[Paper Review] Learning invariant features through local space contraction

Salah Rifai, Xavier Muller+4 more|arXiv (Cornell University)|Apr 21, 2011

Model Reduction and Neural Networks22 references16 citations

TL;DR

This paper introduces the Contracting Auto-Encoder (CAE), a novel regularized auto-encoder that improves feature learning by minimizing the Frobenius norm of the encoder's Jacobian matrix, inducing local space contraction and enhancing invariance to small input perturbations. The method achieves state-of-the-art classification performance when used to pre-train deep neural networks across multiple benchmark datasets.

ABSTRACT

We present in this paper a novel approach for training deterministic auto-encoders. We show that by adding a well chosen penalty term to the classical reconstruction cost function, we can achieve results that equal or surpass those attained by other regularized auto-encoders as well as denoising auto-encoders on a range of datasets. This penalty term corresponds to the Frobenius norm of the Jacobian matrix of the encoder activations with respect to the input. We show that this penalty term results in a localized space contraction which in turn yields robust features on the activation layer. Furthermore, we show how this penalty term is related to both regularized auto-encoders and denoising encoders and how it can be seen as a link between deterministic and non-deterministic auto-encoders. We find empirically that this penalty helps to carve a representation that better captures the local directions of variation dictated by the data, corresponding to a lower-dimensional non-linear manifold, while being more invariant to the vast majority of directions orthogonal to the manifold. Finally, we show that by using the learned features to initialize a MLP, we achieve state of the art classification error on a range of datasets, surpassing other methods of pre-training.

Motivation & Objective

• To develop a representation learning method that enhances robustness to small input variations while preserving data-specific structure.
• To address the lack of theoretical understanding of why certain auto-encoder variants generalize better than others in deep learning.
• To propose a regularization technique that encourages local invariance in learned features without prior assumptions about data factors of variation.
• To bridge deterministic and non-deterministic auto-encoders by showing the proposed penalty connects to both regularized and denoising auto-encoders.
• To empirically validate that the method improves generalization by pre-training deep networks with superior classification accuracy.

Proposed method

• The method introduces a penalty term based on the Frobenius norm of the Jacobian matrix of the encoder's activations with respect to the input.
• This penalty enforces local space contraction, making the representation robust to small input perturbations around training examples.
• The regularization is combined with a standard reconstruction loss to ensure faithful representation of input data.
• The Jacobian's singular value spectrum is used to analyze the local geometry of the learned representation, revealing manifold structure.
• The approach is applied to train deep auto-encoders by stacking multiple CAE layers, enhancing invariance at higher levels.
• The method is evaluated by initializing multi-layer perceptrons with CAE-learned features and measuring test classification error.

Experimental results

Research questions

• RQ1 How can we improve the robustness of learned features to small input perturbations without relying on data augmentation?
• RQ2 What is the geometric effect of minimizing the Jacobian's Frobenius norm on the representation space?
• RQ3 How does the proposed penalty relate to existing methods like denoising auto-encoders and regularized auto-encoders?
• RQ4 To what extent does the method uncover the underlying low-dimensional manifold structure of the data?
• RQ5 Can the learned features, when used to pre-train deep networks, achieve state-of-the-art classification performance?

Key findings

• The CAE consistently outperforms standard and regularized auto-encoders on multiple benchmark datasets in terms of downstream classification accuracy.
• The method achieves performance comparable or superior to denoising auto-encoders, particularly in learning invariant features.
• The singular value spectrum of the Jacobian reveals that CAEs effectively capture the local dimensionality of the data manifold, with more pronounced low-rank structure than standard auto-encoders.
• Stacking CAEs leads to deeper representations with stronger invariance, as shown by flatter contraction profiles over longer distances from training examples.
• The contraction ratio analysis confirms that directions orthogonal to the data manifold are contracted more strongly, while directions along the manifold are preserved.
• When used to pre-train multi-layer perceptrons, the CAE features yield state-of-the-art classification error across all tested datasets.