Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tensor network ranks   In problems involving approximation, completion, denoising, dimension
reduction, estimation, interpolation, modeling, order reduction, regression,
etc, we argue that the near-universal practice of assuming that a function,
matrix, or tensor (which we will see are all the same object in this context)
has \emph{low rank} may be ill-justified. There are many natural instances
where the object in question has high rank with respect to the classical
notions of rank: matrix rank, tensor rank, multilinear rank --- the latter two
being the most straightforward generalizations of the former. To remedy this,
we show that one may vastly expand these classical notions of ranks: Given any
undirected graph $G$, there is a notion of $G$-rank associated with $G$, which
provides us with as many different kinds of ranks as there are undirected
graphs. In particular, the popular tensor network states in physics (e.g.,
\textsc{mps}, \textsc{ttns}, \textsc{peps}) may be regarded as functions of a
specific $G$-rank for various choices of $G$. Among other things, we will see
that a function, matrix, or tensor may have very high matrix, tensor, or
multilinear rank and yet very low $G$-rank for some $G$. In fact the difference
is in the orders of magnitudes and the gaps between $G$-ranks and these
classical ranks are arbitrarily large for some important objects in computer
science, mathematics, and physics. Furthermore, we show that there is a $G$
such that almost every tensor has $G$-rank exponentially lower than its rank or
the dimension of its ambient space.

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The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: physics
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The categories of this paper are: eess
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio