We introduce Sundial, a family of native, flexible, and scalable time series foundation models. To predict the next-patch's distribution, we propose a TimeFlow Loss based on flow-matching, which facilitates native pre-training of Transformers on continuous-valued time series without discrete tokenization. Conditioned on arbitrary-length time series, our models are pre-trained without specifying any prior distribution and can generate multiple probable predictions, achieving more flexibility in representation learning than using parametric densities. Towards time series foundation models, we leverage minimal but crucial adaptations of Transformers and curate TimeBench with one trillion time points, comprising mostly real-world datasets and synthetic data. By mitigating mode collapse via TimeFlow Loss, we pre-train a family of Sundial models on TimeBench, which achieve unprecedented model capacity and generalization performance. In addition to excellent scalability, Sundial achieves state-of-the-art results on both point and probabilistic forecasting benchmarks with a just-in-time inference speed, i.e., making zero-shot predictions within a few milliseconds. We believe that Sundial's pioneering generative forecasting capability can improve model reliability in real-world decision-making. Code is available at: https://github.com/thuml/Sundial.

Many large-scale systems rely on high-quality deep representations (embeddings) to facilitate tasks like retrieval, search, and generative modeling. Matryoshka Representation Learning (MRL) recently emerged as a solution for adaptive embedding lengths, but it requires full model retraining and suffers from noticeable performance degradations at short lengths. In this paper, we show that sparse coding offers a compelling alternative for achieving adaptive representation with minimal overhead and higher fidelity. We propose Contrastive Sparse Representation (CSR), a method that specifies pre-trained embeddings into a high-dimensional but selectively activated feature space. By leveraging lightweight autoencoding and task-aware contrastive objectives, CSR preserves semantic quality while allowing flexible, cost-effective inference at different sparsity levels. Extensive experiments on image, text, and multimodal benchmarks demonstrate that CSR consistently outperforms MRL in terms of both accuracy and retrieval speed—often by large margins—while also cutting training time to a fraction of that required by MRL. Our results establish sparse coding as a powerful paradigm for adaptive representation learning in real-world applications where efficiency and fidelity are both paramount. Code is available at this URL.

Embedding and visualization techniques are essential for analyzing high-dimensional data, but they often struggle with complex data governed by multiple latent variables, potentially distorting key structural characteristics. This paper considers scenarios where the observed features can be partitioned into mutually exclusive subsets, each capturing a different smooth substructure. In such cases, visualizing the data based on each feature partition can better characterize the underlying processes and structures in the data, leading to improved interpretability. To partition the features, we propose solving an optimization problem that promotes graph Laplacian-based smoothness in each partition, thereby prioritizing partitions with simpler geometric structures. Our approach generalizes traditional embedding and visualization techniques, allowing them to learn multiple embeddings simultaneously. We establish that if several independent or partially dependent manifolds are embedded in distinct feature subsets in high-dimensional space, then our framework can reliably identify the correct subsets with theoretical guarantees. Finally, we demonstrate the effectiveness of our approach in extracting multiple low-dimensional structures and partially independent processes from both simulated and real data.

Node equivalence is common in graphs, such as computing networks, encompassing automorphic equivalence (preserving adjacency under node permutations) and attribute equivalence (nodes with identical attributes). Despite their importance for learning node representations, these equivalences are largely ignored by existing graph models. To bridge this gap, we propose a GrAph self-supervised Learning framework with Equivalence (GALE) and analyze its connections to existing techniques. Specifically, we: 1) unify automorphic and attribute equivalence into a single equivalence class; 2) enforce the equivalence principle to make representations within the same class more similar while separating those across classes; 3) introduce approximate equivalence classes with linear time complexity to address the NP-hardness of exact automorphism detection and handle node-feature variation; 4) analyze existing graph encoders, noting limitations in message passing neural networks and graph transformers regarding equivalence constraints; 5) show that graph contrastive learning are a degenerate form of equivalence constraint; and 6) demonstrate that GALE achieves superior performance over baselines.