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In this paper, we contend that the objective of representation learning is to compress and transform the distribution of the data, say sets of tokens, towards a mixture of low-dimensional Gaussian distributions supported on incoherent subspaces. The quality of the final representation can be measured by a unified objective function called sparse rate reduction. From this perspective, popular deep networks such as transformers can be naturally viewed as realizing iterative schemes to optimize this objective incrementally. Particularly, we show that the standard transformer block can be derived from alternating optimization on complementary parts of this objective: the multi-head self-attention operator can be viewed as a gradient descent step to compress the token sets by minimizing their lossy coding rate, and the subsequent multi-layer perceptron can be viewed as attempting to sparsify the representation of the tokens. This leads to a family of white-box transformer-like deep network architectures which are mathematically fully interpretable. Despite their simplicity, experiments show that these networks indeed learn to optimize the designed objective: they compress and sparsify representations of large-scale real-world vision datasets such as ImageNet, and achieve performance very close to thoroughly engineered transformers such as ViT. Code is at https://github. com/Ma-Lab-Berkeley/CRATE. 

This work proposes a minimal computational model for learning structured memories of multiple object classes in an incremental setting. Our approach is based on establishing a closed-loop transcription between the classes and a corresponding set of subspaces, known as a linear discriminative representation, in a lowdimensional feature space. Our method is simpler than existing approaches for incremental learning, and more efficient in terms of model size, storage, and computation: it requires only a single, fixed-capacity autoencoding network with a feature space that is used for both discriminative and generative purposes. Network parameters are optimized simultaneously without architectural manipulations, by solving a constrained minimax game between the encoding and decoding maps over a single rate reduction-based objective. Experimental results show that our method can effectively alleviate catastrophic forgetting, achieving significantly better performance than prior work of generative replay on MNIST, CIFAR-10, and ImageNet-50, despite requiring fewer resources. Source code can be found at https://github.com/tsb0601/i-CTRL 

Modern learning frameworks often train deep neural networks with massive amounts of unlabeled data to learn representations by solving simple pretext tasks, then use the representations as foundations for downstream tasks. These networks are empirically designed; as such, they are usually not interpretable, their representations are not structured, and their designs are potentially redundant. White-box deep networks, in which each layer explicitly identifies and transforms structures in the data, present a promising alternative. However, existing white-box architectures have only been shown to work at scale in supervised settings with labeled data, such as classification. In this work, we provide the first instantiation of the white-box design paradigm that can be applied to large-scale unsupervised representation learning. We do this by exploiting a fundamental connection between diffusion, compression, and (masked) completion, deriving a deep transformer-like masked autoencoder architecture, called CRATE-MAE, in which the role of each layer is mathematically fully interpretable: they transform the data distribution to and from a structured representation. Extensive empirical evaluations confirm our analytical insights. CRATE-MAE demonstrates highly promising performance on large-scale imagery datasets while using only ~30% of the parameters compared to the standard masked autoencoder with the same model configuration. The representations learned by CRATE-MAE have explicit structure and also contain semantic meaning. Code is available at https://github.com/Ma-Lab-Berkeley/CRATE 

Current deep learning architectures suffer from catastrophic forgetting, a failure to retain knowledge of previously learned classes when incrementally trained on new classes. The fundamental roadblock faced by deep learning methods is that the models are optimized as “black boxes,” making it difficult to properly adjust the model parameters to preserve knowledge about previously seen data. To overcome the problem of catastrophic forgetting, we propose utilizing an alternative “white box” architecture derived from the principle of rate reduction, where each layer of the network is explicitly computed without back propagation. Under this paradigm, we demonstrate that, given a pretrained network and new data classes, our approach can provably construct a new network that emulates joint training with all past and new classes. Finally, our experiments show that our proposed learning algorithm observes significantly less decay in classification performance, outperforming state of the art methods on MNIST and CIFAR-10 by a large margin and justifying the use of “white box” algorithms for incremental learning even for sufficiently complex image data. 

This work proposes a new computational framework for learning a structured generative model for real-world datasets. In particular, we propose to learn a Closed-loop Transcriptionbetween a multi-class, multi-dimensional data distribution and a Linear discriminative representation (CTRL) in the feature space that consists of multiple independent multi-dimensional linear subspaces. In particular, we argue that the optimal encoding and decoding mappings sought can be formulated as a two-player minimax game between the encoder and decoderfor the learned representation. A natural utility function for this game is the so-called rate reduction, a simple information-theoretic measure for distances between mixtures of subspace-like Gaussians in the feature space. Our formulation draws inspiration from closed-loop error feedback from control systems and avoids expensive evaluating and minimizing of approximated distances between arbitrary distributions in either the data space or the feature space. To a large extent, this new formulation unifies the concepts and benefits of Auto-Encoding and GAN and naturally extends them to the settings of learning a both discriminative and generative representation for multi-class and multi-dimensional real-world data. Our extensive experiments on many benchmark imagery datasets demonstrate tremendous potential of this new closed-loop formulation: under fair comparison, visual quality of the learned decoder and classification performance of the encoder is competitive and arguably better than existing methods based on GAN, VAE, or a combination of both. Unlike existing generative models, the so-learned features of the multiple classes are structured instead of hidden: different classes are explicitly mapped onto corresponding independent principal subspaces in the feature space, and diverse visual attributes within each class are modeled by the independent principal components within each subspace.