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Structured Tensor Recovery and Decomposition
Tensors, a.k.a. multi-dimensional arrays, arise naturally when modeling higher-order objects and relations. Among ubiquitous applications including image processing, collaborative filtering, demand forecasting and higher-order statistics, there are two recurring themes in general: tensor recovery and tensor decomposition. The first one aims to recover the underlying tensor from incomplete information; the second one is to study a variety of tensor decompositions to represent the array more concisely and moreover to capture the salient characteristics of the underlying data. Both topics are respectively addressed in this thesis.
Chapter 2 and Chapter 3 focus on low-rank tensor recovery (LRTR) from both theoretical and algorithmic perspectives. In Chapter 2, we first provide a negative result to the sum of nuclear norms (SNN) model---an existing convex model widely used for LRTR; then we propose a novel convex model and prove this new model is better than the SNN model in terms of the number of measurements required to recover the underlying low-rank tensor. In Chapter 3, we first build up the connection between robust low-rank tensor recovery and the compressive principle component pursuit (CPCP), a convex model for robust low-rank matrix recovery. Then we focus on developing convergent and scalable optimization methods to solve the CPCP problem. In specific, our convergent method, proposed by combining classical ideas from Frank-Wolfe and proximal methods, achieves scalability with linear per-iteration cost.
Chapter 4 generalizes the successive rank-one approximation (SROA) scheme for matrix eigen-decomposition to a special class of tensors called symmetric and orthogonally decomposable (SOD) tensor. We prove that the SROA scheme can robustly recover the symmetric canonical decomposition of the underlying SOD tensor even in the presence of noise. Perturbation bounds, which can be regarded as a higher-order generalization of the Davis-Kahan theorem, are provided in terms of the noise magnitude
Guaranteed Non-Orthogonal Tensor Decomposition via Alternating Rank- Updates
In this paper, we provide local and global convergence guarantees for
recovering CP (Candecomp/Parafac) tensor decomposition. The main step of the
proposed algorithm is a simple alternating rank- update which is the
alternating version of the tensor power iteration adapted for asymmetric
tensors. Local convergence guarantees are established for third order tensors
of rank in dimensions, when and the tensor
components are incoherent. Thus, we can recover overcomplete tensor
decomposition. We also strengthen the results to global convergence guarantees
under stricter rank condition (for arbitrary constant ) through a simple initialization procedure where the algorithm is
initialized by top singular vectors of random tensor slices. Furthermore, the
approximate local convergence guarantees for -th order tensors are also
provided under rank condition . The guarantees also
include tight perturbation analysis given noisy tensor.Comment: We have added an additional sub-algorithm to remove the (approximate)
residual error left after the tensor power iteratio
Perturbations and Stability of Static Black Holes in Higher Dimensions
In this chapter we consider perturbations and stability of higher dimensional
black holes focusing on the static background case. We first review a
gauge-invariant formalism for linear perturbations in a fairly generic class of
(m+n)-dimensional spacetimes with a warped product metric, including black hole
geometry. We classify perturbations of such a background into three types, the
tensor, vector and scalar-type, according to their tensorial behaviour on the
n-dimensional part of the background spacetime, and for each type of
perturbations, we introduce a set of manifestly gauge invariant variables. We
then introduce harmonic tensors and write down the equations of motion for the
expansion coefficients of the gauge invariant perturbation variables in terms
of the harmonics. In particular, for the tensor-type perturbations a single
master equation is obtained in the (m+n)-dimensional background, which is
applicable for perturbation analysis of not only static black holes but also
some class of rotating black holes as well as black-branes. For the vector and
scalar type, we derive a set of decoupled master equations when the background
is a (2+n)-dimensional static black hole in the Einstein-Maxwell theory with a
cosmological constant. As an application of the master equations, we review the
stability analysis of higher dimensional charged static black holes with a
cosmological constant. We also briefly review the recent results of a
generalisation of the perturbation formulae presented here and stability
analysis to static black holes in generic Lovelock theory.Comment: Invited review for Prog. Theor. Phys. Suppl, 45 pages, 2 figures, 1
table, v2: references added, the notations slightly modified to match PTPS
published versio
Perturbations and Stability of Static Black Holes in Higher Dimensions
In this chapter we consider perturbations and stability of higher dimensional
black holes focusing on the static background case. We first review a
gauge-invariant formalism for linear perturbations in a fairly generic class of
(m+n)-dimensional spacetimes with a warped product metric, including black hole
geometry. We classify perturbations of such a background into three types, the
tensor, vector and scalar-type, according to their tensorial behaviour on the
n-dimensional part of the background spacetime, and for each type of
perturbations, we introduce a set of manifestly gauge invariant variables. We
then introduce harmonic tensors and write down the equations of motion for the
expansion coefficients of the gauge invariant perturbation variables in terms
of the harmonics. In particular, for the tensor-type perturbations a single
master equation is obtained in the (m+n)-dimensional background, which is
applicable for perturbation analysis of not only static black holes but also
some class of rotating black holes as well as black-branes. For the vector and
scalar type, we derive a set of decoupled master equations when the background
is a (2+n)-dimensional static black hole in the Einstein-Maxwell theory with a
cosmological constant. As an application of the master equations, we review the
stability analysis of higher dimensional charged static black holes with a
cosmological constant. We also briefly review the recent results of a
generalisation of the perturbation formulae presented here and stability
analysis to static black holes in generic Lovelock theory.Comment: Invited review for Prog. Theor. Phys. Suppl, 45 pages, 2 figures, 1
table, v2: references added, the notations slightly modified to match PTPS
published versio
On the Use of Group Theoretical and Graphical Techniques toward the Solution of the General N-body Problem
Group theoretic and graphical techniques are used to derive the N-body wave
function for a system of identical bosons with general interactions through
first-order in a perturbation approach. This method is based on the maximal
symmetry present at lowest order in a perturbation series in inverse spatial
dimensions. The symmetric structure at lowest order has a point group
isomorphic with the S_N group, the symmetric group of N particles, and the
resulting perturbation expansion of the Hamiltonian is order-by-order invariant
under the permutations of the S_N group. This invariance under S_N imposes
severe symmetry requirements on the tensor blocks needed at each order in the
perturbation series. We show here that these blocks can be decomposed into a
basis of binary tensors invariant under S_N. This basis is small (25 terms at
first order in the wave function), independent of N, and is derived using
graphical techniques. This checks the N^6 scaling of these terms at first order
by effectively separating the N scaling problem away from the rest of the
physics. The transformation of each binary tensor to the final normal
coordinate basis requires the derivation of Clebsch-Gordon coefficients of S_N
for arbitrary N. This has been accomplished using the group theory of the
symmetric group. This achievement results in an analytic solution for the wave
function, exact through first order, that scales as N^0, effectively
circumventing intensive numerical work. This solution can be systematically
improved with further analytic work by going to yet higher orders in the
perturbation series.Comment: This paper was submitted to the Journal of Mathematical physics, and
is under revie
Second and higher-order perturbations of a spherical spacetime
The Gerlach and Sengupta (GS) formalism of coordinate-invariant, first-order,
spherical and nonspherical perturbations around an arbitrary spherical
spacetime is generalized to higher orders, focusing on second-order
perturbation theory. The GS harmonics are generalized to an arbitrary number of
indices on the unit sphere and a formula is given for their products. The
formalism is optimized for its implementation in a computer algebra system,
something that becomes essential in practice given the size and complexity of
the equations. All evolution equations for the second-order perturbations, as
well as the conservation equations for the energy-momentum tensor at this
perturbation order, are given in covariant form, in Regge-Wheeler gauge.Comment: Accepted for publication in Physical Review
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