Questions tagged [tensor-decomposition]

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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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Special case of CP decomposition

I am reading this article by Kolda. On page 21, it says: ...CP [decomposition] can be viewed as a special case of Tucker [decomposition] where the core tensor is superdiagonal and $P = Q = R$... ...
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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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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 does the outer product of matrics work during approximation of a tensor?

I read a paper about Tensor-SVD, named Factorization strategies for third-order tensors. The section 4.1 provides an approximation strategies based on products of tensors, which is shown as follows: ...
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18 views

rank of a matrix formed from a tensor product

I know that rank of a matrix expressed in term of product of 2 matrices, satisfies the following equation rank(AB) ≤ min(rank(A), rank(B)). When we write a matrix as a product of tensors(of tensor ...
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9 views

Single-factor CP decomposition optimization equivalency with block coordinate descent using the tensor power method

I am trying to understand the objective functions for CP decompositions using the tensor power method as discussed in this paper. In this approach, a series of rank-1 approximations are made using the ...
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11 views

Covariance of Wishart Distribution

Let us consider a random matrix $$W \sim Wishart(n, \Sigma_p),$$ i.e. $W \in \mathbb{R}^{p \times p}$ and $W = Z^T Z$ for $Z \in \mathbb{R}^{n \times p}$ and each row of $Z$ drawn iid from a $N(0, \...
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26 views

The optimal solution of tensor Tucker decomposition problem

I have a question about tensor Tucker deocomposition. Recall that in Higher-Order Singular Value Decomposition (HOSVD), each factor matrices $U_i$ is obtained by taking the top $R_i$ left singular ...
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28 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 ...
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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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What's about a tensor/matrix decomposition that makes it better?

The following question is for tensors, but could also be applied to matrices. I have seen that an N-th order tensor $X \in \mathbb{R}^{I\times...\times I}$ has an original storage complexity of $O(I^N)...
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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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47 views

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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CP-ALS algorithm with Kroonenberg and De Leeuw initialization

I am trying to implement the CANDECOMP/PARAFAC - Alternating Least Square algorithm, with a small improvement in the initial guess. The simple initial guess for A is random in the method==0 if block. ...
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58 views

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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Decomposition of a matrix as a tensor product

Suppose that a squared-matrix $A$ can be decomposed as a tensor product of lower rank squared matrices, i.e. $$ A = M_1\otimes M_2. $$ For simplicity, suppose $A$ is a $N\times N$ matrix where $N/2$ ...
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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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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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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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179 views

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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Component of a irreducible tensor product

Given the operators $\boldsymbol{\alpha}$ and $\boldsymbol{C^{(L)}}$ such that $$ \boldsymbol{\alpha}=\left(\begin{array}{cc} 0 & \boldsymbol{\sigma}_{p} \\ \boldsymbol{\sigma}_{p} & 0 \end{...
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1answer
34 views

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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Dimensionalities of inner Product of tensors

As the inner product of two vectors i.e. tensors of order $1$ results into a reduced-order tensor i.e. scalar (a tensor of order $0$). as the outer product of two vectors results into a matrix. while ...
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60 views

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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1answer
319 views

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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1answer
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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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135 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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1answer
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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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117 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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185 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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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
77 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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61 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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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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297 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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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 ...