Group Equivariant Capsule Networks

By Jan Eric Lenssen, Matthias Fey, Pascal Libuschewski. 2018

119 citations recorded by the source corpus.

We present group equivariant capsule networks, a framework to introduce guaranteed equivariance and invariance properties to the capsule network idea. Our work can be divided into two contributions. First, we present a generic routing by agreement algorithm defined on elements of a group and prove that equivariance of output pose vectors, as well as invariance of output activations, hold under certain conditions. Second, we connect the resulting equivariant capsule networks with work from the field of group convolutional networks. Through this connection, we provide intuitions of how both methods relate and are able to combine the strengths of both approaches in one deep neural network architecture. The resulting framework allows sparse evaluation of the group convolution operator, provides control over specific equivariance and invariance properties, and can use routing by agreement instead of pooling operations. In addition, it is able to provide interpretable and equivariant representation vectors as output capsules, which disentangle evidence of object existence from its pose.

Subfields: Artificial Intelligence

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Citations in the corpus

Stacked Capsule Autoencoders (2019, background): “Lenssen et al., 2018 combined capsule networks with group convolutions to guarantee equiv-ariance and invariance in capsule networks.”
Group Equivariant Stand-Alone Self-Attention For Vision (2020, background): “…Worrall et al., 2017; Weiler et al., 2018b; Li et al., 2018; Cheng et al., 2018; Hoogeboom et al., 2018; Bekkers et al., 2018; Veeling et al., 2018; Lenssen et al., 2018; Graham et al., 2020) to spherical rotations (Cohen et al., 2018; Worrall & Brostow, 2018; Weiler et al., 2018a; Cohen et al.,…”
Attentive Group Equivariant Convolutional Networks (2020, background): “, 1989), planar rotations (Dieleman et al., 2016; Marcos et al., 2017; Worrall et al., 2017; Weiler et al., 2018b; Li et al., 2018; Cheng et al., 2018; Hoogeboom et al., 2018; Bekkers et al., 2018; Veeling et al., 2018; Lenssen et al., 2018; Smets et al., 2020), spherical rotations (Cohen et al.”
Building Deep, Equivariant Capsule Networks (2019, unknown): “Lenssen et al. [2018] proposed a provably-equivariant capsule-network model.”
B-Spline CNNs on Lie Groups (2019, background): “In particular in (Lenssen et al., 2018) the relation to group theory is made explicit with group equivariant capsules that provide a sparse index/value representation of feature maps on groups.”
Wavelet Networks: Scale Equivariant Learning From Raw Waveforms (2020, methodology): “[35] Jan Eric Lenssen, Matthias Fey, and Pascal Libuschewski.”
Probabilistic Symmetries and Invariant Neural Networks (2020, background): “Modifications of the basic convolutional filters have appeared in the form of convolutional capsules (e.g., Sabour et al., 2017; Hinton et al., 2018; Lenssen et al., 2018), and convolution was recently combined with attention mechanisms (Bello et al., 2019; Romero et al., 2020).”
3D Point Capsule Networks (2018, background): “[21] introduced guaranteed equivariance and invariance properties to capsule networks by the use of group convolutions.”
Small Data Challenges in Big Data Era: A Survey of Recent Progress on Unsupervised and Semi-Supervised Methods (2019, methodology): “For example, the group equivariant capsule nets combine the group-equivariant convolutions with the dynamic routing mechanism to train capsule nets [8]; Spherical images are analyzed in the SO(3) group by Spherical CNNs [7], while the equivariance properties of steerable representations have be…”