ReduNet: A White-box Deep Network from the Principle of Maximizing Rate Reduction (2105.10446v3)

Abstract: This work attempts to provide a plausible theoretical framework that aims to interpret modern deep (convolutional) networks from the principles of data compression and discriminative representation. We argue that for high-dimensional multi-class data, the optimal linear discriminative representation maximizes the coding rate difference between the whole dataset and the average of all the subsets. We show that the basic iterative gradient ascent scheme for optimizing the rate reduction objective naturally leads to a multi-layer deep network, named ReduNet, which shares common characteristics of modern deep networks. The deep layered architectures, linear and nonlinear operators, and even parameters of the network are all explicitly constructed layer-by-layer via forward propagation, although they are amenable to fine-tuning via back propagation. All components of so-obtained "white-box" network have precise optimization, statistical, and geometric interpretation. Moreover, all linear operators of the so-derived network naturally become multi-channel convolutions when we enforce classification to be rigorously shift-invariant. The derivation in the invariant setting suggests a trade-off between sparsity and invariance, and also indicates that such a deep convolution network is significantly more efficient to construct and learn in the spectral domain. Our preliminary simulations and experiments clearly verify the effectiveness of both the rate reduction objective and the associated ReduNet. All code and data are available at \url{https://github.com/Ma-Lab-Berkeley}.

• The paper introduces ReduNet, deriving a white-box deep network from maximizing coding rate reduction to achieve sparse, discriminative representations.
• It establishes a forward-designed architecture with explicit, data-driven parameter computation that minimizes reliance on traditional backpropagation.
• Experimental results demonstrate that ReduNet maintains shift invariance and robust performance on benchmarks such as MNIST.

ReduNet: A White-box Deep Network from the Principle of Maximizing Rate Reduction

The paper introduces ReduNet, a novel approach to constructing deep networks from the principle of maximizing coding rate reduction. This approach provides a conceptual framework for interpreting, designing, and optimizing modern deep networks based on principles from information theory, specifically data compression and discriminative representation.

Theoretical Framework

The paper proposes the principle of Maximum Coding Rate Reduction (MCR$^2$) as a means of understanding modern deep neural networks through the lens of data compression. The authors argue that the goal of deep networks, specifically in the context of high-dimensional, multi-class data, should be to learn a linear discriminative representation that maximizes the difference in coding rate between the overall dataset and its constituent classes. This view posits that maximally uncorrelated subspaces should represent different data classes, facilitating efficient compression and robust classification.

The coding rate reduction principle offers a means of balancing intra-class compactness with inter-class diversity in the learned representation. By maximizing the coding rate with respect to this principle, the network can achieve a discriminative representation that is sparse and invariant, aiding classification tasks.

ReduNet Architecture

ReduNet emerges as a realization of the optimization process derived from maximizing the coding rate reduction objective. Constructed entirely in a forward propagation manner, this network architecture encompasses explicit and interpretable layers where network parameters are computed from the training data. Unlike traditional networks that rely predominantly on backpropagation and empirical design, ReduNet provides a systematic, mathematically grounded approach to parameter initialization and network architecture design.

Each layer in ReduNet applies linear and nonlinear transformations derived from the objective's gradient computation, incorporating an explicit mechanism for the manipulation of features through data-driven operators. This approach echoes several key characteristics of widely used architectures like ResNet and CNNs, while differentiating itself by its rigorous foundation in optimization principles.

Shift-Invariance and Multi-Channel Convolutions

In addressing the challenge of shift invariance, ReduNet embeds certain constraints directly into its architecture rather than depending solely on data augmentation techniques. By ensuring that all transformed instances of data are encoded into the same subspace, shift invariance is naturally achieved at a subspace level. This property leads to the emergence of multi-channel convolutional operations, a feature ubiquitous in convolutional neural networks that was traditionally added heuristically.

Furthermore, the authors emphasize the efficiency of constructing and operating these networks in the spectral domain, claiming potential computational benefits and biological plausibility, analogous to processing mechanisms observed in the visual cortex of primates.

Experimental Validation

Experiments conducted demonstrate the effectiveness of ReduNet in achieving stable maximization of the coding rate reduction objective. Notably, simulations indicate that the proposed framework can result in highly discriminative and invariant representations. The experiments on image datasets, such as MNIST, show that ReduNet can feasibly classify data while providing robustness against transformations like translation and rotation.

Implications and Future Directions

ReduNet paves the way for new exploration into forward-designed, interpretable deep networks. Its theoretical underpinning offers significant insights into the design principles of modern networks, suggesting future avenues for designing architectures with efficiency and interpretability.

Practically, ReduNet demonstrates the potential to construct networks with plug-and-play components, minimizing reliance on extensive retroactive optimization via backpropagation. The insights gained from the proposed rate reduction objective could inform the creation of artificial intelligence systems with enhanced robustness and generalizability.

The theoretical contributions of ReduNet hint at broader implications for how deep networks understand and process structured data. Future research might focus on extending these principles to broader classes of transformations, learning efficiency, and scalability to large-scale datasets and network architectures. This framework aligns with the vision of developing a principled, science-driven approach to deep learning that fuses mathematical rigor with empirical success.