Intrinsic dimension of data representations in deep neural networks (1905.12784v2)

Abstract: Deep neural networks progressively transform their inputs across multiple processing layers. What are the geometrical properties of the representations learned by these networks? Here we study the intrinsic dimensionality (ID) of data-representations, i.e. the minimal number of parameters needed to describe a representation. We find that, in a trained network, the ID is orders of magnitude smaller than the number of units in each layer. Across layers, the ID first increases and then progressively decreases in the final layers. Remarkably, the ID of the last hidden layer predicts classification accuracy on the test set. These results can neither be found by linear dimensionality estimates (e.g., with principal component analysis), nor in representations that had been artificially linearized. They are neither found in untrained networks, nor in networks that are trained on randomized labels. This suggests that neural networks that can generalize are those that transform the data into low-dimensional, but not necessarily flat manifolds.

• The paper uses the TwoNN estimator to characterize the intrinsic dimension (ID) across deep neural network layers, revealing it is significantly smaller than layer size and follows a specific profile.
• The intrinsic dimension of the last hidden layer is a strong predictor of test accuracy, showing a negative correlation where lower ID corresponds to better generalization.
• ID analysis suggests data representations form low-dimensional, curved manifolds within DNNs, indicating a fundamental non-linear compression mechanism crucial for generalization.

Intrinsic Dimension of Data Representations in Deep Neural Networks

The paper investigates the intrinsic dimension (ID) of data representations in deep neural networks (DNNs), a crucial geometric property that highlights the minimal number of parameters required to describe these representations without significant information loss. The authors used the TwoNN estimator to characterize the ID across various layers of several state-of-the-art convolutional neural networks (CNNs) for image classification tasks, revealing pivotal insights into the relationship between ID and generalization performance in neural networks.

Key Findings

• Intrinsic Dimension Across Layers: It was observed that the ID in a trained network is significantly smaller than the number of units in each layer. Notably, the ID follows a characteristic shape across different architectures: it increases in the initial layers and then decreases monotonically in the final layers.
• Predictive Power of ID: The ID of the last hidden layer is a strong predictor of classification accuracy on the test set. Results indicate a negative correlation between the ID of the last hidden layer and the network's ability to generalize, emphasizing the lower the ID, the better the generalization.
• Characterization of Manifolds: Unlike linear dimensionality estimates, the ID estimated by TwoNN suggests that data representations are embedded in low-dimensional, curved manifolds. This curvature and low dimensionality remain consistent across various architectures and datasets, underscoring a fundamental operational mechanism of deep networks.
• Effects of Training and Noise: Networks trained on data with random labels or networks with random initializations showed significantly different ID profiles. This suggests that the manifold's dimensionality and its variations are results of effective learning and not an artifact of the network's architecture or initial settings.

Implications and Speculations

These findings illuminate the capacity of DNNs to transform input data into low-dimensional manifolds that are conducive to generalization. This compression is not merely a linear reduction but involves a more complex, non-linear process that preserves crucial features needed for classification. Such insights provide an indirect metric for assessing and optimizing a network's generalization performance and robustness, particularly useful in determining the onset of overfitting.

Moreover, the paper challenges the notion that flattening data manifolds is essential for effective learning. Instead, the reduction in intrinsic dimensionality itself is highlighted as a more critical factor in achieving separability and simplicity in data representations.

Future Directions

The relationship between intrinsic dimension and network capability unveils potential pathways for future research, particularly in developing methods to optimize network structures and training regimes based on ID metrics. An exploration into how intrinsic dimension varies with different types of neural architectures, datasets, and even types of tasks may further elucidate this complex relationship.

Towards practical applications, leveraging the ID as a diagnostic tool for early detection of learning anomalies or inefficiencies in DNNs could enhance the effectiveness and reliability of neural network deployment in real-world scenarios.

In sum, the paper advances our understanding of deep neural networks by offering a robust framework to analyze and interpret the geometric structures of data representations, laying the groundwork for optimized learning paradigms in artificial intelligence.