arXiv subjects

Stéphane Mallat

Publications and source records attributed to Stéphane Mallat.

54 records · Page 3Linked to original sources

arXiv2015

Transformée en scattering sur la spirale temps-chroma-octave

We introduce a scattering representation for the analysis and classification of sounds. It is locally translation-invariant, stable to deformations in time and frequency, and has the ability to capture harmonic structures. The scattering representation can be interpreted as a convolutional neural network which cascades a wavelet transform in time and along a harmonic spiral. We study its application for the analysis of the deformations of the source-filter model.

cs.SD↗

arXiv2015

Deep Learning by Scattering

We introduce general scattering transforms as mathematical models of deep neural networks with l2 pooling. Scattering networks iteratively apply complex valued unitary operators, and the pooling is performed by a complex modulus. An expected scattering defines a contractive representation of a high-dimensional probability distribution, which preserves its mean-square norm. We show that unsupervised learning can be casted as an optimization of the space contraction to preserve the volume occupied by unlabeled examples, at each layer of the network. Supervised learning and classification are performed with an averaged scattering, which provides scattering estimations for multiple classes.

cs.LG↗

arXiv2015

Deep Roto-Translation Scattering for Object Classification

Dictionary learning algorithms or supervised deep convolution networks have considerably improved the efficiency of predefined feature representations such as SIFT. We introduce a deep scattering convolution network, with predefined wavelet filters over spatial and angular variables. This representation brings an important improvement to results previously obtained with predefined features over object image databases such as Caltech and CIFAR. The resulting accuracy is comparable to results obtained with unsupervised deep learning and dictionary based representations. This shows that refining image representations by using geometric priors is a promising direction to improve image classification and its understanding.

cs.CV↗

arXiv2015

Intermittent process analysis with scattering moments

Scattering moments provide nonparametric models of random processes with stationary increments. They are expected values of random variables computed with a nonexpansive operator, obtained by iteratively applying wavelet transforms and modulus nonlinearities, which preserves the variance. First- and second-order scattering moments are shown to characterize intermittency and self-similarity properties of multiscale processes. Scattering moments of Poisson processes, fractional Brownian motions, Lévy processes and multifractal random walks are shown to have characteristic decay. The Generalized Method of Simulated Moments is applied to scattering moments to estimate data generating models. Numerical applications are shown on financial time-series and on energy dissipation of turbulent flows.

stat.ME↗

arXiv2014

Unsupervised Deep Haar Scattering on Graphs

The classification of high-dimensional data defined on graphs is particularly difficult when the graph geometry is unknown. We introduce a Haar scattering transform on graphs, which computes invariant signal descriptors. It is implemented with a deep cascade of additions, subtractions and absolute values, which iteratively compute orthogonal Haar wavelet transforms. Multiscale neighborhoods of unknown graphs are estimated by minimizing an average total variation, with a pair matching algorithm of polynomial complexity. Supervised classification with dimension reduction is tested on data bases of scrambled images, and for signals sampled on unknown irregular grids on a sphere.

cs.LG↗

arXiv2014

Generic Deep Networks with Wavelet Scattering

We introduce a two-layer wavelet scattering network, for object classification. This scattering transform computes a spatial wavelet transform on the first layer and a new joint wavelet transform along spatial, angular and scale variables in the second layer. Numerical experiments demonstrate that this two layer convolution network, which involves no learning and no max pooling, performs efficiently on complex image data sets such as CalTech, with structural objects variability and clutter. It opens the possibility to simplify deep neural network learning by initializing the first layers with wavelet filters.

cs.CV↗

arXiv2014

Rigid-Motion Scattering for Texture Classification

A rigid-motion scattering computes adaptive invariants along translations and rotations, with a deep convolutional network. Convolutions are calculated on the rigid-motion group, with wavelets defined on the translation and rotation variables. It preserves joint rotation and translation information, while providing global invariants at any desired scale. Texture classification is studied, through the characterization of stationary processes from a single realization. State-of-the-art results are obtained on multiple texture data bases, with important rotation and scaling variabilities.

cs.CV↗

arXiv2014

Deep Scattering Spectrum

A scattering transform defines a locally translation invariant representation which is stable to time-warping deformations. It extends MFCC representations by computing modulation spectrum coefficients of multiple orders, through cascades of wavelet convolutions and modulus operators. Second-order scattering coefficients characterize transient phenomena such as attacks and amplitude modulation. A frequency transposition invariant representation is obtained by applying a scattering transform along log-frequency. State-the-of-art classification results are obtained for musical genre and phone classification on GTZAN and TIMIT databases, respectively.

cs.SD↗

arXiv2013

Classification with Scattering Operators

A scattering vector is a local descriptor including multiscale and multi-direction co-occurrence information. It is computed with a cascade of wavelet decompositions and complex modulus. This scattering representation is locally translation invariant and linearizes deformations. A supervised classification algorithm is computed with a PCA model selection on scattering vectors. State of the art results are obtained for handwritten digit recognition and texture classification.

cs.CV↗

arXiv2013

Audio Texture Synthesis with Scattering Moments

We introduce an audio texture synthesis algorithm based on scattering moments. A scattering transform is computed by iteratively decomposing a signal with complex wavelet filter banks and computing their amplitude envelop. Scattering moments provide general representations of stationary processes computed as expected values of scattering coefficients. They are estimated with low variance estimators from single realizations. Audio signals having prescribed scattering moments are synthesized with a gradient descent algorithms. Audio synthesis examples show that scattering representation provide good synthesis of audio textures with much fewer coefficients than the state of the art.

stat.AP↗

arXiv2013

Wavelet methods for shape perception in electro-sensing

This paper aims at presenting a new approach to the electro-sensing problem using wavelets. It provides an efficient algorithm for recognizing the shape of a target from micro-electrical impedance measurements. Stability and resolution capabilities of the proposed algorithm are quantified in numerical simulations.

math.NA↗

arXiv2013

Phase Recovery, MaxCut and Complex Semidefinite Programming

Phase retrieval seeks to recover a signal x from the amplitude |Ax| of linear measurements. We cast the phase retrieval problem as a non-convex quadratic program over a complex phase vector and formulate a tractable relaxation (called PhaseCut) similar to the classical MaxCut semidefinite program. We solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in Gerchberg-Saxton, where each iteration is a matrix vector product. Numerical results show the performance of this approach over three different phase retrieval problems, in comparison with greedy phase retrieval algorithms and matrix completion formulations.

math.OC↗

arXiv2012

Group Invariant Scattering

This paper constructs translation invariant operators on L2(R^d), which are Lipschitz continuous to the action of diffeomorphisms. A scattering propagator is a path ordered product of non-linear and non-commuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz continuous to the action of diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform which is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L2 (G), where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on L2(R^d) and on Ld (SO(d)) defines a translation and rotation invariant scattering on L2(R^d).

math.FA↗

arXiv2012

Invariant Scattering Convolution Networks

A wavelet scattering network computes a translation invariant image representation, which is stable to deformations and preserves high frequency information for classification. It cascades wavelet transform convolutions with non-linear modulus and averaging operators. The first network layer outputs SIFT-type descriptors whereas the next layers provide complementary invariant information which improves classification. The mathematical analysis of wavelet scattering networks explains important properties of deep convolution networks for classification. A scattering representation of stationary processes incorporates higher order moments and can thus discriminate textures having the same Fourier power spectrum. State of the art classification results are obtained for handwritten digits and texture discrimination, using a Gaussian kernel SVM and a generative PCA classifier.

cs.CV↗

arXiv2011

Classification with Invariant Scattering Representations

A scattering transform defines a signal representation which is invariant to translations and Lipschitz continuous relatively to deformations. It is implemented with a non-linear convolution network that iterates over wavelet and modulus operators. Lipschitz continuity locally linearizes deformations. Complex classes of signals and textures can be modeled with low-dimensional affine spaces, computed with a PCA in the scattering domain. Classification is performed with a penalized model selection. State of the art results are obtained for handwritten digit recognition over small training sets, and for texture classification.

cs.CV↗

arXiv2011

Geometric Models with Co-occurrence Groups

A geometric model of sparse signal representations is introduced for classes of signals. It is computed by optimizing co-occurrence groups with a maximum likelihood estimate calculated with a Bernoulli mixture model. Applications to face image compression and MNIST digit classification illustrate the applicability of this model.

cs.CV↗

arXiv2010

Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.

cs.CV↗

arXiv2009

Bandlet Image Estimation with Model Selection

To estimate geometrically regular images in the white noise model and obtain an adaptive near asymptotic minimaxity result, we consider a model selection based bandlet estimator. This bandlet estimator combines the best basis selection behaviour of the model selection and the approximation properties of the bandlet dictionary. We derive its near asymptotic minimaxity for geometrically regular images as an example of model selection with general dictionary of orthogonal bases. This paper is thus also a self contained tutorial on model selection with orthogonal bases dictionary.

math.ST↗