195 research outputs found
Spectral norm of random tensors
We show that the spectral norm of a random tensor (or higher-order array) scales as
under some sub-Gaussian
assumption on the entries. The proof is based on a covering number argument.
Since the spectral norm is dual to the tensor nuclear norm (the tightest convex
relaxation of the set of rank one tensors), the bound implies that the convex
relaxation yields sample complexity that is linear in (the sum of) the number
of dimensions, which is much smaller than other recently proposed convex
relaxations of tensor rank that use unfolding.Comment: 5 page
A Unified View of Graph Regularity via Matrix Decompositions
We prove algorithmic weak and \Szemeredi{} regularity lemmas for several
classes of sparse graphs in the literature, for which only weak regularity
lemmas were previously known. These include core-dense graphs, low threshold
rank graphs, and (a version of) upper regular graphs. More precisely, we
define \emph{cut pseudorandom graphs}, we prove our regularity lemmas for these
graphs, and then we show that cut pseudorandomness captures all of the above
graph classes as special cases.
The core of our approach is an abstracted matrix decomposition, roughly
following Frieze and Kannan [Combinatorica '99] and \Lovasz{} and Szegedy
[Geom.\ Func.\ Anal.\ '07], which can be computed by a simple algorithm by
Charikar [AAC0 '00]. This gives rise to the class of cut pseudorandom graphs,
and using work of Oveis Gharan and Trevisan [TOC '15], it also implies new
PTASes for MAX-CUT, MAX-BISECTION, MIN-BISECTION for a significantly expanded
class of input graphs. (It is NP Hard to get PTASes for these graphs in
general.
On orthogonal tensors and best rank-one approximation ratio
As is well known, the smallest possible ratio between the spectral norm and
the Frobenius norm of an matrix with is and
is (up to scalar scaling) attained only by matrices having pairwise orthonormal
rows. In the present paper, the smallest possible ratio between spectral and
Frobenius norms of tensors of order , also
called the best rank-one approximation ratio in the literature, is
investigated. The exact value is not known for most configurations of . Using a natural definition of orthogonal tensors over the real
field (resp., unitary tensors over the complex field), it is shown that the
obvious lower bound is attained if and only if a
tensor is orthogonal (resp., unitary) up to scaling. Whether or not orthogonal
or unitary tensors exist depends on the dimensions and the
field. A connection between the (non)existence of real orthogonal tensors of
order three and the classical Hurwitz problem on composition algebras can be
established: existence of orthogonal tensors of size
is equivalent to the admissibility of the triple to the Hurwitz
problem. Some implications for higher-order tensors are then given. For
instance, real orthogonal tensors of order
do exist, but only when . In the complex case, the situation is
more drastic: unitary tensors of size with exist only when . Finally, some numerical illustrations
for spectral norm computation are presented
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Marchenko-Pastur law with relaxed independence conditions
We prove the Marchenko-Pastur law for the eigenvalues of sample
covariance matrices in two new situations where the data does not have
independent coordinates. In the first scenario - the block-independent model -
the coordinates of the data are partitioned into blocks in such a way that
the entries in different blocks are independent, but the entries from the same
block may be dependent. In the second scenario - the random tensor model - the
data is the homogeneous random tensor of order , i.e. the coordinates of the
data are all different products of variables chosen from a
set of independent random variables. We show that Marchenko-Pastur law
holds for the block-independent model as long as the size of the largest block
is and for the random tensor model as long as . Our main
technical tools are new concentration inequalities for quadratic forms in
random variables with block-independent coordinates, and for random tensors
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