44 research outputs found
Bispectrum Inversion with Application to Multireference Alignment
We consider the problem of estimating a signal from noisy
circularly-translated versions of itself, called multireference alignment
(MRA). One natural approach to MRA could be to estimate the shifts of the
observations first, and infer the signal by aligning and averaging the data. In
contrast, we consider a method based on estimating the signal directly, using
features of the signal that are invariant under translations. Specifically, we
estimate the power spectrum and the bispectrum of the signal from the
observations. Under mild assumptions, these invariant features contain enough
information to infer the signal. In particular, the bispectrum can be used to
estimate the Fourier phases. To this end, we propose and analyze a few
algorithms. Our main methods consist of non-convex optimization over the smooth
manifold of phases. Empirically, in the absence of noise, these non-convex
algorithms appear to converge to the target signal with random initialization.
The algorithms are also robust to noise. We then suggest three additional
methods. These methods are based on frequency marching, semidefinite relaxation
and integer programming. The first two methods provably recover the phases
exactly in the absence of noise. In the high noise level regime, the invariant
features approach for MRA results in stable estimation if the number of
measurements scales like the cube of the noise variance, which is the
information-theoretic rate. Additionally, it requires only one pass over the
data which is important at low signal-to-noise ratio when the number of
observations must be large
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
Denoising modulo samples: k-NN regression and tightness of SDP relaxation
Many modern applications involve the acquisition of noisy modulo samples of a
function , with the goal being to recover estimates of the original samples
of . For a Lipschitz function , suppose we are
given the samples where
denotes noise. Assuming are zero-mean i.i.d Gaussian's, and
's form a uniform grid, we derive a two-stage algorithm that recovers
estimates of the samples with a uniform error rate holding with high probability. The first stage
involves embedding the points on the unit complex circle, and obtaining
denoised estimates of via a NN (nearest neighbor) estimator.
The second stage involves a sequential unwrapping procedure which unwraps the
denoised mod estimates from the first stage.
Recently, Cucuringu and Tyagi proposed an alternative way of denoising modulo
data which works with their representation on the unit complex circle. They
formulated a smoothness regularized least squares problem on the product
manifold of unit circles, where the smoothness is measured with respect to the
Laplacian of a proximity graph involving the 's. This is a nonconvex
quadratically constrained quadratic program (QCQP) hence they proposed solving
its semidefinite program (SDP) based relaxation. We derive sufficient
conditions under which the SDP is a tight relaxation of the QCQP. Hence under
these conditions, the global solution of QCQP can be obtained in polynomial
time.Comment: 34 pages, 6 figure
On denoising modulo 1 samples of a function
Consider an unknown smooth function , and
say we are given noisy samples of , i.e., for , where denotes 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 representations of mod 1 values over the
unit circle. This amounts to solving a quadratically constrained quadratic
program (QCQP) with non-convex constraints involving points lying on the unit
circle. Our proposed approach is based on solving its relaxation which is a
trust-region sub-problem, and hence solvable efficiently. We demonstrate its
robustness to noise % of our approach via extensive simulations on several
synthetic examples, and provide a detailed theoretical analysis.Comment: 19 pages, 13 figures. To appear in AISTATS 2018. Corrected typos, and
made minor stylistic changes throughout. Main results unchanged. Added
section I (and Figure 13) in appendi
Provably robust estimation of modulo 1 samples of a smooth function with applications to phase unwrapping
International audienceConsider an unknown smooth function , and assume we are given noisy mod 1 samples of mod 1, 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 proposed 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, as well as 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 proposed approach via extensive numerical simulations on synthetic data, along with a simple least-squares based solution for the unwrapping stage, that recovers the original samples of f (up to a global shift). It is shown to perform well at high levels of noise, when taking as input the denoised modulo 1 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 synthetic aperture radar interferometry (InSAR), we are able to solve instances of real-world data with a million sample points in under 10 seconds, on a personal laptop
Denoising modulo samples: k-NN regression and tightness of SDP relaxation
International audienceMany modern applications involve the acquisition of noisy modulo samples of a function f , with the goal being to recover estimates of the original samples of f. For a Lipschitz function f : [0, 1]^d â R, suppose we are given the samples y_i = (f (x_i) + η_i) mod 1; i = 1,. .. , n where η_i denotes noise. Assuming η_i are zero-mean i.i.d Gaussian's, and x_i 's form a uniform grid, we derive a two-stage algorithm that recovers estimates of the samples f (x_i) with a uniform error rate O((log n / n)^{1/(d+2)}) holding with high probability. The first stage involves embedding the points on the unit complex circle, and obtaining denoised estimates of f (x_i) mod 1 via a kNN (nearest neighbor) estimator. The second stage involves a sequential unwrapping procedure which unwraps the denoised mod 1 estimates from the first stage. The estimates of the samples f(x_i) can be subsequently utilized to construct an estimate of the function fâ , with the aforementioned uniform error rate. Recently, Cucuringu and Tyagi [7] proposed an alternative way of denoising modulo 1 data which works with their representation on the unit complex circle. They formulated a smoothness regularized least squares problem on the product manifold of unit circles, where the smoothness is measured with respect to the Laplacian of a proximity graph G involving the x_i 's. This is a nonconvex quadratically constrained quadratic program (QCQP) hence they proposed solving its semidefinite program (SDP) based relaxation. We derive sufficient conditions under which the SDP is a tight relaxation of the QCQP. Hence under these conditions, the global solution of QCQP can be obtained in polynomial time