Powerset Convolutional Neural Networks

TL;DR

The paper extends convolutional neural networks to the domain of set functions indexed by the powerset $2^N$, introducing powerset convolutions that are linear and shift-equivariant with respect to multiple set-based shifts. It defines the mathematical foundation (set functions, powerset convolution, and powerset Fourier transform) and develops Powerset Convolutional Neural Networks (PCNs) with convolutional and pooling layers, implemented in TensorFlow. Empirical evaluation on synthetic tasks (spectral patterns, $k$-juntas, submodularity) and real hypergraph-based tasks (domain classification, simplicial closure) shows PCNs often outperform baselines like MLPs and graph-CNNs, while highlighting the impact of pooling, equivariance, and local vs global shifts. The authors discuss locality, scalability, and complexity, noting feasibility up to $n\approx 30$ ground elements and pointing to future directions for scalable set-function learning and hypergraph applications.

Abstract

We present a novel class of convolutional neural networks (CNNs) for set functions, i.e., data indexed with the powerset of a finite set. The convolutions are derived as linear, shift-equivariant functions for various notions of shifts on set functions. The framework is fundamentally different from graph convolutions based on the Laplacian, as it provides not one but several basic shifts, one for each element in the ground set. Prototypical experiments with several set function classification tasks on synthetic datasets and on datasets derived from real-world hypergraphs demonstrate the potential of our new powerset CNNs.

Figures (1)

• Figure 1: Forward pass of a simple powerset CNN with two convolutional and two pooling layers. Set functions are depicted as signals on the powerset lattice.

Theorems & Definitions (7)

• Definition 1
• Example 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6