Learning Deep Low-Dimensional Models from High-Dimensional Data: From Theory to Practice

The first part will introduce fundamental properties and results for sensing, processing, analyzing, and learning low-dimensional structures from high-dimensional data. We will first discuss classical low-dimensional models, such as sparse coding and low-rank matrix sensing, and motivate these models by applications in computer vision. Based on convex relaxation, we will characterize the conditions, in terms of sample/data complexity, under which the learning problems of recovering such low-dimensional structures become tractable and can be solved efficiently, with guaranteed correctness or accuracy.

Continuing, we focus on the strong conceptual connections between low-dimensional structures and deep models in terms of learned representation. We start with the introduction of an intriguing Neural Collapse phenomenon in the last-layer representation and its universality in deep network, and lays out the mathematical foundations of understanding its cause by studying its optimization landscapes. We then generalize and explain this phenomenon and its implications under data imbalancedness. Furthermore, we demonstrate the practical algorithmic implications of Neural Collapse on training deep neural networks.

We show that low-dimensional structures also emerge in training dynamics of deep networks. Specifically, we show that the evolution of gradient descent only affects a minimal portion of singular vector spaces across all weight matrices. The analysis enables us to considerably improve training efficiency by taking advantage of the low-dimensional structure in learning dynamics. We can construct smaller, equivalent deep linear networks without sacrificing the benefits associated with the wider counterparts. Moreover, it allows us to better understand deep representation learning by elucidating the progressive feature compression and discrimination from shallow to deep layers.

In this part, we will discuss the overall objective of high-dimensional data analysis, that is, learning and transforming the data distribution towards low-dimensional template distributions for downstream tasks (such as linear discriminative representations, expressive mixtures of semantically-meaningful incoherent subspaces), and how it connects to deep representation learning. In particular, we will introduce the principle of maximal coding rate reduction (MCR^2) for learning compact and structured representations. We will discuss how to use MCR^2 as a simple and principled objective to measure the goodness of learned representations. Furthermore, we will discuss the theoretical analysis of the coding reduction objective, including global optima properties and the optimization landscape.

In this part, we will focus on how to construct white-box deep neural network architectures using unrolled optimization. We will review classical methods such as sparse coding through dictionary learning as particular instantiations of this learning paradigm when the underlying signal model is linear or sparse, along with unrolled optimization as a design principle for white-box deep networks (e.g., LISTA network) that are interpretable ab initio. Next, we will discuss how to construct a white-box deep neural network architecture from the principle of data compression. We will show that the basic iterative gradient ascent scheme for optimizing the rate reduction objective leads to a multi-layer deep network, named ReduNet, which shares common characteristics of modern deep networks such as CNN and ResNet.

We demonstrate how combining sparse coding and rate reduction yields sparse linear discriminative representations using an objective called sparse rate reduction. We develop CRATE, a deep network architecture, by unrolling the optimization of this objective and parameterizing feature distribution in each layer. CRATE's operators are mathematically interpretable, with each layer representing an optimization step, making the network a transparent "white box". Remarkably, CRATE closely resembles the transformer architecture, suggesting that the interpretability gained from such networks might also improve our understanding of current, practical deep architectures. Experiments show that CRATE, despite its simplicity, indeed learns to compress and sparsify representations of large-scale real-world image and text datasets. It achieves performance very close to highly engineered transformer-based models. We will also discuss recent results on scaling up such white-box transformers as well as more efficient white-box architecture designs.