Hien Dang

Hien Dang, Tho Tran, Stanley Osher et al.

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified "unconstrained feature model". In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

Hien Dang, Tho Tran, Tan Nguyen et al.

The posterior collapse phenomenon in variational autoencoder (VAE), where the variational posterior distribution closely matches the prior distribution, can hinder the quality of the learned latent variables. As a consequence of posterior collapse, the latent variables extracted by the encoder in VAE preserve less information from the input data and thus fail to produce meaningful representations as input to the reconstruction process in the decoder. While this phenomenon has been an actively addressed topic related to VAE performance, the theory for posterior collapse remains underdeveloped, especially beyond the standard VAE. In this work, we advance the theoretical understanding of posterior collapse to two important and prevalent yet less studied classes of VAE: conditional VAE and hierarchical VAE. Specifically, via a non-trivial theoretical analysis of linear conditional VAE and hierarchical VAE with two levels of latent, we prove that the cause of posterior collapses in these models includes the correlation between the input and output of the conditional VAE and the effect of learnable encoder variance in the hierarchical VAE. We empirically validate our theoretical findings for linear conditional and hierarchical VAE and demonstrate that these results are also predictive for non-linear cases with extensive experiments.

Hien Dang, Pratik Patil, Alessandro Rinaldo

Self-distillation (SD) is the process of retraining a student on a mixture of ground-truth labels and the teacher's own predictions using the same architecture and training data. Although SD has been empirically shown to often improve generalization, its formal guarantees remain limited. We study SD for ridge regression in unconstrained setting in which the mixing weight $ξ$ may be outside the unit interval. Conditioned on the training data and without any distributional assumptions, we prove that for any squared prediction risk (including out-of-distribution), the optimally mixed student strictly improves upon the ridge teacher for every regularization level $λ> 0$ at which the teacher ridge risk $R(λ)$ is nonstationary (i.e., $R'(λ) \neq 0$). We obtain a closed-form expression for the optimal mixing weight $ξ^\star(λ)$ for any value of $λ$ and show that it obeys the sign rule: $\operatorname{sign}(ξ^\star(λ))=-\operatorname{sign}(R'(λ))$. In particular, $ξ^\star(λ)$ can be negative, which is the case in over-regularized regimes. To quantify the risk improvement due to SD, we derive exact deterministic equivalents for the optimal SD risk in the proportional asymptotics regime (where the sample and feature sizes $n$ and $p$ both diverge but their aspect ratio $p/n$ converges) under general anisotropic covariance and deterministic signals. Our asymptotic analysis extends standard second-order ridge deterministic equivalents to their fourth-order analogs using block linearization, which may be of independent interest. From a practical standpoint, we propose a consistent one-shot tuning method to estimate $ξ^\star$ without grid search, sample splitting, or refitting. Experiments on real-world datasets and pretrained neural network features support our theory and the one-shot tuning method.

Tung-Long Vuong, Julien Monteil, Hien Dang et al.

Variational Autoencoders (VAEs) are a powerful alternative to matrix factorization for recommendation. A common technique in VAE-based collaborative filtering (CF) consists in applying binary input masking to user interaction vectors, which improves performance but remains underexplored theoretically. In this work, we analyze how collaboration arises in VAE-based CF and show it is governed by latent proximity: we derive a latent sharing radius that informs when an SGD update on one user strictly reduces the loss on another user, with influence decaying as the latent Wasserstein distance increases. We further study the induced geometry: with clean inputs, VAE-based CF primarily exploits \emph{local} collaboration between input-similar users and under-utilizes global collaboration between far-but-related users. We compare two mechanisms that encourage \emph{global} mixing and characterize their trade-offs: (1) $β$-KL regularization directly tightens the information bottleneck, promoting posterior overlap but risking representational collapse if too large; (2) input masking induces stochastic geometric contractions and expansions, which can bring distant users onto the same latent neighborhood but also introduce neighborhood drift. To preserve user identity while enabling global consistency, we propose an anchor regularizer that aligns user posteriors with item embeddings, stabilizing users under masking and facilitating signal sharing across related items. Our analyses are validated on the Netflix, MovieLens-20M, and Million Song datasets. We also successfully deployed our proposed algorithm on an Amazon streaming platform following a successful online experiment.

Hien Dang, Tho Tran, Tan Nguyen et al.

The current paradigm of training deep neural networks for classification tasks includes minimizing the empirical risk that pushes the training loss value towards zero, even after the training error has been vanished. In this terminal phase of training, it has been observed that the last-layer features collapse to their class-means and these class-means converge to the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is termed as Neural Collapse (NC). To theoretically understand this phenomenon, recent works employ a simplified unconstrained feature model to prove that NC emerges at the global solutions of the training problem. However, when the training dataset is class-imbalanced, some NC properties will no longer be true. For example, the class-means geometry will skew away from the simplex ETF when the loss converges. In this paper, we generalize NC to imbalanced regime for cross-entropy loss under the unconstrained ReLU feature model. We prove that, while the within-class features collapse property still holds in this setting, the class-means will converge to a structure consisting of orthogonal vectors with different lengths. Furthermore, we find that the classifier weights are aligned to the scaled and centered class-means with scaling factors depend on the number of training samples of each class, which generalizes NC in the class-balanced setting. We empirically prove our results through experiments on practical architectures and dataset.

Nghiem T. Diep, Hien Dang, Tuan Truong et al.

Parameter-efficient fine-tuning (PEFT) methods have become the standard paradigm for adapting large-scale models. Among these techniques, Weight-Decomposed Low-Rank Adaptation (DoRA) has been shown to improve both the learning capacity and training stability of the vanilla Low-Rank Adaptation (LoRA) method by explicitly decomposing pre-trained weights into magnitude and directional components. In this work, we propose DoRAN, a new variant of DoRA designed to further stabilize training and boost the sample efficiency of DoRA. Our approach includes two key stages: (i) injecting noise into the denominator of DoRA's weight decomposition, which serves as an adaptive regularizer to mitigate instabilities; and (ii) replacing static low-rank matrices with auxiliary networks that generate them dynamically, enabling parameter coupling across layers and yielding better sample efficiency in both theory and practice. Comprehensive experiments on vision and language benchmarks show that DoRAN consistently outperforms LoRA, DoRA, and other PEFT baselines. These results underscore the effectiveness of combining stabilization through noise-based regularization with network-based parameter generation, offering a promising direction for robust and efficient fine-tuning of foundation models.

Long-Tung Vuong, Vy Vo, Hien Dang et al.

Despite a strong theoretical foundation, empirical experiments reveal that existing domain generalization (DG) algorithms often fail to consistently outperform the ERM baseline. We argue that this issue arises because most DG studies focus on establishing theoretical guarantees for generalization under unrealistic assumptions, such as the availability of sufficient, diverse (or even infinite) domains or access to target domain knowledge. As a result, the extent to which domain generalization is achievable in scenarios with limited domains remains largely unexplored. This paper seeks to address this gap by examining generalization through the lens of the conditions necessary for its existence and learnability. Specifically, we systematically establish a set of necessary and sufficient conditions for generalization. Our analysis highlights that existing DG methods primarily act as regularization mechanisms focused on satisfying sufficient conditions, while often neglecting necessary ones. However, sufficient conditions cannot be verified in settings with limited training domains. In such cases, regularization targeting sufficient conditions aims to maximize the likelihood of generalization, whereas regularization targeting necessary conditions ensures its existence. Using this analysis, we reveal the shortcomings of existing DG algorithms by showing that, while they promote sufficient conditions, they inadvertently violate necessary conditions. To validate our theoretical insights, we propose a practical method that promotes the sufficient condition while maintaining the necessary conditions through a novel subspace representation alignment strategy. This approach highlights the advantages of preserving the necessary conditions on well-established DG benchmarks.