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 ca
n 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 can be constructed to be either real-valued or complex-valued and have good concentration in both time and frequency.
We study the problem of phase retrieval in which one aims to recover a function $f$ from the magnitude of its wavelet transform $|mathcal{W}_psi f|$. We consider bandlimited functions and derive new uniqueness results for phase retrieval, where the w
avelet itself can be complex-valued. In particular, we prove the first uniqueness result for the case that the wavelet $psi$ has a finite number of vanishing moments. In addition, we establish the first result on unique reconstruction from samples of the wavelet transform magnitude when the wavelet coefficients are complex-valued
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 ambig
uity 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.
Given two symmetric and positive semidefinite square matrices $A, B$, is it true that any matrix given as the product of $m$ copies of $A$ and $n$ copies of $B$ in a particular sequence must be dominated in the spectral norm by the ordered matrix pro
duct $A^m B^n$? For example, is $$ | AABAABABB | leq | AAAAABBBB | ? $$ Drury has characterized precisely which disordered words have the property that an inequality of this type holds for all matrices $A,B$. However, the $1$-parameter family of counterexamples Drury constructs for these characterizations is comprised of $3 times 3$ matrices, and thus as stated the characterization applies only for $N times N$ matrices with $N geq 3$. In contrast, we prove that for $2 times 2$ matrices, the general rearrangement inequality holds for all disordered words. We also show that for larger $N times N$ matrices, the general rearrangement inequality holds for all disordered words, for most $A,B$ (in a sense of full measure) that are sufficiently small perturbations of the identity.
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 alw
ays unstable in infinite-dimensional Hilbert spaces [7] and possibly severely ill-conditioned in finite-dimensional Hilbert spaces [7]. Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll functions [1]. In finite dimensions, we present first evidence that this semi-global reconstruction regime allows one to do phase retrieval from measurements of bandlimited signals induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales like a low order polynomial in the space dimension. To this end, we utilise reconstruction formulae which have become common tools in recent years [6,12,18,20].
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.
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 adversa
rial 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 convolutional neural networks and on ImageNet with Inception-v3 and ResNet-101.
The problem of phase retrieval is to determine a signal $fin mathcal{H}$, with $mathcal{H}$ a Hilbert space, from intensity measurements $|F(omega)|$, where $F(omega):=langle f , varphi_omegarangle$ are measurements of $f$ with respect to a measureme
nt system $(varphi_omega)_{omegain Omega}subset mathcal{H}$. Although phase retrieval is always stable in the finite dimensional setting whenever it is possible (i.e. injectivity implies stability for the inverse problem), the situation is drastically different if $mathcal{H}$ is infinite-dimensional: in that case phase retrieval is never uniformly stable [8, 4]; moreover the stability deteriorates severely in the dimension of the problem [8]. On the other hand, all empirically observed instabilities are of a certain type: they occur whenever the function $|F|$ of intensity measurements is concentrated on disjoint sets $D_jsubset Omega$, i.e., when $F= sum_{j=1}^k F_j$ where each $F_j$ is concentrated on $D_j$ (and $k geq 2$). 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 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.
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 state
s 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 $mathbb{K}=mathbb{C}$) 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 $mathbb{K}=mathbb{C}$ 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.
In this paper we consider the following problem of phase retrieval: Given a collection of real-valued band-limited functions ${psi_{lambda}}_{lambdain Lambda}subset L^2(mathbb{R}^d)$ that constitutes a semi-discrete frame, we ask whether any real-val
ued function $f in L^2(mathbb{R}^d)$ can be uniquely recovered from its unsigned convolutions ${{|f ast psi_lambda|}_{lambda in Lambda}}$. We find that under some mild assumptions on the semi-discrete frame and if $f$ has exponential decay at $infty$, it suffices to know $|f ast psi_lambda|$ on suitably fine lattices to uniquely determine $f$ (up to a global sign factor). We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of $L^2(mathbb{R}^d)$, $d=1,2$, we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.
