arXiv subjects

Felix Krahmer

Publications and source records attributed to Felix Krahmer.

At least 55 records · Page 3Linked to original sources

arXiv2020

Complex phase retrieval from subgaussian measurements

Phase retrieval refers to the problem of reconstructing an unknown vector $x_0 \in \mathbb{C}^n$ or $x_0 \in \mathbb{R}^n $ from $m$ measurements of the form $y_i = \big\vert \langle ξ^{\left(i\right)}, x_0 \rangle \big\vert^2 $, where $ \left\{ ξ^{\left(i\right)} \right\}^m_{i=1} \subset \mathbb{C}^m $ are known measurement vectors. While Gaussian measurements allow for recovery of arbitrary signals provided the number of measurements scales at least linearly in the number of dimensions, it has been shown that ambiguities may arise for certain other classes of measurements $ \left\{ ξ^{\left(i\right)} \right\}^{m}_{i=1}$ such as Bernoulli measurements or Fourier measurements. In this paper, we will prove that even when a subgaussian vector $ ξ^{\left(i\right)} \in \mathbb{C}^m $ does not fulfill a small-ball probability assumption, the PhaseLift method is still able to reconstruct a large class of signals $x_0 \in \mathbb{R}^n$ from the measurements. This extends recent work by Krahmer and Liu from the real-valued to the complex-valued case. However, our proof strategy is quite different and we expect some of the new proof ideas to be useful in several other measurement scenarios as well. We then extend our results $x_0 \in \mathbb{C}^n $ up to an additional assumption which, as we show, is necessary.

cs.IT↗

arXiv2020

A sample efficient sparse FFT for arbitrary frequency candidate sets in high dimensions

In this paper a sublinear time algorithm is presented for the reconstruction of functions that can be represented by just few out of a potentially large candidate set of Fourier basis functions in high spatial dimensions, a so-called high-dimensional sparse fast Fourier transform. In contrast to many other such algorithms, our method works for arbitrary candidate sets and does not make additional structural assumptions on the candidate set. Our transform significantly improves upon the other approaches available for such a general framework in terms of the scaling of the sample complexity. Our algorithm is based on sampling the function along multiple rank-1 lattices with random generators. Combined with a dimension-incremental approach, our method yields a sparse Fourier transform whose computational complexity only grows mildly in the dimension and can hence be efficiently computed even in high dimensions. Our theoretical analysis establishes that any Fourier $s$-sparse function can be accurately reconstructed with high probability. This guarantee is complemented by several numerical tests demonstrating the high efficiency and versatile applicability for the exactly sparse case and also for the compressible case.

math.NA↗

arXiv2020

Sparse Harmonic Transforms: A New Class of Sublinear-time Algorithms for Learning Functions of Many Variables

We develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications including, e.g., various Uncertainty Quantification(UQ) problems involving the solution of parametric PDE that are approximately sparse in Chebyshev or Legendre product bases. We expect that our results provide a starting point for a new line of research on sublinear-time solution techniques for UQ applications of the type above which will eventually be able to scale to significantly higher-dimensional problems than what are currently computationally feasible. More concretely, let $B$ be a finite Bounded Orthonormal Product Basis (BOPB) of cardinality $|B|=N$. We will develop methods that approximate any function $f$ that is sparse in the BOPB, that is, $f:\mathcal{D}\subset R^D\rightarrow C$ of the form $f(\mathbf{x})=\sum_{b\in S}c_b\cdot b(\mathbf{x})$ with $S\subset B$ of cardinality $|S| =s\ll N$. Our method has a runtime of just $(s\log N)^{O(1)}$, uses only $(s\log N)^{O(1)}$ function evaluations on a fixed and nonadaptive grid, and not more than $(s\log N)^{O(1)}$ bits of memory. For $s\ll N$, the runtime $(s\log N)^{O(1)}$ will be less than what is required to simply enumerate the elements of the basis $B$; thus our method is the first approach applicable in a general BOPB framework that falls into the class referred to as "sublinear-time". This and the similarly reduced sample and memory requirements set our algorithm apart from previous works based on standard compressive sensing algorithms such as basis pursuit which typically store and utilize full intermediate basis representations of size $Ω(N)$.

math.NA↗

arXiv2020

Optimal Fast Johnson-Lindenstrauss Embeddings for Large Data Sets

Johnson-Lindenstrauss embeddings are widely used to reduce the dimension and thus the processing time of data. To reduce the total complexity, also fast algorithms for applying these embeddings are necessary. To date, such fast algorithms are only available either for a non-optimal embedding dimension or up to a certain threshold on the number of data points. We address a variant of this problem where one aims to simultaneously embed larger subsets of the data set. Our method follows an approach by Nelson: A subsampled Hadamard transform maps points into a space of lower, but not optimal dimension. Subsequently, a random matrix with independent entries projects to an optimal embedding dimension. For subsets whose size scales at least polynomially in the ambient dimension, the complexity of this method comes close to the number of operations just to read the data under mild assumptions on the size of the data set that are considerably less restrictive than in previous works. We also prove a lower bound showing that subsampled Hadamard matrices alone cannot reach an optimal embedding dimension. Hence, the second embedding cannot be omitted.

cs.DS↗

arXiv2020

On the convex geometry of blind deconvolution and matrix completion

Low-rank matrix recovery from structured measurements has been a topic of intense study in the last decade and many important problems like matrix completion and blind deconvolution have been formulated in this framework. An important benchmark method to solve these problems is to minimize the nuclear norm, a convex proxy for the rank. A common approach to establish recovery guarantees for this convex program relies on the construction of a so-called approximate dual certificate. However, this approach provides only limited insight in various respects. Most prominently, the noise bounds exhibit seemingly suboptimal dimension factors. In this paper we take a novel, more geometric viewpoint to analyze both the matrix completion and the blind deconvolution scenario. We find that for both these applications the dimension factors in the noise bounds are not an artifact of the proof, but the problems are intrinsically badly conditioned. We show, however, that bad conditioning only arises for very small noise levels: Under mild assumptions that include many realistic noise levels we derive near-optimal error estimates for blind deconvolution under adversarial noise.

cs.IT↗

arXiv2020

Well conditioned ptychograpic imaging via lost subspace completion

Ptychography, a special case of the phase retrieval problem, is a popular method in modern imaging. Its measurements are based on the shifts of a locally supported window function. In general, direct recovery of an object from such measurements is known to be an ill-posed problem. Although for some windows the conditioning can be controlled, for a number of important cases it is not possible, for instance for Gaussian windows. In this paper we develop a subspace completion algorithm, which enables stable reconstruction for a much wider choice of windows, including Gaussian windows. The combination with a regularization technique leads to improved conditioning and better noise robustness.

math.NA↗

arXiv2020

Predicting sparse circle maps from their dynamics

The problem of identifying a dynamical system from its dynamics is of great importance for many applications. Recently it has been suggested to impose sparsity models for improved recovery performance. In this paper, we provide recovery guarantees for such a scenario. More precisely, we show that ergodic systems on the circle described by sparse trigonometric polynomials can be recovered from a number of samples scaling near-linearly in the sparsity.

math.DS↗

arXiv2019

One-Bit Sigma-Delta modulation on the circle

Manifold models in data analysis and signal processing have become more prominent in recent years. In this paper, we will look at one of the main tasks of modern signal processing, namely, at analog-to-digital (A/D) conversion in connection with a simple manifold model - the circle. We will focus on Sigma-Delta modulation which is a popular method for A/D conversion of bandlimited signals that employs coarse quantization coupled with oversampling. Classical Sigma-Delta schemes provide mismatches and large errors at the initialization point if the signal to be converted is defined on the circle. In this paper, our goal is to get around these problems for Sigma-Delta schemes. Our results show how to design an update for the first and the second order schemes based on the reconstruction error analysis such that for the updated scheme the reconstruction error is improved.

eess.SP↗

arXiv2019

On the geometry of polytopes generated by heavy-tailed random vectors

We study the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in $\mathbb{R}^n$. We show that under minimal assumptions on $X$, for $N \gtrsim n$ and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector---namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on $X$ we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when $X$ is $q$-stable or when $X$ has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing---noise blind sparse recovery.

math.PR↗

arXiv2018

Are good local minima wide in sparse recovery?

The idea of compressed sensing is to exploit representations in suitable (overcomplete) dictionaries that allow to recover signals far beyond the Nyquist rate provided that they admit a sparse representation in the respective dictionary. The latter gives rise to the sparse recovery problem of finding the best sparse linear approximation of given data in a given generating system. In this paper we analyze the iterative hard thresholding (IHT) algorithm as one of the most popular greedy methods for solving the sparse recovery problem, and demonstrate that systematically perturbing the IHT algorithm by adding noise to intermediate iterates yields improved results. Further improvements can be obtained by entirely rephrasing the problem as a parametric deep-learning-type of optimization problem. By introducing perturbations via dropout, we demonstrate to significantly outperform the classical IHT algorithm, obtaining $3$ to $6$ times lower average objective errors.

cs.CV↗

arXiv2018

A Quotient Property for Matrices with Heavy-Tailed Entries and its Application to Noise-Blind Compressed Sensing

For a large class of random matrices $A$ with i.i.d. entries we show that the $\ell_1$-quotient property holds with probability exponentially close to 1. In contrast to previous results, our analysis does not require concentration of the entrywise distributions. We provide a unified proof that recovers corresponding previous results for (sub-)Gaussian and Weibull distributions. Our findings generalize known results on the geometry of random polytopes, providing lower bounds on the size of the largest Euclidean ball contained in the centrally symmetric polytope spanned by the columns of $A$. At the same time, our results establish robustness of noise-blind $\ell_1$-decoders for recovering sparse vectors $x$ from underdetermined, noisy linear measurements $y = Ax + w$ under the weakest possible assumptions on the entrywise distributions that allow for recovery with optimal sample complexity even in the noiseless case. Our analysis predicts superior robustness behavior for measurement matrices with super-Gaussian entries, which we confirm by numerical experiments.

math.PR↗

arXiv2018

Sparse Power Factorization: Balancing peakiness and sample complexity

In many applications, one is faced with an inverse problem, where the known signal depends in a bilinear way on two unknown input vectors. Often at least one of the input vectors is assumed to be sparse, i.e., to have only few non-zero entries. Sparse Power Factorization (SPF), proposed by Lee, Wu, and Bresler, aims to tackle this problem. They have established recovery guarantees for a somewhat restrictive class of signals under the assumption that the measurements are random. We generalize these recovery guarantees to a significantly enlarged and more realistic signal class at the expense of a moderately increased number of measurements.

cs.IT↗

arXiv2017

On Unlimited Sampling

Shannon's sampling theorem provides a link between the continuous and the discrete realms stating that bandlimited signals are uniquely determined by its values on a discrete set. This theorem is realized in practice using so called analog--to--digital converters (ADCs). Unlike Shannon's sampling theorem, the ADCs are limited in dynamic range. Whenever a signal exceeds some preset threshold, the ADC saturates, resulting in aliasing due to clipping. The goal of this paper is to analyze an alternative approach that does not suffer from these problems. Our work is based on recent developments in ADC design, which allow for ADCs that reset rather than to saturate, thus producing modulo samples. An open problem that remains is: Given such modulo samples of a bandlimited function as well as the dynamic range of the ADC, how can the original signal be recovered and what are the sufficient conditions that guarantee perfect recovery? In this paper, we prove such sufficiency conditions and complement them with a stable recovery algorithm. Our results are not limited to certain amplitude ranges, in fact even the same circuit architecture allows for the recovery of arbitrary large amplitudes as long as some estimate of the signal norm is available when recovering. Numerical experiments that corroborate our theory indeed show that it is possible to perfectly recover function that takes values that are orders of magnitude higher than the ADC's threshold.

cs.IT↗

arXiv2017

Total Variation Minimization in Compressed Sensing

This chapter gives an overview over recovery guarantees for total variation minimization in compressed sensing for different measurement scenarios. In addition to summarizing the results in the area, we illustrate why an approach that is common for synthesis sparse signals fails and different techniques are necessary. Lastly, we discuss a generalizations of recent results for Gaussian measurements to the subgaussian case.

cs.IT↗

arXiv2017

Phase Retrieval Without Small-Ball Probability Assumptions

In the context of the phase retrieval problem, it is known that certain natural classes of measurements, such as Fourier measurements and random Bernoulli measurements, do not lead to the unique reconstruction of all possible signals, even in combination with certain practically feasible random masks. To avoid this difficulty, the analysis is often restricted to measurement ensembles (or masks) that satisfy a small-ball probability condition, in order to ensure that the reconstruction is unique. This paper shows a complementary result: for random Bernoulli measurements, there is still a large class of signals that can be reconstructed uniquely, namely those signals that are "non-peaky." In fact, this result is much more general: it holds for random measurements sampled from any subgaussian distribution D, without any small-ball conditions. This is demonstrated in two ways: first, a proof of stability and uniqueness, and second, a uniform recovery guarantee for the PhaseLift algorithm. In all of these cases, the number of measurements m approaches the information-theoretic lower bound. Finally, for random Bernoulli measurements with erasures, it is shown that PhaseLift achieves uniform recovery of all signals (including peaky ones).

cs.IT↗

arXiv2017

Spectral Methods for Passive Imaging: Non-asymptotic Performance and Robustness

We study the problem of passive imaging through convolutive channels. A scene is illuminated with an unknown, unstructured source, and the measured response is the convolution of this source with multiple channel responses, each of which is time-limited. Spectral methods based on the commutativity of convolution, first proposed and analyzed in the 1990s, provide an elegant mathematical framework for attacking this problem. However, these now classical methods are very sensitive to noise, especially when working from relatively small sample sizes. In this paper, we show that a linear subspace model on the coefficients of the impulse responses of the channels can make this problem well-posed. We derive non-asymptotic error bounds for the generic subspace model by analyzing the spectral gap of the cross-correlation matrix of the channels relative to the perturbation introduced by noise. Numerical results show that this modified spectral method offers significant improvements over the classical method and outperforms other competing methods for multichannel blind deconvolution.

cs.IT↗

arXiv2017

Blind Demixing and Deconvolution at Near-Optimal Rate

We consider simultaneous blind deconvolution of r source signals from their noisy superposition, a problem also referred to blind demixing and deconvolution. This signal processing problem occurs in the context of the Internet of Things where a massive number of sensors sporadically communicate only short messages over unknown channels. We show that robust recovery of message and channel vectors can be achieved via convex optimization when random linear encoding using i.i.d. complex Gaussian matrices is used at the devices and the number of required measurements at the receiver scales with the degrees of freedom of the overall estimation problem. Since the scaling is linear in r our result significantly improves over recent works.

cs.IT↗

arXiv2017

Quantized Compressed Sensing for Partial Random Circulant Matrices

We provide the first analysis of a non-trivial quantization scheme for compressed sensing measurements arising from structured measurements. Specifically, our analysis studies compressed sensing matrices consisting of rows selected at random, without replacement, from a circulant matrix generated by a random subgaussian vector. We quantize the measurements using stable, possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based on convex optimization. We show that the part of the reconstruction error due to quantization decays polynomially in the number of measurements. This is in line with analogous results on Sigma-Delta quantization associated with random Gaussian or subgaussian matrices, and significantly better than results associated with the widely assumed memoryless scalar quantization. Moreover, we prove that our approach is stable and robust; i.e., the reconstruction error degrades gracefully in the presence of non-quantization noise and when the underlying signal is not strictly sparse. The analysis relies on results concerning subgaussian chaos processes as well as a variation of McDiarmid's inequality.

cs.IT↗