Data Manifold Research Articles

Cartan moving frames and the data manifolds

Cartan moving frames and the data manifolds

Spontaneous symmetry breaking in generative diffusion models*

Abstract Generative diffusion models have recently emerged as a leading approach for generating high-dimensional data. In this paper, we show that the dynamics of these models exhibit a spontaneous symmetry breaking phenomenon that divides the generative dynamics into two distinct phases: (1) a linear steady-state dynamics around a central fixed-point and, (2) an attractor dynamics directed towards the data manifold. These two ‘phases’ are separated by the change in stability of the central fixed-point, with the resulting window of instability being responsible for the diversity of the generated samples. Using both theoretical and empirical evidence, we show that an accurate simulation of the early dynamics does not significantly contribute to the final generation, since early fluctuations are reverted to the central fixed point. To leverage this insight, we propose a Gaussian late initialization scheme, which significantly improves model performance, achieving up to 3× Fréchet inception distance improvements on fast samplers, while also increasing sample diversity (e.g. racial composition of generated CelebA images). Our work offers a new way to understand the generative dynamics of diffusion models that has the potential to bring about higher performance and less biased fast-samplers.

Understanding adversarial robustness against on-manifold adversarial examples

Deep neural networks (DNNs) are shown to be vulnerable to adversarial examples. A well-trained model can be easily attacked by adding small perturbations to the original data. One of the hypotheses of the existence of the adversarial examples is the off-manifold assumption: adversarial examples lie off the data manifold. However, recent researches showed that on-manifold adversarial examples also exist. In this paper, we revisit the off-manifold assumption and study a question: at what level is the poor adversarial robustness of neural networks due to on-manifold adversarial examples? Since the true data manifold is unknown in practice, we consider two approximated on-manifold adversarial examples on both real and synthesis datasets. On real datasets, we show that on-manifold adversarial examples have greater attack rates than off-manifold adversarial examples on both standard-trained and adversarially-trained models. On synthetic datasets, theoretically, we prove that on-manifold adversarial examples are powerful, yet adversarial training focuses on off-manifold directions and ignores the on-manifold adversarial examples. Furthermore, we provide analysis to show that the properties derived theoretically can also be observed in practice. Our analysis suggests that on-manifold adversarial examples are important. We should pay more attention to on-manifold adversarial examples to train robust models.

Multi-kernel clustering with tensor fusion on Grassmann manifold for high-dimensional genomic data

The high dimensionality and noise challenges in genomic data make it difficult for traditional clustering methods. Existing multi-kernel clustering methods aim to improve the quality of the affinity matrix by learning a set of base kernels, thereby enhancing clustering performance. However, directly learning from the original base kernels presents challenges in handling errors and redundancies when dealing with high-dimensional data, and there is still a lack of feasible multi-kernel fusion strategies. To address these issues, we propose a Multi-Kernel Clustering method with Tensor fusion on Grassmann manifolds, called MKCTM. Specifically, we maximize the clustering consensus among base kernels by imposing tensor low-rank constraints to eliminate noise and redundancy. Unlike traditional kernel fusion approaches, our method fuses learned base kernels on the Grassmann manifold, resulting in a final consensus matrix for clustering. We integrate tensor learning and fusion processes into a unified optimization model and propose an effective iterative optimization algorithm for solving it. Experimental results on ten datasets, comparing against 12 popular baseline clustering methods, confirm the superiority of our approach. Our code is available at https://github.com/foureverfei/MKCTM.git.

Learning on manifolds without manifold learning

Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional Euclidean space. A great deal of research deals with obtaining information about this manifold, such as the eigendecomposition of the Laplace–Beltrami operator or coordinate charts, and using this information for function approximation. This two-step approach implies some extra errors in the approximation stemming from estimating the basic quantities of the data manifold in addition to the errors inherent in function approximation. In this paper, we project the unknown manifold as a submanifold of an ambient hypersphere and study the question of constructing a one-shot approximation using a specially designed sequence of localized spherical polynomial kernels on the hypersphere. Our approach does not require preprocessing of the data to obtain information about the manifold other than its dimension. We give optimal rates of approximation for relatively “rough” functions.

Adaptive Learning of the Latent Space of Wasserstein Generative Adversarial Networks

Generative models based on latent variables, such as generative adversarial networks (GANs) and variational auto-encoders (VAEs), have gained lots of interests due to their impressive performance in many fields. However, many data such as natural images usually do not populate the ambient Euclidean space but instead reside in a lower-dimensional manifold. Thus an inappropriate choice of the latent dimension fails to uncover the structure of the data, possibly resulting in mismatch of latent representations and poor generative qualities. Towards addressing these problems, we propose a novel framework called the latent Wasserstein GAN (LWGAN) that fuses the Wasserstein auto-encoder and the Wasserstein GAN so that the intrinsic dimension of the data manifold can be adaptively learned by a modified informative latent distribution. We prove that there exist an encoder network and a generator network in such a way that the intrinsic dimension of the learned encoding distribution is equal to the dimension of the data manifold. We theoretically establish that our estimated intrinsic dimension is a consistent estimate of the true dimension of the data manifold. Meanwhile, we provide an upper bound on the generalization error of LWGAN, implying that we force the synthetic data distribution to be similar to the real data distribution from a population perspective. Comprehensive empirical experiments verify our framework and show that LWGAN is able to identify the correct intrinsic dimension under several scenarios, and simultaneously generate high-quality synthetic data by sampling from the learned latent distribution.

Implications of data topology for deep generative models

Many deep generative models, such as variational autoencoders (VAEs) and generative adversarial networks (GANs), learn an immersion mapping from a standard normal distribution in a low-dimensional latent space into a higher-dimensional data space. As such, these mappings are only capable of producing simple data topologies, i.e., those equivalent to an immersion of Euclidean space. In this work, we demonstrate the limitations of such latent space generative models when trained on data distributions with non-trivial topologies. We do this by training these models on synthetic image datasets with known topologies (spheres, torii, etc.). We then show how this results in failures of both data generation as well as data interpolation. Next, we compare this behavior to two classes of deep generative models that in principle allow for more complex data topologies. First, we look at chart autoencoders (CAEs), which construct a smooth data manifold from multiple latent space chart mappings. Second, we explore score-based models, e.g., denoising diffusion probabilistic models, which estimate gradients of the data distribution without resorting to an explicit mapping to a latent space. Our results show that these models do demonstrate improved ability over latent space models in modeling data distributions with complex topologies, however, challenges still remain.

Manifold-based approach for neural network robustness analysis

It is important to understand the mathematical foundations of neural networks and to include robustness in model evaluation. Here, we introduce algorithms based on manifold curvature estimation to assess neural network robustness. These algorithms rely solely on training data and do not require regular or adversarial test data. Initially, a metric is proposed to measure the curvature of discrete data manifolds by introducing weighted angles concept between subspaces. Following this, a robustness measure is introduced that is independent of network architecture or model parameters. Lastly, two additional methods are introduced, utilizing curvature estimation of special manifolds formed by using gradient vectors between output and input network layers, alongside manifold curvature estimation. A comprehensive evaluation is provided on multiple network models using the CIFAR-10 dataset. Manifold geometry-based robustness analysis may lead to the development of not only accurate but also robust neural network models.

Manifold-based Shapley explanations for high dimensional correlated features

Explainable artificial intelligence (XAI) holds significant importance in enhancing the reliability and transparency of network decision-making. SHapley Additive exPlanations (SHAP) is a game-theoretic approach for network interpretation, attributing confidence to inputs features to measure their importance. However, SHAP often relies on a flawed assumption that the model’s features are independent, leading to incorrect results when dealing with correlated features. In this paper, we introduce a novel manifold-based Shapley explanation method, termed Latent SHAP. Latent SHAP transforms high-dimensional data into low-dimensional manifolds to capture correlations among features. We compute Shapley values on the data manifold and devise three distinct gradient-based mapping methods to transfer them back to the high-dimensional space. Our primary objectives include: (1) correcting misinterpretations by SHAP in certain samples; (2) addressing the challenge of feature correlations in high-dimensional data interpretation; and (3) reducing algorithmic complexity through Manifold SHAP for application in complex network interpretations. Code is available at https://github.com/Teriri1999/Latent-SHAP.

Ensuring Toplogical Data-Structure Preservation under Autoencoder Compression Due to Latent Space Regularization in Gauss–Legendre Nodes

We formulate a data-independent latent space regularization constraint for general unsupervised autoencoders. The regularization relies on sampling the autoencoder Jacobian at Legendre nodes, which are the centers of the Gauss–Legendre quadrature. Revisiting this classic allows us to prove that regularized autoencoders ensure a one-to-one re-embedding of the initial data manifold into its latent representation. Demonstrations show that previously proposed regularization strategies, such as contractive autoencoding, cause topological defects even in simple examples, as do convolutional-based (variational) autoencoders. In contrast, topological preservation is ensured by standard multilayer perceptron neural networks when regularized using our approach. This observation extends from the classic FashionMNIST dataset to (low-resolution) MRI brain scans, suggesting that reliable low-dimensional representations of complex high-dimensional datasets can be achieved using this regularization technique.

Heterogeneous domain adaptation via incremental discriminative knowledge consistency

Heterogeneous domain adaptation is a challenging problem in transfer learning since samples from the source and target domains reside in different feature spaces with different feature dimensions. The key problem is how to minimize some gaps (e.g., data distribution mismatch) presented in two heterogeneous domains and produce highly discriminative representations for the target domain. In this paper, we attempt to address these challenges with the proposed incremental discriminative knowledge consistency (IDKC) method, which integrates cross-domain mapping, distribution matching, discriminative knowledge preservation, and domain-specific geometry structure consistency into a unified learning model. Specifically, we attempt to learn a domain-specific projection to project original samples into a common subspace in which the marginal distribution is well aligned and the discriminative knowledge consistency is preserved by leveraging the labeled samples from both domains. Moreover, domain-specific structure consistency is enforced to preserve the data manifold from the original space to the common feature space in each domain. Meanwhile, we further apply pseudo labeling to unlabeled target samples based on the feature correlation and retain pseudo labels with high feature correlation coefficients for the next iterative learning. Our pseudo-labeling strategy expands the number of labeled target samples in each category and thus enforces class-discriminative knowledge consistency to produce more discriminative feature representations for the target domain. Extensive experiments on several standard benchmarks for object recognition, cross-language text classification, and digit classification tasks verify the effectiveness of our method.

Extended Poisson Gaussian-Process Latent Variable Model for Unsupervised Neural Decoding.

Dimension reduction on neural activity paves a way for unsupervised neural decoding by dissociating the measurement of internal neural pattern reactivation from the measurement of external variable tuning. With assumptions only on the smoothness of latent dynamics and of internal tuning curves, the Poisson gaussian-process latent variable model (P-GPLVM; Wu etal., 2017) is a powerful tool to discover the low-dimensional latent structure for high-dimensional spike trains. However, when given novel neural data, the original model lacks a method to infer their latent trajectories in the learned latent space, limiting its ability for estimating the neural reactivation. Here, we extend the P-GPLVM to enable the latent variable inference of new data constrained by previously learned smoothness and mapping information. We also describe a principled approach for the constrained latent variable inference for temporally compressed patterns of activity, such as those found in population burst events during hippocampal sharp-wave ripples, as well as metrics for assessing the validity of neural pattern reactivation and inferring the encoded experience. Applying these approaches to hippocampal ensemble recordings during active maze exploration, we replicate the result that P-GPLVM learns a latent space encoding the animal's position. We further demonstrate that this latent space can differentiate one maze context from another. By inferring the latent variables of new neural data during running, certain neural patterns are observed to reactivate, in accordance with the similarity of experiences encoded by its nearby neural trajectories in the training data manifold. Finally, reactivation of neural patterns can be estimated for neural activity during population burst events as well, allowing the identification for replay events of versatile behaviors and more general experiences. Thus, our extension of the P-GPLVM framework for unsupervised analysis of neural activity can be used to answer critical questions related to scientific discovery.

Robust anomaly detection via adversarial counterfactual generation

The capability to devise robust outlier and anomaly detection tools is an important research topic in machine learning and data mining. Recent techniques have been focusing on reinforcing detection with sophisticated data generation tools that successfully refine the learning process by generating variants of the data that expand the recognition capabilities of the outlier detector. In this paper, we propose ARN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ARN}$$\\end{document}, a semi-supervised anomaly detection and generation method based on adversarial counterfactual reconstruction. ARN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ARN}$$\\end{document} exploits a regularized autoencoder to optimize the reconstruction of variants of normal examples with minimal differences that are recognized as outliers. The combination of regularization and counterfactual reconstruction helps to stabilize the learning process, which results in both realistic outlier generation and substantially extended detection capability. In fact, the counterfactual generation enables a smart exploration of the search space by successfully relating small changes in all the actual samples from the true distribution to high anomaly scores. Experiments on several benchmark datasets show that our model improves the current state of the art by valuable margins because of its ability to model the true boundaries of the data manifold.

Explaining neural scaling laws

The population loss of trained deep neural networks often follows precise power-law scaling relations with either the size of the training dataset or the number of parameters in the network. We propose a theory that explains the origins of and connects these scaling laws. We identify variance-limited and resolution-limited scaling behavior for both dataset and model size, for a total of four scaling regimes. The variance-limited scaling follows simply from the existence of a well-behaved infinite data or infinite width limit, while the resolution-limited regime can be explained by positing that models are effectively resolving a smooth data manifold. In the large width limit, this can be equivalently obtained from the spectrum of certain kernels, and we present evidence that large width and large dataset resolution-limited scaling exponents are related by a duality. We exhibit all four scaling regimes in the controlled setting of large random feature and pretrained models and test the predictions empirically on a range of standard architectures and datasets. We also observe several empirical relationships between datasets and scaling exponents under modifications of task and architecture aspect ratio. Our work provides a taxonomy for classifying different scaling regimes, underscores that there can be different mechanisms driving improvements in loss, and lends insight into the microscopic origin and relationships between scaling exponents.

Evaluating Generative Models in Medical Imaging.

Data synthesis can address important data availability challenges in biomedical informatics. Quantitative evaluation of generative models may help understand their applications to synthesizing biomedical data. This poster paper examines state-of-the-art generative models used in medical imaging, such as StyleGAN and DDPM models, and evaluates their performance in learning data manifolds and in the visible features of generated samples. Results show that existing generative models have much to improve based on the studied measures.

Understanding the vulnerability of skeleton-based Human Activity Recognition via black-box attack

Human Activity Recognition (HAR) has been employed in a wide range of applications, e.g. self-driving cars, where safety and lives are at stake. Recently, the robustness of skeleton-based HAR methods have been questioned due to their vulnerability to adversarial attacks. However, the proposed attacks require the full-knowledge of the attacked classifier, which is overly restrictive. In this paper, we show such threats indeed exist, even when the attacker only has access to the input/output of the model. To this end, we propose the very first black-box adversarial attack approach in skeleton-based HAR called BASAR. BASAR explores the interplay between the classification boundary and the natural motion manifold. To our best knowledge, this is the first time data manifold is introduced in adversarial attacks on time series. Via BASAR, we find on-manifold adversarial samples are extremely deceitful and rather common in skeletal motions, in contrast to the common belief that adversarial samples only exist off-manifold. Through exhaustive evaluation, we show that BASAR can deliver successful attacks across classifiers, datasets, and attack modes. By attack, BASAR helps identify the potential causes of the model vulnerability and provides insights on possible improvements. Finally, to mitigate the newly identified threat, we propose a new adversarial training approach by leveraging the sophisticated distributions of on/off-manifold adversarial samples, called mixed manifold-based adversarial training (MMAT). MMAT can successfully help defend against adversarial attacks without compromising classification accuracy.

Manifold Peaks Nonnegative Matrix Factorization.

Nonnegative matrix factorization (NMF) has attracted increasing interest for its high interpretability in recent years. It is shown that the NMF is closely related to fuzzy k -means clustering, where the basis matrix represents the cluster centroids. However, most of the existing NMF-based clustering algorithms often have their decomposed centroids deviate away from the data manifold, which potentially undermines the clustering results, especially when the datasets lie on complicated geometric structures. In this article, we present a manifold peaks NMF (MPNMF) for data clustering. The proposed approach has the following advantages: 1) it selects a number of MPs to characterize the backbone of the data manifold; 2) it enforces the centroids to lie on the original data manifold, by restricting each centroid to be a conic combination of a small number of nearby MPs; 3) it generalizes the graph smoothness regularization to guide the local graph construction; and 4) it solves a general problem of quadratic regularized nonnegative least squares (NNLSs) with group l0 -norm constraint and further develops an efficient optimization algorithm to solve the objective function of the MPNMF. Extensive experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed approach.

Deep Spectral Clustering With Constrained Laplacian Rank.

Spectral clustering (SC) is a well-performed and prevalent technique for data processing and analysis, which has attracted significant attention in the field of clustering. While the scalability and generalization ability of this method make it prohibitive for the large-scale dataset and the out-of-sample-extension problem. In this work, we propose a new efficient deep clustering architecture based on SC, named deep SC (DSC) with constrained Laplacian rank (DSCCLR). DSCCLR develops a self-adaptive affinity matrix with a clustering-friendly structure by constraining the Laplacian rank, which greatly mines the intrinsic relationships. Meanwhile, by introducing a simple fully connected network with an orthogonality constraint on the last layer, DSCCLR learns discriminative representations in a short training time. The proposed method has the following salient properties: 1) it overcomes limited generalization ability and scalability of the existing DSC methods; 2) it explores the intrinsic relationship between samples in the affinity matrix, which maintains the latent manifold of data as much as possible; and 3) it alleviates the complexity of eigendecomposition via a simple but effective fully connected network. The extensive empirical results demonstrate the superiorities of DSCCLR over other 17 clustering methods.

Bt-GAN: Generating Fair Synthetic Healthdata via Bias-transforming Generative Adversarial Networks

Synthetic data generation offers a promising solution to enhance the usefulness of Electronic Healthcare Records (EHR) by generating realistic de-identified data. However, the existing literature primarily focuses on the quality of synthetic health data, neglecting the crucial aspect of fairness in downstream predictions. Consequently, models trained on synthetic EHR have faced criticism for producing biased outcomes in target tasks. These biases can arise from either spurious correlations between features or the failure of models to accurately represent sub-groups. To address these concerns, we present Bias-transforming Generative Adversarial Networks (Bt-GAN), a GAN-based synthetic data generator specifically designed for the healthcare domain. In order to tackle spurious correlations (i), we propose an information-constrained Data Generation Process (DGP) that enables the generator to learn a fair deterministic transformation based on a well-defined notion of algorithmic fairness. To overcome the challenge of capturing exact sub-group representations (ii), we incentivize the generator to preserve sub-group densities through score-based weighted sampling. This approach compels the generator to learn from underrepresented regions of the data manifold. To evaluate the effectiveness of our proposed method, we conduct extensive experiments using the Medical Information Mart for Intensive Care (MIMIC-III) database. Our results demonstrate that Bt-GAN achieves state-of-the-art accuracy while significantly improving fairness and minimizing bias amplification. Furthermore, we perform an in-depth explainability analysis to provide additional evidence supporting the validity of our study. In conclusion, our research introduces a novel and professional approach to addressing the limitations of synthetic data generation in the healthcare domain. By incorporating fairness considerations and leveraging advanced techniques such as GANs, we pave the way for more reliable and unbiased predictions in healthcare applications.

Generative Model Perception Rectification Algorithm for Trade-Off between Diversity and Quality

How to balance the diversity and quality of results from generative models through perception rectification poses a significant challenge. Abnormal perception in generative models is typically caused by two factors: inadequate model structure and imbalanced data distribution. In response to this issue, we propose the dynamic model perception rectification algorithm (DMPRA) for generalized generative models. The core idea is to gain a comprehensive perception of the data in the generative model by appropriately highlighting the low-density samples in the perception space, also known as the minor group samples. The entire process can be summarized as "search-evaluation-adjustment". To identify low-density regions in the data manifold within the perception space of generative models, we introduce a filtering method based on extended neighborhood sampling. Based on the informational value of samples from low-density regions, our proposed mechanism generates informative weights to assess the significance of these samples in correcting the models' perception. By using dynamic adjustment, DMPRA ensures simultaneous enhancement of diversity and quality in the presence of imbalanced data distribution. Experimental results indicate that the algorithm has effectively improved Generative Adversarial Nets (GANs), Normalizing Flows (Flows), Variational Auto-Encoders (VAEs), and Diffusion Models (Diffusion).