ICLR 2026 Papers

Time-series forecasting increasingly demands not only accurate observational predictions but also causal forecasting under interventional and counterfactual queries in multivariate systems. We present DoFlow, a flow-based generative model defined over a causal Directed Acyclic Graph (DAG) that delivers coherent observational and interventional predictions, as well as counterfactuals through the natural encoding–decoding mechanism of continuous normalizing flows (CNFs). We also provide a supporting counterfactual recovery theory under certain assumptions. Beyond forecasting, DoFlow provides explicit likelihoods of future trajectories, enabling principled anomaly detection. Experiments on synthetic datasets with various causal DAG structures and real-world hydropower and cancer-treatment time series show that DoFlow achieves accurate system-wide observational forecasting, enables causal forecasting over interventional and counterfactual queries, and effectively detects anomalies. This work contributes to the broader goal of unifying causal reasoning and generative modeling for complex dynamical systems.

We study best-of-$N$ for large language models (LLMs) where the selection is based on majority voting. In particular, we analyze the limit $N \to \infty$, which we denote as best-of-$\infty$. While this approach achieves impressive performance in the limit, it requires an infinite test-time budget. To address this, we propose an adaptive generation scheme that selects $N$ based on answer agreement, thereby efficiently allocating inference-time computation. Beyond adaptivity, we extend the framework to weighted ensembles of multiple LLMs, showing that such mixtures can outperform any individual model. The optimal ensemble weighting is formulated and efficiently computed as a mixed-integer linear program. Extensive experiments demonstrate the effectiveness of our approach.

Multivariate time-series anomaly detection (MTSAD) faces a critical trade-off between detection performance and model transparency. We propose PGRF-Net, a novel architecture designed to achieve competitive performance while providing structured evidence to support diagnostic insights. At its core, PGRF-Net uses a Multi-Faceted Evidence Extractor that combines prototype learning with the discovery of dynamic relational structures between variables. This extractor generates four distinct types of anomaly evidence: predictive deviation, structural changes in learned variable dependencies, contextual deviation from normal-behavior prototypes, and the magnitude of localized spike events. This evidence is then processed by a Gated Evidence Fusion Network, which learns to weigh each source via data-driven gating. PGRF-Net is trained via a two-stage unsupervised strategy for robust extractor learning and subsequent fusion tuning. Extensive experiments on five public MTSAD benchmarks demonstrate its competitive or superior detection performance. Importantly, by decomposing the final anomaly score into these four evidence types, our model facilitates diagnostic analysis, offering a practical step towards more interpretable, evidence-based MTSAD.

Test-time scaling aims to improve language model performance by leveraging additional compute during inference. Many works have empirically studied techniques such as Best-of-N (BoN) and Rejection Sampling (RS) that make use of a verifier to enable test-time scaling. However, to date there is little theoretical understanding of how verifier *imperfection* affects performance — a gap we address in this work. Specifically, we prove that the instance-level accuracy of these methods is precisely characterized by the geometry of the verifier’s ROC curve. Our theory has two important takeaways, confirmed by experiments with Qwen and LLama models on GSM8K and MATH500. First, RS outperforms BoN for fixed compute, while both methods converge to the same accuracy in the infinite-compute limit. Second, it is generally impossible to predict the high-compute performance of either method based on observations in the low-compute regime.

Reasoning LLMs suffer from quadratic compute growth as their context length increases, making reinforcement learning with verifiable rewards (RLVR) and test-time scaling prohibitively expensive. Prior work has tried to lighten the computational burden by shortening reasoning traces through pruning, summarization, or multi-stage training, but these methods remain bound to quadratic costs. We introduce Delethink, a thinking algorithm that realizes the Markovian Thinking Paradigm. Instead of producing one long monolithic reasoning trace, Delethink thinks in a sequence of chunks, the Delethink trace. Each chunk continues reasoning by referring only to a fixed number of prior tokens, which functions as a Markovian state sufficient for progressing reasoning, while deleting the rest. This preserves continuity without carrying the quadratic baggage. As a result, compute scales linearly and peak memory remains constant. In experiments, we show that Delethink can be applied directly to off-the-shelf reasoning models ranging from $1.5\textnormal{B}$ to $30\textnormal{B}$ parameters, with no loss in performance. Extended reasoning becomes possible under fixed memory and linear compute, while enabling efficient RL training on new tasks. On the DeepScaleR dataset, Delethink trains R1DistillQwen1.5B to the same benchmark performance as a standard long chain-of-thought (LongCoT) approach, where both models generate up to $24\textnormal{k}$ thinking tokens. The difference is efficiency. Delethink reasons $40\%$ faster with $70\%$ less memory footprint. By decoupling reasoning length from context length, the Markovian Thinking paradigm opens the door to next-generation reasoning LLMs that can scale to millions of tokens with linear compute and constant memory.

For real-world BCI applications, lightweight Electroencephalography (EEG) systems offer the best cost–deployment balance. However, such spatial sparsity of EEG limits spatial fidelity, hurting learning and introducing bias. EEG spatial super-resolution methods aim to recover high-density EEG signals from sparse measurements, yet is often hindered by distribution shift and signal distortion and thus reducing fidelity and usability for EEG analysis and visualization. To overcome these challenges, we introduce SRGDiff, a step-aware residual-guided diffusion model that formulates EEG spatial super-resolution as dynamic conditional generation. Our key idea is to learn a dynamic residual condition from the low-density input that predicts the step-wise temporal and spatial details to add and uses the evolving cue to steer the denoising process toward high density reconstructions. At each denoising step, the proposed residual condition is additively fused with the previous denoiser feature maps, then a step-dependent affine modulation scales and shifts the activation to produce the current features. This iterative procedure dynamically extracts step-wise temporal rhythms and spatial-topographic cues to steer high-density recovery and maintain a fidelity–consistency balance. We adopt a comprehensive evaluation protocol spanning signal-, feature-, and downstream-level metrics across SEED, SEED-IV, and Localize-MI and multiple upsampling scales. SRGDiff achieves consistent gains of up to 40\% over strong baselines, proving its superiority in the task of EEG spatial super-resolution. Moreover, topographic visualizations comparison and substantial EEG-FID gains jointly indicate that our SR EEG mitigates the spatial–spectral shift between low- and high-density recordings. Our code is available at https://anonymous.4open.science/r/DDPM-VAE-6F6B.

Foundation models are reshaping EEG analysis, yet an important problem of EEG tokenization remains a challenge. This paper presents TFM-Tokenizer, a novel tokenization framework that learns a vocabulary of time-frequency motifs from *single-channel* EEG signals and encodes them into discrete tokens. We propose a dual-path architecture with time–frequency masking to capture robust motif representations, and it is model-agnostic, supporting both lightweight transformers and existing foundation models for downstream tasks. Our study demonstrates three key benefits: *Accuracy:* Experiments on four diverse EEG benchmarks demonstrate consistent performance gains across both single- and multi-dataset pretraining settings, achieving up to 11% improvement in Cohen’s Kappa over strong baselines. *Generalization:* Moreover, as a plug-and-play component, it consistently boosts the performance of diverse foundation models, including BIOT and LaBraM. *Scalability:* By operating at the single-channel level rather than relying on the strict 10–20 EEG system, our method has the potential to be device-agnostic. Experiments on ear-EEG sleep staging, which differs from the pretraining data in signal format, channel configuration, recording device, and task, show that our tokenizer outperforms baselines by 14%. A comprehensive token analysis reveals strong class-discriminative, frequency-aware, and consistent structure, enabling improved representation quality and interpretability. Code is available at https://anonymous.4open.science/r/TFM-Token-FE33.

Learning in high-dimensional MDPs with complex state dynamics became possible with the progress achieved in reinforcement learning research. At the same time, deep neural policies have been observed to be highly unstable with respect to the minor variations in their state space, causing volatile and unpredictable behaviour. To alleviate these volatilities, a line of work suggested techniques to cope with this problem via explicitly regularizing the temporal difference loss to ensure local $\epsilon$-invariance in the state space. In this paper, we provide theoretical foundations on the impact of $\epsilon$-local invariance training on the deep neural policy manifolds. Our comprehensive theoretical and experimental analysis reveals that standard reinforcement learning inherently learns counterfactual values while recent training techniques that focus on explicitly enforcing $\epsilon$-local invariance cause policies to lose counterfactuality, and further result in learning misaligned and inconsistent values. In connection to this analysis, we further highlight that this line of training methods break the core intuition and the true biological inspiration of reinforcement learning, and introduce an intrinsic gap between how natural intelligence understands and interacts with an environment in contrast to AI agents trained via $\epsilon$-local invariance methods. The misalignment, inaccuracy and the loss of counterfactuality revealed in our paper further demonstrate the need to rethink the approach in establishing truly reliable and generalizable reinforcement learning policies.

Many methods aim to enhance time series forecasting by decomposing the series through intricate model structures and prior knowledge, yet they are inevitably limited by computational complexity and the robustness of the assumptions. Our research uncovers that in the complex domain and higher-order hypercomplex spaces, the characteristic frequencies of time series naturally decrease. Leveraging this insight, we propose Numerion, a time series forecasting model based on multiple hypercomplex spaces. Specifically, grounded in theoretical support, we generalize linear layers and activation functions to hypercomplex spaces of arbitrary power-of-two dimensions and introduce a novel Real-Hypercomplex-Real Domain Multi-Layer Perceptron (RHR-MLP) architecture. Numerion utilizes multiple RHR-MLPs to map time series into hypercomplex spaces of varying dimensions, naturally decomposing and independently modeling the series, and adaptively fuses the latent patterns exhibited in different spaces through a dynamic fusion mechanism. Experiments validate the model’s performance, achieving state-of-the-art results on multiple public datasets. Visualizations and quantitative analyses comprehensively demonstrate the ability of multi-dimensional RHR-MLPs to naturally decompose time series and reveal the tendency of higher-dimensional hypercomplex spaces to capture lower-frequency features.

Univariate time series (UTS), where each timestamp records a single variable, serve as crucial indicators in web systems and cloud servers. Anomaly detection in UTS plays an essential role in both data mining and system reliability management. However, existing reconstruction-based and prediction-based methods struggle to capture certain subtle anomalies, particularly small point anomalies and slowly rising anomalies. To address these challenges, we propose a novel prediction-based framework named Contextual and Seasonal LSTMs (CS-LSTMs). CS-LSTMs are built upon a noise decomposition strategy and jointly leverage contextual dependencies and seasonal patterns, thereby strengthening the detection of subtle anomalies. By integrating both time-domain and frequency-domain representations, CS-LSTMs achieve more accurate modeling of periodic trends and anomaly localization. Extensive evaluations on public benchmark datasets demonstrate that CS-LSTMs consistently outperform state-of-the-art methods, highlighting their effectiveness and practical value in robust time series anomaly detection.

Linear dynamical systems (LDSs) have been powerful tools for modeling high-dimensional time-series data across many domains, including neuroscience. However, a single LDS often struggles to capture the heterogeneity of neural data, where trajectories recorded under different conditions can have variations in their dynamics. Mixtures of linear dynamical systems (MoLDS) provide a path to model these variations in temporal dynamics for different observed trajectories. However, MoLDS remains difficult to apply in complex and noisy settings, limiting its practical use in neural data analysis. Tensor-based moment methods can provide global identifiability guarantees for MoLDS, but their performance degrades under realistic noise and complexity. Commonly used expectation-maximization (EM) methods offer flexibility in fitting latent models but are highly sensitive to initialization and prone to poor local minima. Here, we propose a tensor-based moment method that provides identifiability guarantees for learning MoLDS, which can be followed by EM updates to combine the strengths of both approaches. The novelty in our approach lies in the construction of moment tensors using the input–output data, on which we then apply Simultaneous Matrix Diagonalization (SMD) to recover globally consistent estimates of mixture weights and system parameters. These estimates can then be refined through a full Kalman filter-smoother EM algorithm, with closed-form updates for all LDS parameters. We validate our framework on synthetic benchmarks and real-world datasets. On synthetic data, the proposed Tensor-EM method achieves more reliable recovery and improved robustness compared to either pure tensor or randomly initialized EM methods. We then analyze neural recordings from the primate somatosensory cortex while a non-human primate performs reaches in different directions. In this setting, our method successfully models and clusters different conditions as separate subsystems, which is consistent with supervised single-LDS fits for each condition. Finally, we apply this approach to another neural dataset where monkeys perform a sequential reaching task. These results demonstrate that MoLDS provides an effective framework for modeling complex neural data in different brain regions, and that Tensor-EM is a principled and reliable approach to MoLDS learning for these applications.

Given the query, key and value matrices $Q, K, V\in \mathbb{R}^{n\times d}$, the attention matrix is defined as $\mathrm{Att}(Q, K, V)=D^{-1}AV$ where $A=\exp(QK^\top/\sqrt{d})$ with $\exp(\cdot)$ applied entrywise, $D=\mathrm{diag}(A{\bf 1}_n)$. The attention matrix is the backbone of modern transformers and large language models, but explicitly forming the softmax matrix $D^{-1}A$ incurs $\Omega(n^2)$, motivating numerous approximation schemes that reduce runtime to $\widetilde O(nd)$ via sparsity or low-rank factorization. We propose a quantum data structure that approximates any row of $\mathrm{Att}(Q, K, V)$ using only row queries to $Q, K, V$. Our algorithm preprocesses these matrices in $\widetilde{O}\left( \epsilon^{-1} n^{0.5} \left( s_\lambda^{2.5} + s_\lambda^{1.5} d + \alpha^{0.5} d \right) \right)$ time, where $\epsilon$ is the target accuracy, $s_\lambda$ is the $\lambda$-statistical dimension of the exponential kernel defined by $Q$ and $K$, and $\alpha$ measures the row distortion of $V$ that is at most $d/{\rm srank}(V)$, the stable rank of $V$. Each row query can be answered in $\widetilde{O}(s_\lambda^2 + s_\lambda d)$ time. To our knowledge, this is the first quantum data structure that approximates rows of the attention matrix in sublinear time with respect to $n$. Our approach relies on a quantum Nystr{\"o}m approximation of the exponential kernel, quantum multivariate mean estimation for computing $D$, and quantum leverage score sampling for the multiplication with $V$.

Scaling test-time compute has emerged as an effective strategy for improving the performance of large language models. However, existing methods typically allocate compute uniformly across all queries, overlooking variation in query difficulty. To address this inefficiency, we formulate test-time compute allocation as a novel bandit learning problem and propose adaptive algorithms that estimate query difficulty on the fly and allocate compute accordingly. Compared to uniform allocation, our algorithms allocate more compute to challenging queries while maintaining accuracy on easier ones. Among challenging queries, our algorithms further learn to prioritize solvable instances, effectively reducing excessive computing on unsolvable queries. We theoretically prove that our algorithms achieve better compute efficiency than uniform allocation and empirically validate their effectiveness on math and code benchmarks. Specifically, our algorithms achieve up to an 11.10\% performance improvement (15.04\% relative) on the MATH-500 dataset, up to 10.82\% (14.44\% relative) on the AIME25 dataset, and up to an 11.23\% performance improvement (15.29\% relative) on the LiveCodeBench dataset.

Due to tractable analysis and control, linear state-space models (LSSMs) provide a fundamental mathematical tool for time-series data modeling in various disciplines. In particular, many LSSMs have sparse system matrices because interactions among variables are limited or only a few significant relationships exist. However, current learning algorithms for LSSMs lack the ability to learn system matrices with the sparsity constraint due to the similarity transformation. To address this issue, we impose sparsity-promoting priors on system matrices to balance modeling error and model complexity. By taking hidden states of LSSMs as latent variables, we then explore the expectation-maximization (EM) algorithm to derive a maximum a posteriori (MAP) estimate of both hidden states and system matrices from noisy observations. Based on the Global Convergence Theorem, we further demonstrate that the proposed learning algorithm yields a sequence converging to a local maximum or saddle point of the joint posterior distribution. Finally, experimental results on simulation and real-world problems illustrate that the proposed algorithm can preserve the inherent topological structure among variables and significantly improve prediction accuracy over classical learning algorithms.

Estimating heterogeneous treatment effects (HTEs) in time-varying settings is particularly challenging, as the probability of observing certain treatment sequences decreases exponentially with longer prediction horizons. Thus, the observed data contain little support for many plausible treatment sequences, which creates severe overlap problems. Existing meta-learners for the time-varying setting typically assume adequate treatment overlap, and thus suffer from exploding estimation variance when the overlap is low. To address this problem, we introduce a novel overlap-weighted orthogonal WO meta-learner for estimating HTEs that targets regions in the observed data with high probability of receiving the interventional treatment sequences. This offers a fully data-driven approach through which our WO-learner can counteract instabilities as in existing meta-learners and thus obtain more reliable HTE estimates. Methodologically, we develop a novel Neyman-orthogonal population risk function that minimizes the overlap-weighted oracle risk. We show that our WO-learner has the favorable property of Neyman-orthogonality, meaning that it is robust against misspecification in the nuisance functions. Further, our WO-learner is fully model-agnostic and can be applied to any machine learning model. Through extensive experiments with both transformer and LSTM backbones, we demonstrate the benefits of our novel WO-learner.

Existing time series forecasting methods primarily rely on the numerical data itself. However, real-world time series exhibit complex patterns associated with multimodal information, making them difficult to predict with numerical data alone. While several multimodal time series forecasting methods have emerged, they either utilize text with limited supplementary information or focus merely on representation extraction, extracting minimal textual information for forecasting. To unlock the Value of Text, we propose VoT, a method with Event-driven Reasoning and Multi-level Alignment. Event-driven Reasoning combines the rich information in exogenous text with the powerful reasoning capabilities of LLMs for time series forecasting. To guide the LLMs in effective reasoning, we propose the Historical In-context Learning that retrieves and applies historical examples as in-context guidance. To maximize the utilization of text, we propose Multi-level Alignment. At the representation level, we utilize the Endogenous Text Alignment to integrate the endogenous text information with the time series. At the prediction level, we design the Adaptive Frequency Fusion to fuse the frequency components of event-driven prediction and numerical prediction to achieve complementary advantages. Experiments on real-world datasets across 10 domains demonstrate significant improvements over existing methods, validating the effectiveness of our approach in the utilization of text. The code is made available at https://anonymous.4open.science/r/VoT-465C.

When continual test-time adaptation (TTA) persists over the long term, errors accumulate in a model and further lead it to predict only a few classes regardless of the input, known as model collapse. Recent studies have explored reset strategies that erase these accumulated errors completely. However, their periodic resets lead to suboptimal adaptation, as they occur independently of collapse. Also, their full resets cause the catastrophic loss of knowledge acquired over time, even though it could be beneficial in future. To this end, we propose 1) an Adaptive and Selective Reset (ASR) scheme that dynamically determines when and where to reset, 2) an importance-aware regularizer to recover essential knowledge lost from reset, and 3) an on-the-fly adaptation adjustment scheme to enhance adaptability under challenging domain shifts. Extensive experiments across long-term TTA benchmarks demonstrate the effectiveness of our approach, particularly under challenging conditions. Our code will be released.

We aim to learn wavefunctions simulated by time-dependent density functional theory (TDDFT), which can be efficiently represented as linear combination coefficients of atomic orbitals. In real-time TDDFT, the electronic wavefunctions of a molecule evolve over time in response to an external excitation, enabling first-principles predictions of physical properties such as optical absorption, electron dynamics, and high-order response. However, conventional real-time TDDFT relies on time-consuming propagation of all occupied states with fine time steps. In this work, we propose OrbEvo, which is based on an equivariant graph transformer architecture and learns to evolve the full electronic wavefunction coefficients across time steps. First, to account for external field, we design an equivariant conditioning to encode both strength and direction of external electric field and break the symmetry from SO(3) to SO(2). Furthermore, we design two OrbEvo models, OrbEvo-WF and OrbEvo-DM, using wavefunction pooling and density matrix as interaction method, respectively. Motivated by the central role of the density functional in TDDFT, OrbEvo-DM encodes the density matrix aggregated from all occupied electronic states into feature vectors via tensor contraction, providing a more intuitive approach to learn the time evolution operator. We adopt a training strategy specifically tailored to limit the error accumulation of time-dependent wavefunctions over autoregressive rollout. To evaluate our approach, we generate TDDFT datasets consisting of 5,000 different molecules in the QM9 dataset and 1,500 molecular configurations of the malonaldehyde molecule in the MD17 dataset. Results show that our OrbEvo model accurately captures quantum dynamics of excited states under external field, including time-dependent wavefunctions, time-dependent dipole moment, and optical absorption spectra characterized by dipole oscillator strength. It also shows strong generalization capability on the diverse molecules in the QM9 dataset.

Sudden concept drift makes previously trained predictors unreliable, yet deciding when to retrain and what post-drift data size is sufficient is rarely addressed. We propose CALIPER —a detector- and model-agnostic, data-only test that estimates the post-drift data size required for stable retraining. CALIPER exploits state dependence in streams generated by dynamical systems: we run a single-pass weighted local regression over the post-drift window and track a one-step proxy error as a function of a locality parameter $\theta$. When an effective sample size gate is satisfied, a monotonically non-increasing trend in this error with increasing a locality parameter indicates that the data size is sufficiently informative for retraining. We also provide a theoretical analysis of our method, and we show that the algorithm has a low per-update time and memory. Across datasets from four heterogeneous domains, three learner families, and two detectors, CALIPER consistently matches or exceeds the best fixed data size for retraining while incurring negligible overhead and often outperforming incremental updates. CALIPER closes the gap between drift detection and data-sufficient adaptation in streaming learning.