Questions tagged [tensor]
The tensor tag has no usage guidance.
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Combination of simple tensors - II
This is a follow-up question to Combination of simple tensors.
I am interested in devising an alternative norm (I mean, other than the usual $\pi$ or $\epsilon$ norms) in the tensor product of two ...
2
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Combination of simple tensors
I aksed this question on Math Stack Exchange 6 days ago, with no answer: https://math.stackexchange.com/q/4875445/1297919
Let $X$ and $Y$ two Banach spaces and let $X\otimes Y$ their tensor product. ...
5
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Bochner Laplacian in coordinates
Sorry if this is a too basic question, but I didn't find an answer anywhere:
The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
4
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Is the asymptotic rank of a tensor bounded by (naive) border rank?
Write $R(T)$ for the rank of an order-$3$ tensor $T \in \mathbb C^{m \times n \times p}$ over the complex numbers. If $T' \in \mathbb {C}^{m' \times n' \times p'}$ is another such tensor then let $T \...
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Best approximation with tensors of rank $\ge2$
Let $k\in\mathbb N$, $H_i$ be a (finite-dimensional, if necessary) $\mathbb R$-Hilbert space for $i\in I:=\{1,\ldots,k\}$, $H:=\bigotimes_{i\in I}H_i$ denote the tensor product$^1$ of $(H_i)_{i\in I}$ ...
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What is the best known bound for the bilinear complexity of $4\times 4$ matrices product
Assume we work on the complex field $\mathbb{C}$. And we use $\langle p,q,r\rangle$ to denote the bilinear complexity of product of a $p\times q$ matrix and a $q\times r$. Recently I read a paper on ...
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How far is the slice rank of a tensor from its CP rank
Assume we work on any infinite field and 3-ordered tensor. Clearly for any tensor $T$, we have $\operatorname{srk}(T)\le \operatorname{rk}(T)$. Here, $\operatorname{srk}(T)$ (resp. $\operatorname{rk}(...
5
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What is expected (border) rank of the knonecker product of 3-tensors
Given two three order tensors $T$ and $S$ in $F^{m\times n\times p}$ and $F^{a\times b\times c}$. Clearly $\operatorname{rk}(T\otimes S)\le \operatorname{rk}(T)\operatorname{rk}(S)$. Does the equality ...
1
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1
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Diagonalizability, orthogonal diagonalizability of higher order tensors and their being or not being dense in some suitable topology
For our discussion, we'll assume that we're working with $\mathbb{R}^m$ only, but much or all of the following discussion should be carried over immediately to any finite dimensional inner product ...
4
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Is there a fast way to do this tensor power/trace operation?
Well, I asked this question on Math SE, and didn't get any responses, so I'm trying it here.
Given an $M*N*P$ tensor $T$, is there a fast way of computing the following "eighth-power trace"?
...
4
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Construct a 4th-order tensor with matricization ranks $r$ that is not rank $r$
I ask you for this possibly not so simple task:
Explicitly construct a 4th-order tensor $A \in \mathbb{C}^{n_1 \times \ldots \times n_4}$ that does not have (border) rank $r$, but for which each ...
1
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0
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303
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About "residual" scalar curvature in Einstein warped product manifold
I have an Einstein warped product manifold $M=B \times_f F$ with warped metric $g=g^B+f^2g^F$, where $F$ is Ricci flat, then its scalar curvature is $S^F=0$.
It is well known that the scalar curvature ...
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Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?
My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ...
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0
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119
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Some kind of product of two 2d tensors to create a 3d tensor?
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays):
given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
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Exponentiation of vector spaces?
This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...
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$1$-parameter family of metrics preserving the normal direction
Let $(M^n,g)$ be a compact Riemannian manifold with boundary, $n \geq 2$, and let $N$ be the unit outward normal to $\partial M$. Denote by $S^2(M)$ the symmetric covariant $2$-tensors, by $S^2_0(M)$ ...
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Is 'weak' Strassen Conjecture true?
$\newcommand{\rank}{\mathop{\mathrm{rank}}}$Strassen conjectured for two tensors $T_{1}$ and $T_{2}$, $\rank(T_{1}\oplus T_{2})=\rank(T_{1})+\rank(T_{2})$. This is not generally true according to ...
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3
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Proving the graded structure of the tensor algebra from only the universal property
When the tensor algebra is presented, it is usually constructed as the direct sum of all tensor powers of the space. By this construction, the graded structure of the tensor algebra is easy to prove. ...
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tensor dimension/reshaping group
Consider an $N$ dimensional tensor $T$ using the strided view representation used by PyTorch, i.e. we have a storage vector $S$ projected into $N$ dimensions using a size tuple $s$ and a stride tuple $...
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Decomposition of tensor field on hypersurface
Let $(\mathcal{M},g)$ be a Lorentzian manifold, which is globally of the form $\mathcal{M}\cong I\times\Sigma$, where $I\subset\mathbb{R}$ ("time") and $\Sigma$ ("space") is some $...
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Example of a curvature with no associated metric
Is there a concrete example of a $4$ tensor $R_{ijkl}$ with the same symmetries as the Riemannian curvature tensor, i.e.
\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} =...
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Does every ‘curvature’ tensor induce a metric? [duplicate]
So we know that given a Riemannian manifold $(M,g)$ we can calculate its Riemannian curvature $R$. This covariant $4$-tensor field then satisfies some important symmetries
\begin{gather*}
R_{ijkl} = - ...
0
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1
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Tensor nuclear norm for a binary 3rd-order tensor
I am interested in the low-rank approximation of a binary(01) third-order tensor. Does anyone know how to define its tensor nuclear norm based on whatever tensor decomposition methods? Could anyone ...
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Bounds for metric in normal coordinate
Let $M$ be a Riemannian $n$-manifold and $x \in M$. For the metric tensor $g_{ij}$ of geodesic normal coordinates at $x$, there is a formula $g_{ij}(y) = \delta_{ij} + \frac13 R_{kijl} y^k y^l + O(\|y\...
5
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Waring rank of monomials, and how it depends on the ground field
The Waring rank of a degree-$d$ homogeneous polynomial $p$ is the least integer $r$ such that you can write $p$ as a linear combination of $r$ $d$-th powers of linear forms $\{\ell_k\}$:
$$
p = \sum_{...
4
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2
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What is the current status of representation theory of $n$-ary groups in terms of hypermatrices?
An $n$-ary group is a generalization of the usual concept of a group where the binary operation (2-argument operation) is instead an $n$-ary ($n$-argument) operation. More info here on Wikipedia.
I ...
4
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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 ...
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15
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Why are matrices ubiquitous but hypermatrices rare?
I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, hypermatrices.
Of course ...
2
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0
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What are examples of "perfect tensors"?
A "perfect tensor" is defined on the nLab very abstractly as "its tensor/hom-adjuncts $V^{\otimes k} \to V^{\otimes n - k}$ for $k \le n/2$ are isometries". The only example I'm ...
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Local diagonalisation of a degenerated 2d metric tensor
Consider a smooth 2d-manifold $M$ and let $g$ be a smooth $(0,2)$-tensor satisfying $rk(g)\geq1$ everywhere. Obviously if $rk(g)=2$ at a point $p\in M$ then $g$ is locally diagonalisable (i.e. there ...
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What resource do Markov and Shi mean when they estimate tensor contraction complexity?
Markov and Shi in their paper Simulating quantum computation by contracting tensor networks define the contraction complexity as follows (page 10):
The complexity of π is the maximum degree of a ...
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1
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Tensor matricizations and their decompositions
Suppose we have a 4-index tensor $t_{ijkl}$ (all 4 dimensions are equal size). We can make a matrix out of it by taking first and last two indexes as new indexes: $t_{ijkl} \rightarrow M_{ij, kl}$. ...
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Decomposition of tensors
It is well known that every traceless, symmetric $2$-tensor can be decomposed uniquely into a Lie derivative part and a Codazzi part. Is there an analog for totally symmetric $k$-tensors?
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Tucker decompositions over arbitrary fields
Given an $n$-mode tensor $\mathcal{T}\in\mathbb{R}^{d_1\times\dotsb\times d_n}$, there exists a Tucker decomposition of $\mathcal T$ of the form
$$\mathcal{T} = \mathcal{X}\times_1 W_1\times_2\dotsb\...
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Is there a geometric intepretation of the trace of tensor on a Riemannian manifold?
For a long time I thought the trace for matrices were just an elementary function with nice properties. But it is much more than that and really should be think of as a geometrical object (see ...
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How is this product of tensors defined?
I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following:
Here, $\Delta_{-2}$ denotes the usual Laplacian ...
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1
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Multi-variable expansion of $e^{u^T X}$ as a power series in terms of tensors [closed]
I have the following question related to multivariable moment generating functions. Given two vectors $u,X\in\mathbb{R}^d$, I want to characterize the quantity $e^{u^T X}$ in terms of "powers&...
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0
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The Ricci curvature is bounded below by scalar curvature
So I have more questions coming from Dr Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature. In theorem 9.4,...
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A question from Richard Hamilton's paper "A matrix Harnack estimate for the heat equation"
Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126."
On page 125, at the end of the proof of Theorem 4.3, I abstract ...
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Is the kernel $\vert d_X - d_Y \vert^p$ conditionally negative definite?
Given two finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, for $p > 0$, define the kernel ($4$-D tensor) $K$ on $(X \times Y)^2$ by:
$$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...
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Solving nonlinear differential multi-variable equation with block-matrices
Here is the problem:
Given a formula $f:\mathbb{R}^{n+k}\rightarrow\mathbb{R}^n$, written as $f(x_1(t),x_2(t),...,x_n(t);a1,...ak)$ with $k$ real unknown parameters $a_1,...,a_k$. For any $(a1,...ak)$,...
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Characterization of all-orthogonal tensors
In the paper [1], it is proven in Theorem 2 that any $n$-tensor $\mathcal{A}\in\mathbb{R}^{d_1\times...\times d_n}$ can be decomposed as
$$
\mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n
$$
...
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Generalization of a standard algebraic group theory result for a tensor problem
$\DeclareMathOperator\GL{GL}$Let $X$, $Y$, $Z$ be $\mathbb{C}$-vector spaces, and let $A\subseteq X$ and $B\subseteq Y$ and $C\subseteq Z$ be linear subspaces. Let $V=X \otimes Y \otimes Z$, acted on ...
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Adjacency matrix/tensor operations for graph sequences?
Consider a graph $G=(V,E)$. Its adjacency matrix $A$ is defined by $A_{u,v} = 1$ if $(u,v)\in E$, $0$ otherwise.
Consider a vector $x$ that associates a value $x_v$ to each vertex of $G$. Consider the ...
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Derivative of eigenpair with respect to matrix
Suppose that $A$ is real and symmetric matrix (or tensor) of dimension $3 \times 3$, with its spectral decomposition
$$A = \sum_{i=1}^3 \lambda_i\ n_i\otimes n_i$$
where $\lambda_i$, $n_i$ and $\...
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How to count the number of tensors over a finite field of tensor rank $r$?
For simplicity, work over $\mathbb F_2$ and only consider order-$3$ equilateral tensors. For $r\in\mathbb Z_{>0}$, how many tensors $\mathfrak{T}\in\mathsf{Ten}_{n}^{\otimes 3}(\mathbb F_2)$ are ...
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References Request: Bach tensor
Recently I want to study about Bach tensor in detail. From the fundemental definition and properties to conformal invariant. Is there any references to me about Bach tensor? If there are some origins ...
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Pseudo-tensor- and tensor-densities: Sections of what bundle?
Let $\mathcal{M}$ be a smooth manifold. A tensor field is then usually defined to be a section of the tensor bundle
$$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$
...
3
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1
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277
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Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$
I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction:
First, I take the following partial ...
2
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1
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Eigenvalues of large symmetric random tensors
I am studying the eigenvalues of large random tensors and realise that very little is known about it. I was wondering what is already known and what could be potential leads to find their limiting ...