Dimension reduction is useful for exploratory data analysis. In many applications, it is of interest to discover variation that is enriched in a "foreground" dataset relative to a "background" dataset. Recently, contrastive principal component analysis (CPCA) was proposed for this setting. However, the lack of a formal probabilistic model makes it difficult to reason about CPCA and to tune its hyperparameter. In this work, we propose probabilistic contrastive principal component analysis (PCPCA), a model-based alternative to CPCA. We discuss how to set the hyperparameter in theory and in practice, and we show several of PCPCA's advantages over CPCA, including greater interpretability, uncertainty quantification and principled inference, robustness to noise and missing data, and the ability to generate data from the model. We demonstrate PCPCA's performance through a series of simulations and case-control experiments with datasets of gene expression, protein expression, and images.
National University of Singapore（新加坡国立大学）；Youtu Lab Tencent（腾讯优图实验室）
This paper proposes a probabilistic contrastive loss function for self-supervised learning. The well-known contrastive loss is deterministic and involves a temperature hyperparameter that scales the inner product between two normed feature embeddings. By reinterpreting the temperature hyperparameter as a quantity related to the radius of the hypersphere, we derive a new loss function that involves a confidence measure which quantifies uncertainty in a mathematically grounding manner. Some intriguing properties of the proposed loss function are empirically demonstrated, which agree with human-l…
Flatiron Institute（熨斗研究院）；NYU Langone Medical School（纽约大学朗格尼医学院）
Finding informative low-dimensional representations that can be computed efficiently in large datasets is an important problem in data analysis. Recently, contrastive Principal Component Analysis (cPCA) was proposed as a more informative generalization of PCA that takes advantage of contrastive learning. However, the performance of cPCA is sensitive to hyper-parameter choice and there is currently no online algorithm for implementing cPCA. Here, we introduce a modified cPCA method, which we denote cPCA*, that is more interpretable and less sensitive to the choice of hyper-parameter. We derive …
Toyota Motor Europe（丰田汽车欧洲公司）；Czech Technical University in Prague（捷克布拉格技术大学）
Cross entropy loss has served as the main objective function for classification-based tasks. Widely deployed for learning neural network classifiers, it shows both effectiveness and a probabilistic interpretation. Recently, after the success of self supervised contrastive representation learning methods, supervised contrastive methods have been proposed to learn representations and have shown superior and more robust performance, compared to solely training with cross entropy loss. However, cross entropy loss is still needed to train the final classification layer. In this work, we investigate…
We present a new technique called contrastive principal component analysis (cPCA) that is designed to discover low-dimensional structure that is unique to a dataset, or enriched in one dataset relative to other data. The technique is a generalization of standard PCA, for the setting where multiple datasets are available -- e.g. a treatment and a control group, or a mixed versus a homogeneous population -- and the goal is to explore patterns that are specific to one of the datasets. We conduct a wide variety of experiments in which cPCA identifies important dataset-specific patterns that are mi…
Principal Component Analysis (PCA) is a popular tool for dimensionality reduction and feature extraction in data analysis. There is a probabilistic version of PCA, known as Probabilistic PCA (PPCA). However, standard PCA and PPCA are not robust, as they are sensitive to outliers. To alleviate this problem, this paper introduces the Self-Paced Learning mechanism into PPCA, and proposes a novel method called Self-Paced Probabilistic Principal Component Analysis (SP-PPCA). Furthermore, we design the corresponding optimization algorithm based on the alternative search strategy and the expectation-…
PRCL: Probabilistic Representation Contrastive Learning for Semi-Supervised Semantic Segmentation
School of Computer Science and Engineering, Northeastern University（东北大学计算机科学与工程学院）；Intelligent Game and Decision Laboratory（智能游戏与决策实验室）；Alibaba DAMO Academy, Alibaba Group（阿里巴巴集团达摩院）；Zhejiang University（浙江大学）
Lehigh University（理海大学）；University of Virginia（弗吉尼亚大学）
Dimensionality reduction on Riemannian manifolds is challenging due to the complex nonlinear data structures. While probabilistic principal geodesic analysis~(PPGA) has been proposed to generalize conventional principal component analysis (PCA) onto manifolds, its effectiveness is limited to data with a single modality. In this paper, we present a novel Gaussian latent variable model that provides a unique way to integrate multiple PGA models into a maximum-likelihood framework. This leads to a well-defined mixture model of probabilistic principal geodesic analysis (MPPGA) on sub-populations, …
Mixtures of probabilistic principal component analysis (MPPCA) is a well-known mixture model extension of principal component analysis (PCA). Similar to PCA, MPPCA assumes the data samples in each mixture contain homoscedastic noise. However, datasets with heterogeneous noise across samples are becoming increasingly common, as larger datasets are generated by collecting samples from several sources with varying noise profiles. The performance of MPPCA is suboptimal for data with heteroscedastic noise across samples. This paper proposes a heteroscedastic mixtures of probabilistic PCA technique …
Principal component analysis (PCA) is a well-established tool in machine learning and data processing. The principal axes in PCA were shown to be equivalent to the maximum marginal likelihood estimator of the factor loading matrix in a latent factor model for the observed data, assuming that the latent factors are independently distributed as standard normal distributions. However, the independence assumption may be unrealistic for many scenarios such as modeling multiple time series, spatial processes, and functional data, where the outcomes are correlated. In this paper, we introduce the gen…
Probabilistic Contrastive Learning with Explicit Concentration on the Hypersphere
Athinoula A. Martinos Center（阿西努拉·A.马蒂诺斯中心）；MGH, HMS, Harvard University（哈佛医学院、麻省总医院、哈佛大学）；Imperial College London（伦敦帝国学院）；Technical University of Munich（慕尼黑工业大学）；University of Zurich（苏黎世大学）
Quanzhou Institute of Equipment Manufacturing, Haixi Institutes, Chinese Academy of Sciences（中国科学院海西研究院泉州装备制造研究所）；University of Reading（雷丁大学）
An improved mixture of probabilistic principal component analysis (PPCA) has been introduced for nonlinear data-driven process monitoring in this paper. To realize this purpose, the technique of a mixture of probabilistic principal component analysers is utilized to establish the model of the underlying nonlinear process with local PPCA models, where a novel composite monitoring statistic is proposed based on the integration of two monitoring statistics in modified PPCA-based fault detection approach. Besides, the weighted mean of the monitoring statistics aforementioned is utilised as a metri…
Establishing a low-dimensional representation of the data leads to efficient data learning strategies. In many cases, the reduced dimension needs to be explicitly stated and estimated from the data. We explore the estimation of dimension in finite samples as a constrained optimization problem, where the estimated dimension is a maximizer of a penalized profile likelihood criterion within the framework of a probabilistic principal components analysis. Unlike other penalized maximization problems that require an "optimal" penalty tuning parameter, we propose a data-averaging procedure whereby th…
High dimensional data has introduced challenges that are difficult to address when attempting to implement classical approaches of statistical process control. This has made it a topic of interest for research due in recent years. However, in many cases, data sets have underlying structures, such as in advanced manufacturing systems. If extracted correctly, efficient methods for process control can be developed. This paper proposes a robust sparse dimensionality reduction approach for correlated high-dimensional process monitoring to address the aforementioned issues. The developed monitoring …
We provide a probabilistic and infinitesimal view of how the principal component analysis procedure (PCA) can be generalized to analysis of nonlinear manifold valued data. Starting with the probabilistic PCA interpretation of the Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an intrinsic way that does not resort to linearization of the data space. The underlying probability model is constructed by mapping a Euclidean stochastic process to the manifold using stochastic development of Euclidean semimartingales. The construction uses a connection and bundles of covar…
Aston University（阿斯顿大学）；Radboud University（拉德堡德大学）；Massachussetts Institute of Technology（麻省理工学院）
Using the linear Gaussian latent variable model as a starting point we relax some of the constraints it imposes by deriving a nonparametric latent feature Gaussian variable model. This model introduces additional discrete latent variables to the original structure. The Bayesian nonparametric nature of this new model allows it to adapt complexity as more data is observed and project each data point onto a varying number of subspaces. The linear relationship between the continuous latent and observed variables make the proposed model straightforward to interpret, resembling a locally adaptive pr…
This is a tutorial and survey paper on factor analysis, probabilistic Principal Component Analysis (PCA), variational inference, and Variational Autoencoder (VAE). These methods, which are tightly related, are dimensionality reduction and generative models. They assume that every data point is generated from or caused by a low-dimensional latent factor. By learning the parameters of distribution of latent space, the corresponding low-dimensional factors are found for the sake of dimensionality reduction. For their stochastic and generative behaviour, these models can also be used for generatio…
University of Tsukuba（筑波大学）；National Institute of Advanced Industrial Science and Technology（产业技术综合研究所）；University of Electro-Communication（电气通信大学）
Modeling of textures in natural images is an important task to make a microscopic model of natural images. Portilla and Simoncelli proposed a generative texture model, which is based on the mechanism of visual systems in brains, with a set of texture features and a feature matching. On the other hand, the texture features, used in Portillas' model, have redundancy between its components came from typical natural textures. In this paper, we propose a contracted texture model which provides a dimension reduction for the Portillas' feature. This model is based on a hierarchical principal componen…
In this paper, we develop new statistical theory for probabilistic principal component analysis models in high dimensions. The focus is the estimation of the noise variance, which is an important and unresolved issue when the number of variables is large in comparison with the sample size. We first unveil the reasons of a widely observed downward bias of the maximum likelihood estimator of the variance when the data dimension is high. We then propose a bias-corrected estimator using random matrix theory and establish its asymptotic normality. The superiority of the new (bias-corrected) estimat…
JD AI Research（京东人工智能研究院）；SUN YAT-SEN UNIVERSITY（中山大学）；University of Science and Technology of China（中国科学技术大学）
Unsupervised learning is just at a tipping point where it could really take off. Among these approaches, contrastive learning has seen tremendous progress and led to state-of-the-art performance. In this paper, we construct a novel probabilistic graphical model that effectively incorporates the low rank promoting prior into the framework of contrastive learning, referred to as LORAC. In contrast to the existing conventional self-supervised approaches that only considers independent learning, our hypothesis explicitly requires that all the samples belonging to the same instance class lie on the…
We propose a probabilistic interpretation of exponential dot product attention of transformers and contrastive learning based off of exponential families. The attention sublayer of transformers is equivalent to a gradient ascent step of the log normalizer, which is the log-sum-exp term in the Hopfield theory of attention. This ascent step induces a parallel expansion of points, which is counterbalanced by a contraction from layer normalization. We also state theoretical limitations of our theory and the Hopfield theory and suggest directions for resolution.
We propose a novel model for nonlinear dimension reduction motivated by the probabilistic formulation of principal component analysis. Nonlinearity is achieved by specifying different transformation matrices at different locations of the latent space and smoothing the transformation using a Markov random field type prior. The computation is made feasible by the recent advances in sampling from von Mises-Fisher distributions.
Accurate predictions of pollutant concentrations at new locations are often of interest in air pollution studies on fine particulate matters (PM$_{2.5}$), in which data is usually not measured at all study locations. PM$_{2.5}$ is also a mixture of many different chemical components. Principal component analysis (PCA) can be incorporated to obtain lower-dimensional representative scores of such multi-pollutant data. Spatial prediction can then be used to estimate these scores at new locations. Recently developed predictive PCA modifies the traditional PCA algorithm to obtain scores with spatia…
User representation is essential for providing high-quality commercial services in industry. Universal user representation has received many interests recently, with which we can be free from the cumbersome work of training a specific model for each downstream application. In this paper, we attempt to improve universal user representation from two points of views. First, a contrastive self-supervised learning paradigm is presented to guide the representation model training. It provides a unified framework that allows for long-term or short-term interest representation learning in a data-driven…
Multi-view feature extraction is an efficient approach for alleviating the issue of dimensionality in highdimensional multi-view data. Contrastive learning (CL), which is a popular self-supervised learning method, has recently attracted considerable attention. Most CL-based methods were constructed only from the sample level. In this study, we propose a novel multiview feature extraction method based on dual contrastive head, which introduce structural-level contrastive loss into sample-level CL-based method. Structural-level CL push the potential subspace structures consistent in any two cros…
Missing Not At Random (MNAR) values lead to significant biases in the data, since the probability of missingness depends on the unobserved values.They are ''not ignorable'' in the sense that they often require defining a model for the missing data mechanism, which makes inference or imputation tasks more complex. Furthermore, this implies a strong \textit{a priori} on the parametric form of the distribution.However, some works have obtained guarantees on the estimation of parameters in the presence of MNAR data, without specifying the distribution of missing data \citep{mohan2018estimation, ta…
We show that Contrastive Learning (CL) under a broad family of loss functions (including InfoNCE) has a unified formulation of coordinate-wise optimization on the network parameter $\boldsymbolθ$ and pairwise importance $α$, where the \emph{max player} $\boldsymbolθ$ learns representation for contrastiveness, and the \emph{min player} $α$ puts more weights on pairs of distinct samples that share similar representations. The resulting formulation, called $α$-CL, unifies not only various existing contrastive losses, which differ by how sample-pair importance $α$ is constructed, but also is able …
Variational autoencoders (VAEs) suffer from posterior collapse, where the powerful neural networks used for modeling and inference optimize the objective without meaningfully using the latent representation. We introduce inference critics that detect and incentivize against posterior collapse by requiring correspondence between latent variables and the observations. By connecting the critic's objective to the literature in self-supervised contrastive representation learning, we show both theoretically and empirically that optimizing inference critics increases the mutual information between ob…
While supervised learning has enabled great progress in many applications, unsupervised learning has not seen such widespread adoption, and remains an important and challenging endeavor for artificial intelligence. In this work, we propose a universal unsupervised learning approach to extract useful representations from high-dimensional data, which we call Contrastive Predictive Coding. The key insight of our model is to learn such representations by predicting the future in latent space by using powerful autoregressive models. We use a probabilistic contrastive loss which induces the latent s…
University of Science and Technology of China（中国科学技术大学）；JD AI Research（京东人工智能研究院）
This paper explores useful modifications of the recent development in contrastive learning via novel probabilistic modeling. We derive a particular form of contrastive loss named Joint Contrastive Learning (JCL). JCL implicitly involves the simultaneous learning of an infinite number of query-key pairs, which poses tighter constraints when searching for invariant features. We derive an upper bound on this formulation that allows analytical solutions in an end-to-end training manner. While JCL is practically effective in numerous computer vision applications, we also theoretically unveil the ce…
University of Toronto（多伦多大学）；LG AI Research（LG人工智能研究院）
Contrastive learning has led to substantial improvements in the quality of learned embedding representations for tasks such as image classification. However, a key drawback of existing contrastive augmentation methods is that they may lead to the modification of the image content which can yield undesired alterations of its semantics. This can affect the performance of the model on downstream tasks. Hence, in this paper, we ask whether we can augment image data in contrastive learning such that the task-relevant semantic content of an image is preserved. For this purpose, we propose to leverag…
A Mathematical Perspective On Contrastive Learning
Stores Foundational AI, Amazon, Palo Alto CA 94301 and Pasadena CA 91125（亚马逊公司、帕洛阿尔托加州94301和帕萨迪纳加州91125）；Computing and Mathematical Sciences, California Institute of Technology, Pasadena CA 91125（计算与数学科学系，加州理工学院，帕萨迪纳加州91125）
Conditional Neural Processes~(CNPs) bridge neural networks with probabilistic inference to approximate functions of Stochastic Processes under meta-learning settings. Given a batch of non-{\it i.i.d} function instantiations, CNPs are jointly optimized for in-instantiation observation prediction and cross-instantiation meta-representation adaptation within a generative reconstruction pipeline. There can be a challenge in tying together such two targets when the distribution of function observations scales to high-dimensional and noisy spaces. Instead, noise contrastive estimation might be able …
A Supervised Contrastive Learning Pretrain-Finetune Approach for Time Series
School of Operations Research and Information Engineering, Cornell University（康奈尔大学运筹学与信息工程学院）；IBM Research, Thomas J. Watson Research Center（IBM研究部托马斯·J·沃森研究中心）
To extract robust deep representations from long sequential modeling of speech data, we propose a self-supervised learning approach, namely Contrastive Separative Coding (CSC). Our key finding is to learn such representations by separating the target signal from contrastive interfering signals. First, a multi-task separative encoder is built to extract shared separable and discriminative embedding; secondly, we propose a powerful cross-attention mechanism performed over speaker representations across various interfering conditions, allowing the model to focus on and globally aggregate the most…
