Questions tagged [tensor-decomposition]

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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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18 views

Meaning of a tensor that belongs to X(M)⊗X^{2}(M).

I have a question about the notation used for a tensor field. I know that if I have a 3-form $R$, then we say that $R \in \Omega^{3}(M)$ where M is the base manifold. In particular if the 3-form $R$ ...
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25 views

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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1answer
21 views

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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10 views

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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1answer
53 views

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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16 views

Finding the irreducible components of a rank 3 tensor

In 3 dimensions, a rank-3 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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1answer
107 views

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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1answer
39 views

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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71 views

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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2answers
46 views

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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38 views

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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What are the irreducible representations of $\mathbf{10}\otimes \mathbf{3}$ in $SU(3)$

Since, a rank-3 tensor has 10 components and a rank-1 tensor has 3 components in $SU(3)$, I know that we are searching for the different irreducible representations of the tensor $v_{ijk}w_{l}$. The ...
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42 views

Exterior power of symmetric product of GL(2,R) tensor representations

A paper I am reading uses an identification of $GL(2,\mathbb{R})$ representations: \begin{align} \Lambda^2(\text{Sym}^3\mathbb{R}^2) = (\text{Sym}^4\mathbb{R}^2 \otimes \Lambda^2\mathbb{R}^2 ) \...
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68 views

Maximal Ideals in Tensor product of Algebras

Let A, B be algebras over algebraically closed field $\mathbb F$, and m be a maximal ideal in $A\otimes_\mathbb{F} B$ such that $A\otimes_\mathbb{F} B/m \cong \mathbb F$. Show that there are maximal ...
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1answer
51 views

Readings on Tensors, Tensor Algebra and Tensor Decomposition

I have just started my Masters Degree in Mathematics and I will be focussing on Tensors (Viewed as Multidimensional Arrays) and Tensor decompositions. My professor is by no means an expert on Tensors ...
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30 views

Problem in odd-even decomposition of a generic metric

The metric of the unit two-sphere is given by $ \Omega_{\mu \nu} = \begin{equation} \begin{pmatrix} 1 & 0 \\ 0 & \sin^2 \theta \end{pmatrix}. \end{...
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40 views

Tensor ring decomposition of a tensor with cyclic symmetry

Tensor ring decomposition of a given tensor $\boldsymbol{X}\in \mathbb{R}^{n_1\times\cdots\times n_d}$ is to find tensors $G^1,\cdots,G^d\in \mathbb{R}^{r\times n_i\times r}$ which minimize $$\sum_{\...
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33 views

Missing value imputation by tensor decomposition

As a beginner in linear algebra and tensor decomposition techniques, I may be posting a naive question. I have a tensor A that has some missing values. I am trying to predict the missing values using ...
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41 views

Symmetrizing an arbitrary rank tensor by pairwise permutations

Given a rank two tensor $T_{ab}$, we can symmetrize and antisymmetrize it as $T_{(ab)} = \frac{1}{2}(T_{ab}+T_{ba}), \quad T_{[ab]} = \frac{1}{2}(T_{ab}-T_{ba})$. In this case it is trivial to see ...
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1answer
50 views

Does the kernel of every alternating form contain a decomposable element?

Let $V$ be a real $n$-dimensional vector space, and let $1 < k < n$. Let $\alpha \in \bigwedge^k (V^*) \cong (\bigwedge^k V)^*$. Thinking of $\alpha$ as a linear functional $\bigwedge^k V \to \...
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139 views

Can every closed differential form be expressed via constant coefficients?

Let $M$ be a smooth $n$ dimensional manifold, and let $1 \le k < n$. Let $\omega \in \Omega^k(M)$ be a closed $k$-form on $M$. Let $p \in M$. Do there exist coordinates around $p$, such that $\...
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2answers
100 views

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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40 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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1answer
74 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 ...
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1answer
29 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 $\bigwedge^k V$....
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53 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 $\bigwedge^2V$, ...
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1answer
66 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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1answer
54 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 $(k,d)$,...
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1answer
111 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 ...
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1answer
56 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 ...
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1answer
367 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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72 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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1answer
57 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
69 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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2answers
391 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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114 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
84 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^...
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1answer
23 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 ...
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261 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 ...
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1answer
176 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
89 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 ...
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1answer
121 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 $p \in M$, $\omega_p=e^1_p \wedge e^2_p \wedge \dots \wedge e^k_p$ ...
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1answer
93 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 ...
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1answer
3k 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 ...
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1answer
265 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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57 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. ...
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1answer
842 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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83 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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42 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 ...