72 research outputs found
Eigenvector Synchronization, Graph Rigidity and the Molecule Problem
The graph realization problem has received a great deal of attention in
recent years, due to its importance in applications such as wireless sensor
networks and structural biology. In this paper, we extend on previous work and
propose the 3D-ASAP algorithm, for the graph realization problem in
, given a sparse and noisy set of distance measurements. 3D-ASAP
is a divide and conquer, non-incremental and non-iterative algorithm, which
integrates local distance information into a global structure determination.
Our approach starts with identifying, for every node, a subgraph of its 1-hop
neighborhood graph, which can be accurately embedded in its own coordinate
system. In the noise-free case, the computed coordinates of the sensors in each
patch must agree with their global positioning up to some unknown rigid motion,
that is, up to translation, rotation and possibly reflection. In other words,
to every patch there corresponds an element of the Euclidean group Euc(3) of
rigid transformations in , and the goal is to estimate the group
elements that will properly align all the patches in a globally consistent way.
Furthermore, 3D-ASAP successfully incorporates information specific to the
molecule problem in structural biology, in particular information on known
substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a
faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a
preprocessing step for dividing the initial graph into smaller subgraphs. Our
extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very
robust to high levels of noise in the measured distances and to sparse
connectivity in the measurement graph, and compare favorably to similar
state-of-the art localization algorithms.Comment: 49 pages, 8 figure
Returns-Driven Macro Regimes and Characteristic Lead-Lag Behaviour between Asset Classes
We define data-driven macroeconomic regimes by clustering the relative
performance in time of indices belonging to different asset classes. We then
investigate lead-lag relationships within the regimes identified. Our study
unravels market features characteristic of different windows in time and
leverages on this knowledge to highlight market trends or risks that can be
informative with respect to recurrent market developments. The framework
developed also lays the foundations for multiple possible extensions.Comment: 9 pages, 8 figure
Provably robust estimation of modulo 1 samples of a smooth function with applications to phase unwrapping
Consider an unknown smooth function , and
say we are given noisy mod 1 samples of , i.e., , for , where denotes the noise. Given
the samples , our goal is to recover smooth, robust
estimates of the clean samples . We formulate a natural
approach for solving this problem, which works with angular embeddings of the
noisy mod 1 samples over the unit circle, inspired by the angular
synchronization framework. This amounts to solving a smoothness regularized
least-squares problem -- a quadratically constrained quadratic program (QCQP)
-- where the variables are constrained to lie on the unit circle. Our approach
is based on solving its relaxation, which is a trust-region sub-problem and
hence solvable efficiently. We provide theoretical guarantees demonstrating its
robustness to noise for adversarial, and random Gaussian and Bernoulli noise
models. To the best of our knowledge, these are the first such theoretical
results for this problem. We demonstrate the robustness and efficiency of our
approach via extensive numerical simulations on synthetic data, along with a
simple least-squares solution for the unwrapping stage, that recovers the
original samples of (up to a global shift). It is shown to perform well at
high levels of noise, when taking as input the denoised modulo samples.
Finally, we also consider two other approaches for denoising the modulo 1
samples that leverage tools from Riemannian optimization on manifolds,
including a Burer-Monteiro approach for a semidefinite programming relaxation
of our formulation. For the two-dimensional version of the problem, which has
applications in radar interferometry, we are able to solve instances of
real-world data with a million sample points in under 10 seconds, on a personal
laptop.Comment: 68 pages, 32 figures. arXiv admin note: text overlap with
arXiv:1710.1021
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