Questions tagged [tensor-decomposition]

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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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Tensorial decomposition and compressed core tensor

By curiosity, there are in the scientific literature tensor decompositions that return a compressed version of the original tensor called core tensor, except for Tucker decomposition? More exactly, I ...
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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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Extend tensor mode-$n$ product to multivariate functions

Given a tensor $\mathcal{X}\in\mathbb{R}^{I_1\times I_2\times\cdots I_N}$ and a matrix $U\in\mathbb{R}^{U_n\times I_n}$, we can define the mode-$n$ product $\mathcal{Y}=\mathcal{X}\times_n U$ by $$ y_{...
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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 ...
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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 "...
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1 answer
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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 ...
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Decomposing the tensor product $10⊗24$ 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 ...
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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 ...
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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 ...
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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 ...
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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}...
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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\...
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Coordinate Transformation of Stress Tensor

I Encountered a problem in a textbook I have been working on which goes through a simple coordinate transformation using covariant and contravariant components of the stress-strain tensor. I cannot ...
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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 ...
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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] ...
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An svd-like tensor decomposition splitting a tensor into two lower-dimensional tensors and a singular vector

I have also posted this question on mathoverflow, but it seems there are a lot of questions related to SVDs here and a tag "tensor-decomposition", so I will give it a shot. I am looking for ...
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2 votes
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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 − 13x_2 \wedge x_3 − 10x_2 \...
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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 ...
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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 ...
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Schouten theorem and doubts in the calculation of the tensor derivative

I am working on the following theorem Theorem (J. A. Schouten) For $n\geq 4$ the metric $g$ is conformally flat if and only if $W=0$. For $n=3$, the metric $g$ is conformally flat if and only if the ...
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References to Q-curvature

I am working on Q-curvature: $$ Q^n_g= \frac{1}{2(n-1)} \Delta_g R_g+\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}\:R_g-\frac{2}{(n-2)^2}\:| Ric |^2_g $$ See, for example, the expression (3) in https://...
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SVD in scipy and numpy for tensors

Can someone explain to me the difference between SVD of numpy and scipy for Multidimensional arrays (Tensors)? ...
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Block metric tensor and Fermi coordinates.

Consider a manifold $M^2\subset\mathbb{R}^4$ locally parametrized by $Y(\xi^1,\xi^2)$ and let $X$ be a Fermi coordinates parametrization of a neighbourhood of $M$ (which we call $\bar{M}$), that is $$ ...
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Why in tensors is the identity $\int{A_{\nu}\partial_{\mu}}\partial^{\nu}A^{\mu} = \int A_{\mu}\partial^{\mu}\partial^{\nu}A_{\nu}$ valid

I am trying to understand the and there is a step in the Derivation of something and there is a step which I cannot understand. Its basically a lack of understanding of tensors so any help would be ...
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Condition for endomorphism to be an element of tensor product of Lie representations

I'm coming from a physics background, so apologies if this question is technically ill-phrased. Consider an operator $H\in \text{End}(\mathcal{H}_{A}\otimes \mathcal{H}_{B})$, where $\mathcal{H}_{X}$ ...
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Proving $R/I \otimes_R M \cong_R M/IM$ using UMP and a biadditive map not with exact sequences.

I want to Understand a paragraph of the proof of $R/I \otimes_R M \cong_R M/IM.$ in example$(8)$ on pg. 370 of Dummit and Foote (third edition) Here is the example from Dummit & Foote: My ...
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What are the sufficient and necessary conditions for a unitary matrix to be written in a tensor product of 2x2 matrices?

I understand an arbitrary unitary matrix cannot be always written in a tensor product of 2x2 matrices. But, if a unitary matrix meets some conditions, can it be done? If so, what are those conditions?
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6 votes
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Is every harmonic polynomial a linear combination of these?

In $N$-dimensional space, we can show by direct calculation that the polynomial $$ r^{2K+N-2}\nabla_a\nabla_b\nabla_c\cdots \frac{1}{r^{N-2}} \hspace{1cm} \text{(with $K$ derivatives)} $$ is ...
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3 votes
1 answer
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How to show this tensor is not decomposable

Let $V$ a vector space over $\mathbb{F}$ with basis $B=\{e_1,\dots, e_n \}$ and $B'=\{e^1,\dots,e^n \}$ adjoint basis in $V^*$, Consider the following tensor: $$t^{ij}=(i+j) $$ I need to see that is ...
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How can I express a tensor in terms of a basis different to canonical?

Considerthe canonical basis $(e_1,e_2,e_3)$ and its dual basis $e^1 , e^2 , e^3$. Let $A = (a_{ij})$ a matrix in $M_{3 \times 3}(\mathbb{R})$ given by: $$A = \begin{bmatrix} 1 & -3 & 1\\ -3 &...
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A tensor over $\mathbb R$ is sum of two decomposable tensors

Let $W$ be a vector space over $\mathbb R$ with basis $\{a,b\}$. Consider the tensor $$a \otimes a \otimes a - b \otimes b \otimes a + a \otimes b \otimes b + b \otimes a \otimes b.$$ a) Show that ...
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Tensor Notation ambiguity

Sorry for this silly question, but this notation keeps popping up in my face in too many different places without any clarification, so I will be so grateful if someone can help and answer my question:...
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Frobenius norm optimization problem

How to solve the following optimization problem which arises in the field of Higher Order SVD? Let $A \in \mathbb{R}^{m\times n}$, $W \in \mathbb{R}^{m\times l}$. find $\arg \max _{A}\|A^T W\|_F$ s.t. ...
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2 votes
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How to algebraically express the condition for a tensor to be of rank $1$?

Let $W$ be a finite-dimensional complex vector space and let $V = W\otimes \cdots \otimes W$ ($m$ times). A non-zero element $v \in V$ is said to be of rank $1$ if $v$ can be written as $$v = w_1 \...
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How many projections $\pi: \mathbb{F}_3^2 \to \mathbb{F}_3^2 $ there? [closed]

How many projections $\pi: \mathbb{F}_3^2 \to \mathbb{F}_3^2 $ there? Justify your answer! If someone can give me a tipps on this question.D
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Irreducible representation of $SO(3)$ by rank-3 tensors and higher?

I read that any rank-2 tensor can be decomposed into the sum of a traceless symmetric tensor, an anti-symmetric tensor and a unit tensor, all closed under $SO(3)$. The three form an irreducible ...
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1 answer
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Given $\sum_i\sum_j \sigma_{ij} a_i b_j^T$ and orthogonal $\{a_i\}$, find orthogonal $\{b_j\}$

Suppose I have a matrix $T \in \mathbb{R}^{n \times n}$ of the form: $$T = \sum_{i = 1}^{n_A} \sum_{j = 1}^{n_B} \sigma_{ij} a_i b_j^T$$ where $1 \leq n_A, n_B \leq n$, $\sigma_{ij} \in \mathbb{R}\...
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Integrating unitary vector in calculating stress resultant

The problem defines: velocity in the local base $\{e_r , e_\theta , e_z\}$, the gradient and acceleration in the local curvilignear base $e_{i} \otimes e_{j}$ avec $i, j \in\{r, \theta, z\}$ ...
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Find orthonormal $\{b_i\}$ such that $\mathcal{A} = \sum_i \sum_j \lambda_i \mu_j b_i b_j^T$

Suppose $\{b_i\}_{i = 1}^d \subset \mathbb{R}^n$ is an orthonormal set of vectors, $d \leq n$, and assume that a matrix $\mathcal{A} \in \mathbb{R}^{n \times n}$ has the following form: $$\mathcal{A} ...
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Gradient Descent for Tensor Decomposition - Find low rank dimensions

I encountered this paper on travel time estimation using Tensor Decomposition and at Page 4, Figure 5 there is an algorithm to decompose a tensor using Gradient Descent. The first line of this ...
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linear independence of symmetric tensors

I am reading a paper that incidentally uses a bit of theory of symmetric tensor spaces. I came across the following claim: If we're given linearly independent vectors $x_1, \ldots, x_n \in \mathbb{...
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Tucker and TT decompositions in 3D case

I’m wondering if Tucker factorization and Tensor train decomposition are equivalent in case of 3-rd order tensor and how to show it?
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Decomposition of tensor product of defining representation with itself for $G=\mathrm{SO}(5)$

I want to decompose the tensor product $V\otimes V$, where $V=\Bbb C^5$ denotes the defining representation of $\mathrm{SO}(5)$, into irreducible representations using the following formula for the ...
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Finding the irreducible components of a rank 3 tensor

In 3 dimensions, a rank-2 tensor can be identified via a scalar, a vector and a (symmetric, traceless) tensor component by contracting it with $\delta_{ij}$ or $\epsilon_{ijk}$: $$ T_{ij} = \...
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Proof of 3-fold Tensor Uniqueness

In the notes linked below Professor Roughgarden states Theorem 3.1 without proof. http://timroughgarden.org/s17/l/l10.pdf Any ideas how one would go about proving this statement or references to ...
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Tensor product raised to power $N$.

Can the following quantity be reduced to further $[A \otimes B + (\mathbb{1} - A) \otimes D]^N$, for positive interger $N$? Here, $\otimes$ denotes the Kronecker (tensor) product and the matrices $A$ ...
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What are the matricization and vectorization of tensor products?

I'm trying to understand the concepts of matricization (matrix unfolding) and vectorization of tensor products. In the past, I've only dealt with tensor products of infinite-dimensional Banach and ...
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If a tensor's multilinear rank is $(R,R,R)$, then is its canonical/CP rank also $R$?

Given an order $2$ tensor (i.e. a matrix), one always has that row rank is equal to the column rank, so that its multilinear rank or $n$-ranks is always equal to $(R,R)$ for some $R$. Moreover that $R$...
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non totally decomposability of vectors in Grassmannian

In "Algebraic geometry: a first course", by Harris, Grassmannian is described, under the Plucker embedding, as the locus of totally decomposable vectors in the projectivization of the exterior power $\...
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