[Paper Review] Deep Signature Transforms

Patric Bonnier, Patrick Kidger+3 more|arXiv (Cornell University)|May 21, 2019

Generative Adversarial Networks and Image SynthesisComputer Science48 references53 citations

TL;DR

This paper integrates the signature transform with deep learning by learning data-dependent augmentations and stacking signature layers as neural-network components, enabling stream-preserving, multi-layer architectures for sequential data tasks.

ABSTRACT

The signature is an infinite graded sequence of statistics known to characterise a stream of data up to a negligible equivalence class. It is a transform which has previously been treated as a fixed feature transformation, on top of which a model may be built. We propose a novel approach which combines the advantages of the signature transform with modern deep learning frameworks. By learning an augmentation of the stream prior to the signature transform, the terms of the signature may be selected in a data-dependent way. More generally, we describe how the signature transform may be used as a layer anywhere within a neural network. In this context it may be interpreted as a pooling operation. We present the results of empirical experiments to back up the theoretical justification. Code available at https://github.com/patrick-kidger/Deep-Signature-Transforms.

Motivation & Objective

• Motivate and formalize the signature transform as a learnable layer within neural networks.
• Propose data-dependent augmentation before the signature transform to capture higher-order information.
• Introduce the concept of lifts to stack multiple signature layers while preserving stream structure.
• Show how signature layers can function as pooling operations and be composed in deep architectures.
• Demonstrate practical efficacy across generative, supervised, and reinforcement learning tasks.

Proposed method

• Define Sig^N as the truncated signature of depth N on path-like data.
• Introduce Phi^theta as learnable feature maps that augment inputs prior to signature calculation.
• Describe stream-preserving strategies for Phi^theta to maintain path structure before Sig^N.
• Explain lifts (ell) to create streams of signatures, enabling multiple signature layers in a network.
• Present DeepSig models combining Phi^theta, Sig^N, lifts, and neural networks (Figure 2).
• Show how to train and backpropagate through signature layers with practical implementation notes (Signatory).

Experimental results

Research questions

• RQ1 Can the signature transform be effectively embedded as a differentiable layer in neural networks?
• RQ2 Does learning the augmentation Phi^theta improve information capture beyond fixed augmentations?
• RQ3 How can lifts and multiple signature layers be composed to model non-Markovian sequential data?
• RQ4 Do deep signature models outperform traditional sequential models on generative, supervised, and reinforcement tasks?

Key findings

• The signature transform acts as a universal nonlinearity suitable for neural networks when used as a layer (with differentiable backpropagation).
• Learning data-dependent augmentations before the signature improves capture of higher-order information in the first N signature terms.
• Signature layers can be stacked via lifts to form deep signature models, enabling multi-layer streaming representations with preserved path structure.
• Deep signature models outperform baselines (LSTM, GRU, RNN, etc.) on Hurst parameter estimation tasks by substantial margins (as shown in Table 1).
• A generative model using deep signature architectures can produce paths statistically indistinguishable from Ornstein–Uhlenbeck data; the discriminator uses signature-based kernels to compare distributions.
• The methodology extends to non-Markovian reinforcement learning by using a signature-based memory mechanism that outperforms a comparable RNN in Mountain Car tasks.