William Leeb
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View article: Toward Single Particle Reconstruction without Particle Picking: Breaking the Detection Limit
Toward Single Particle Reconstruction without Particle Picking: Breaking the Detection Limit Open
Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method to resolve biological macromolecules. In a cryo-EM experiment, the microscope produces…
View article: Matrix Denoising with Partial Noise Statistics: Optimal Singular Value Shrinkage of Spiked F-Matrices
Matrix Denoising with Partial Noise Statistics: Optimal Singular Value Shrinkage of Spiked F-Matrices Open
We study the problem of estimating a large, low-rank matrix corrupted by additive noise of unknown covariance, assuming one has access to additional side information in the form of noise-only measurements. We study the Whiten-Shrink-reColo…
View article: Dihedral Multi-Reference Alignment
Dihedral Multi-Reference Alignment Open
We study the dihedral multi-reference alignment problem of estimating the orbit of a signal from multiple noisy observations of the signal, acted on by random elements of the dihedral group. We show that if the group elements are drawn fro…
View article: On the robustness of certain norms.
On the robustness of certain norms. Open
We study a family of norms defined for functions on an interval. These norms are obtained by taking the $p$-norm of the Volterra operator applied to the function. The corresponding distances have been previously studied in the context of c…
View article: On metrics robust to noise and deformations
On metrics robust to noise and deformations Open
We study the properties of a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures on the line; special ca…
View article: Super-resolution multi-reference alignment
Super-resolution multi-reference alignment Open
We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled and noisy observations. We focus on the low SNR regime, and show that a signal in ${\mathbb{R}}^M$ is unique…
View article: Matrix Denoising for Weighted Loss Functions and Heterogeneous Signals
Matrix Denoising for Weighted Loss Functions and Heterogeneous Signals Open
We consider the problem of estimating a low-rank matrix from a noisy observed matrix. Previous work has shown that the optimal method depends crucially on the choice of loss function. In this paper, we use a family of weighted loss functio…
View article: Optimal singular value shrinkage for operator norm loss
Optimal singular value shrinkage for operator norm loss Open
We study the denoising of low-rank matrices by singular value shrinkage. Recent work of Gavish and Donoho constructs a framework for finding optimal singular value shrinkers for a wide class of loss functions. We use this framework to deri…
View article: Optimal prediction in the linearly transformed spiked model
Optimal prediction in the linearly transformed spiked model Open
We consider the linearly transformed spiked model, where the observations $Y_{i}$ are noisy linear transforms of unobserved signals of interest $X_{i}$: \\begin{equation*}Y_{i}=A_{i}X_{i}+\\varepsilon_{i},\\end{equation*} for $i=1,\\ldots …
View article: On the Numerical Solution of Fourth-Order Linear Two-Point Boundary Value Problems
On the Numerical Solution of Fourth-Order Linear Two-Point Boundary Value Problems Open
This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our metho…
View article: Properties of Laplacian Pyramids for Extension and Denoising
Properties of Laplacian Pyramids for Extension and Denoising Open
We analyze the Laplacian pyramids algorithm of Rabin and Coifman for extending and denoising a function sampled on a discrete set of points. We provide mild conditions under which the algorithm converges, and prove stability bounds on the …
View article: Evaluating the boundary and Stieltjes transform of limiting spectral distributions for random matrices with a separable variance profile
Evaluating the boundary and Stieltjes transform of limiting spectral distributions for random matrices with a separable variance profile Open
We present numerical algorithms for solving two problems encountered in random matrix theory and its applications. First, we compute the boundary of the limiting spectral distribution for random matrices with a separable variance profile. …
View article: Rapid evaluation of the spectral signal detection threshold and\n Stieltjes transform
Rapid evaluation of the spectral signal detection threshold and\n Stieltjes transform Open
Accurate detection of signal components is a frequently-encountered challenge\nin statistical applications with low signal-to-noise ratio. This problem is\nparticularly challenging in settings with heteroscedastic noise. In certain\nsignal…
View article: Multireference Alignment Is Easier With an Aperiodic Translation Distribution
Multireference Alignment Is Easier With an Aperiodic Translation Distribution Open
In the multireference alignment model, a signal is observed by the action of a random circular translation and the addition of Gaussian noise. The goal is to recover the signal's orbit by accessing multiple independent observations. Of par…
View article: Toward single particle reconstruction without particle picking: Breaking the detection limit
Toward single particle reconstruction without particle picking: Breaking the detection limit Open
Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method to resolve biological macromolecules. In a cryo-EM experiment, the microscope produces…
View article: On the Numerical Solution of Fourth-Order Linear Two-Point Boundary\n Value Problems
On the Numerical Solution of Fourth-Order Linear Two-Point Boundary\n Value Problems Open
This paper introduces a fast and numerically stable algorithm for the\nsolution of fourth-order linear boundary value problems on an interval. This\ntype of equation arises in a variety of settings in physics and signal\nprocessing. Our me…
View article: Multireference Alignment is Easier with an Aperiodic Translation\n Distribution
Multireference Alignment is Easier with an Aperiodic Translation\n Distribution Open
In the multireference alignment model, a signal is observed by the action of\na random circular translation and the addition of Gaussian noise. The goal is\nto recover the signal's orbit by accessing multiple independent observations.\nOf …
View article: Optimal prediction in the linearly transformed spiked model
Optimal prediction in the linearly transformed spiked model Open
We consider the linearly transformed spiked model, where observations $Y_i$ are noisy linear transforms of unobserved signals of interest $X_i$: \begin{align*} Y_i = A_i X_i + \varepsilon_i, \end{align*} for $i=1,\ldots,n$. The transform m…
View article: PCA from noisy, linearly reduced data: the diagonal case
PCA from noisy, linearly reduced data: the diagonal case Open
Suppose we observe data of the form $Y_i = D_i (S_i + \varepsilon_i) \in \mathbb{R}^p$ or $Y_i = D_i S_i + \varepsilon_i \in \mathbb{R}^p$, $i=1,\ldots,n$, where $D_i \in \mathbb{R}^{p\times p}$ are known diagonal matrices, $\varepsilon_i$…