Contrastive learning methods enforce label distance relationships in feature space to improve representation capability for regression models. However, these methods highly depend on label information to correctly recover ordinal relationships of features, limiting their applications to semi-supervised regression. In this work, we extend contrastive regression methods to allow unlabeled data to be used in the semi-supervised setting, thereby reducing the dependence on costly annotations. Particularly we construct the feature similarity matrix with both labeled and unlabeled samples in a mini-batch to reflect inter-sample relationships, and an accurate ordinal ranking of involved unlabeled samples can be recovered through spectral seriation algorithms if the level of error is within certain bounds. The introduction of labeled samples above provides regularization of the ordinal ranking with guidance from the ground-truth label information, making the ranking more reliable. To reduce feature perturbations, we further utilize the dynamic programming algorithm to select robust features for the matrix construction. The recovered ordinal relationship is then used for contrastive learning on unlabeled samples, and we thus allow more data to be used for feature representation learning, thereby achieving more robust results. The ordinal rankings can also be used to supervise predictions on unlabeled samples, serving as an additional training signal. We provide theoretical guarantees and empirical verification through experiments on various datasets, demonstrating that our method can surpass existing state-of-the-art semi-supervised deep regression methods.

The probabilistic forecasting of time series is a well-recognized challenge, particularly in disentangling correlations among interacting time series and addressing the complexities of distribution modeling. By treating time series as temporal dynamics, we introduce **KooNPro**, a novel probabilistic time series forecasting model that combines variance-aware deep **Koo**pman model with **N**eural **Pro**cess. KooNPro introduces a variance-aware continuous spectrum using Gaussian distributions to capture complex temporal dynamics with improved stability. It further integrates the Neural Process to capture fine dynamics, enabling enhanced dynamics capture and prediction. Extensive experiments on nine real-world datasets demonstrate that KooNPro consistently outperforms state-of-the-art baselines. Ablation studies highlight the importance of the Neural Process component and explore the impact of key hyperparameters. Overall, KooNPro presents a promising novel approach for probabilistic time series forecasting.

Neural ordinary differential equations (NODEs) have demonstrated strong capabilities in modeling time series. However, existing NODE- based methods often focus solely on the surface-level dynamics derived from observed states, which limits their ability to capture more complex underlying behaviors. To overcome this challenge, we propose KoNODE, a Koopman-driven NODE framework that explicitly models the evolution of ODE parameters over time to encode deep-level information. KoNODE captures the essential yet simple intrinsic linear dynamics that govern the surface dynamics by employing Koopman operators. Our framework operates at three hierarchical levels: the observed state dynamics, the parameter dynamics, and the Koopman linear dynamics, representing the fundamental driving rules of the state dynamics. The proposed approach offers significant improvements in two critical time series tasks: long-term prediction (enabled by the simple linear dynamics) and generalization to new data (driven by the evolving ODE parameters). We validate KoNODE through experiments on synthetic data from complex dynamic systems and real-world datasets, demonstrating its effectiveness in practical scenarios.

Diffusion models have achieved impressive success in high-fidelity image generation but suffer from slow sampling due to their inherently iterative denoising process. While recent one-step methods accelerate inference by learning direct noise-to-image mappings, they sacrifice the interpretability and fine-grained control intrinsic to diffusion dynamics, key advantages that enable applications like editable generation. To resolve this dichotomy, we introduce **Hierarchical Koopman Diffusion**, a novel framework that achieves both one-step sampling and interpretable generative trajectories. Grounded in Koopman operator theory, our method lifts the nonlinear diffusion dynamics into a latent space where evolution is governed by globally linear operators, enabling closed-form trajectory solutions. This formulation not only eliminates iterative sampling but also provides full access to intermediate states, allowing manual intervention during generation. To model the multi-scale nature of images, we design a hierarchical architecture that disentangles generative dynamics across spatial resolutions via scale-specific Koopman subspaces, capturing coarse-to-fine details systematically. We empirically show that the Hierarchical Koopman Diffusion not only achieves competitive one-step generation performance but also provides a principled mechanism for interpreting and manipulating the generative process through spectral analysis. Our framework bridges the gap between fast sampling and interpretability in diffusion models, paving the way for explainable image synthesis in generative modeling.

Contrastive learning methods can be applied to deep regression by enforcing label distance relationships in feature space. However, these methods are limited to labeled data only unlike for classification, where unlabeled data can be used for contrastive pretraining. In this work, we extend contrastive regression methods to allow unlabeled data to be used in a semi-supervised setting, thereby reducing the reliance on manual annotations. We observe that the feature similarity matrix between unlabeled samples still reflect inter-sample relationships, and that an accurate ordinal relationship can be recovered through spectral seriation algorithms if the level of error is within certain bounds. By using the recovered ordinal relationship for contrastive learning on unlabeled samples, we can allow more data to be used for feature representation learning, thereby achieve more robust results. The ordinal rankings can also be used to supervise predictions on unlabeled samples, which can serve as an additional training signal. We provide theoretical guarantees and empirical support through experiments on different datasets, demonstrating that our method can surpass existing state-of-the-art semi-supervised deep regression methods. To the best of our knowledge, this work is the first to explore using unlabeled data to perform contrastive learning for regression.