arXiv subjects

Rémi Gribonval

Publications and source records attributed to Rémi Gribonval.

At least 73 records · Page 4Linked to original sources

arXiv2018

Is the 1-norm the best convex sparse regularization?

The 1-norm is a good convex regularization for the recovery of sparse vectors from under-determined linear measurements. No other convex regularization seems to surpass its sparse recovery performance. How can this be explained? To answer this question, we define several notions of "best" (convex) regulariza-tion in the context of general low-dimensional recovery and show that indeed the 1-norm is an optimal convex sparse regularization within this framework.

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arXiv2018

Instance Optimal Decoding and the Restricted Isometry Property

In this paper, we address the question of information preservation in ill-posed, non-linear inverse problems, assuming that the measured data is close to a low-dimensional model set. We provide necessary and sufficient conditions for the existence of a so-called instance optimal decoder, i.e., that is robust to noise and modelling error. Inspired by existing results in compressive sensing, our analysis is based on a (Lower) Restricted Isometry Property (LRIP), formulated in a non-linear fashion. We also provide sufficient conditions for non-uniform recovery with random measurement operators, with a new formulation of the LRIP. We finish by describing typical strategies to prove the LRIP in both linear and non-linear cases, and illustrate our results by studying the invertibility of a one-layer neural net with random weights.

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arXiv2017

Learning a Complete Image Indexing Pipeline

To work at scale, a complete image indexing system comprises two components: An inverted file index to restrict the actual search to only a subset that should contain most of the items relevant to the query; An approximate distance computation mechanism to rapidly scan these lists. While supervised deep learning has recently enabled improvements to the latter, the former continues to be based on unsupervised clustering in the literature. In this work, we propose a first system that learns both components within a unifying neural framework of structured binary encoding.

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arXiv2017

A modeling and algorithmic framework for (non)social (co)sparse audio restoration

We propose a unified modeling and algorithmic framework for audio restoration problem. It encompasses analysis sparse priors as well as more classical synthesis sparse priors, and regular sparsity as well as various forms of structured sparsity embodied by shrinkage operators (such as social shrinkage). The versatility of the framework is illustrated on two restoration scenarios: denoising, and declipping. Extensive experimental results on these scenarios highlight both the speedups of 20% or even more offered by the analysis sparse prior, and the substantial declipping quality that is achievable with both the social and the plain flavor. While both flavors overall exhibit similar performance, their detailed comparison displays distinct trends depending whether declipping or denoising is considered.

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arXiv2017

Analyzing the Approximation Error of the Fast Graph Fourier Transform

The graph Fourier transform (GFT) is in general dense and requires O(n^2) time to compute and O(n^2) memory space to store. In this paper, we pursue our previous work on the approximate fast graph Fourier transform (FGFT). The FGFT is computed via a truncated Jacobi algorithm, and is defined as the product of J Givens rotations (very sparse orthogonal matrices). The truncation parameter, J, represents a trade-off between precision of the transform and time of computation (and storage space). We explore further this trade-off and study, on different types of graphs, how is the approximation error distributed along the spectrum.

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arXiv2017

SUBIC: A supervised, structured binary code for image search

For large-scale visual search, highly compressed yet meaningful representations of images are essential. Structured vector quantizers based on product quantization and its variants are usually employed to achieve such compression while minimizing the loss of accuracy. Yet, unlike binary hashing schemes, these unsupervised methods have not yet benefited from the supervision, end-to-end learning and novel architectures ushered in by the deep learning revolution. We hence propose herein a novel method to make deep convolutional neural networks produce supervised, compact, structured binary codes for visual search. Our method makes use of a novel block-softmax non-linearity and of batch-based entropy losses that together induce structure in the learned encodings. We show that our method outperforms state-of-the-art compact representations based on deep hashing or structured quantization in single and cross-domain category retrieval, instance retrieval and classification. We make our code and models publicly available online.

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arXiv2017

Compressed sensing in Hilbert spaces

In many linear inverse problems, we want to estimate an unknown vector belonging to a high-dimensional (or infinite-dimensional) space from few linear measurements. To overcome the ill-posed nature of such problems, we use a low-dimension assumption on the unknown vector: it belongs to a low-dimensional model set. The question of whether it is possible to recover such an unknown vector from few measurements then arises. If the answer is yes, it is also important to be able to describe a way to perform such a recovery. We describe a general framework where appropriately chosen random measurements guarantee that recovery is possible. We further describe a way to study the performance of recovery methods that consist in the minimization of a regularization function under a data-fit constraint.

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arXiv2017

Beyond Moore-Penrose Part I: Generalized Inverses that Minimize Matrix Norms

This is the first paper of a two-long series in which we study linear generalized inverses that minimize matrix norms. Such generalized inverses are famously represented by the Moore-Penrose pseudoinverse (MPP) which happens to minimize the Frobenius norm. Freeing up the degrees of freedom associated with Frobenius optimality enables us to promote other interesting properties. In this Part I, we look at the basic properties of norm-minimizing generalized inverses, especially in terms of uniqueness and relation to the MPP. We first show that the MPP minimizes many norms beyond those unitarily invariant, thus further bolstering its role as a robust choice in many situations. We then concentrate on some norms which are generally not minimized by the MPP, but whose minimization is relevant for linear inverse problems and sparse representations. In particular, we look at mixed norms and the induced $\ell^p \rightarrow \ell^q$ norms. An interesting representative is the sparse pseudoinverse which we study in much more detail in Part II. Next, we shift attention from norms to matrices with interesting behaviors. We exhibit a class whose generalized inverse is always the MPP-even for norms that normally result in different inverses-and a class for which many generalized inverses coincide, but not with the MPP. Finally, we discuss efficient computation of norm-minimizing generalized inverses.

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arXiv2017

Beyond Moore-Penrose Part II: The Sparse Pseudoinverse

This is the second part of a two-paper series on generalized inverses that minimize matrix norms. In Part II we focus on generalized inverses that are minimizers of entrywise p norms whose main representative is the sparse pseudoinverse for $p = 1$. We are motivated by the idea to replace the Moore-Penrose pseudoinverse by a sparser generalized inverse which is in some sense well-behaved. Sparsity implies that it is faster to apply the resulting matrix; well-behavedness would imply that we do not lose much in stability with respect to the least-squares performance of the MPP. We first address questions of uniqueness and non-zero count of (putative) sparse pseu-doinverses. We show that a sparse pseudoinverse is generically unique, and that it indeed reaches optimal sparsity for almost all matrices. We then turn to proving our main stability result: finite-size concentration bounds for the Frobenius norm of p-minimal inverses for $1 $\le$ p $\le$ 2$. Our proof is based on tools from convex analysis and random matrix theory, in particular the recently developed convex Gaussian min-max theorem. Along the way we prove several results about sparse representations and convex programming that were known folklore, but of which we could find no proof.

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arXiv2017

Approximate fast graph Fourier transforms via multi-layer sparse approximations

The Fast Fourier Transform (FFT) is an algorithm of paramount importance in signal processing as it allows to apply the Fourier transform in O(n log n) instead of O(n 2) arithmetic operations. Graph Signal Processing (GSP) is a recent research domain that generalizes classical signal processing tools, such as the Fourier transform, to situations where the signal domain is given by any arbitrary graph instead of a regular grid. Today, there is no method to rapidly apply graph Fourier transforms. We propose in this paper a method to obtain approximate graph Fourier transforms that can be applied rapidly and stored efficiently. It is based on a greedy approximate diagonalization of the graph Laplacian matrix, carried out using a modified version of the famous Jacobi eigenvalues algorithm. The method is described and analyzed in detail, and then applied to both synthetic and real graphs, showing its potential.

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arXiv2017

Sketching for Large-Scale Learning of Mixture Models

Learning parameters from voluminous data can be prohibitive in terms of memory and computational requirements. We propose a "compressive learning" framework where we estimate model parameters from a sketch of the training data. This sketch is a collection of generalized moments of the underlying probability distribution of the data. It can be computed in a single pass on the training set, and is easily computable on streams or distributed datasets. The proposed framework shares similarities with compressive sensing, which aims at drastically reducing the dimension of high-dimensional signals while preserving the ability to reconstruct them. To perform the estimation task, we derive an iterative algorithm analogous to sparse reconstruction algorithms in the context of linear inverse problems. We exemplify our framework with the compressive estimation of a Gaussian Mixture Model (GMM), providing heuristics on the choice of the sketching procedure and theoretical guarantees of reconstruction. We experimentally show on synthetic data that the proposed algorithm yields results comparable to the classical Expectation-Maximization (EM) technique while requiring significantly less memory and fewer computations when the number of database elements is large. We further demonstrate the potential of the approach on real large-scale data (over 10 8 training samples) for the task of model-based speaker verification. Finally, we draw some connections between the proposed framework and approximate Hilbert space embedding of probability distributions using random features. We show that the proposed sketching operator can be seen as an innovative method to design translation-invariant kernels adapted to the analysis of GMMs. We also use this theoretical framework to derive information preservation guarantees, in the spirit of infinite-dimensional compressive sensing.

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arXiv2017

Compressive K-means

The Lloyd-Max algorithm is a classical approach to perform K-means clustering. Unfortunately, its cost becomes prohibitive as the training dataset grows large. We propose a compressive version of K-means (CKM), that estimates cluster centers from a sketch, i.e. from a drastically compressed representation of the training dataset. We demonstrate empirically that CKM performs similarly to Lloyd-Max, for a sketch size proportional to the number of cen-troids times the ambient dimension, and independent of the size of the original dataset. Given the sketch, the computational complexity of CKM is also independent of the size of the dataset. Unlike Lloyd-Max which requires several replicates, we further demonstrate that CKM is almost insensitive to initialization. For a large dataset of 10^7 data points, we show that CKM can run two orders of magnitude faster than five replicates of Lloyd-Max, with similar clustering performance on artificial data. Finally, CKM achieves lower classification errors on handwritten digits classification.

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arXiv2017

Recipes for stable linear embeddings from Hilbert spaces to R^m

We consider the problem of constructing a linear map from a Hilbert space $\mathcal{H}$ (possibly infinite dimensional) to $\mathbb{R}^m$ that satisfies a restricted isometry property (RIP) on an arbitrary signal model $\mathcal{S} \subset \mathcal{H}$. We present a generic framework that handles a large class of low-dimensional subsets but also unstructured and structured linear maps. We provide a simple recipe to prove that a random linear map satisfies a general RIP on $\mathcal{S}$ with high probability. We also describe a generic technique to construct linear maps that satisfy the RIP. Finally, we detail how to use our results in several examples, which allow us to recover and extend many known compressive sampling results.

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arXiv2016

Stable recovery of low-dimensional cones in Hilbert spaces: One RIP to rule them all

Many inverse problems in signal processing deal with the robust estimation of unknown data from underdetermined linear observations. Low dimensional models, when combined with appropriate regularizers, have been shown to be efficient at performing this task. Sparse models with the 1-norm or low rank models with the nuclear norm are examples of such successful combinations. Stable recovery guarantees in these settings have been established using a common tool adapted to each case: the notion of restricted isometry property (RIP). In this paper, we establish generic RIP-based guarantees for the stable recovery of cones (positively homogeneous model sets) with arbitrary regularizers. These guarantees are illustrated on selected examples. For block structured sparsity in the infinite dimensional setting, we use the guarantees for a family of regularizers which efficiency in terms of RIP constant can be controlled, leading to stronger and sharper guarantees than the state of the art.

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arXiv2016

Approximate search with quantized sparse representations

This paper tackles the task of storing a large collection of vectors, such as visual descriptors, and of searching in it. To this end, we propose to approximate database vectors by constrained sparse coding, where possible atom weights are restricted to belong to a finite subset. This formulation encompasses, as particular cases, previous state-of-the-art methods such as product or residual quantization. As opposed to traditional sparse coding methods, quantized sparse coding includes memory usage as a design constraint, thereby allowing us to index a large collection such as the BIGANN billion-sized benchmark. Our experiments, carried out on standard benchmarks, show that our formulation leads to competitive solutions when considering different trade-offs between learning/coding time, index size and search quality.

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arXiv2016

Random sampling of bandlimited signals on graphs

We study the problem of sampling k-bandlimited signals on graphs. We propose two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure, and its performance depends on a parameter called the graph coherence. On the contrary, the second strategy is adaptive but yields optimal results. Indeed, no more than O(k log(k)) measurements are sufficient to ensure an accurate and stable recovery of all k-bandlimited signals. This second strategy is based on a careful choice of the sampling distribution, which can be estimated quickly. Then, we propose a computationally efficient decoder to reconstruct k-bandlimited signals from their samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we conduct several experiments to test these techniques.

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arXiv2016

Flexible Multi-layer Sparse Approximations of Matrices and Applications

The computational cost of many signal processing and machine learning techniques is often dominated by the cost of applying certain linear operators to high-dimensional vectors. This paper introduces an algorithm aimed at reducing the complexity of applying linear operators in high dimension by approximately factorizing the corresponding matrix into few sparse factors. The approach relies on recent advances in non-convex optimization. It is first explained and analyzed in details and then demonstrated experimentally on various problems including dictionary learning for image denoising, and the approximation of large matrices arising in inverse problems.

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arXiv2015

Learning Co-Sparse Analysis Operators with Separable Structures

In the co-sparse analysis model a set of filters is applied to a signal out of the signal class of interest yielding sparse filter responses. As such, it may serve as a prior in inverse problems, or for structural analysis of signals that are known to belong to the signal class. The more the model is adapted to the class, the more reliable it is for these purposes. The task of learning such operators for a given class is therefore a crucial problem. In many applications, it is also required that the filter responses are obtained in a timely manner, which can be achieved by filters with a separable structure. Not only can operators of this sort be efficiently used for computing the filter responses, but they also have the advantage that less training samples are required to obtain a reliable estimate of the operator. The first contribution of this work is to give theoretical evidence for this claim by providing an upper bound for the sample complexity of the learning process. The second is a stochastic gradient descent (SGD) method designed to learn an analysis operator with separable structures, which includes a novel and efficient step size selection rule. Numerical experiments are provided that link the sample complexity to the convergence speed of the SGD algorithm.

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