
64 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 ...
2
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1answer
693 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 ...
1
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1answer
110 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 ...
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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. ...
0
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1answer
257 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 ...
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0answers
62 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 ...
7
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1answer
368 views

Tensor product and linear dependence of vectors

Let $V_1, \ldots, V_k$ be complex vector spaces. Given $k$ vectors $v_1 \in V_1, \ldots, v_k \in V_k$, we define that the tensor product $v_1 \otimes \ldots \otimes v_k$ has rank 1. For any tensor $T \...
2
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1answer
165 views

A decomposable element in the tensor product $V \otimes W$

In the book of A Course in Algebra by E. B. Vinberg, at page $298$, it is given that, An element $z \in V \otimes W$, i.e in the tensor product of the space $V$ and $W$, is called decomposable ...
2
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0answers
63 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 ...
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 =\...
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0answers
87 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 $A_k\otimes B_k$, ...
0
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1answer
235 views

Decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields

I am trying to solve the following problem: For each rational prime $p$, describe the decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields, ...
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0answers
130 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 ...
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1answer
46 views

Decomposable tensors in $\Lambda^2 {\mathbb{C^n}}$

Let $V=C^n$. How can I describe all decomposable tensors in $\Lambda^2 {V}$ for $n=3$ and $n \ge 4$?
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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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2answers
1k views

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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0answers
147 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 ...
1
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1answer
430 views

n mode product and kronecker product relation

I am reading tensor decompositions and applications, by Tamara Kolda. There she mentions a property of $\mathscr{Y} = \mathscr{X} \times_1 A^{(1)} \times_2 A^{(2)} \ldots \times_N A^{(N)} \iff$ $ Y_{...
3
votes
2answers
230 views

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 ...