Phase Harmonic Correlations and Convolutional Neural Networks

Mallat, Stéphane; Zhang, Sixin; Rochette, Gaspar

doi:10.48550/arxiv.1810.12136

Cited by 9 publications

(21 citation statements)

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“…In previous years, it has been shown that particular low-variance representations inspired from deep neural networks can efficiently characterize non-Gaussian fields. Based on the multi-scale decomposition achieved by the wavelet transform, these representations are built from successive applications of the so-called scattering operator on the field under study (convolution by a wavelet followed by a modulus operator, Mallat ( 2012)), and/or from phase harmonics of its wavelet coefficients (multiplication of their phase by an integer, Mallat et al (2018)). They can then be analyzed directly as well as from their covariance matrix, and have obtained state-ofthe-art classification results when applied to handwritten and texture discrimination (Bruna & Mallat 2013;Sifre & Mallat 2013).…”

Section: New Non-gaussian Statisticsmentioning

confidence: 99%

“…The middle image in Fig. 8 was sampled from a nearly maximum entropy distribution conditioned by wavelet harmonic covariance coefficients (Mallat et al 2018). One can observe from this figure that the image obtained from wavelet harmonic covariances captures better the statistics of the original, including the geometry of high amplitude outliers and filaments, although it uses fewer moments than the Gaussian model.…”

Section: New Non-gaussian Statisticsmentioning

confidence: 99%

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The Quijote simulations

Villaescusa-Navarro,

Hahn,

Massara

et al. 2019

Preprint

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The Quijote simulations are a set of 43100 full N-body simulations spanning more than 7000 cosmological models in the {Ω m , Ω b , h, n s , σ 8 , M ν , w} hyperplane. At a single redshift the simulations contain more than 8.5 trillions of particles over a combined volume of 43100 (h −1 Gpc) 3 . Billions of dark matter halos and cosmic voids have been identified in the simulations, whose runs required more than 35 million core hours. The Quijote simulations have been designed for two main purposes: 1) to quantify the information content on cosmological observables, and 2) to provide enough data to train machine learning algorithms. In this paper we describe the simulations and show a few of their applications. We also release the Petabyte of data generated, comprising hundreds of thousands of simulation snapshots at multiple redshifts, halo and void catalogs, together with millions of summary statistics such as power spectra, bispectra, correlation functions, marked power spectra, and estimated probability density functions.

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Section: New Non-gaussian Statisticsmentioning

confidence: 99%

Section: New Non-gaussian Statisticsmentioning

confidence: 99%

The Quijote simulations

Villaescusa-Navarro,

Hahn,

Massara

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Mallat et al [43] consider a continuous analogy of our convolutional construction (Definition 3). They show that a ReLU activation function acts as a phase filter and that the layer is bi-Lipschitz, and hence injective, provided that the filters have a sufficiently diverse phase and form a frame.…”

Section: Related Workmentioning

confidence: 99%

“…which does hold, suggesting that the lower bound in Proposition 1 is not pessimistic enough. + E.2 Relationship to Mallat et al [43] In [43] the authors consider a construction analogous to our convolutional construction (in Definition 3) defined on a continuum (i.e. infinite-dimensional function defined on an interval) rather than on a vector (i.e.…”

Section: Appendix C Proofs From Section 4 C1 Proof Of Theoremmentioning

confidence: 99%

“…The frame condition is given by Proposition 2.6 in [43] that the weight matrix must satisfy in order ensure that W is invertible and stable. In the notation of [43] the filters ψλ are analogous to the Fourier transform of the kernels in Definition 2, and the condition in (2.25) in [43] is one natural way to generalize the notion of a basis to a continuous signal. Hence, Proposition 2.6 in [43] can be loosely interpreted as a statement that the kernels in a given layer of width P form a basis of R P .…”

Section: Appendix C Proofs From Section 4 C1 Proof Of Theoremmentioning

confidence: 99%

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Globally Injective ReLU Networks

Puthawala,

Kothari,

Lassas

et al. 2020

Preprint

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We study injective ReLU neural networks. Injectivity plays an important role in generative models where it facilitates inference; in inverse problems with generative priors it is a precursor to well posedness. We establish sharp conditions for injectivity of ReLU layers and networks, both fully connected and convolutional. We make no architectural assumptions beyond the ReLU activations so our results apply to a very general class of neural networks. We show through a layer-wise analysis that an expansivity factor of two is necessary for injectivity; we also show sufficiency by constructing weight matrices which guarantee injectivity. Further, we show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires considerably larger expansivity which might seem counterintuitive. We then derive the inverse Lipschitz constants and study the approximation-theoretic properties of injective neural networks. Using arguments from differential topology we prove that, under mild technical conditions, any Lipschitz map can be approximated by an injective neural network. This justifies the use of injective neural networks in problems which a priori do not require injectivity. Our results establish a theoretical basis for the study of nonlinear inverse and inference problems using neural networks.Preprint. Under review.

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Non-Gaussian modelling and statistical denoising of Planck dust polarisation full-sky maps using scattering transforms

Delouis

Allys

Gauvrit

et al. 2022

A&A

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Scattering transforms have been successfully used to describe dust polarisation for flat-sky images. This paper expands this framework to noisy observations on the sphere with the aim of obtaining denoised Stokes Q and U all-sky maps at 353 GHz, as well as a non-Gaussian model of dust polarisation, from the Planck data. To achieve this goal, we extended the computation of scattering coefficients to the HEALPix pixelation and introduced cross-statistics that allowed us to make use of half-mission maps as well as the correlation between dust temperature and polarisation. Introducing a general framework, we developed an algorithm that uses the scattering statistics to separate dust polarisation from data noise. The separation was validated on mock data before it was applied to the SRoll2 Planck maps at N side = 256. The validation shows that the statistics of the dust emission, including its non-Gaussian properties, are recovered until max ∼ 700, where, at high Galactic latitudes, the dust power is weaker than that of the dust by two orders of magnitude. On scales where the dust power is weaker than one-tenth of the power of the noise, structures in the output maps have comparable statistics, but are not spatially coincident with those of the input maps. Our results on Planck data are significant milestones opening new perspectives for statistical studies of dust polarisation and for the simulation of Galactic polarised foregrounds. The Planck denoised maps are available (see ftp://sroll20.ias.u-psud.fr/sroll40_353) together with results from our validation on mock data, which may be used to quantify uncertainties.

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Phase Harmonic Correlations and Convolutional Neural Networks

Cited by 9 publications

References 0 publications

The Quijote simulations

The Quijote simulations

Globally Injective ReLU Networks

Non-Gaussian modelling and statistical denoising of Planck dust polarisation full-sky maps using scattering transforms

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