Heart Sound Classification Using Deep Structured Features

Tschannen, Michael; Krämer, Thomas; Marti, Gian; Heinzmann, Matthias; Wiatowski, Thomas

doi:10.22489/cinc.2016.162-186

Cited by 61 publications

(41 citation statements)

References 7 publications

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“…Since its introduction in [8], the scattering transform has found successful applications in, for example, audio genre, visual textures or medical data classification [3,11,12]. …”

Section: The Scattering Transform Of F Ismentioning

confidence: 99%

Wavelet transform modulus: phase retrieval and scattering

Waldspurger

2018

Journées Équations Aux Dérivées Partielles

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We discuss the problem that consists in reconstructing a function from the modulus of its wavelet transform. In the case where the wavelets are Cauchy wavelets, all analytic functions are uniquely determined by this modulus. Additionally, although it is not uniformly continuous, the reconstruction operator enjoys a form of local stability. We describe these two results, and give an idea of the proof of the first one. To conclude, we present a related result on a more sophisticated operator, based on the wavelet transform modulus: the scattering transform.

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“…Since its introduction in [8], the scattering transform has found successful applications in, for example, audio genre, visual textures or medical data classification [3,11,12]. …”

Section: The Scattering Transform Of F Ismentioning

confidence: 99%

Wavelet transform modulus: phase retrieval and scattering

Waldspurger

2018

Journées Équations Aux Dérivées Partielles

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“…The lower bound in (8) guarantees that at least ((1 − ε) · 100)% of the input signal energy are contained in the feature vector {Φ n Ω (f )} N n=0 generated in the first N network layers. We note that establishing the upper bound in (8) does not pose any significant difficulties as it follows straight from the results in [6, Appendix E].…”

Section: Depth-constrained Scattering Networkmentioning

confidence: 99%

“…The lower bound in (8) implies a trivial null-set for the feature extractor Φ Ω and thereby ensures that the only signal f that is mapped to the all-zeros feature vector is f = 0. We emphasize that the energy decay results in Theorem 3.1 pertain to the feature maps U [q]f , whereas energy conservation according to (8) applies to the feature vector {Φ n Ω (f )} N n=0 . The next result explains how to choose R in the WH and r in the wavelet case so as to satisfy (8).…”

Section: Depth-constrained Scattering Networkmentioning

confidence: 99%

“…We emphasize that the energy decay results in Theorem 3.1 pertain to the feature maps U [q]f , whereas energy conservation according to (8) applies to the feature vector {Φ n Ω (f )} N n=0 . The next result explains how to choose R in the WH and r in the wavelet case so as to satisfy (8). In particular, we shall see that for every (possibly small) ε > 0 and every N ∈ N, say ε = 0.01 and N = 1, there exist R > 0 and r > 1 such that (8)…”

Section: Depth-constrained Scattering Networkmentioning

confidence: 99%

“…We address this question for the class of scattering networks as introduced in [5] and extended in [6]. Scattering network-based feature extractors were shown to yield classification performance competitive with the state-of-the-art on various data sets [7][8][9]. Moreover, a mathematical theory exists, which allows to establish formally that such feature extractors are-under certain technical conditions-horizontally [5] or vertically [6] translation-invariant, energy-conserving [10][11][12], deformation-stable in the sense of [5] or exhibit limited sensitivity to deformations on input signal classes such as band-limited functions [6], cartoon functions [13], and Lipschitz functions [13].…”

Section: Introductionmentioning

confidence: 99%

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Topology reduction in deep convolutional feature extraction networks

Wiatowski¹,

Bölcskei²

2017

Wavelets and Sparsity XVII

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Deep convolutional neural networks (CNNs) used in practice employ potentially hundreds of layers and 10,000s of nodes. Such network sizes entail significant computational complexity due to the large number of convolutions that need to be carried out; in addition, a large number of parameters needs to be learned and stored. Very deep and wide CNNs may therefore not be well suited to applications operating under severe resource constraints as is the case, e.g., in low-power embedded and mobile platforms. This paper aims at understanding the impact of CNN topology, specifically depth and width, on the network's feature extraction capabilities. We address this question for the class of scattering networks that employ either Weyl-Heisenberg filters or wavelets, the modulus non-linearity, and no pooling. The exponential feature map energy decay results in Wiatowski et al., 2017, are generalized to O(a −N ), where an arbitrary decay factor a > 1 can be realized through suitable choice of the Weyl-Heisenberg prototype function or the mother wavelet. We then show how networks of fixed (possibly small) depth N can be designed to guarantee that ((1 − ε) · 100)% of the input signal's energy are contained in the feature vector. Based on the notion of operationally significant nodes, we characterize, partly rigorously and partly heuristically, the topology-reducing effects of (effectively) band-limited input signals, band-limited filters, and feature map symmetries. Finally, for networks based on Weyl-Heisenberg filters, we determine the prototype function bandwidth that minimizes-for fixed network depth N -the average number of operationally significant nodes per layer.

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