arXiv subjects

George Papanicolaou

Publications and source records attributed to George Papanicolaou.

At least 19 recordsLinked to original sources

arXiv2026

Iterative Atom Refinement: A Monotonicity Principle for Dictionary Learning

Dictionary learning seeks to recover an unknown dictionary $A$ from observations ${\bf y}_i = A{\bf x}_i$ with sparse coefficient vectors ${\bf x}_i$. We introduce the \emph{Iterative Atom Refinement} (IAR) algorithm, a simple procedure for recovering individual dictionary atoms. Starting from a random direction, IAR repeatedly selects the observations most strongly correlated with the current iterate and updates the direction by averaging the selected data. Our main contribution is a rigorous convergence theory of IAR. Using high-dimensional probabilistic estimates and a novel monotonicity principle for atom-selection probabilities, we show that a small initial advantage of one atom is amplified until that atom is isolated. Under our model assumptions, IAR identifies a generating atom after only three refinement steps. Numerical experiments support the theory and show that the resulting dynamics accurately capture the behavior observed in dictionary refinement.

cs.LG↗

arXiv2026

Imaging with super-resolution in changing random media

We develop an imaging algorithm that exploits strong scattering to achieve super-resolution in changing random media. The method processes large and diverse array datasets using sparse dictionary learning, clustering, and multidimensional scaling. Starting from random initializations, the algorithm reliably extracts the unknown medium properties necessary for accurate imaging using back-propagation, $\ell_2$ or $\ell_1$ methods. Remarkably, scattering enhances resolution beyond homogeneous medium limits. When abundant data are available, the algorithm allows the realization of super-resolution in imaging.

physics.optics↗

arXiv2024

Super-resolution in disordered media using neural networks

We propose a methodology that exploits large and diverse data sets to accurately estimate the ambient medium's Green's functions in strongly scattering media. Given these estimates, obtained with and without the use of neural networks, excellent imaging results are achieved, with a resolution that is better than that of a homogeneous medium. This phenomenon, also known as super-resolution, occurs because the ambient scattering medium effectively enhances the physical imaging aperture. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.

cs.LG↗

arXiv2023

Wave-informed dictionary learning for high-resolution imaging in complex media

We propose an approach for imaging in scattering media when large and diverse data sets are available. It has two steps. Using a dictionary learning algorithm the first step estimates the true Green's function vectors as columns in an unordered sensing matrix. The array data comes from many sparse sets of sources whose location and strength are not known to us. In the second step, the columns of the estimated sensing matrix are ordered for imaging using Multi-Dimensional Scaling with connectivity information derived from cross-correlations of its columns, as in time reversal. For these two steps to work together we need data from large arrays of receivers so the columns of the sensing matrix are incoherent for the first step, as well as from sub-arrays so that they are coherent enough to obtain the connectivity needed in the second step. Through simulation experiments, we show that the proposed approach is able to provide images in complex media whose resolution is that of a homogeneous medium.

cs.CV↗

arXiv2022

Quantitative phase and absorption contrast imaging

We present an algorithm for coherent diffractive imaging with phaseless measurements. It treats the forward model as a combination of coherent and incoherent waves. The algorithm reconstructs absorption and phase contrast that quantifies the attenuation and the refraction of the waves propagating through an object. It requires coherent or partially coherent illuminations, and several detectors to record the intensity of the distorted wave that passes through the object under inspection. The diversity of illuminations, obtained by putting masks between the source and the object, provides enough information for imaging. The computational cost of our algorithm is linear in the number of pixels of the image. Therefore, it is efficient for high-resolution imaging. Our algorithm guarantees exact recovery if the image is sparse for a given basis. Numerical experiments in the setting of phaseless diffraction imaging of sparse objects validate the efficiency and the precision of the suggested algorithm.

physics.comp-ph↗

arXiv2021

Correlation based Imaging for rotating satellites

We consider imaging of fast moving small objects in space, such as low earth orbit satellites, which are also rotating around a fixed axis. The imaging system consists of ground based, asynchronous sources of radiation and several passive receivers above the dense atmosphere. We use the cross-correlation of the received signals to reduce distortions from ambient medium fluctuations. Imaging with correlations also has the advantage of not requiring any knowledge about the probing pulse and depends weakly on the emitter positions. We account for the target's orbital velocity by introducing the necessary Doppler compensation. To image a fast rotating object we also need to compensate for the rotation. We show that the rotation parameters can be extracted directly from the auto-correlation of the data before the formation of the image. We then investigate and analyze an imaging method that relies on backpropagating the cross-correlation data structure to two points rather than one, thus forming an interference matrix. The proposed imaging method consists of estimating the reflectivity as the top eigenvector of the migrated cross-correlation data interference matrix. We call this the rank-1 image and show that it provides superior image resolution compared to the usual single-point migration scheme for fast moving and rotating objects. Moreover, we observe a significant improvement in resolution due to the rotation leading to a diffraction limited resolution. We carry out a theoretical analysis that illustrates the role of the two point migration method as well as that of the inverse aperture and rotation in improving resolution. Extensive numerical simulations support the theoretical results.

eess.SP↗

arXiv2020

Fast signal recovery from quadratic measurements

We present a novel approach for recovering a sparse signal from cross-correlated data. Cross-correlations naturally arise in many fields of imaging, such as optics, holography and seismic interferometry. Compared to the sparse signal recovery problem that uses linear measurements, the unknown is now a matrix formed by the cross correlation of the unknown signal. Hence, the bottleneck for inversion is the number of unknowns that grows quadratically. The main idea of our proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal of the unknown matrix, whose dimension grows linearly with the size of the problem. The keystone of the methodology is the use of an efficient {\em Noise Collector} that absorbs the data that come from the off-diagonal elements of the unknown matrix and that do not carry extra information about the support of the signal. This results in a linear problem whose cost is similar to the one that uses linear measurements. Our theory shows that the proposed approach provides exact support recovery when the data is not too noisy, and that there are no false positives for any level of noise. Moreover, our theory also demonstrates that when using cross-correlated data, the level of sparsity that can be recovered increases, scaling almost linearly with the number of data. The numerical experiments presented in the paper corroborate these findings.

eess.SP↗

arXiv2020

Generalized correlation based Imaging for satellites

We consider imaging of fast moving small objects in space, such as low earth orbit satellites. The imaging system consists of ground based, asynchronous sources of radiation and several passive receivers above the dense atmosphere. We use the cross correlation of the received signals to reduce distortions from ambient medium fluctuations. Imaging with correlations also has the advantage of not requiring any knowledge about the probing pulse and depends weakly on the emitter positions. We account for the target's orbital velocity by introducing the necessary Doppler compensation. We show that over limited imaging regions, a constant Doppler factor can be used, resulting in an efficient data structure for the correlations of the recorded signals. We then investigate and analyze different imaging methods using the cross-correlation data structure. Specifically, we show that using a generalized two point migration of the cross correlation data, the top eigenvector of the migrated data matrix provides superior image resolution compared to the usual single-point migration scheme. We carry out a theoretical analysis that illustrates the role of the two point migration methods as well as that of the inverse aperture in improving resolution. Extensive numerical simulations support the theoretical results and assess the scope of the imaging methodology.

eess.SP↗

arXiv2020

Traveling waves in a mean field learning model

Lucas and Moll have proposed a system of forward-backward partial differential equations that model knowledge diffusion and economic growth. It arises from a microscopic model of learning for a mean-field type interacting system of individual agents. In this paper, we prove existence of traveling wave solutions to this system. They correspond to what is known in economics as balanced growth path solutions. We also study the dependence of the solutions and their propagation speed on various economic parameters of the system.

math.AP↗

arXiv2020

PCA for Implied Volatility Surfaces

Principal component analysis (PCA) is a useful tool when trying to construct factor models from historical asset returns. For the implied volatilities of U.S. equities there is a PCA-based model with a principal eigenportfolio whose return time series lies close to that of an overarching market factor. The authors show that this market factor is the index resulting from the daily compounding of a weighted average of implied-volatility returns, with weights based on the options' open interest (OI) and Vega. The authors also analyze the singular vectors derived from the tensor structure of the implied volatilities of S&P500 constituents, and find evidence indicating that some type of OI and Vega-weighted index should be one of at least two significant factors in this market.

q-fin.ST↗

arXiv2020

Synthetic aperture imaging and motion estimation using tensor methods

We consider a synthetic aperture imaging configuration, such as synthetic aperture radar (SAR), where we want to first separate reflections from moving targets from those coming from a stationary background, and then to image separately the moving and the stationary reflectors. For this purpose, we introduce a representation of the data as a third order tensor formed from data coming from partially overlapping sub-apertures. We then apply a tensor robust principal component analysis (TRPCA) to the tensor data which separates them into the parts coming from the stationary and moving reflectors. Images are formed with the separated data sets. Our analysis shows a distinctly improved performance of TRPCA, compared to the usual matrix case. In particular, the tensor decomposition can identify motion features that are undetectable when using the conventional motion estimation methods, including matrix RPCA. We illustrate the performance of the method with numerical simulations in the X-band radar regime.

eess.SP↗

arXiv2019

The Noise Collector for sparse recovery in high dimensions

The ability to detect sparse signals from noisy high-dimensional data is a top priority in modern science and engineering. A sparse solution of the linear system $A ρ= b_0$ can be found efficiently with an $l_1$-norm minimization approach if the data is noiseless. Detection of the signal's support from data corrupted by noise is still a challenging problem, especially if the level of noise must be estimated. We propose a new efficient approach that does not require any parameter estimation. We introduce the Noise Collector (NC) matrix $C$ and solve an augmented system $A ρ+ C η= b_0 + e$, where $ e$ is the noise. We show that the $l_1$-norm minimal solution of the augmented system has zero false discovery rate for any level of noise and with probability that tends to one as the dimension of $ b_0$ increases to infinity. We also obtain exact support recovery if the noise is not too large, and develop a Fast Noise Collector Algorithm which makes the computational cost of solving the augmented system comparable to that of the original one. Finally, we demonstrate the effectiveness of the method in applications to passive array imaging.

eess.SP↗

arXiv2019

Imaging with highly incomplete and corrupted data

We consider the problem of imaging sparse scenes from a few noisy data using an $l_1$-minimization approach. This problem can be cast as a linear system of the form $A \, ρ=b$, where $A$ is an $N\times K$ measurement matrix. We assume that the dimension of the unknown sparse vector $ρ\in {\mathbb{C}}^K$ is much larger than the dimension of the data vector $b \in {\mathbb{C}}^N$, i.e, $K \gg N$. We provide a theoretical framework that allows us to examine under what conditions the $\ell_1$-minimization problem admits a solution that is close to the exact one in the presence of noise. Our analysis shows that $l_1$-minimization is not robust for imaging with noisy data when high resolution is required. To improve the performance of $l_1$-minimization we propose to solve instead the augmented linear system $ [A \, | \, C] ρ=b$, where the $N \times Σ$ matrix $C$ is a noise collector. It is constructed so as its column vectors provide a frame on which the noise of the data, a vector of dimension $N$, can be well approximated. Theoretically, the dimension $Σ$ of the noise collector should be $e^N$ which would make its use not practical. However, our numerical results illustrate that robust results in the presence of noise can be obtained with a large enough number of columns $Σ\approx 10 K$.

eess.IV↗

arXiv2019

Low rank plus sparse decomposition of synthetic aperture radar data for target imaging and tracking

We analyze synthetic aperture radar (SAR) imaging of complex ground scenes that contain both stationary and moving targets. In the usual SAR acquisition scheme, we consider ways to preprocess the data so as to separate the contributions of the moving targets from those due to stationary background reflectors. Both components of the data, that is, reflections from stationary and moving targets, are considered as signal that is needed for target imaging and tracking, respectively. The approach we use is to decompose the data matrix into a low rank and a sparse part. This decomposition enables us to capture the reflections from moving targets into the sparse part and those from stationary targets into the low rank part of the data. The computational tool for this is robust principal component analysis (RPCA) applied to the SAR data matrix. We also introduce a lossless baseband transformation of the data, which simplifies the analysis and improves the performance of the RPCA algorithm. Our main contribution is a theoretical analysis that determines an optimal choice of parameters for the RPCA algorithm so as to have an effective and stable separation of SAR data coming from moving and stationary targets. This analysis gives also a lower bound for detectable target velocities. We show in particular that the rank of the sparse matrix is proportional to the square root of the target's speed in the direction that connects the SAR platform trajectory to the imaging region. The robustness of the approach is illustrated with numerical simulations in the X-band SAR regime.

eess.SP↗

arXiv2019

Synthetic aperture imaging with intensity-only data

We consider imaging the reflectivity of scatterers from intensity-only data recorded by a single moving transducer that both emits and receives signals, forming a synthetic aperture. By exploiting frequency illumination diversity, we obtain multiple intensity measurements at each location, from which we determine field cross-correlations using an appropriate phase controlled illumination strategy and the inner product polarization identity. The field cross-correlations obtained this way do not, however, provide all the missing phase information because they are determined up to a phase that depends on the receiver's location. The main result of this paper is an algorithm with which we recover the field cross-correlations up to a single phase that is common to all the data measured over the synthetic aperture, so all the data are synchronized. Thus, we can image coherently with data over all frequencies and measurement locations as if full phase information was recorded.

physics.comp-ph↗

arXiv2019

Robust multifrequency imaging with MUSIC

In this paper, we study the MUltiple SIgnal Classification (MUSIC) algorithm often used to image small targets when multiple measurement vectors are available. We show that this algorithm may be used when the imaging problem can be cast as a linear system that admits a special factorization. We discuss several active array imaging configurations where this factorization is exact, as well as other configurations where the factorization only holds approximately and, hence, the results provided by MUSIC deteriorate. We give special attention to the most general setting where an active array with an arbitrary number of transmitters and receivers uses signals of multiple frequencies to image the targets. This setting provides all the possible diversity of information that can be obtained from the illuminations. We give a theorem that shows that MUSIC is robust with respect to additive noise provided that the targets are well separated. The theorem also shows the relevance of using appropriate sets of controlled parameters, such as excitations, to form the images with MUSIC robustly. We present numerical experiments that support our theoretical results.

math.NA↗

arXiv2018

Mean Field Model for Collective Motion Bistability

We consider the Czirók model for collective motion of locusts along a one-dimensional torus. In the model, each agent's velocity locally interacts with other agents' velocities in the system, and there is also exogenous randomness to each agent's velocity. The interaction tends to create the alignment of collective motion. By analyzing the associated nonlinear Fokker-Planck equation, we obtain the condition for the existence of stationary order states and the conditions for their linear stability. These conditions depend on the noise level, which should be strong enough, and on the interaction between the agent's velocities, which should be neither too small, nor too strong. We carry out the fluctuation analysis of the interacting system and describe the large deviation principle to calculate the transition probability from one order state to the other. Numerical simulations confirm our analytical findings.

math.AP↗

arXiv2017

Random matrix approach to estimation of high-dimensional factor models

In dealing with high-dimensional data sets, factor models are often useful for dimension reduction. The estimation of factor models has been actively studied in various fields. In the first part of this paper, we present a new approach to estimate high-dimensional factor models, using the empirical spectral density of residuals. The spectrum of covariance matrices from financial data typically exhibits two characteristic aspects: a few spikes and bulk. The former represent factors that mainly drive the features and the latter arises from idiosyncratic noise. Motivated by these two aspects, we consider a minimum distance between two spectrums; one from a covariance structure model and the other from real residuals of financial data that are obtained by subtracting principal components. Our method simultaneously provides estimators of the number of factors and information about correlation structures in residuals. Using free random variable techniques, the proposed algorithm can be implemented and controlled effectively. Monte Carlo simulations confirm that our method is robust to noise or the presence of weak factors. Furthermore, the application to financial time-series shows that our estimators capture essential aspects of market dynamics.

q-fin.ST↗