A Lie Group Approach to Riemannian Batch Normalization

Manifold-valued measurements exist in numerous applications within computer vision and machine learning. Recent studies have extended Deep Neural Networks (DNNs) to manifolds, and concomitantly, normalization techniques have also been adapted to several manifolds, referred to as Riemannian normalization. Nonetheless, most of the existing Riemannian normalization methods have been derived in an ad hoc manner and only apply to specific manifolds. This paper establishes a unified framework for Riemannian Batch Normalization (RBN) techniques on Lie groups. Our framework offers the theoretical guarantee of controlling both the Riemannian mean and variance. Empirically, we focus on Symmetric Positive Definite (SPD) manifolds, which possess three distinct types of Lie group structures. Using the deformation concept, we generalize the existing Lie groups on SPD manifolds into three families of parameterized Lie groups. Specific normalization layers induced by these Lie groups are then proposed for SPD neural networks. We demonstrate the effectiveness of our approach through three sets of experiments: radar recognition, human action recognition, and electroencephalography (EEG) classification. The code is available at https://github.com/GitZH-Chen/LieBN.git.

ASVD: Activation-aware Singular Value Decomposition for Compressing Large Language Models

This paper explores a new post-hoc training-free compression paradigm for compressing Large Language Models (LLMs) to facilitate their wider adoption in various computing environments. We delve into the challenges of LLM compression, notably their dependency on extensive training data and computational resources. We propose a training-free approach dubbed Activation-aware Singular Value Decomposition (ASVD) to address these limitations. ASVD effectively manages activation outliers by adjusting the weight matrix based on the activation distribution, improving decomposition accuracy and efficiency. Our method also addresses the varying sensitivity of different LLM layers to decomposition, with an iterative calibration process for optimal layer-specific decomposition. Experiments demonstrate that ASVD can compress network by 10%-20% without losing reasoning capacities. Additionally, it can be seamlessly integrated with other LLM compression paradigms, showcasing its flexible compatibility. Code and compressed models are available at https://github.com/hahnyuan/ASVD4LLM.

RankFeat&RankWeight: Rank-1 Feature/Weight Removal for Out-of-distribution Detection

The task of out-of-distribution (OOD) detection is crucial for deploying machine learning models in real-world settings. In this paper, we observe that the singular value distributions of the in-distribution (ID) and OOD features are quite different: the OOD feature matrix tends to have a larger dominant singular value than the ID feature, and the class predictions of OOD samples are largely determined by it. This observation motivates us to propose \texttt{RankFeat}, a simple yet effective \emph{post hoc} approach for OOD detection by removing the rank-1 matrix composed of the largest singular value and the associated singular vectors from the high-level feature. \texttt{RankFeat} achieves \emph{state-of-the-art} performance and reduces the average false positive rate (FPR95) by 17.90\% compared with the previous best method. The success of \texttt{RankFeat} motivates us to investigate whether a similar phenomenon would exist in the parameter matrices of neural networks. We thus propose \texttt{RankWeight} which removes the rank-1 weight from the parameter matrices of a single deep layer. Our \texttt{RankWeight}is also \emph{post hoc} and only requires computing the rank-1 matrix once. As a standalone approach, \texttt{RankWeight} has very competitive performance against other methods across various backbones. Moreover, \texttt{RankWeight} enjoys flexible compatibility with a wide range of OOD detection methods. The combination of \texttt{RankWeight} and \texttt{RankFeat} refreshes the new \emph{state-of-the-art} performance, achieving the FPR95 as low as 16.13\% on the ImageNet-1k benchmark. Extensive ablation studies and comprehensive theoretical analyses are presented to support the empirical results.

Flow Factorized Representation Learning

A prominent goal of representation learning research is to achieve representations which are factorized in a useful manner with respect to the ground truth factors of variation. The fields of disentangled and equivariant representation learning have approached this ideal from a range of complimentary perspectives; however, to date, most approaches have proven to either be ill-specified or insufficiently flexible to effectively separate all realistic factors of interest in a learned latent space. In this work, we propose an alternative viewpoint on such structured representation learning which we call Flow Factorized Representation Learning, and demonstrate it to learn both more efficient and more usefully structured representations than existing frameworks. Specifically, we introduce a generative model which specifies a distinct set of latent probability paths that define different input transformations. Each latent flow is generated by the gradient field of a learned potential following dynamic optimal transport. Our novel setup brings new understandings to both \textit{disentanglement} and \textit{equivariance}. We show that our model achieves higher likelihoods on standard representation learning benchmarks while simultaneously being closer to approximately equivariant models. Furthermore, we demonstrate that the transformations learned by our model are flexibly composable and can also extrapolate to new data, implying a degree of robustness and generalizability approaching the ultimate goal of usefully factorized representation learning.

Householder Projector for Unsupervised Latent Semantics Discovery

Generative Adversarial Networks (GANs), especially the recent style-based generators (StyleGANs), have versatile semantics in the structured latent space. Latent semantics discovery methods emerge to move around the latent code such that only one factor varies during the traversal. Recently, an unsupervised method proposed a promising direction to directly use the eigenvectors of the projection matrix that maps latent codes to features as the interpretable directions. However, one overlooked fact is that the projection matrix is non-orthogonal and the number of eigenvectors is too large. The non-orthogonality would entangle semantic attributes in the top few eigenvectors, and the large dimensionality might result in meaningless variations among the directions even if the matrix is orthogonal. To avoid these issues, we propose Householder Projector, a flexible and general low-rank orthogonal matrix representation based on Householder transformations, to parameterize the projection matrix. The orthogonality guarantees that the eigenvectors correspond to disentangled interpretable semantics, while the low-rank property encourages that each identified direction has meaningful variations. We integrate our projector into pre-trained StyleGAN2/StyleGAN3 and evaluate the models on several benchmarks. Within only $1\%$ of the original training steps for fine-tuning, our projector helps StyleGANs to discover more disentangled and precise semantic attributes without sacrificing image fidelity.

Riemannian Multiclass Logistics Regression for SPD Neural Networks

Deep neural networks for learning symmetric positive definite (SPD) matrices are gaining increasing attention in machine learning. Despite the significant progress, most existing SPD networks use traditional Euclidean classifiers on approximated spaces rather than intrinsic classifiers that accurately capture the geometry of SPD manifolds. Inspired by the success of hyperbolic neural networks (HNNs), we propose Riemannian multiclass logistics regression (RMLR) for SPD networks. We introduce a general unified framework for a family of Riemannian metrics on SPD manifolds and showcase the specific $\orth{n}$-invariant Log-Euclidean Metrics for SPD networks. Moreover, we encompass the most popular classifier in existing SPD networks as a special case of our framework. Extensive experiments on popular SPD learning benchmarks demonstrate the superiority of our classifiers.

Latent Traversals in Generative Models as Potential Flows

Despite the significant recent progress in deep generative models, the underlying structure of their latent spaces is still poorly understood, thereby making the task of performing semantically meaningful latent traversals an open research challenge. Most prior work has aimed to solve this challenge by modeling latent structures linearly, and finding corresponding linear directions which result in `disentangled' generations. In this work, we instead propose to model latent structures with a learned dynamic potential landscape, thereby performing latent traversals as the flow of samples down the landscape's gradient. Inspired by physics, optimal transport, and neuroscience, these potential landscapes are learned as physically realistic partial differential equations, thereby allowing them to flexibly vary over both space and time. To achieve disentanglement, multiple potentials are learned simultaneously, and are constrained by a classifier to be distinct and semantically self-consistent. Experimentally, we demonstrate that our method achieves both more qualitatively and quantitatively disentangled trajectories than state-of-the-art baselines. Further, we demonstrate that our method can be integrated as a regularization term during training, thereby acting as an inductive bias towards the learning of structured representations, ultimately improving model likelihood on similarly structured data.

Adaptive Riemannian Metrics on SPD Manifolds

Symmetric Positive Definite (SPD) matrices have received wide attention in machine learning due to their intrinsic capacity of encoding underlying structural correlation in data. To reflect the non-Euclidean geometry of SPD manifolds, many successful Riemannian metrics have been proposed. However, existing fixed metric tensors might lead to sub-optimal performance for SPD matrices learning, especially for SPD neural networks. To remedy this limitation, we leverage the idea of pullback and propose adaptive Riemannian metrics for SPD manifolds. Moreover, we present comprehensive theories for our metrics. Experiments on three datasets demonstrate that equipped with the proposed metrics, SPD networks can exhibit superior performance.

Orthogonal SVD Covariance Conditioning and Latent Disentanglement

Inserting an SVD meta-layer into neural networks is prone to make the covariance ill-conditioned, which could harm the model in the training stability and generalization abilities. In this paper, we systematically study how to improve the covariance conditioning by enforcing orthogonality to the Pre-SVD layer. Existing orthogonal treatments on the weights are first investigated. However, these techniques can improve the conditioning but would hurt the performance. To avoid such a side effect, we propose the Nearest Orthogonal Gradient (NOG) and Optimal Learning Rate (OLR). The effectiveness of our methods is validated in two applications: decorrelated Batch Normalization (BN) and Global Covariance Pooling (GCP). Extensive experiments on visual recognition demonstrate that our methods can simultaneously improve covariance conditioning and generalization. The combinations with orthogonal weight can further boost the performance. Moreover, we show that our orthogonality techniques can benefit generative models for better latent disentanglement through a series of experiments on various benchmarks. Code is available at: \href{https://github.com/KingJamesSong/OrthoImproveCond}{https://github.com/KingJamesSong/OrthoImproveCond}.

RankFeat: Rank-1 Feature Removal for Out-of-distribution Detection

The task of out-of-distribution (OOD) detection is crucial for deploying machine learning models in real-world settings. In this paper, we observe that the singular value distributions of the in-distribution (ID) and OOD features are quite different: the OOD feature matrix tends to have a larger dominant singular value than the ID feature, and the class predictions of OOD samples are largely determined by it. This observation motivates us to propose \texttt{RankFeat}, a simple yet effective \texttt{post hoc} approach for OOD detection by removing the rank-1 matrix composed of the largest singular value and the associated singular vectors from the high-level feature (\emph{i.e.,} $\mathbf{X}{-} \mathbf{s}_{1}\mathbf{u}_{1}\mathbf{v}_{1}^{T}$). \texttt{RankFeat} achieves the \emph{state-of-the-art} performance and reduces the average false positive rate (FPR95) by 17.90\% compared with the previous best method. Extensive ablation studies and comprehensive theoretical analyses are presented to support the empirical results.