The task of causal representation learning aims to uncover latent higher-level causal variables that affect lower-level observations. Identifying the true latent causal variables from observed data, while allowing instantaneous causal relations among latent variables, remains a challenge, however. To this end, we start with the analysis of three intrinsic indeterminacies in identifying latent variables from observations: transitivity, permutation indeterminacy, and scaling indeterminacy. We find that transitivity acts as a key role in impeding the identifiability of latent causal variables. To address the unidentifiable issue due to transitivity, we introduce a novel identifiability condition where the underlying latent causal model satisfies a linear-Gaussian model, in which the causal coefficients and the distribution of Gaussian noise are modulated by an additional observed variable. Under certain assumptions, including the existence of a reference condition under which latent causal influences vanish, we can show that the latent causal variables can be identified up to trivial permutation and scaling, and that partial identifiability results can still be obtained when this reference condition is violated for a subset of latent variables. Furthermore, based on these theoretical results, we propose a novel method, termed Structural caUsAl Variational autoEncoder (SuaVE), which directly learns causal representations and causal relationships among them, together with the mapping from the latent causal variables to the observed ones. Experimental results on synthetic and real data demonstrate the identifiability and consistency results and the efficacy of SuaVE in learning causal representations.

Recovering the underlying Directed Acyclic Graph (DAG) structures from observational data presents a formidable challenge, partly due to the combinatorial nature of the DAG-constrained optimization problem. Recently, researchers have identified gradient vanishing as one of the primary obstacles in differentiable DAG learning and have proposed several DAG constraints to mitigate this issue. By developing the necessary theory to establish a connection between analytic functions and DAG constraints, we demonstrate that analytic functions from the set $\\{f(x) = c_0 + \\sum_{i=1}^{\infty}c_ix^i | \\forall i > 0, c_i > 0; r = \\lim_{i\\rightarrow \\infty}c_{i}/c_{i+1} > 0\\}$ can be employed to formulate effective DAG constraints. Furthermore, we establish that this set of functions is closed under several functional operators, including differentiation, summation, and multiplication. Consequently, these operators can be leveraged to create novel DAG constraints based on existing ones. Using these properties, we design a series of DAG constraints and develop an efficient algorithm to evaluate them. Experiments in various settings demonstrate that our DAG constraints outperform previous state-of-the-art comparators. Our implementation is available at https://github.com/zzhang1987/AnalyticDAGLearning.

Rehearsal-based continual learning (CL) aims to mitigate catastrophic forgetting by maintaining a subset of samples from previous tasks and replaying them. The rehearsal memory can be naturally constructed as a coreset, designed to form a compact subset that enables training with performance comparable to using the full dataset. The coreset selection task can be formulated as bilevel optimization that solves for the subset to minimize the outer objective of the learning task. Existing methods primarily rely on inefficient probabilistic sampling or local gradient-based scoring to approximate sample importance through an iterative process that can be susceptible to ambiguity or noise. Specifically, non-representative samples like ambiguous or noisy samples are difficult to learn and incur high loss values even when training on the full dataset. However, existing methods relying on local gradient tend to highlight these samples in an attempt to minimize the outer loss, leading to a suboptimal coreset. To enhance coreset selection, especially in CL where high-quality samples are essential, we propose a coreset selection method that measures sample importance using reducible loss (ReL) that quantifies the impact of adding a sample to model performance. By leveraging ReL and a process derived from bilevel optimization, we identify and retain samples that yield the highest performance gain. They are shown to be informative and representative. Furthermore, ReL requires only forward computation, making it significantly more efficient than previous methods. To better apply coreset selection in CL, we extend our method to address key challenges such as task interference, streaming data, and knowledge distillation. Experiments on data summarization and continual learning demonstrate the effectiveness and efficiency of our approach.

Multimodal representation learning, exemplified by multimodal contrastive learning (MMCL) using image-text pairs, aims to learn powerful representations by aligning cues across modalities. This approach relies on the core assumption that the exemplar image-text pairs constitute two representations of an identical concept. However, recent research has revealed that real-world datasets often exhibit cross-modal misalignment. There are two distinct viewpoints on how to address this issue: one suggests mitigating the misalignment, and the other leveraging it. We seek here to reconcile these seemingly opposing perspectives, and to provide a practical guide for practitioners. Using latent variable models we thus formalize cross-modal misalignment by introducing two specific mechanisms: Selection bias, where some semantic variables are absent in the text, and perturbation bias, where semantic variables are altered—both leading to misalignment in data pairs. Our theoretical analysis demonstrates that, under mild assumptions, the representations learned by MMCL capture exactly the information related to the subset of the semantic variables invariant to selection and perturbation biases. This provides a unified perspective for understanding misalignment. Based on this, we further offer actionable insights into how misalignment should inform the design of real-world ML systems. We validate our theoretical findings via extensive empirical studies on both synthetic data and real image-text datasets, shedding light on the nuanced impact of cross-modal misalignment on multimodal representation learning.

Continual learning (CL) aims to incrementally train a model to a sequence of tasks while maintaining performance on previously seen ones. Despite effectiveness in mitigating forgetting, data storage and replay may be infeasible due to privacy or security constraints, and are impractical or unavailable for arbitrary pre-trained models. Data-free or examplar-free CL aims to continually update models with new tasks without storing previous data. In addition to regularizing updates, we employ model inversion to synthesize data from the trained model, anchoring learned knowledge through replay without retaining old data. However, model inversion in predictive models faces two key challenges. First, generating inputs (e.g., images) solely from highly compressed output labels (e.g., classes) often causes drift between synthetic and real data. Replaying on such synthetic data can contaminate and erode knowledge learned from real data, further degrading inversion quality over time. Second, performing inversion is usually computationally expensive, as each iteration requires backpropagation through the entire model and many steps are needed for convergence. These problems are more severe with large pre-trained models such as Contrastive Language-Image Pre-training (CLIP) models. To improve model inversion efficiency, we propose Per-layer Model Inversion (PMI) approach inspired by the faster convergence of single-layer optimization. The inputs optimized from PMI provide strong initialization for full-model inversion, significantly reducing the number of iterations required for convergence. To address feature distribution shift, we model class-wise feature distribution using a Gaussian distribution and preserve distributional information with a contrastive model. Sampling features for inversion ensures alignment between synthetic and real feature distributions. Combining PMI and feature modeling, we demonstrate the feasibility of incrementally training models on new classes by generating data from pseudo image features mapped through semantic-aware feature projection. Our method shows strong effectiveness and compatibility across multiple CL settings.

Causal representation learning aims to unveil latent high-level causal representations from observed low-level data. One of its primary tasks is to provide reliable assurance of identifying these latent causal models, known as \textit{identifiability}. A recent breakthrough explores identifiability by leveraging the change of causal influences among latent causal variables across multiple environments \citep{liu2022identifying}. However, this progress rests on the assumption that the causal relationships among latent causal variables adhere strictly to linear Gaussian models. In this paper, we extend the scope of latent causal models to involve nonlinear causal relationships, represented by polynomial models, and general noise distributions conforming to the exponential family. Additionally, we investigate the necessity of imposing changes on all causal parameters and present partial identifiability results when part of them remains unchanged. Further, we propose a novel empirical estimation method, grounded in our theoretical finding, that enables learning consistent latent causal representations. Our experimental results, obtained from both synthetic and real-world data, validate our theoretical contributions concerning identifiability and consistency.

Continual Learning (CL) methods aim to enable machine learning models to learn new tasks without catastrophic forgetting of those that have been previously mastered. Existing CL approaches often keep a buffer of previously-seen samples, perform knowledge distillation, or use regularization techniques towards this goal. Despite their performance, they still suffer from interference across tasks which leads to catastrophic forgetting. To ameliorate this problem, we propose to only activate and select sparse neurons for learning current and past tasks at any stage. More parameters space and model capacity can thus be reserved for the future tasks. This minimizes the interference between parameters for different tasks. To do so, we propose a Sparse neural Network for Continual Learning (SNCL), which employs variational Bayesian sparsity priors on the activations of the neurons in all layers. Full Experience Replay (FER) provides effective supervision in learning the sparse activations of the neurons in different layers. A loss-aware reservoir-sampling strategy is developed to maintain the memory buffer. The proposed method is agnostic as to the network structures and the task boundaries. Experiments on different datasets show that SNCL achieves state-of-the-art result for mitigating forgetting.

Recovering underlying Directed Acyclic Graph (DAG) structures from observational data is highly challenging due to the combinatorial nature of the DAG-constrained optimization problem. Recently, DAG learning has been cast as a continuous optimization problem by characterizing the DAG constraint as a smooth equality one, generally based on polynomials over adjacency matrices. Existing methods place very small coefficients on high-order polynomial terms for stabilization, since they argue that large coefficients on the higher-order terms are harmful due to numeric exploding. On the contrary, we discover that large coefficients on higher-order terms are beneficial for DAG learning, when the spectral radiuses of the adjacency matrices are small, and that larger coefficients for higher-order terms can approximate the DAG constraints much better than the small counterparts. Based on this, we propose a novel DAG learning method with efficient truncated matrix power iteration to approximate geometric series based DAG constraints. Empirically, our DAG learning method outperforms the previous state-of-the-arts in various settings, often by a factor of $3$ or more in terms of structural Hamming distance.

Variational dropout (VD) is a generalization of Gaussian dropout, which aims at inferring the posterior of network weights based on a log-uniform prior on them to learn these weights as well as dropout rate simultaneously. The log-uniform prior not only interprets the regularization capacity of Gaussian dropout in network training, but also underpins the inference of such posterior. However, the log-uniform prior is an improper prior (i.e., its integral is infinite), which causes the inference of posterior to be ill-posed, thus restricting the regularization performance of VD. To address this problem, we present a new generalization of Gaussian dropout, termed variational Bayesian dropout (VBD), which turns to exploit a hierarchical prior on the network weights and infer a new joint posterior. Specifically, we implement the hierarchical prior as a zero-mean Gaussian distribution with variance sampled from a uniform hyper-prior. Then, we incorporate such a prior into inferring the joint posterior over network weights and the variance in the hierarchical prior, with which both the network training and dropout rate estimation can be cast into a joint optimization problem. More importantly, the hierarchical prior is a proper prior which enables the inference of posterior to be well-posed. In addition, we further show that the proposed VBD can be seamlessly applied to network compression. Experiments on classification and network compression demonstrate the superior performance of the proposed VBD in regularizing network training.

Non-negative matrix factorization (NMF) is a challenging problem due to its ill-posed nature. The key for the success of NMF is to exploit appropriate prior models for those two decomposed factor matrices. Although lots of effective sparsity-inducing prior models have been developed for NMF, they are often rooted in either ℓpregularization with p > 0, which only provide an approximation to the ℓ0sparsity, ultimately resulting in a sub-optimal solution. To address this problem, we propose a novel truncated spike-and-slab prior based Bayesian NMF method. Through integrating a Bernoulli distribution with a truncated Gaussian distribution together, the proposed prior is capable of imposing the exact ℓ0regularization as well as the non-negativity constraint on the factor matrices. Further, the proposed prior can be extended to robust NMF problem. Experimental results in blind source separation, face images representation and image denoising demonstrate the advantage of the proposed method.

Variational Bayesian (VB) learning has been successfully applied to instantaneous blind source separation. However, the traditional VB learning is restricted to the separation of independent source signals. Moreover, it has the difficulty to recover source signals with a sizable number of samples because of its rapidly increasing computational requirement. To overcome such shortcomings, frame-based VB (FVB) learning is proposed to address both independent and dependent source separation with a large number of samples in this paper. Specifically, a Gaussian process (GP) is employed to model independent or dependent source signals. To our knowledge, GP has been only used to model each of independent source signals. For dependent source signals, this paper proposes a novel modeling process: initial source signals are zigzag concatenated into a long serial and GP is then used to model it. In order to obtain a reliable covariance function for GP, first, we apply singular value decomposition to give initial estimated source signals and then we select an appropriate covariance function with which GP can perfectly fit them. In order to alleviate the computational burden of VB learning, we split observed signals into frames, and then model and infer source signals for each frame. Compared with the state-of-the-art algorithms, the experimental results show that the FVB learning has potential to provide improvement in separation performance not only for independent source signals but also for dependent ones, especially for long data records.