CKConv: Continuous Kernel Convolution For Sequential Data (2102.02611v3)

Abstract: Conventional neural architectures for sequential data present important limitations. Recurrent networks suffer from exploding and vanishing gradients, small effective memory horizons, and must be trained sequentially. Convolutional networks are unable to handle sequences of unknown size and their memory horizon must be defined a priori. In this work, we show that all these problems can be solved by formulating convolutional kernels in CNNs as continuous functions. The resulting Continuous Kernel Convolution (CKConv) allows us to model arbitrarily long sequences in a parallel manner, within a single operation, and without relying on any form of recurrence. We show that Continuous Kernel Convolutional Networks (CKCNNs) obtain state-of-the-art results in multiple datasets, e.g., permuted MNIST, and, thanks to their continuous nature, are able to handle non-uniformly sampled datasets and irregularly-sampled data natively. CKCNNs match or perform better than neural ODEs designed for these purposes in a faster and simpler manner.

• The paper introduces CKConv, which reformulates CNN kernels as continuous functions to overcome RNN limitations and preset memory horizons.
• It demonstrates that CKCNNs enable parallel training and deliver state-of-the-art performance on datasets like permuted MNIST and various time-series benchmarks.
• The approach efficiently handles irregular and non-uniform data, opening new avenues in fields such as healthcare monitoring and real-time analysis.

Continuous Kernel Convolution for Sequential Data

The paper explores the limitations of traditional neural network architectures like RNNs and CNNs in handling sequential data. Recurrent neural networks face challenges such as exploding and vanishing gradients, small effective memory horizons, and sequential training constraints. Conversely, convolutional neural networks have fixed memory horizons defined a priori and struggle with sequences of variable sizes.

The authors introduce a novel approach called Continuous Kernel Convolution (CKConv), which formulates CNN kernels as continuous functions. This formulation allows for processing arbitrarily long sequences in parallel through a single operation, without relying on recurrent mechanisms. CKConvs can define memory horizons independently of network depth or size, enabling efficient handling of non-uniformly and irregularly sampled datasets.

The paper demonstrates that Continuous Kernel Convolutional Networks (CKCNNs) achieve state-of-the-art results across multiple datasets, including permuted MNIST and others, comparing favorably against neural ODEs designed to tackle similar tasks.

Key Numerical Results and Claims

1. Memory Horizons: CKConvs can define arbitrarily large memory horizons, providing unmatched flexibility compared to both RNNs and traditional CNNs.
2. Parallel Training: CKCNNs eliminate the vanishing/exploding gradient problem associated with RNNs and can be trained much faster due to parallelization.
3. Irregular and Non-uniform Data: CKCNNs handle irregular and varying resolution datasets more effectively than models with discrete convolutional kernels.
4. Performance:
   • On sMNIST, a small CKCNN model (100k parameters) achieved superior accuracy compared to significantly larger models.
   • Similarly, in pMNIST, CKCNNs bettered the performance of the largest competing models by 0.8%.
   • On time-series datasets like CharacterTrajectories and SpeechCommands, CKCNNs outperformed continuous-time sequential models such as NCDE and GRU-ODE, highlighting their versatility.

Implications and Future Directions

The CKConv approach offers considerable enhancements in deep learning for sequential data. Practically, this opens new avenues in fields requiring handling of large sequences or irregular data sampling, such as healthcare data analyses and real-time monitoring systems. The flexibility of CKConvs could considerably reduce architecture sensitivity, allowing more robust modeling across diverse tasks.

This research hints at a shift towards models that leverage continuous kernel functions for enhanced memory capability and data handling robustness. When combined with further attention-like mechanisms, CKConv could pave the way for better sequence prediction and autoregressive modeling, potentially disrupting current paradigms in audio, video, and reinforcement learning applications.

While CKConvs demonstrate promising results, exploring other potential kernel parameterizations, improving memory efficiency, and understanding their interaction with conventional deep learning architectures remain crucial areas of interest for future work. These innovations could yield even more powerful and efficient models, driving forward the capabilities of AI in sequential data handling.