Questions tagged [tensor]

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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. ...
Lorenzo Guglielmi's user avatar
5 votes
1 answer
221 views

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^{\...
B.Hueber's user avatar
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4 votes
1 answer
152 views

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 \...
Sean Eberhard's user avatar
3 votes
1 answer
60 views

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 ...
Nick Chen's user avatar
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1 answer
169 views

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 ...
Nick Chen's user avatar
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1 answer
91 views

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}(...
Nick Chen's user avatar
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1 vote
1 answer
129 views

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 ...
Learning math's user avatar
4 votes
0 answers
119 views

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"? ...
Craig's user avatar
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0 answers
303 views

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 ...
MathDG's user avatar
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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 $B_{n\times p}$, ...
Min Wu's user avatar
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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 $S^2_0(M)$ ...
Eduardo Longa's user avatar
2 votes
0 answers
82 views

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 ...
Nick Chen's user avatar
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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 $...
mikeyd's user avatar
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1 answer
177 views

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 $...
B.Hueber's user avatar
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5 votes
2 answers
314 views

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} =...
Matthew Lou's user avatar
6 votes
1 answer
573 views

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} = - ...
Matthew Lou's user avatar
0 votes
1 answer
61 views

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 ...
Yang Ming's user avatar
4 votes
2 answers
390 views

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\...
Uzu Lim's user avatar
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1 answer
354 views

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_{...
Nathaniel Johnston's user avatar
4 votes
1 answer
182 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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2 votes
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252 views

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 ...
unknown's user avatar
  • 451
5 votes
2 answers
387 views

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 ...
Athena's user avatar
  • 275
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1 answer
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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 ...
Grwlf's user avatar
  • 135
2 votes
1 answer
220 views

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?
AMath91's user avatar
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0 answers
62 views

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\...
Keaton Hamm's user avatar
2 votes
1 answer
924 views

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 ...
RaphaelB4's user avatar
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98 views

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 ...
Eduardo Longa's user avatar
-1 votes
1 answer
89 views

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&...
Ruby K's user avatar
  • 9
2 votes
0 answers
113 views

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,...
James Chiu's user avatar
0 votes
0 answers
118 views

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 ...
Geom Zari's user avatar
0 votes
0 answers
174 views

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(...
SiXUlm's user avatar
  • 101
2 votes
3 answers
408 views

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. ...
sudgy's user avatar
  • 121
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0 answers
52 views

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 $(a1,...ak)$,...
TomJunior's user avatar
4 votes
1 answer
109 views

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 ...
Sebastian K.'s user avatar
1 vote
0 answers
122 views

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 ...
Matthieu Latapy's user avatar
1 vote
1 answer
151 views

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 $\...
TARS's user avatar
  • 13
1 vote
0 answers
96 views

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 ...
Grantsome's user avatar
4 votes
0 answers
215 views

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 $$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$ ...
B.Hueber's user avatar
  • 987
3 votes
1 answer
276 views

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 ...
B.K-Theory's user avatar
8 votes
0 answers
264 views

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 ...
Ben's user avatar
  • 1,010
2 votes
1 answer
204 views

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 ...
Matt's user avatar
  • 97
1 vote
0 answers
75 views

Spectral theorem for symmetric real tensors

Is there a definition of eigenvalues that allows to use a spectral theorem? Let $\mathbf{T}$ be a real fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}\...
Matt's user avatar
  • 97
0 votes
0 answers
124 views

Is it possible to reduce eigenvalues of tensors to an matrix eigenvalue problem?

Can we construct a larger matrix $M$ such that its eigenvalues are the same as the eigenvalues of a tensor $T$ of order 3? Let $\mathbf{T}$ be a fully symmetric tensor of order $3$ and size $N$. Its ...
Matt's user avatar
  • 97
2 votes
2 answers
505 views

Can the eigenvalues of a real symmetric tensor be complex?

Let $T$ be a fully symmetric tensor of rank $3$ and size $N$. Using the following definition of eigenvalues, let $x\in \mathbb{C}^N$ and $\lambda\in\mathbb{C}$ such that: \begin{equation} \sum_{jk}^...
Matt's user avatar
  • 97
2 votes
1 answer
136 views

Mean Gaussian curvature from non-unit vector

Pg.248 of "Textbook in Tensor Calculus and Differential Geometry" by Prasun Nayak. Let us suppose that $\lambda_{h|}^i$ is not a unit vector and therefore, the mean curvature $M_h$ in this ...
Chern-Simons's user avatar
3 votes
1 answer
346 views

Quaternions as eigenvalues of rank 3 tensors

Let us consider a matrix $M^{(a)}$ of size $N \times N$, having $N$ eigenvalues $\lambda_i \in \mathbb{C}$. Considering a rank-3 tensor, we can informally think of it as a sequence of $N$ matrices $M^{...
Matt's user avatar
  • 97
4 votes
2 answers
405 views

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 ...
Sidharth Ghoshal's user avatar
3 votes
1 answer
211 views

A different notion of a decomposable symmetric tensor (besides Veronese)

$\DeclareMathOperator{\complex}{\mathbb{C}}$ Let $\bigvee^m(\complex^n)\subseteq (\complex^n)^{\otimes m}$ denote the space of symmetric tensors, i.e. the set of $x \in (\complex^n)^{\otimes m}$ that ...
Ben's user avatar
  • 1,010
-1 votes
1 answer
475 views

Christoffel symbols as the expansion coefficients of covariant/contravariant derivatives

Page 155 of Vector and Tensor Analysis with Applications, by A.I Borishenko and I.E. Tarapov, the authors state that Christoffel symbols of the second kind are expansions of $\frac{\partial {\bf e}_j}{...
Olumide's user avatar
  • 661
0 votes
0 answers
337 views

What is the physical meaning of torsion

The torsion tensor in 4 dimensions $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-...
Eris's user avatar
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