ICLR 2026 Papers

This work presents DCFlow, a novel self-supervised cross-modal flow estimation framework that integrates a decoupled optimization strategy and a cross-modal consistency constraint. Unlike previous unsupervised approaches that implicitly learn flow estimation solely from appearance similarity, we introduce a decoupled optimization strategy with task-specific supervision to address modality discrepancy and geometric misalignment distinctly. This is achieved by collaboratively training a modality transfer network and a flow estimation network. To enable reliable motion supervision without ground-truth flow, we propose a geometry-aware data synthesis pipeline combined with an outlier-robust loss. Additionally, we introduce a cross-modal consistency constraint to jointly optimize both networks, significantly improving flow prediction accuracy. For evaluation, we construct a comprehensive cross-modal flow benchmark by repurposing public datasets. Experimental results demonstrate that DCFlow can be integrated with various flow estimation networks and achieves state-of-the-art performance among unsupervised approaches.

The powerful cooperation of federated learning (FL) and differential privacy (DP) provides a promising paradigm for the large-scale private clients. However, existing analyses in FL-DP mostly rely on the composition theorem and cannot tightly quantify the privacy leakage challenges, which is tight for a few communication rounds but yields an arbitrarily loose and divergent bound eventually. This also implies a counterintuitive judgment, suggesting that FL-DP may not provide adequate privacy support during long-term training under constant-level noisy perturbations, yielding discrepancy between the theoretical and experimental results. To further investigate the convergent privacy and reliability of the FL-DP framework, in this paper, we comprehensively evaluate the worst privacy of two classical methods under the non-convex and smooth objectives based on the $f$-DP analysis. With the aid of the shifted interpolation technique, we successfully prove that privacy in Noisy-FedAvg has a tight convergent bound. Moreover, with the regularization of the proxy term, privacy in Noisy-FedProx has a stable constant lower bound. Our analysis further demonstrates a solid theoretical foundation for the reliability of privacy in FL-DP. Meanwhile, our conclusions can also be losslessly converted to other classical DP analytical frameworks, e.g. $(\epsilon,\delta)$-DP and R$\'{e}$nyi-DP (RDP), to provide more fine-grained understandings for the FL-DP frameworks.

Dynamic feature transformation (the rich regime) does not always align with predictive performance (better representation), yet accuracy is often used as a proxy for richness, limiting analysis of their relationship. We propose a computationally efficient, performance-independent metric of richness grounded in the low-rank bias of rich dynamics, which recovers neural collapse as a special case. The metric is empirically more stable than existing alternatives and captures known lazy-to-rich transitions (e.g., grokking) without relying on accuracy. We further use it to examine how training factors (e.g., learning rate) relate to richness, confirming recognized assumptions and highlighting new observations (e.g., batch normalization promote rich dynamics). An eigendecomposition-based visualization is also introduced to support interpretability, together providing a diagnostic tool for studying the relationship between training factors, dynamics, and representations.

Reinforcement Learning with Verifiable Reward (RLVR) has equipped large language models (LLMs) with the capability of reasoning over complicated logical problems through policy optimization. However, conventional methods require complete annotation of the entire dataset and allocate computation uniformly over all samples. We articulate the lottery sample hypothesis in policy optimization of LLMs: a large training set contains a small subset that, when trained alone, yields performance comparable to that of the full dataset. This paper therefore explores the following question: How can we identify these lottery-winning samples from the original dataset without access to answers? Unlike prior efforts that analyze the effect of different samples in the training set with complete annotation, this paper focuses on the unsupervised discovery of critical instances for LLM reasoning and proposes a novel framework termed Complementary Conformal Selection (CONST). Specifically, CONST evaluates the importance of samples by considering two complementary components: procedural volatility and outcome volatility. Procedural volatility measures the potential variations during the LLM’s reasoning process, while outcome volatility captures inconsistencies in the final answer. Subsequently, conformal prediction is used to obtain a prediction set whose cardinality serves as the criterion for selecting the lottery-winning samples for annotation. We also provide a theoretical analysis, showing that CONST can effectively approximate the optimal policy. Extensive experiments on various LLMs across different datasets demonstrate the effectiveness of CONST.

Functional data, representing curves or trajectories, are ubiquitous in fields like biomedicine and motion analysis. A fundamental challenge is phase variability—temporal misalignments that obscure underlying patterns and degrade model performance. Current methods often address registration (alignment) and classification as separate, sequential tasks. This paper introduces DeepFRC, an end-to-end deep learning framework that jointly learns diffeomorphic warping functions and a classifier within a unified architecture. DeepFRC combines a neural deformation operator for elastic alignment, a spectral representation using Fourier basis for smooth functional embedding, and a class-aware contrastive loss that promotes both intra-class coherence and inter-class separation. We provide the first theoretical guarantees for such a joint model, proving its ability to approximate optimal warpings and establishing a data-dependent generalization bound that formally links registration fidelity to classification performance. Extensive experiments on synthetic and real-world datasets demonstrate that DeepFRC consistently outperforms state-of-the-art methods in both alignment quality and classification accuracy, while ablation studies validate the synergy of its components. DeepFRC also shows notable robustness to noise, missing data, and varying dataset scales. Code is available at https://github.com/Drivergo-93589/DeepFRC.

Causal representation learning has attracted significant research interest during the past few years, as a means for improving model generalization and robustness. Causal representations of interventional image pairs (also called ``actionable counterfactuals'' in the literature), have the property that only variables corresponding to scene elements affected by the intervention / action are changed between the start state and the end state. While most work in this area has focused on identifying and representing the variables of the scene under a causal model, fewer efforts have focused on representations of the interventions themselves. In this work, we show that an effective strategy for improving out of distribution (OOD) robustness is to focus on the representation of actionable counterfactuals in the latent space. Specifically, we propose that an intervention can be represented by a Causal Delta Embedding that is invariant to the visual scene and sparse in terms of the causal variables it affects. Leveraging this insight, we propose a method for learning causal representations from image pairs, without any additional supervision. Experiments in the Causal Triplet challenge demonstrate that Causal Delta Embeddings are highly effective in OOD settings, significantly exceeding baseline performance in both synthetic and real-world benchmarks.

Based on the prevalent intermediate fusion (IF) and late fusion (LF) frameworks, multimodal representation learning (MML) demonstrates its superiority over unimodal representation learning. To investigate the intrinsic factors underlying the empirical success of MML, research grounded in theoretical justifications from the perspective of generalization error has emerged. However, these provable MML studies derive the theoretical findings based on LF, while theoretical exploration based on IF remains scarce. This naturally gives rise to a question: **Can we design a comprehensive MML approach supported by the sufficient theoretical analysis across fusion types?** To this end, we revisit the IF and LF paradigms from a fine-grained dimensional perspective. The derived theoretical evidence sufficiently establishes the superiority of IF over LF under a specific constraint. Based on a general $K$-Lipschitz continuity assumption, we derive the generalization error upper bound of the IF-based methods, indicating that eliminating the distribution incoherence can improve the generalizability of IF-based MML methods. Building upon these theoretical insights, we establish a novel IF-based MML method, which introduces the informatic constraint and performs distribution cohering. Extensive experimental results on multiple widely adopted datasets verify the effectiveness of the proposed method.

Semi-supervised federated learning (SSFL) has emerged as an effective approach to leverage unlabeled data distributed across multiple data owners for improving model generalization. Existing SSFL methods typically assume that labeled and unlabeled data share the same label space. However, in realistic federated scenarios, unlabeled data often contain categories absent from the labeled set, i.e., outliers, which can severely degrade the performance of SSFL algorithms. In this paper, we address this under-explored issue, formally propose the open-set semi-supervised federated learning (OSSFL) problem, and develop the first OSSFL framework, FedOpenMatch. Our method adopts a one-vs-all (OVA) classifier as the outlier detector, equipped with logit adjustment to mitigate inlier-outlier imbalance and a gradient stop mechanism to reduce feature interference between the OVA and inlier classifiers. In addition, we introduce the logit consistency regularization loss, yielding more robust performance. Extensive experiments on standard benchmarks across diverse data settings demonstrate the effectiveness of FedOpenMatch, which significantly outperforms the baselines.

Positive-unlabeled (PU) learning is a weakly supervised binary classification problem, in which the goal is to learn a binary classifier from only positive and unlabeled data, without access to negative data. In recent years, many PU learning algorithms have been developed to improve model performance. However, experimental settings are highly inconsistent, making it difficult to identify which algorithm performs better. In this paper, we propose the first PU learning benchmark to systematically compare PU learning algorithms. During our implementation, we identify subtle yet critical factors that affect the realistic and fair evaluation of PU learning algorithms. On the one hand, many PU learning algorithms rely on a validation set that includes negative data for model selection. This is unrealistic in traditional PU learning settings, where no negative data are available. To handle this problem, we systematically investigate model selection criteria for PU learning. On the other hand, the problem settings and solutions of PU learning have different families, i.e., the one-sample and two-sample settings. However, existing evaluation protocols are heavily biased towards the one-sample setting and neglect the significant difference between them. We identify the internal label shift problem of unlabeled training data for the one-sample setting and propose a simple yet effective calibration approach to ensure fair comparisons within and across families. We hope our framework will provide an accessible, realistic, and fair environment for evaluating PU learning algorithms in the future.

Traditional multimodal learners find unified representations for tasks like visual question answering, but rely heavily on large paired datasets. However, an overlooked yet potentially powerful question is: can one leverage auxiliary $\textit{unpaired}$ multimodal data to directly enhance representation learning in a $\textit{target}$ modality? We introduce $\textbf{UML}$: $\textbf{U}$npaired $\textbf{M}$ultimodal $\textbf{L}$earner, a modality-agnostic training paradigm in which a single model alternately processes inputs from different modalities while sharing parameters across them. This design exploits the assumption that different modalities are projections of a shared underlying reality, allowing the model to benefit from cross-modal structure without requiring explicit pairs. Theoretically, under linear data-generating assumptions, we show that unpaired auxiliary data can yield representations strictly more informative about the world than unimodal training. Empirically, we show that incorporating unpaired data that share underlying semantic information from auxiliary modalities—such as text, audio, or images—consistently improves downstream performance across diverse unimodal targets such as image and audio.

Unsupervised contrastive learning has shown significant performance improvements in recent years, often approaching or even rivaling supervised learning in various tasks. However, its learning mechanism is fundamentally different from supervised learning. Previous works have shown that difficult examples (well-recognized in supervised learning as examples around the decision boundary), which are essential in supervised learning, contribute minimally in unsupervised settings. In this paper, perhaps surprisingly, we find that the direct removal of difficult examples, although reduces the sample size, can boost the downstream classification performance of contrastive learning. To uncover the reasons behind this, we develop a theoretical framework modeling the similarity between different pairs of samples. Guided by this framework, we conduct a thorough theoretical analysis revealing that the presence of difficult examples negatively affects the generalization of contrastive learning. Furthermore, we demonstrate that the removal of these examples, and techniques such as margin tuning and temperature scaling can enhance its generalization bounds, thereby improving performance. Empirically, we propose a simple and efficient mechanism for selecting difficult examples and validate the effectiveness of the aforementioned methods, which substantiates the reliability of our proposed theoretical framework.

We introduce Universal Beta Splatting (UBS), a unified framework that generalizes 3D Gaussian Splatting to N-dimensional anisotropic Beta kernels for explicit radiance field rendering. Unlike fixed Gaussian primitives, Beta kernels enable controllable dependency modeling across spatial, angular, and temporal dimensions within a single representation. Our unified approach captures complex light transport effects, handles anisotropic view-dependent appearance, and models scene dynamics without requiring auxiliary networks or specific color encodings. UBS maintains backward compatibility by approximating to Gaussian Splatting as a special case, guaranteeing plug-in usability and lower performance bounds. The learned Beta parameters naturally decompose scene properties into interpretable without explicit supervision: spatial (surface vs. texture), angular (diffuse vs. specular), and temporal (static vs. dynamic). Our CUDA-accelerated implementation achieves real-time rendering while consistently outperforming existing methods across static, view-dependent, and dynamic benchmarks, establishing Beta kernels as a scalable universal primitive for radiance field rendering.

The transformer architecture has demonstrated remarkable capabilities in modern artificial intelligence, among which the capability of implicitly learning an internal model during inference time is widely believed to play a key role in the understanding of pre-trained large language models. However, most recent works have been focusing on studying supervised learning topics such as in-context learning, leaving the field of unsupervised learning largely unexplored. This paper investigates the capabilities of transformers in solving Gaussian Mixture Models (GMMs), a fundamental unsupervised learning problem through the lens of statistical estimation. We propose a transformer-based learning framework called Transformer for Gaussian Mixture Models (TGMM) that simultaneously learns to solve multiple GMM tasks using a shared transformer backbone. The learned models are empirically demonstrated to effectively mitigate the limitations of classical methods such as Expectation-Maximization (EM) or spectral algorithms, at the same time exhibit reasonable robustness to distribution shifts. Theoretically, we prove that transformers can efficiently approximate both the Expectation-Maximization (EM) algorithm and a core component of spectral methods—namely, cubic tensor power iterations. These results not only improve upon prior work on approximating the EM algorithm, but also provide, to our knowledge, the first theoretical guarantee that transformers can approximate high-order tensor operations. Our study bridges the gap between practical success and theoretical understanding, positioning transformers as versatile tools for unsupervised learning.

In recent years, Contrastive Learning (CL) has garnered significant attention due to its efficacy across various domains, spanning from visual and textual modalities. A fundamental aspect of CL is aligning the representations of anchor instances with relevant positive samples while simultaneously separating them from negative ones. Prior studies have extensively explored diverse strategies for generating and sampling contrastive (i.e., positive/negative) pairs. Despite the empirical success, the theoretical understanding of the CL approach remains under-explored, leaving questions such as the rationale behind contrastive-pair sampling and its contributions to the model performance unclear. This paper addresses this gap by providing a comprehensive theoretical analysis from the angle of optimality conditions and introducing the SnaPhArd Contrast Learning (SPACL). Specifically, SPACL prioritizes hard positive and hard negative samples during constructing contrastive pairs and computing the contrastive loss, rather than treating all samples equally. Experimental results across two downstream tasks demonstrate that SPACL consistently outperforms or competes favorably with state-of-the-art methods, showcasing its robustness and efficacy. A comprehensive ablation study further examines the effectiveness of SPACL's individual components to verify the theoretic findings.

Unsupervised 3D representation learning reduces the burden of labeling multimodal 3D data for fusion perception tasks. Among different pre-training paradigms, differentiable-rendering-based methods have shown most promise. However, existing works separately conduct pre-training for each modalities due to computational costs of processing large point clouds with images. As such, mutual benefit of high-level semantics (from image) and 3D structure (from point cloud) has not been exploited. To address this gap, we propose a joint unsupervised differentiable-rendering-based pre-training method for images and point clouds, termed CLAP, short for Curvature sampLing and leArnable Prototype. Specifically, our method overcomes the computational hurdle by Curvature Sampling to select the more informative points/pixels for pre-training. To uncover the performance benefits brought by their complementarity, we propose to use learnable prototypes to represent parts of the 3D scenes in a common feature space and an Expectation-Maximization training scheme to associate embeddings of each modality to prototypes. We further propose a swapping prediction loss that explores their interplay through prototypes along with a Gram Matrix Regularization term to maintain training stability. Experiments on NuScenes and Waymo datasets show that CLAP achieves up to 100% more performance gain as compared to previous SOTA pre-training methods. Codes and models will be released.

As an important branch of unsupervised learning, deep clustering has seen substantial progress in recent years. However, the majority of current deep clustering methods operate under the assumption of balanced or near-balanced cluster distributions. This assumption contradicts the common long-tailed class distributions in real-world data, leading to severe performance degradation in deep clustering. Although many long-tailed learning methods have been proposed, these approaches typically rely on label information to differentiate treatment across different classes, which renders them inapplicable to deep clustering scenarios. How to re-weight the training of deep clustering models in an unsupervised setting remains an open challenge. To address this, we propose a mini-cluster guided long-tailed deep clustering method, termed MiniClustering. We introduce a specialized clustering head that divide data into much more clusters than the target number of clusters. These predicted clusters are referred to as mini-clusters. The mini-cluster-level predictions serve as the guide for estimating the appropriate weights for classes with varying degrees of long-tailedness. The weights are then incorporated to re-weight the self-training loss in model training. In this way, we can mitigate model bias by re-weighting gradients from different classes. We evaluate our method on multiple benchmark datasets with different imbalance ratios to demonstrate its effectiveness. Further, our method can be readily applied to the downstream of existing unsupervised representation learning frameworks for long-tailed deep clustering. It can also adapt label-dependent long-tailed learning methods to unsupervised clustering tasks by leveraging the estimated weights. *The code is available in the Supplementary Material.*

Image clustering is a fundamental task in visual machine learning. A key research direction in this field is the incorporation of prior knowledge. Recently, such prior knowledge has evolved from internal compactness constraints to external textual guidance. In particular, the introduction of textual modalities through CLIP has demonstrated impressive performance. However, CLIP is designed primarily for image–text alignment and may not be sufficient to capture clustering structures. Moreover, existing approaches often assume that textual features are universally beneficial, overlooking their varying suitability for different datasets. To address these issues, we propose to use spatial structure and selective text to jointly facilitate image clustering (SATC). Specifically, we design a graph attention network (GAT)-based encoder to capture relational dependencies among image patches, thereby extracting spatial features to facilitate clustering. In addition, we introduce a textual feature selector that uses the potential clustering compactness of textual features as the selection criterion and adaptively integrates them into the clustering process. Theoretical guidance is provided for this selector. Finally, the cluster assignment is produced through Tri-modal mutual distillation. Extensive experiments on 18 benchmark datasets demonstrate the effectiveness of SATC. The experimental results further verify the rationality of the textual feature selector. The code will be published.

Flow map models such as Consistency Models (CM) and Mean Flow (MF) enable few-step generation by learning the long jump of the ODE solution of diffusion models, yet training remains unstable, sensitive to hyperparameters, and costly. Initializing from a pre-trained diffusion model helps, but still requires converting infinitesimal steps into a long-jump map, leaving instability unresolved. We introduce *mid-training*, the first concept and practical method that inserts a lightweight intermediate stage between the (diffusion) pre-training and the final flow map training (i.e., post-training) for vision generation. Concretely, *Consistency Mid-Training* (CMT) is a compact and principled stage that trains a model to map points along a solver trajectory from a pre-trained model, starting from a prior sample, directly to the solver-generated clean sample. It yields a trajectory-consistent and stable initialization. This initializer outperforms random and diffusion-based baselines and enables fast, robust convergence without heuristics. Initializing post-training with CMT weights further simplifies flow map learning. Empirically, CMT achieves state-of-the-art two-step FIDs of 1.97 (CIFAR-10), 1.32 (ImageNet $64\times64$), and 1.84 (ImageNet $512\times512$), using up to $98$\% less training data and GPU time than CMs. On ImageNet $256\times256$, it attains 1-step FID 3.34 with $\sim50$\% less training than MF from scratch (FID 3.43). On MSCOCO T2I, CMT reaches the best FID with $\sim47$\% less training. This establishes CMT as a principled, efficient, and general framework for training flow map models. Code and models are available at https://github.com/sony/cmt.

This paper proposes Kernelized Dynamics Pruning (KDP), a novel layer pruning method from the perspective of simplifying representation dynamics within large language models (LLMs). Motivated by the high similarity between consecutive layer representations, we view the LLM's forward pass as a discrete-time dynamical system. We speculate that this phenomenon indicates the model's internal dynamics have entered a ``slow manifold'', which exhibits computational redundancy. Based on this insight, we project the representations into a kernel space where the complex, non-linear transformation between them is simplified to an approximately linear one. Then, a simple network learns the inverse kernel transformation, thereby enabling the pruning of the entire layer block. Both theoretical analysis and extensive experiments validate the effectiveness of KDP, demonstrating its superiority over existing pruning baselines. Code is available at https://anonymous.4open.science/r/draft-123abc.