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Albert Fannjiang

Publications and source records attributed to Albert Fannjiang.

At least 19 recordsLinked to original sources
arXiv2026

Certified Spherical MUSIC for 3D Localization under Adversarial Subspace Perturbations

We study an oracle subspace-perturbation model in which the 3D localization procedure is given an \(s\)-dimensional subspace \(\widetilde{\mathcal U}\) and a deterministic error bound $\eps_{\rm sub}$ measuring the sine-theta distance between $\widetilde{\mathcal U}$ and an $s$-dimensional subspace $\cU$ of far-field patterns with wavenumber $\kappa$. Under explicit arbitrary-cloud separation and conditioning hypotheses, we prove that the perturbed spherical MUSIC objective \[ \widetilde q(\bz) = 1-\|P_{\widetilde{\mathcal U}}\varphi_\bz\|_2^2 \] has a unique strongly convex well in each ball \(B_{\gamma/\kappa}(x_j)\) and the objective has a uniform value gap outside the union of the certified wells. A fixed-step gradient map with \(h\asymp\kappa^{-2}\) leaves every certified well invariant and converges linearly to its unique minimizer. Consequently, thresholding on an \(O(\kappa^{-1})\)-mesh, followed by gradient descent from all accepted grid points and duplicate removal, recovers all relevant minima. The arbitrary-cloud frame analysis gives the sufficient condition \[ \kappa\delta_X\gtrsim s^{2/3} \] through an absolute coherence row sum and Gershgorin's theorem. We also construct lower-frame counterexamples below the \(s^{1/6}\) scale, upper-frame counterexamples below the \(s^{1/3}\) scale, and examples showing that the exponent \(2/3\) is optimal for the absolute-row-sum argument. The latter is a sharpness result for the proof method and is not a spectral necessity claim. Finally, for parameter classes containing a uniformly admissible one-point displacement path, we prove that the deterministic oracle localization modulus is \[ \mathfrak R(\eps) \asymp \frac{\eps}{\kappa}. \]

math.CA↗
arXiv2026

Multidimensional Gradient-MUSIC: A Global Nonconvex Optimization Framework for Optimal Resolution

We develop a multidimensional version of Gradient-MUSIC for estimating the frequencies of a nonharmonic signal from noisy samples. The guiding principle is that frequency recovery should be based only on the signal subspace determined by the data. From this viewpoint, the MUSIC functional is an economical nonconvex objective encoding the relevant information, and the problem becomes one of understanding the geometry of its perturbed landscape. Our main contribution is a general structural theory showing that, under explicit conditions on the measurement kernel and the perturbation of the signal subspace, the perturbed MUSIC function is an admissible optimization landscape: suitable initial points can be found efficiently by coarse thresholding, gradient descent converges to the relevant local minima, and these minima obey quantitative error bounds. Thus the theory is not merely existential; it provides a constructive global optimization framework for multidimensional optimal resolution. We verify the abstract conditions in detail for two canonical sampling geometries: discrete samples on a cube and continuous samples on a ball. In both cases we obtain uniform, nonasymptotic recovery guarantees under deterministic as well as stochastic noise. In particular, for lattice samples in a cube of side length $4m$, if the true frequencies are separated by at least $\beta_d/m$ and the noise has $\ell^\infty$ norm at most $\varepsilon$, then Gradient-MUSIC recovers the frequencies with error at most \[ C_d \frac{\varepsilon}{m}, \] where $C_d, \beta_d>0$ depend only on the dimension. This scaling is minimax optimal in $m$ and $\varepsilon$. Under stationary Gaussian noise, the error improves to \[ C_d\frac{\sigma\sqrt{\log(m)}}{m^{1+d/2}}. \] This is the noisy super-resolution scaling: (see paper for rest of abstract)

math.OC↗
arXiv2025

Optimal structured approximation of Fourier subspaces, Toeplitz matrices, and exponential sums

This paper studies three structured approximation problems: (1) Recovering the range of a Fourier matrix from a single observation, (2) Recovering a corrupted low-rank Toeplitz/Hankel matrix, and (3) Recovering a finite exponential sum from noisy samples. All three problems are computationally challenging because their structural constraints are difficult to enforce directly. We show that all three tasks can be solved efficiently and optimally by applying the Gradient-MUSIC algorithm for spectral estimation. To provide an example, for a rank-$r$ Toeplitz matrix $T\in {\mathbb C}^{n\times n}$ that satisfies a regularity assumption and is corrupted by an arbitrary $E\in {\mathbb C}^{n\times n}$ such that $\|E\|_2\leq \alpha n$, our algorithm outputs a Toeplitz matrix $T_\sharp$ of rank exactly $r$ such that $\|T-T_\sharp\|_2 \leq C \|E\|_2$, where $C,\alpha>0$ are absolute constants. This performance guarantee is minimax optimal in $n$, $r$, and $\|E\|_2$. For the other two structured approximation problems, we also provide algorithms that are minimax optimal in the number of samples, rank/sparsity, and noise level. At the heart of this paper is a quantitative transference principle which shows how to convert computational methods and theory for spectral estimation into corresponding methods and theory for the other three problems.

cs.IT↗
arXiv2025

Optimality of Gradient-MUSIC for spectral estimation

We introduce the Gradient-MUSIC algorithm for estimating the unknown frequencies and amplitudes of a nonharmonic signal from noisy time samples. While the classical MUSIC algorithm performs a computationally expensive search over a fine grid, Gradient-MUSIC is significantly more efficient and eliminates the need for discretization over a fine grid by using optimization techniques. It coarsely scans the 1D landscape to find initialization simultaneously for all frequencies followed by parallelizable local refinement via gradient descent. We also analyze its performance when the noise level is sufficiently small and the signal frequencies are separated by at least $8\pi/m$, where $\pi/m$ is the standard resolution of this problem. Even though the 1D landscape is nonconvex, we prove a global convergence result for Gradient-MUSIC: coarse scanning provably finds suitable initialization and gradient descent converges at a linear rate. In addition to convergence results, we also upper bound the error between the true signal frequencies and amplitudes with those found by Gradient-MUSIC. For example, if the noise has $\ell^\infty$ norm at most $\varepsilon$, then the frequencies and amplitudes are recovered up to error at most $C\varepsilon/m$ and $C\varepsilon$ respectively for a universal $C>0$, which are minimax optimal in $m$, $\varepsilon$, and number of frequencies. Our theory can also handle stochastic noise with performance guarantees under nonstationary independent Gaussian noise. Our main approach is a comprehensive geometric analysis of the landscape, a perspective that has not been explored before.

cs.IT↗
arXiv2023

3D Tomographic Phase Retrieval and Unwrapping

This paper develops uniqueness theory for 3D phase retrieval with finite, discrete measurement data for strong phase objects and weak phase objects, including: (i) {\em Unique determination of (phase) projections from diffraction patterns} -- General measurement schemes with coded and uncoded apertures are proposed and shown to ensure unique reduction of diffraction patterns to the phase projection for a strong phase object (respectively, the projection for a weak phase object) in each direction separately without the knowledge of relative orientations and locations. (ii) {\em Uniqueness for 3D phase unwrapping} -- General conditions for unique determination of a 3D strong phase object from its phase projection data are established, including, but not limited to, random tilt schemes densely sampled from a spherical triangle of vertexes in three orthogonal directions and other deterministic tilt schemes. (iii) {\em Uniqueness for projection tomography} -- Unique determination of an object of $n^3$ voxels from generic $n$ projections or $n+1$ coded diffraction patterns is proved. This approach of reducing 3D phase retrieval to the problem of (phase) projection tomography has the practical implication of enabling classification and alignment, when relative orientations are unknown, to be carried out in terms of (phase) projections, instead of diffraction patterns. The applications with the measurement schemes such as single-axis tilt, conical tilt, dual-axis tilt, random conical tilt and general random tilt are discussed.

cs.IT↗
arXiv2023

Noise-Robust One-Bit Diffraction Tomography and Optimal Dose Fractionation

This study presents a noise-robust framework for 1-bit diffraction tomography, a novel imaging approach that relies on intensity-only binary measurements obtained through coded apertures. The proposed reconstruction scheme leverages random matrix theory and iterative algorithms to effectively recover 3D object structures under high-noise conditions. A key contribution is the numerical investigation of dose fractionation, revealing optimal performance at a signal-to-noise ratio near 1, {\em independent of the total dose}. This finding addresses the question: How to distribute a given level of total radiation energy among different tomographic views in order to optimize the quality of reconstruction?

cs.IT↗
arXiv2022

Uniqueness Theorems for Tomographic Phase Retrieval with Few Diffraction Patterns

3D tomographic phase retrieval under the Born approximation for discrete objects supported on a $n\times n\times n$ grid is analyzed. It is proved that $n$ projections are sufficient and necessary for unique determination by computed tomography (CT) with full projected field measurements and that $n+1$ coded projected diffraction patterns are sufficient for unique determination, up to a global phase factor, in tomographic phase retrieval. Hence $n+1$ is nearly, if not exactly, the minimum number of diffractions patterns needed for 3D tomographic phase retrieval under the Born approximation.

cs.IT↗
arXiv2020

The Numerics of Phase Retrieval

Phase retrieval, i.e., the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications such as X-ray crystallography, diffraction imaging, optics, quantum mechanics, and astronomy. This problem has confounded engineers, physicists, and mathematicians for many decades. Recently, phase retrieval has seen a resurgence in research activity, ignited by new imaging modalities and novel mathematical concepts. As our scientific experiments produce larger and larger datasets and we aim for faster and faster throughput, it becomes increasingly important to study the involved numerical algorithms in a systematic and principled manner. Indeed, the last decade has witnessed a surge in the systematic study of computational algorithms for phase retrieval. In this paper we will review these recent advances from a numerical viewpoint.

eess.IV↗
arXiv2019

Super-resolution limit of the ESPRIT algorithm

The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its $M+1$ consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is $1/M$ and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than $1/M$ apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT is an efficient method that does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like ${\rm SRF}^{2λ+2} \times {\rm Noise}$, where the Super-Resolution Factor (${\rm SRF}$) governs the difficulty of the problem and $λ$ is the cardinality of the largest clump. {If the support contains one clump of closely spaced atoms, the min-max error is ${\rm SRF}^{2λ+2} \times {\rm Noise}/M$. Our error bound matches the min-max rate up to a factor of $M$ in the small noise regime. Our results therefore establishes the near-optimality of ESPRIT,} and our theory is validated by numerical experiments.

cs.IT↗
arXiv2018

Raster Grid Pathology and the Cure

Blind ptychography is a phase retrieval method using multiple coded diffraction patterns from different, overlapping parts of the unknown extended object illuminated with an unknown window function. The window function is also known as the probe in the optics literature. As such blind ptychography is an inverse problem of simultaneous recovery of the object and the window function given the intensities of the windowed Fourier transform and has a multi-scale set-up in which the probe has an intermediate scale between the pixel scale and the macro-scale of the extended object. Uniqueness problem for blind ptychography is analyzed rigorously for the raster scan (of a constant step size τ) and its variants, in which another scale comes into play: the overlap between adjacent blocks (the shifted windows). The block phases are shown to form an arithmetic progression and the complete characterization of the raster scan ambiguities is given, including: First, the periodic raster grid pathology of degrees of freedom proportional to τ^2 and, second, a non-periodic, arithmetically progressing phase shift from block to block. Finally irregularly perturbed raster scans are shown to remove all ambiguities other than the inherent ambiguities of the scaling factor and the affine phase ambiguity under the minimum requirement of roughly 50% overlap ratio.

eess.IV↗
arXiv2017

Coded Aperture Ptychography: Uniqueness and Reconstruction

Uniqueness of solution is proved for any ptychographic scheme with a random masks under a minimum overlap condition and local geometric convergence analysis is given for the alternating projection (AP) and Douglas-Rachford (DR) algorithms. DR is shown to possess a unique fixed point in the object domain and for AP a simple criterion for distinguishing the true solution among possibly many fixed points is given. A minimalist scheme is proposed where the adjacent masks overlap 50\% of area and each pixel of the object is illuminated by exactly four times during the whole measurement process. Such a scheme is conveniently parametrized by the number $q$ of shifted masks in each direction. The lower bound $1-C/q^2$ is proved for the geometric convergence rate of the minimalist scheme, predicting a poor performance with large $q$ which is confirmed by numerical experiments. Extensive numerical experiments are performed to explore what the general features of a well-performing mask are like, what the best-performing values of $q$ for a given mask are, how robust the minimalist scheme is with respect to measurement noise and what the significant factors affecting the noise stability are.

math.NA↗
arXiv2016

Compressive Spectral Estimation with Single-Snapshot ESPRIT: Stability and Resolution

In this paper Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) is developed for spectral estimation with single-snapshot measurement. Stability and resolution analysis with performance guarantee for Single-Snapshot ESPRIT (SS-ESPRIT) is the main focus. In the noise-free case, exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise and under the assumption that the true frequencies are separated by at least two times Rayleigh's Resolution Length, an explicit error bound for frequency reconstruction is given in terms of the dynamic range and the separation of the frequencies. The separation and sparsity constraint compares favorably with those of the leading approaches to compressed sensing in the continuum.

cs.IT↗
arXiv2016

Fourier-Domain Fixed Point Algorithms with Coded Diffraction Patterns

Fourier-domain Difference Map (FDM) for phase retrieval with two oversampled coded diffraction patterns are proposed. FDM is a 3-parameter family of fixed point algorithms including Fourier-domain Hybrid-Projection-Reflection (FHPR) and Douglas-Rachford (FDR) algorithm. For generic complex objects without any object constraint, FDM yields a unique fixed point, after proper projection back to the object domain, which is the true solution to the phase retrieval problem up to a global phase factor.

physics.data-an↗
arXiv2015

Phase Retrieval with One or Two Diffraction Patterns by Alternating Projection with Null Initialization

Alternating projection (AP) of various forms, including the Parallel AP (PAP), Real-constrained AP (RAP) and the Serial AP (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition. To compensate for the local nature of convergence, the null initialization is proposed for initial guess and proved to produce asymptotically accurate initialization for the case of Gaussian random measurement. Numerical experiments show that the null initialization produces more accurate initial guess than the spectral initialization and that AP converges faster to the true object than other iterative schemes for non-convex optimization such as the Wirtinger Flow. In numerical experiments, AP with the null initialization converges globally to the true object.

physics.data-an↗
arXiv2014

MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution

This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurements is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent much smaller than an upper bound established by Donoho.

cs.IT↗
arXiv2013

Fourier phasing with phase-uncertain mask

Fourier phasing is the problem of retrieving Fourier phase information from Fourier intensity data. The standard Fourier phase retrieval (without a mask) is known to have many solutions which cause the standard phasing algorithms to stagnate and produce wrong or inaccurate solutions. In this paper Fourier phase retrieval is carried out with the introduction of a randomly fabricated mask in measurement and reconstruction. Highly probable uniqueness of solution, up to a global phase, was previously proved with exact knowledge of the mask. Here the uniqueness result is extended to the case where only rough information about the mask's phases is assumed. The exponential probability bound for uniqueness is given in terms of the uncertainty-to-diversity ratio (UDR) of the unknown mask. New phasing algorithms alternating between the object update and the mask update are systematically tested and demonstrated to have the capability of recovering both the object and the mask (within the object support) simultaneously, consistent with the uniqueness result. Phasing with a phase-uncertain mask is shown to be robust with respect to the correlation in the mask as well as the Gaussian and Poisson noises.

physics.optics↗