arXiv subjects

Holger Rauhut

Publications and source records attributed to Holger Rauhut.

At least 37 records · Page 2Linked to original sources

arXiv2022

Path classification by stochastic linear recurrent neural networks

We investigate the functioning of a classifying biological neural network from the perspective of statistical learning theory, modelled, in a simplified setting, as a continuous-time stochastic recurrent neural network (RNN) with identity activation function. In the purely stochastic (robust) regime, we give a generalisation error bound that holds with high probability, thus showing that the empirical risk minimiser is the best-in-class hypothesis. We show that RNNs retain a partial signature of the paths they are fed as the unique information exploited for training and classification tasks. We argue that these RNNs are easy to train and robust and back these observations with numerical experiments on both synthetic and real data. We also exhibit a trade-off phenomenon between accuracy and robustness.

stat.ML↗

arXiv2021

Star DGT: a Robust Gabor Transform for Speech Denoising

In this paper, we address the speech denoising problem, where Gaussian, pink and blue additive noises are to be removed from a given speech signal. Our approach is based on a redundant, analysis-sparse representation of the original speech signal. We pick an eigenvector of the Zauner unitary matrix and -- under certain assumptions on the ambient dimension -- we use it as window vector to generate a spark deficient Gabor frame. The analysis operator associated with such a frame, is a (highly) redundant Gabor transform, which we use as a sparsifying transform in denoising procedure. We conduct computational experiments on real-world speech data, using as baseline three Gabor transforms generated by state-of-the-art window vectors in time-frequency analysis and compare their performance to the proposed Gabor transform. The results show that our proposed redundant Gabor transform outperforms all others, consistently for all signals.

cs.SD↗

arXiv2021

Convergence of gradient descent for learning linear neural networks

We study the convergence properties of gradient descent for training deep linear neural networks, i.e., deep matrix factorizations, by extending a previous analysis for the related gradient flow. We show that under suitable conditions on the step sizes gradient descent converges to a critical point of the loss function, i.e., the square loss in this article. Furthermore, we demonstrate that for almost all initializations gradient descent converges to a global minimum in the case of two layers. In the case of three or more layers we show that gradient descent converges to a global minimum on the manifold matrices of some fixed rank, where the rank cannot be determined a priori.

cs.LG↗

arXiv2021

Overparameterization and generalization error: weighted trigonometric interpolation

Motivated by surprisingly good generalization properties of learned deep neural networks in overparameterized scenarios and by the related double descent phenomenon, this paper analyzes the relation between smoothness and low generalization error in an overparameterized linear learning problem. We study a random Fourier series model, where the task is to estimate the unknown Fourier coefficients from equidistant samples. We derive exact expressions for the generalization error of both plain and weighted least squares estimators. We show precisely how a bias towards smooth interpolants, in the form of weighted trigonometric interpolation, can lead to smaller generalization error in the overparameterized regime compared to the underparameterized regime. This provides insight into the power of overparameterization, which is common in modern machine learning.

cs.LG↗

arXiv2021

Spark Deficient Gabor Frames for Inverse Problems

In this paper, we apply star-Digital Gabor Transform in analysis Compressed Sensing and speech denoising. Based on assumptions on the ambient dimension, we produce a window vector that generates a spark deficient Gabor frame with many linear dependencies among its elements. We conduct computational experiments on both synthetic and real-world signals, using as baseline three Gabor transforms generated by state-of-the-art window vectors and compare their performance to star-Gabor transform. Results show that the proposed star-Gabor transform outperforms all others in all signal cases.

math.NA↗

arXiv2021

Spark Deficient Gabor Frame Provides a Novel Analysis Operator for Compressed Sensing

The analysis sparsity model is a very effective approach in modern Compressed Sensing applications. Specifically, redundant analysis operators can lead to fewer measurements needed for reconstruction when employing the analysis $l_1$-minimization in Compressed Sensing. In this paper, we pick an eigenvector of the Zauner unitary matrix and -- under certain assumptions on the ambient dimension -- we build a spark deficient Gabor frame. The analysis operator associated with such a spark deficient Gabor frame, is a new (highly) redundant Gabor transform, which we use as a sparsifying transform in Compressed Sensing. We conduct computational experiments -- on both synthetic and real-world data -- solving the analysis $l_1$-minimization problem of Compressed Sensing, with four different choices of analysis operators, including our Gabor analysis operator. The results show that our proposed redundant Gabor transform outperforms -- in all cases -- Gabor transforms generated by state-of-the-art window vectors of time-frequency analysis.

cs.IT↗

arXiv2021

Compressive Sensing and Neural Networks from a Statistical Learning Perspective

Various iterative reconstruction algorithms for inverse problems can be unfolded as neural networks. Empirically, this approach has often led to improved results, but theoretical guarantees are still scarce. While some progress on generalization properties of neural networks have been made, great challenges remain. In this chapter, we discuss and combine these topics to present a generalization error analysis for a class of neural networks suitable for sparse reconstruction from few linear measurements. The hypothesis class considered is inspired by the classical iterative soft-thresholding algorithm (ISTA). The neural networks in this class are obtained by unfolding iterations of ISTA and learning some of the weights. Based on training samples, we aim at learning the optimal network parameters via empirical risk minimization and thereby the optimal network that reconstructs signals from their compressive linear measurements. In particular, we may learn a sparsity basis that is shared by all of the iterations/layers and thereby obtain a new approach for dictionary learning. For this class of networks, we present a generalization bound, which is based on bounding the Rademacher complexity of hypothesis classes consisting of such deep networks via Dudley's integral. Remarkably, under realistic conditions, the generalization error scales only logarithmically in the number of layers, and at most linear in number of measurements.

math.ST↗

arXiv2021

New challenges in covariance estimation: multiple structures and coarse quantization

In this self-contained chapter, we revisit a fundamental problem of multivariate statistics: estimating covariance matrices from finitely many independent samples. Based on massive Multiple-Input Multiple-Output (MIMO) systems we illustrate the necessity of leveraging structure and considering quantization of samples when estimating covariance matrices in practice. We then provide a selective survey of theoretical advances of the last decade focusing on the estimation of structured covariance matrices. This review is spiced up by some yet unpublished insights on how to benefit from combined structural constraints. Finally, we summarize the findings of our recently published preprint "Covariance estimation under one-bit quantization" to show how guaranteed covariance estimation is possible even under coarse quantization of the samples.

math.ST↗

arXiv2020

Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizers

We study the convergence of gradient flows related to learning deep linear neural networks (where the activation function is the identity map) from data. In this case, the composition of the network layers amounts to simply multiplying the weight matrices of all layers together, resulting in an overparameterized problem. The gradient flow with respect to these factors can be re-interpreted as a Riemannian gradient flow on the manifold of rank-$r$ matrices endowed with a suitable Riemannian metric. We show that the flow always converges to a critical point of the underlying functional. Moreover, we establish that, for almost all initializations, the flow converges to a global minimum on the manifold of rank $k$ matrices for some $k\leq r$.

math.OC↗

arXiv2020

Unfolding recurrence by Green's functions for optimized reservoir computing

Cortical networks are strongly recurrent, and neurons have intrinsic temporal dynamics. This sets them apart from deep feed-forward networks. Despite the tremendous progress in the application of feed-forward networks and their theoretical understanding, it remains unclear how the interplay of recurrence and non-linearities in recurrent cortical networks contributes to their function. The purpose of this work is to present a solvable recurrent network model that links to feed forward networks. By perturbative methods we transform the time-continuous, recurrent dynamics into an effective feed-forward structure of linear and non-linear temporal kernels. The resulting analytical expressions allow us to build optimal time-series classifiers from random reservoir networks. Firstly, this allows us to optimize not only the readout vectors, but also the input projection, demonstrating a strong potential performance gain. Secondly, the analysis exposes how the second order stimulus statistics is a crucial element that interacts with the non-linearity of the dynamics and boosts performance.

cond-mat.dis-nn↗

arXiv2020

Sparse recovery in bounded Riesz systems with applications to numerical methods for PDEs

We study sparse recovery with structured random measurement matrices having independent, identically distributed, and uniformly bounded rows and with a nontrivial covariance structure. This class of matrices arises from random sampling of bounded Riesz systems and generalizes random partial Fourier matrices. Our main result improves the currently available results for the null space and restricted isometry properties of such random matrices. The main novelty of our analysis is a new upper bound for the expectation of the supremum of a Bernoulli process associated with a restricted isometry constant. We apply our result to prove new performance guarantees for the CORSING method, a recently introduced numerical approximation technique for partial differential equations (PDEs) based on compressive sensing.

cs.IT↗

arXiv2019

Jointly Low-Rank and Bisparse Recovery: Questions and Partial Answers

We investigate the problem of recovering jointly $r$-rank and $s$-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that $m \asymp r s \ln(en/s)$ measurements make the recovery possible in theory, meaning via a nonpractical algorithm. In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterative-hard-thresholding algorithm when $m \asymp r s^γ\ln(en/s)$ for some exponent $γ> 0$. We show that this is feasible for $γ= 2$, and that the proposed analysis cannot cover the case $γ\leq 1$. The precise value of the optimal exponent $γ\in [1,2]$ is the object of a question, raised but unresolved in this paper, about head projections for the jointly low-rank and bisparse structure. Some related questions are partially answered in passing. For rank-one measurements, we suggest on arcane grounds an iterative-hard-thresholding algorithm modified to exploit the nonstandard restricted isometry property obeyed by this type of measurements.

math.NA↗

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

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

Improved bounds for sparse recovery from subsampled random convolutions

We study the recovery of sparse vectors from subsampled random convolutions via $\ell_1$-minimization. We consider the setup in which both the subsampling locations as well as the generating vector are chosen at random. For a subgaussian generator with independent entries, we improve previously known estimates: if the sparsity $s$ is small enough, i.e., $s \lesssim \sqrt{n/\log(n)}$, we show that $m \gtrsim s \log(en/s)$ measurements are sufficient to recover $s$-sparse vectors in dimension $n$ with high probability, matching the well-known condition for recovery from standard Gaussian measurements. If $s$ is larger, then essentially $m \geq s \log^2(s) \log(\log(s)) \log(n)$ measurements are sufficient, again improving over previous estimates. Our results are shown via the so-called robust null space property which is weaker than the standard restricted isometry property. Our method of proof involves a novel combination of small ball estimates with chaining techniques {which should be of independent interest.

cs.IT↗

arXiv2017

Multi-level Compressed Sensing Petrov-Galerkin discretization of high-dimensional parametric PDEs

We analyze a novel multi-level version of a recently introduced compressed sensing (CS) Petrov-Galerkin (PG) method from [H. Rauhut and Ch. Schwab: Compressive Sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations, Math. Comp. 304(2017) 661-700] for the solution of many-parametric partial differential equations. We propose to use multi-level PG discretizations, based on a hierarchy of nested finite dimensional subspaces, and to reconstruct parametric solutions at each level from level-dependent random samples of the high-dimensional parameter space via CS methods such as weighted l1-minimization. For affine parametric, linear operator equations, we prove that our approach allows to approximate the parametric solution with (almost) optimal convergence order as specified by certain summability properties of the coefficient sequence in a general polynomial chaos expansion of the parametric solution and by the convergence order of the PG discretization in the physical variables. The computations of the parameter samples of the PDE solution is "embarrassingly parallel", as in Monte-Carlo Methods. Contrary to other recent approaches, and as already noted in [A. Doostan and H. Owhadi: A non-adapted sparse approximation of PDEs with stochastic inputs. JCP 230(2011) 3015-3034] the optimality of the computed approximations does not require a-priori assumptions on ordering and structure of the index sets of the largest gpc coefficients (such as the "downward closed" property). We prove that under certain assumptions work versus accuracy of the new algorithms is asymptotically equal to that of one PG solve for the corresponding nominal problem on the finest discretization level up to a constant.

math.NA↗

arXiv2017

One-bit compressed sensing with partial Gaussian circulant matrices

In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $δ$, $m\sim δ^{-4} s\log(N/sδ)$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $δ$ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an $\ell_1/\ell_2$ RIP-property. Under a slightly worse dependence on $δ$, we establish stability with respect to approximate sparsity, as well as full vector recovery results.

cs.IT↗

arXiv2017

Masked Toeplitz covariance estimation

The problem of estimating the covariance matrix $Σ$ of a $p$-variate distribution based on its $n$ observations arises in many data analysis contexts. While for $n>p$, the classical sample covariance matrix $\hatΣ_n$ is a good estimator for $Σ$, it fails in the high-dimensional setting when $n\ll p$. In this scenario one requires prior knowledge about the structure of the covariance matrix in order to construct reasonable estimators. Under the common assumption that $Σ$ is sparse, a refined estimator is given by $M\cdot\hatΣ_n$, where $M$ is a suitable symmetric mask matrix indicating the nonzero entries of $Σ$ and $\cdot$ denotes the entrywise product of matrices. In the present work we assume that $Σ$ has Toeplitz structure corresponding to stationary signals. This suggests to average the sample covariance $\hatΣ_n$ over the diagonals in order to obtain an estimator $\tildeΣ_n$ of Toeplitz structure. Assuming in addition that $Σ$ is sparse suggests to study estimators of the form $M\cdot\tildeΣ_n$. For Gaussian random vectors and, more generally, random vectors satisfying the convex concentration property, our main result bounds the estimation error in terms of $n$ and $p$ and shows that accurate estimation is indeed possible when $n \ll p$. The new bound significantly generalizes previous results by Cai, Ren and Zhou and provides an alternative proof. Our analysis exploits the connection between the spectral norm of a Toeplitz matrix and the supremum norm of the corresponding spectral density function.

cs.IT↗