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Questions tagged [tensor-decomposition]

For questions related to tensor decomposition. A tensor decomposition is any scheme for expressing a tensor as a sequence of elementary operations acting on other, often simpler tensors.

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What if a CP/PARAFAC tensor decomposition can be further decomposed, "recursively"?

Standard CP/PARAFAC decomposition: A tensor $\mathcal{T}$ in shape $(I_1,\dots,I_N)$ is produced by $N$ matrices $\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}$ where each $\mathbf{A}^{(n)}$ is in shape $(...
graphitump's user avatar
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Decomposing matrix as tensor product

I have a matrix $F\in \mathbb{R}^{n\times n^2}$ for some integer $n>1$. I know in some cases, I can decompose $F$ as $$F = \mathbf{v}^\dagger\otimes \mathcal{F}$$ where $\mathbf{v}\in \mathbb{R}^n$ ...
confusion's user avatar
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Antisymmetric tensor fo rank four in three dimensional space

I'm trying to find an expression for a tensor of fourth rank (in three dimensional space) that is antisymmetric with respect to its indices pairs, ie. a tensor with the following property: $$ \frac{1}{...
BitterDecoction's user avatar
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Proof that the tensor product of $n$ vector spaces and a vector space can be written as the tensor product of $n+1$ vector spaces

Given $n + 1$ vector spaces $V_1, \ldots, V_{n + 1}$, a tensor product $\langle V_1 \otimes \cdots \otimes V_n, \otimes_n \rangle$ of $V_1, \ldots, V_n$ and a tensor product $\langle (V_1 \otimes \...
Marc Mertens's user avatar
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Einstein summation for matricized tensor times Khatri Rao product (mttkrp) for use in tensor decomposition

I am trying to compute the matricized tensor times Khatri-Rao product (mttkrp) as part of an optimization problem for tensor factorization, shown below: $A = \mathcal{X}_{(1)}(B \otimes C)$ Where $\...
pigtowndandy's user avatar
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How to solve the matrix equation $\sum_iA_i^TA_iXB_iB_i^T=\sum_iA_i^TC_iB_i^T$ efficiently?

How would I go about solving the matrix equation $\sum_iA_i^TA_iXB_iB_i^T=\sum_iA_i^TC_iB_i^T$ for $X$? The simplest thing to do would be to, of course, consider $Y=\sum_iA_i^TC_iB_i^T$, vectorise and ...
Rahul Bordoloi's user avatar
5 votes
2 answers
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Gradients for partially symmetric CP decomposition of 3rd order tensor

I am interested in computing a rank-$R$ CP decomposition of a 3rd order tensor that is partially symmetric about the first 2 modes. The factorization of a vanilla CP decomposition is given below as a ...
pigtowndandy's user avatar
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Is there a fast way to do this tensor power/trace operation?

Given an $M*N*P$ tensor $T$, is there a fast way of computing the following "eighth-power trace"? $$ f(T) = \sum_{m,n,p} T_{m_{00},n_{00},p_{00}} T_{m_{01},n_{01},p_{00}} T_{m_{10},n_{00},p_{...
Craig's user avatar
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Tensor product of basis and dual basis

In my textbook is stated the following expression: $$\hat{\delta}=\delta^i_j(\vec{a}_i\otimes\vec{a}^j)=\vec{a}_i\otimes\vec{a}^i$$ And I just can't understand what the logic here is. First of all ...
Krum Kutsarov's user avatar
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Totally symmetric rank $n$ generalization of Helmholtz decomposition

Suppose $F_{i_1\cdots i_n}(x)$ is a totally symmetric rank $n$ tensor field in 3 dimensions ($i=1,2,3$). Is it possible to decompose $F_{i_1\cdots i_n}(x)$ in a way which generalizes the Helmholtz ...
fewfew4's user avatar
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Tensor product decomposition

This question is a generalization of this one. For the sake of this question, a tensor product of $𝑉$ and $π‘Š$ is a couple $(𝑇,β„Ž)$ where $𝑇$ is a vector space and $β„Ž:π‘‰Γ—π‘Šβ†’π‘‡$ is a bilinear map ...
Laurent Claessens's user avatar
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unicity of tensor product decomposition

For the sake of this question, a tensor product of two vector spaces $V$ and $W$ over a field $K$ is a couple $(T,h)$ where $T$ is a vector space over $K$ and $h:V\times W\to T$ is a bilinear map ...
Laurent Claessens's user avatar
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Tensor decomposition of a rank-3 tensor used for low-rank construction of a 'similar' tensor

I am a computational physicist, trying to construct a rank-3 tensor $ \textbf{g}_{mnr} \in \mathbb{C}^{n_m\times n_n\times n_r}$, where each element represents the scattering amplitude from a state $m$...
user2188518's user avatar
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Algebra equation for 3 rank tensor

Suppose I work in $4$ dimensions. I have an algebraic equation in the following form, which contains a 3 rank tensor $X ^{\alpha \lambda \mu }$ \begin{equation} X ^{\alpha \lambda \mu }\eta ^{\beta \...
A.D's user avatar
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Understanding the Definition of a Rank-1 Tensor

A tensor is nothing but a multidimensional array. We can think of an $n-mode$ tensor as a structure whose each element has to be referred with the help of $n$ indices or $n$ axes. Now, while reading ...
Thomas Finley's user avatar
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calculation rules of unit vector

I cannot figure out how to derive a formula in a scientific paper (LINDBORG, 2007, DOI: 10.1175/JAS3864.1). I will list all the information needed below: The starting point of the derivation is this ...
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How to construct such Matrix A

Given a non singular and symmetric $4 \times 4 $ matrix $G$ over the field of real numbers. Is it always possible to find another non singular $ 4 \times 4$ matrix $A$ such that $\sum_j \sum_k G_{jk}...
ibnAbu's user avatar
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Higher order Tensor computations

I have a $D$-way tensor H of dimension $I \times I \times \dots \times I$ ($D$ times), that represent the coefficients of a polynomial. For better understanding, I provided an image of 3-way tensor ...
Neuling's user avatar
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How to calculate $\text{End}(V^{\otimes n})$

Let $\mathfrak g$ be a complex semisimple Lie algebra, and $V$ the fundamental $\mathfrak g$-module. Then we can decompose $V^{\otimes n}$ into the direct sum of irreducibles. For example, in the case ...
William Leynoid's user avatar
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fundamental questions regarding approximations of tensors

My situation: I'm an absolute beginner regarding tensors and Ive currently working on a manuscript about a format which approximate / represent tensors of possibly high order. But before I can start ...
AsaMitaka's user avatar
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Do tensors $n_in_jn_kn_l$ where $n_i$ is a vector span all 4th-order symmetric tensors?

A real tensor $T$ of order $r$ in dimension $n$ is a collection of real coefficients $T_{i_1i_2\dots i_r}$, $1\leq i_1, i_2, \dots, i_r\leq n$. A tensor $T$ is symmetric if the coefficients $T_{i_1i_2\...
Hussein's user avatar
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Hilbert space decomposition through tensor product

I'm trying to understand how to decompose a high $N$-dimensional Hilbert space $V$ thinking it as a subspace of the tensor product space $V_1 \otimes V_2$ of two smaller Hilbert spaces $V_1$ and $V_2$,...
Philap's user avatar
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Elliptic Decomposition Theorem

The famous Hodge Decomposition Theorem can be generalized in the context of elliptic operators. The general decomposition theorem can be found in Theorem 5.5 of Spin Geometry of H. Blaine Lawson and M....
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Tensoring over $k$ and over $A$ in Exercise 3 on pg.31 of Atiyah and MacDonald

Here is the question I am trying to solve: Let $A$ be a local ring, $M$ and $N$ finitely generated $A$ modules. Prove that if $M \otimes N = 0,$ then $M = 0$ or $N = 0.$ And here is the hint given in ...
weird's user avatar
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What means "way" of an "m-way, n-dimension tensor"?

In a preprint the author uses the term m-way in the context of tensors without defining it. It cannot be the tensor dimension as the whole sentence says: "Let $\mathcal{A}$ represent an $m$-way, $...
granular_bastard's user avatar
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Decompose polynomial of degree 3 in a sum of cubed orthogonal linear forms

Given is a homogenous polynomial $p$ of degree $3$ with $n$ terms where the coefficients can only be $\pm1$. The polynomial has $m$ variables and in each term the variables have degree $1$, i.e., $$p(...
granular_bastard's user avatar
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Tensor-matrix mode-$n$ product with matrix as right hand operand

Kolda et al. and De Lathauwer et al. talk about the mode-$n$ tensor-matrix product where if $X,Y \in \mathbb{R}^{N \times N \times N}$ and $C \in \mathbb{R}^{N \times N}$, we have \begin{equation*} Y =...
Nitin Malapally's user avatar
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Tensor decomposition along a world-line

I was looking into this general relativity paper from Barone, F., Facchi, P., & Tulczyjew, W. M. (2011). and there's a part where they talk about decomposing a tensor in its parallel and ...
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Contraction of only one of the indexes of a symmetric tensor with an antissymetric tensor is zero?

So guys I was trying to show some expressions in general relativity and I have the contraction of two tensor objects, those being: $$\sigma_{\mu\nu} = \sigma_{\nu\mu},$$ and, $$\omega_{\mu\nu} = -\...
Alan de Gois's user avatar
2 votes
1 answer
38 views

Decomposing the all $1$s matrix in tensor products

Consider the following matrices $$I = \begin{pmatrix}1& 0\\ 0 & 1\end{pmatrix},\quad X = \begin{pmatrix}0& 1\\ 1 & 0\end{pmatrix}$$ Define the indexed quantity $P^n_k(I, X)$ to be ...
user1936752's user avatar
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3 votes
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Are all $v \in \Lambda ^k V$ decomposable if $k > \frac{1}{2} \dim V$?

Update: It seems I proved only the $(\dim S)$-decomposibility of the sum of two decomposables, since $s_1 + s_2$ is not decomposable (which I'm not quite sure of). Original post: Let $V$ be any $n$-...
Physor's user avatar
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Decomposable tensors

A tensor $t \in V \otimes W$ is called decomposable if $t = v \otimes w$ for some $v \in V$ and $w \in W$. The set of decomposable tensors is the image of the map \begin{equation} f : \mathbb{R}^2 \...
Orb's user avatar
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Relation between symmetric outer product decomposition and symmetric multilinear decomposition

Suppose tensor $\mathcal{A}$ is a symmetric real tensor of order $k$. Then, symmetric outer product decomposition of $\mathcal{A}$ is $$ \mathcal{A} = \sum_{i=1}^p \lambda_i v_i^{\bigotimes k}, $$ ...
abcd's user avatar
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Every $w\in \Omega^2 (V)$ is decomposable if $\operatorname{dim}(V) =3$

This question was asked in my assignment on tensors and I am stuck on this question. Question: Let $V$ be a vector space. An element $ w\in A^k (V)$ is called decomposable if $w = \phi_1 \wedge \...
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1 answer
118 views

Calculate the coordinates of $x \otimes y \in V \otimes W.$

Here is the question I want to answer: Let $x = (1,1) \in V = \mathbb R^2$ and $y = (1, 2, 1) \in W = \mathbb R^3.$ Calculate the coordinates of $x \otimes y \in V \otimes W$ with respect to the ...
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2 votes
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25 views

Determine orthotropic axes of Stiffness Tensor

Consider a fourth oder stiffness tensor $\mathbb{C}$. The components $C_{ijkl}$ are given with respect to a global coordinate system. Is there a way to determine the orthotropic axes from this ...
hm1212's user avatar
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Matrix-rank nonincreasing unitary tensor operations

I have a multidimensional array $A_{ijkl}$ $\in\mathbb{C}^{m\times n\times o \times p}$ indexed by four integers $i,j,k,l$. I will call $i$ and $j$ the "left" indices, $j$ and $k$ the "...
Jordan Taylor's user avatar
1 vote
1 answer
171 views

Singular value decomposition for tensor

I am looking at (the limitation of) the extension of the singular value decomposition to tensors. I would like to show that there is a tensor $A_{i,j,k}$ that cannot be decomposed in the following ...
Hans's user avatar
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315 views

Rank 1 tensors, how to describe them? (specific case)

I want to undestand a specific case. I consider two $\mathbb{C}$-vectorial spaces, $\mathbb{C}^2$ both. Then, I want to work with $\mathbb{C}^2\otimes \mathbb{C}^2$. Now, I consider basis for each ...
Gyadso's user avatar
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Decomposing the tensor product $10βŠ—24$ in term of tensors under SU(5) group

Suppose that the rank two tensor $C_{ij}=10$ is an antisymmetric representation of SU(5) group and $24$ is the adjoint representation which written as rank two tensor $D_{kl}=24$ I understand that the ...
Nouri's user avatar
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Trying to effect permutating a tensor on its rank

I am reading through Fast Matrix Multiplication by Markus Blaser. I am trying to prove Lemma 5.3 from page 19. It states the following: For any tensor $T\in \mathbb{F}^{n\times m\times t}$, and any ...
Louie_the_unsolver's user avatar
4 votes
0 answers
138 views

Decomposition of Hodge Operator

Given a decomposition of a vector space $V \simeq U \oplus W$. Then as taking the exterior algebra preserves coproducts (it is left adjoint to the forgetful functor from graded-commutative graded ...
HDB's user avatar
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2 votes
1 answer
227 views

How can the isotropic and deviatoric parts of fourth- and fifth-rank tensors be obtained? [closed]

For a second-order rank, we know that the isotropic and deviatoric parts is given by $$ \frac{1}{2} \left( A_{ij} + A_{ji} \right) - \frac{1}{3} A_{ss} \delta_{ij} \, , $$ whereas for a third-rank ...
keynes's user avatar
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Tensor product: given the result and tensor multiplication, find the tensor decomposition?

I am trying to code this problem: Suppose there are four elements to use when tensoring: $$ \left\{ I = \begin{bmatrix}1&0\\0&1\end{bmatrix},\, X = \begin{bmatrix}0&1\\1&0\end{bmatrix}...
M. Al Jumaily's user avatar
2 votes
0 answers
51 views

Is there a way to represent electrostatics tensors in tensor (possibly tensor product) way?

I'm working with electrostatic interaction tensors, which are defined as follows: \begin{align} T &= \frac{1}{r} \\ T^\alpha &\equiv \nabla T = -\frac{r^\alpha}{r^3} \\ T^{\alpha\...
Bbllaaddee's user avatar
1 vote
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47 views

What is the maximum of $\operatorname{Tr}(EUAU^\dagger)$ when $U$ is unitary?

Given matrices $E$ and $A$, is there a good way to obtain the maximizing unitary $U$ for the objective function $\operatorname{Tr}(EUAU^*)$? (I know that the maximizer of $\operatorname{Tr}(EU)$ is ...
Yantao Wu's user avatar
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1 answer
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CP decomposition as a special case of Tucker decomposition

I am reading this article "Tensor Decompositions and Applications" by Kolda and Bader. On page 21, it says: ...CP [decomposition] can be viewed as a special case of Tucker [decomposition] ...
CaTx's user avatar
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2 votes
1 answer
110 views

Determine wether a skew-symmetric tensor is decomposable or not.

For $x_1,x_2,x_3,x_4$ elements of a vector space $V$, I would like to find out wheter the skew-symmetric tensor $11x_1 \wedge x_2 + 10x_1 \wedge x_3 + 17x_1 \wedge x_4 βˆ’ 13x_2 \wedge x_3 βˆ’ 10x_2 \...
Rhaena's user avatar
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1 answer
186 views

How to build 2nd rank tensors from set of vectors?

Suppose we have 3 vectors: $A_i$, $B_i$, $C_i$, where $i=1,2,3$. How to build all possible dependent 2nd rank tensors based on these vectors? I believe, that at first we need to obtain all possible ...
Akru Bas's user avatar
1 vote
1 answer
59 views

Choosing an Irreducible Tensor Operator Basis where the Singular Values of Each Basis Element are the Same

Let $\mathcal{B(H)}$ be the space of all bounded linear operators on the Hilbert space $\mathcal{H}$. Let $g \rightarrow \mathcal{U}_g$, where $\mathcal{U}_g (\cdot):= U(g) (\cdot) U(g)^\dagger $, be ...
Si Chen's user avatar
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