The Ridgelet transform of distributions

Kostadinova, Sanja; Pilipović, Stevan; Санева, Катерина; Vindas, Jasson

doi:10.1080/10652469.2013.853057

Cited by 30 publications

(58 citation statements)

References 17 publications

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“…In the late 1990s, Candès [23,24], Rubin [25], and Murata [1] independently proposed the so-called ridgelet transform, which has since been investigated by a number of authors [26,27,28,29,30,31].…”

Section: Integral Representation Of Neural Network and Ridgelet Transmentioning

confidence: 99%

“…Although many researchers have investigated the ridgelet transform [26,29,30,31], in all the settings Z does not directly admit some fundamental activation functions, namely the sigmoidal function and the ReLU. One of the challenges we faced is to define the ridgelet transform for W " S 1 0 , which admits the sigmoidal function and the ReLU.…”

Section: Our Goalmentioning

confidence: 99%

“…Murata [1] originally posed s " 0, which is suitable for the Euclidean formulation. Other authors such as Candès [24] used s " 1{2, Rubin [25] used s " m, and Kostadinova et al [30] used s " 1 . When f P L 1 pR m q and ψ P L 8 pRq, by using Hölder's inequality, the ridgelet transform is absolutely convergent at every pa, bq P Y m`1 .…”

Section: An Overviewmentioning

confidence: 99%

“…Extension of the ridgelet transform of non-integrable functions requires more sophisticated approaches, because a direct computation of the Radon transform may diverge. For instance, Kostadinova et al [30] extend X " S 1 0 by using a duality technique. In § 5.3 we extend the ridgelet transform to L 2 pR m q, by using the bounded extension procedure.…”

Section: Definition and Well-definednessmentioning

confidence: 99%

“…In this section we discuss the admissibility condition and the reconstruction formula, not only in the Fourier domain as many authors did [23,24,1,30,31], but also in the real domain and in the Radon domain. Both domains are key to the constructive formulation.…”

Section: Reconstruction Formula For Weak Ridgelet Transformmentioning

confidence: 99%

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Neural network with unbounded activation functions is universal approximator

Sonoda

Motomura²

2017

Applied and Computational Harmonic Analysis

255

177

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This paper presents an investigation of the approximation property of neural networks with unbounded activation functions, such as the rectified linear unit (ReLU), which is the new de-facto standard of deep learning. The ReLU network can be analyzed by the ridgelet transform with respect to Lizorkin distributions. By showing three reconstruction formulas by using the Fourier slice theorem, the Radon transform, and Parseval's relation, it is shown that a neural network with unbounded activation functions still satisfies the universal approximation property. As an additional consequence, the ridgelet transform, or the backprojection filter in the Radon domain, is what the network learns after backpropagation. Subject to a constructive admissibility condition, the trained network can be obtained by simply discretizing the ridgelet transform, without backpropagation. Numerical examples not only support the consistency of the admissibility condition but also imply that some non-admissible cases result in low-pass filtering.

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Section: Integral Representation Of Neural Network and Ridgelet Transmentioning

confidence: 99%

Section: Our Goalmentioning

confidence: 99%

Section: An Overviewmentioning

confidence: 99%

Section: Definition and Well-definednessmentioning

confidence: 99%

Section: Reconstruction Formula For Weak Ridgelet Transformmentioning

confidence: 99%

See 3 more Smart Citations

Neural network with unbounded activation functions is universal approximator

Sonoda

Motomura²

2017

Applied and Computational Harmonic Analysis

255

177

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Continuity properties of the shearlet transform and the shearlet synthesis operator on the Lizorkin type spaces

Bartolucci

Pilipović

Teofanov

2022

Mathematische Nachrichten

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We develop a distributional framework for the shearlet transform Sψ0.16em:0.16emS0true(R2true)0.16em→0.16emscriptS(S)${\mathcal {S}}_{\psi }\,{:}\, {\mathcal {S}}_0\big (\mathbb {R}^2\big )\,{\rightarrow }\,{\mathcal {S}}(\mathbb {S})$ and the shearlet synthesis operator Sψt-0.16em:-0.16emscriptS(S)-0.16em→-0.16emS0true(R2true)${\mathcal {S}}^t_{\psi }\!: \!{\mathcal {S}}(\mathbb {S})\!\rightarrow \!{\mathcal {S}}_0\big (\mathbb {R}^2\big )$, where S0true(R2true)${\mathcal {S}}_0\big (\mathbb {R}^2\big )$ is the Lizorkin test function space and scriptSfalse(double-struckSfalse)${\mathcal {S}}(\mathbb {S})$ is the space of highly localized test functions on the standard shearlet group S$\mathbb {S}$. These spaces and their duals S0′(double-struckR2),0.16emS′(S)$\mathcal {S}_0^\prime {\big(\mathbb {R}^2\big)},\, \mathcal {S}^\prime (\mathbb {S})$ are called Lizorkin type spaces of test functions and distributions. We analyze the continuity properties of these transforms when the admissible vector ψ belongs to S0true(R2true)${\mathcal {S}}_0\big (\mathbb {R}^2\big )$. Then, we define the shearlet transform and the shearlet synthesis operator of Lizorkin type distributions as transpose mappings of the shearlet synthesis operator and the shearlet transform, respectively. They yield continuous mappings from S0′(double-struckR2)$\mathcal {S}_0^\prime {\big(\mathbb {R}^2\big)}$ to S′(S)$\mathcal {S}^\prime (\mathbb {S})$ and from S′(S)$\mathcal {S}^\prime (\mathbb {S})$ to S0′true(R2true)$\mathcal {S}_0^\prime \big (\mathbb {R}^2\big )$. Furthermore, we show the consistency of our definition with the shearlet transform defined by direct evaluation of a distribution on the shearlets. The same can be done for the shearlet synthesis operator. Finally, we give a reconstruction formula for Lizorkin type distributions, from which follows that the action of such generalized functions can be written as an absolutely convergent integral over the standard shearlet group.

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