1,436 research outputs found
Linear embeddings of low-dimensional subsets of a Hilbert space to Rm
International audienceWe consider the problem of embedding a low-dimensional set, M, from an infinite-dimensional Hilbert space, H, to a finite-dimensional space. Defining appropriate random linear projections, we propose two constructions of linear maps that have the restricted isometry property (RIP) on the secant set of M with high probability. The first one is optimal in the sense that it only needs a number of projections essentially proportional to the intrinsic dimension of M to satisfy the RIP. The second one, which is based on a variable density sampling technique, is computationally more efficient, while potentially requiring more measurements
Dimensionality reduction with subgaussian matrices: a unified theory
We present a theory for Euclidean dimensionality reduction with subgaussian
matrices which unifies several restricted isometry property and
Johnson-Lindenstrauss type results obtained earlier for specific data sets. In
particular, we recover and, in several cases, improve results for sets of
sparse and structured sparse vectors, low-rank matrices and tensors, and smooth
manifolds. In addition, we establish a new Johnson-Lindenstrauss embedding for
data sets taking the form of an infinite union of subspaces of a Hilbert space
Impossibility of dimension reduction in the nuclear norm
Let (the Schatten--von Neumann trace class) denote the Banach
space of all compact linear operators whose nuclear norm
is finite, where
are the singular values of . We prove that
for arbitrarily large there exists a subset
with that cannot be
embedded with bi-Lipschitz distortion into any -dimensional
linear subspace of . is not even a -Lipschitz
quotient of any subset of any -dimensional linear subspace of
. Thus, does not admit a dimension reduction
result \'a la Johnson and Lindenstrauss (1984), which complements the work of
Harrow, Montanaro and Short (2011) on the limitations of quantum dimension
reduction under the assumption that the embedding into low dimensions is a
quantum channel. Such a statement was previously known with
replaced by the Banach space of absolutely summable sequences via the
work of Brinkman and Charikar (2003). In fact, the above set can
be taken to be the same set as the one that Brinkman and Charikar considered,
viewed as a collection of diagonal matrices in . The challenge is
to demonstrate that cannot be faithfully realized in an arbitrary
low-dimensional subspace of , while Brinkman and Charikar
obtained such an assertion only for subspaces of that consist of
diagonal operators (i.e., subspaces of ). We establish this by proving
that the Markov 2-convexity constant of any finite dimensional linear subspace
of is at most a universal constant multiple of
Distance Geometry in Quasihypermetric Spaces. III
Let be a compact metric space and let denote the
space of all finite signed Borel measures on . Define by
and set , where ranges over the collection of signed
measures in of total mass 1. This paper, with two earlier
papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric
spaces. I and II], investigates the geometric constant and its
relationship to the metric properties of and the functional-analytic
properties of a certain subspace of when equipped with a
natural semi-inner product. Specifically, this paper explores links between the
properties of and metric embeddings of , and the properties of
when is a finite metric space.Comment: 20 pages. References [10] and [11] are arXiv:0809.0740v1 [math.MG]
and arXiv:0809.0744v1 [math.MG
The Role of Topology in Quantum Tomography
We investigate quantum tomography in scenarios where prior information
restricts the state space to a smooth manifold of lower dimensionality. By
considering stability we provide a general framework that relates the topology
of the manifold to the minimal number of binary measurement settings that is
necessary to discriminate any two states on the manifold. We apply these
findings to cases where the subset of states under consideration is given by
states with bounded rank, fixed spectrum, given unitary symmetry or taken from
a unitary orbit. For all these cases we provide both upper and lower bounds on
the minimal number of binary measurement settings necessary to discriminate any
two states of these subsets
On metric Ramsey-type phenomena
The main question studied in this article may be viewed as a nonlinear
analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey
theory in combinatorics. Given a finite metric space on n points, we seek its
subspace of largest cardinality which can be embedded with a given distortion
in Hilbert space. We provide nearly tight upper and lower bounds on the
cardinality of this subspace in terms of n and the desired distortion. Our main
theorem states that for any epsilon>0, every n point metric space contains a
subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space
with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the
distortion is tight up to the log(1/\epsilon) factor. We further include a
comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio
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