Phase Retrieval in the General Setting of Continuous Frames for Banach Spaces

Localized adversarial artifacts for compressed sensing MRI

arXiv preprint, 2022

Abstract. As interest in deep neural networks (DNNs) for image reconstruction tasks grows, their reliability has been called into question (Antun et al., 2020; Gottschling et al., 2020). However, recent work has shown that compared to total variation (TV) minimization, they show similar robustness to adversarial noise in terms of $\ell^2$ -reconstruction error (Genzel et al., 2022). We consider a different notion of robustness, using the $\ell^\infty$ -norm, and argue that localized reconstruction artifacts are a more relevant defect than the $\ell^2$ -error. We create adversarial perturbations to undersampled MRI measurements which induce severe localized artifacts in the TV-regularized reconstruction. The same attack method is not as effective against DNN based reconstruction. Finally, we show that this phenomenon is inherent to reconstruction methods for which exact recovery can be guaranteed, as with compressed sensing reconstructions with $\ell^1$ - or TV-minimization.

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On the connection between uniqueness from samples and stability in Gabor phase retrieval

arXiv preprint, 2022

Rima Alaifari, Francesca Bartolucci, Stefan Steinerberger, Matthias Wellershoff

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Ill-Posed Problems: From Linear to Nonlinear and Beyond

Harmonic and Applied Analysis. Birkhäuser, Cham, 101-148, 2021

Rima Alaifari

Abstract. Inverse (ill-posed) problems appear in many applications such as medical imaging, astronomy, seismic imaging, nondestructive testing, signal processing, etc. Typically, these problems cannot be solved by conventional methods as they suffer from instabilities and regularization is required. This chapter has evolved from a mini-course taught at the Summer School on Applied Harmonic Analysis and Machine Learning at the University of Genoa in 2019. It offers an overview of the theory of inverse problems and discusses three ill-posed problems that have been studied rather recently in the literature: limited data reconstruction in computerized tomography, phase retrieval, and image classification with DNNs. The selection highlights that for modern problems, the usefulness of standard theory of regularization can be limited.

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Phase retrieval from sampled Gabor transform magnitudes: Counterexamples

Journal of Fourier Analysis and Applications 28 (1), 1-8, 2021

Rima Alaifari, Matthias Wellershoff

Abstract. We consider the recovery of square-integrable signals from discrete, equidistant samples of their Gabor transform magnitude and show that, in general, signals can not be recovered from such samples. In particular, we show that for any lattice, one can construct functions in $L^2(\mathbb{R})$ which do not agree up to global phase but whose Gabor transform magnitudes sampled on the lattice agree. These functions have good concentration in both time and frequency and can be constructed to be real-valued for rectangular lattices.

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Phase retrieval of bandlimited functions for the wavelet transform

arXiv preprint, 2020

Rima Alaifari, Francesca Bartolucci, Matthias Wellershoff

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Uniqueness of STFT phase retrieval for bandlimited functions

Applied and Computational Harmonic Analysis 50, 34-48, 2021

Rima Alaifari, Matthias Wellershoff

Abstract. We consider the problem of phase retrieval from magnitudes of short-time Fourier transform (STFT) measurements. It is well-known that signals are uniquely determined (up to global phase) by their STFT magnitude when the underlying window has an ambiguity function that is nowhere vanishing. It is less clear, however, what can be said in terms of unique phase-retrievability when the ambiguity function of the underlying window vanishes on some of the time-frequency plane. In this short note, we demonstrate that by considering signals in Paley-Wiener spaces, it is possible to prove new uniqueness results for STFT phase retrieval. Among those, we establish a first uniqueness theorem for STFT phase retrieval from magnitude-only samples in a real-valued setting.

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On Matrix Rearrangement Inequalities

Proceedings of the AMS, Vol. 148, Iss. 5, 2020

Rima Alaifari, Xiuyuan Cheng, Lillian B. Pierce, Stefan Steinerberger

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Stability estimates for phase retrieval from discrete Gabor measurements

Journal of Fourier Analysis and Applications 27 (2), 1-31, 2021

Rima Alaifari, Matthias Wellershoff

Abstract. Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that, in the deterministic setting, phase retrieval from frame coefficients is always unstable in infinite dimensional Hilbert spaces [5] and possibly severely ill-conditioned in finite dimensional Hilbert spaces [5].
Recently, it was also shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable when one is willing to accept a more relaxed semi-global stability regime [1].
We present first evidence that this semi-global stability regime allows one to do phase retrieval from measurements induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales linearly in the space dimension. To this end, we utilise well-known reconstruction formulae which have been used repeatedly in recent years [4], [6-8].

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Gabor phase retrieval is severely ill-posed

Applied and Computational Harmonic Analysis 50, 401-419, 2021

Rima Alaifari, Philipp Grohs

Abstract. The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This result also holds for frames that are possibly continuous [2]. On the other hand, the problem is always stable in finite-dimensional settings. A prominent example of such a phase retrieval problem is the recovery of a signal from the modulus of its Gabor transform. In this paper, we study Gabor phase retrieval and ask how the stability degrades on a natural family of finite-dimensional subspaces of the signal domain $L^2(\mathbb{R})$ . We prove that the stability constant scales at least quadratically exponentially in the dimension of the subspaces. Our construction also shows that typical priors such as sparsity or smoothness promoting penalties do not constitute regularization terms for phase retrieval.

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ADef: an Iterative Algorithm to Construct Adversarial Deformations

International Conference on Learning Representations (ICLR) 2019

Rima Alaifari, Giovanni S. Alberti, Tandri Gauksson

Abstract. While deep neural networks have proven to be a powerful tool for many recognition and classification tasks, their stability properties are still not well understood. In the past, image classifiers have been shown to be vulnerable to so-called adversarial attacks, which are created by additively perturbing the correctly classified image.
In this paper, we propose the ADef algorithm to construct a different kind of adversarial attack created by iteratively applying small deformations to the image, found through a gradient descent step. We demonstrate our results on MNIST with a convolutional neural network and on ImageNet with Inception-v3 and ResNet-101.

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Stable phase retrieval in infinite dimensions

Foundations of Computational Mathematics, Vol. 19, Issue 4, 2019

Rima Alaifari, Ingrid Daubechies, Philipp Grohs, Rujie Yin

Abstract. The problem of phase retrieval is to determine a signal $f\in \mathcal{H}$ , with $\mathcal{H}$ a Hilbert space, from intensity measurements
$|F(\omega)|:=|\langle f , \varphi_\omega\rangle|$ associated with a measurement system $(\varphi_\omega)_{\omega\in \Omega}\subset \mathcal{H}$ . Such problems can be seen in a wide variety of applications, ranging from X-ray crystallography, microscopy to audio processing and deep learning algorithms and accordingly, a large body of literature treating the mathematical and algorithmic solution of phase retrieval problems has emerged in recent years.

Recent work [9,3] has shown that, whenever $\mathcal{H}$ is infinite-dimensional, phase retrieval is never uniformly stable, and that, although it is always stable in the finite dimensional setting, the stability deteriorates severely in the dimension of the problem [9]. Any finite dimensional approximation of an infinite dimensional problem has to take into account this phenomenon which makes one wonder whether phase retrieval is even advisable in these situations.

On the other hand, all observed instabilities are of a certain type: they occur whenever the function $|F|$ of intensity measurements is concentrated on disjoint sets $D_j\subset \Omega$ , i.e., when $F= \sum_{j=1}^k F_j$ where $F_j$ is concentrated on $D_j$ (and $k \geq 2$ ). Indeed, it is easy to see that intensity measurements of any function $\sum_{j=1}^k e^{\i \alpha_j} F_j$ will be close to those of $F$ while the functions themselves need not be close at all.

Motivated by these considerations we propose a new paradigm for stable phase retrieval by considering the problem of reconstructing $F$ up to a phase factor that is not global, but that can be different for each of the subsets $D_j$ , i.e., recovering $F$ up to the equivalence
$F \sim \sum_{j=1}^k e^{\i \alpha_j} F_j.$

We present concrete applications (for example in X-ray diffraction imaging or audio processing) where this new notion of stability is natural and meaningful and show that in this setting stable phase retrieval can actually be achieved, for instance if the measurement system is a Gabor frame or a frame of Cauchy wavelets.

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Phase retrieval in the general setting of continuous frames for Banach spaces

SIAM Math Analysis Vol. 49 Issue 3, 2017

Rima Alaifari, Philipp Grohs

Abstract. We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main result states that also in this framework, the problem of phase retrieval is never uniformly stable in infinite dimensions. On the other hand, we show weak stability of the problem. This complements recent work [9], where it has been shown that phase retrieval is always unstable for the setting of discrete frames in Hilbert spaces. In particular, our result implies that the stability properties cannot be improved by oversampling the underlying discrete frame.

We generalize the notion of complement property (CP) to the setting of continuous frames for Banach spaces (over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{R}$ ) and verify that it is a necessary condition for uniqueness of the phase retrieval problem; when $\mathbb{K}=\mathbb{R}$ the CP is also sufficient for uniqueness. In our general setting, we also prove a conjecture posed by Bandeira et al. [5], which was originally formulated for finite-dimensional spaces: for the case $\mathbb{K}=\mathbb{R}$ the strong complement property (SCP) is a necessary condition for stability. To prove our main result, we show that the SCP can never hold for frames of infinite-dimensional Banach spaces.

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Reconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames

Journal of Fourier Analysis and Applications Vol. 23 Issue 6, 2017

Rima Alaifari, Ingrid Daubechies, Philipp Grohs, Gaurav Thakur

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Phase Retrieval in the General Setting of Continuous Frames for Banach Spaces

Localized adversarial artifacts for compressed sensing MRI

arXiv preprint, 2022

Open PDF

On the connection between uniqueness from samples and stability in Gabor phase retrieval

arXiv preprint, 2022

Rima Alaifari, Francesca Bartolucci, Stefan Steinerberger, Matthias Wellershoff

Open PDF

Ill-Posed Problems: From Linear to Nonlinear and Beyond

Harmonic and Applied Analysis. Birkhäuser, Cham, 101-148, 2021

Rima Alaifari

Open PDF

Phase retrieval from sampled Gabor transform magnitudes: Counterexamples

Journal of Fourier Analysis and Applications 28 (1), 1-8, 2021

Rima Alaifari, Matthias Wellershoff

Open PDF

Phase retrieval of bandlimited functions for the wavelet transform

arXiv preprint, 2020

Rima Alaifari, Francesca Bartolucci, Matthias Wellershoff

Open PDF

Uniqueness of STFT phase retrieval for bandlimited functions

Applied and Computational Harmonic Analysis 50, 34-48, 2021

Rima Alaifari, Matthias Wellershoff

Open PDF

On Matrix Rearrangement Inequalities

Proceedings of the AMS, Vol. 148, Iss. 5, 2020

Rima Alaifari, Xiuyuan Cheng, Lillian B. Pierce, Stefan Steinerberger

Open PDF

Stability estimates for phase retrieval from discrete Gabor measurements

Journal of Fourier Analysis and Applications 27 (2), 1-31, 2021

Rima Alaifari, Matthias Wellershoff

Open PDF

Gabor phase retrieval is severely ill-posed

Applied and Computational Harmonic Analysis 50, 401-419, 2021

Rima Alaifari, Philipp Grohs

Open PDF

ADef: an Iterative Algorithm to Construct Adversarial Deformations

International Conference on Learning Representations (ICLR) 2019

Rima Alaifari, Giovanni S. Alberti, Tandri Gauksson

Open PDF

Stable phase retrieval in infinite dimensions

Foundations of Computational Mathematics, Vol. 19, Issue 4, 2019

Rima Alaifari, Ingrid Daubechies, Philipp Grohs, Rujie Yin

Open PDF

Phase retrieval in the general setting of continuous frames for Banach spaces

SIAM Math Analysis Vol. 49 Issue 3, 2017

Rima Alaifari, Philipp Grohs

Open PDF

Reconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames

Journal of Fourier Analysis and Applications Vol. 23 Issue 6, 2017

Rima Alaifari, Ingrid Daubechies, Philipp Grohs, Gaurav Thakur

Open PDF

Stability estimates for the regularized inversion of the truncated Hilbert transform

Inverse Problems Vol. 32, 2016

Rima Alaifari, Michel Defrise, Alexander Katsevich

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Lower bounds for the truncated Hilbert transform

Revista Matemática Iberoamericana Vol. 32 Issue 1, 2016

Rima Alaifari, Lillian B. Pierce, Stefan Steinerberger

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Asymptotic analysis of the SVD of the truncated Hilbert transform with overlap

SIAM Math Analysis Vol. 47 Issue 1, 2015

Rima Alaifari, Michel Defrise, Alexander Katsevich

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Spectral analysis of the truncated Hilbert transform with overlap

SIAM Math Analysis Vol. 46 Issue 1, 2014

Reema Al-Aifari, Alexander Katsevich

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The continuous Procrustes distance between two surfaces

Communications on Pure and Applied Mathematics Vol. 66 Issue 6, 2013

Yaron Lipman, Reema Al-Aifari, Ingrid Daubechies Princeton University

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Yaron Lipman, Reema Al-Aifari, Ingrid Daubechies
Princeton University