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While overparameterization in machine learning models offers great benefits in terms of optimization and generalization, it also leads to increased computational requirements as model sizes grow. In this work, we show that by leveraging the inherent low-dimensional structures of data and compressible dynamics within the model parameters, we can reap the benefits of overparameterization without the computational burdens. In practice, we demonstrate the effectiveness of this approach for deep low-rank matrix completion as well as fine-tuning language models. Our approach is grounded in theoretical findings for deep overparameterized low-rank matrix recovery, where we show that the learning dynamics of each weight matrix are confined to an invariant low-dimensional subspace. Consequently, we can construct and train compact, highly compressed factorizations possessing the same benefits as their overparameterized counterparts. In the context of deep matrix completion, our technique substantially improves training efficiency while retaining the advantages of overparameterization. For language model fine-tuning, we propose a method called "Deep LoRA", which improves the existing low-rank adaptation (LoRA) technique, leading to reduced overfitting and a simplified hyperparameter setup, while maintaining comparable efficiency. We validate the effectiveness of Deep LoRA on natural language tasks, particularly when fine-tuning with limited data. 

We introduce Optimal Eye Surgeon (OES), a framework for pruning and training deep image generator networks. Typically, untrained deep convolutional networks, which include image sampling operations, serve as effective image priors (Ulyanov et al., 2018). However, they tend to overfit to noise in image restoration tasks due to being overparameterized. OES addresses this by adaptively pruning networks at random initialization to a level of underparameterization. This process effectively captures low-frequency image components even without training, by just masking. When trained to fit noisy images, these pruned subnetworks, which we term Sparse-DIP, resist overfitting to noise. This benefit arises from underparameterization and the regularization effect of masking, constraining them in the manifold of image priors. We demonstrate that subnetworks pruned through OES surpass other leading pruning methods, such as the Lottery Ticket Hypothesis, which is known to be suboptimal for image recovery tasks (Wu et al., 2023). Our extensive experiments demonstrate the transferability of OES-masks and the characteristics of sparse-subnetworks for image generation. 

Diffusion models, emerging as powerful deep generative tools, excel in various applications. They operate through a two-steps process: introducing noise into training samples and then employing a model to convert random noise into new samples (e.g., images). However, their remarkable generative performance is hindered by slow training and sampling. This is due to the necessity of tracking extensive forward and reverse diffusion trajectories, and employing a large model with numerous parameters across multiple timesteps (i.e., noise levels). To tackle these challenges, we present a multi-stage framework inspired by our empirical findings. These observations indicate the advantages of employing distinct parameters tailored to each timestep while retaining universal parameters shared across all time steps. Our approach involves segmenting the time interval into multiple stages where we employ custom multi-decoder U-net architecture that blends time-dependent models with a universally shared encoder. Our framework enables the efficient distribution of computational resources and mitigates inter-stage interference, which substantially improves training efficiency. Extensive numerical experiments affirm the effectiveness of our framework, showcasing significant training and sampling efficiency enhancements on three state-of-the-art diffusion models, including large-scale latent diffusion models. Furthermore, our ablation studies illustrate the impact of two important components in our framework: (i) a novel timestep clustering algorithm for stage division, and (ii) an innovative multi-decoder U-net architecture, seamlessly integrating universal and customized hyperparameters. 

The maximal coding rate reduction (MCR2) objective for learning structured and compact deep representations is drawing increasing attention, especially after its recent usage in the derivation of fully explainable and highly effective deep network architectures. However, it lacks a complete theoretical justification: only the properties of its global optima are known, and its global landscape has not been studied. In this work, we give a complete characterization of the properties of all its local and global optima, as well as other types of critical points. Specifically, we show that each (local or global) maximizer of the MCR2 problem corresponds to a low-dimensional, discriminative, and diverse representation, and furthermore, each critical point of the objective is either a local maximizer or a strict saddle point. Such a favorable landscape makes MCR2 a natural choice of objective for learning diverse and discriminative representations via first-order optimization methods. To validate our theoretical findings, we conduct extensive experiments on both synthetic and real data sets.