Invariant Representation Learning

Camera recapture introduces complex optical degradations, such as perspective warping, illumination shifts, and Moiré interference, that remain challenging for deep watermarking systems. We present TIACam, a text-anchored invariant feature learning framework with auto-augmentation for camera-robust zero-watermarking. The method integrates three key innovations: (1) a learnable auto-augmentor that discovers camera-like distortions through differentiable geometric, photometric, and Moiré operators; (2) a text-anchored invariant feature learner that enforces semantic consistency via cross-modal adversarial alignment between image and text; and (3) a zero-watermarking head that binds binary messages in the invariant feature space without modifying image pixels. This unified formulation jointly optimizes invariance, semantic alignment, and watermark recoverability. Extensive experiments on both synthetic and real-world camera captures demonstrate that TIACam achieves state-of-the-art feature stability and watermark extraction accuracy, establishing a principled bridge between multimodal invariance learning and physically robust zero-watermarking.

Nonstationary time series forecasting suffers from the distribution shift issue due to the different distributions that produce the training and test data. Existing methods attempt to alleviate the dependence by, e.g., removing low-order moments from each individual sample. These solutions fail to capture the underlying time-evolving structure across samples and do not model the complex time structure. In this paper, we aim to address the distribution shift in the frequency space by considering all possible time structures. To this end, we propose a Time-Invariant Frequency Operator (TIFO), which learns stationarity-aware weights over the frequency spectrum across the entire dataset. The weight representation highlights stationary frequency components while suppressing non-stationary ones, thereby mitigating the distribution shift issue in time series. To justify our method, we show that the Fourier transform of time series data implicitly induces eigen-decomposition in the frequency space. TIFO is a plug-and-play approach that can be seamlessly integrated into various forecasting models. Experiments demonstrate our method achieves 18 top-1 and 6 top-2 results out of 28 forecasting settings. Notably, it yields 33.3% and 55.3% improvements in average MSE on the ETTm2 dataset. In addition, TIFO reduces computational costs by 60% -70% compared to baseline methods, demonstrating strong scalability across diverse forecasting models.

Data augmentation is one of the most widely used techniques to improve generalization in modern machine learning, often justified by its ability to promote invariance to label-irrelevant transformations. However, its theoretical role remains only partially understood. In this work, we propose an information-theoretic framework that systematically accounts for the effect of augmentation on generalization and invariance learning. Our approach builds upon mutual information-based bounds, which relate the generalization gap to the amount of information a learning algorithm retains about its training data. We extend this framework by modeling the augmented distribution as a composition of the original data distribution with a distribution over transformations, which naturally induces an orbit-averaged loss function. Under mild sub-Gaussian assumptions on the loss function and the augmentation process, we derive a new generalization bound that decompose the expected generalization gap into three interpretable terms: (1) a distributional divergence between the original and augmented data, (2) a stability term measuring the algorithm dependence on training data, and (3) a sensitivity term capturing the effect of augmentation variability. To connect our bounds to the geometry of the augmentation group, we introduce the notion of group diameter, defined as the maximal perturbation that augmentations can induce in the input space. The group diameter provides a unified control parameter that bounds all three terms and highlights an intrinsic trade-off: small diameters preserve data fidelity but offer limited regularization, while large diameters enhance stability at the cost of increased bias and sensitivity. We validate our theoretical bounds with numerical experiments, demonstrating that it reliably tracks and predicts the behavior of the true generalization gap.

Joint-embedding self-supervised learning (SSL), the key paradigm for unsupervised representation learning from visual data, learns from invariances between semantically-related data pairs. We study the one-to-many mapping problem in SSL, where each datum may be mapped to multiple valid targets. This arises when data pairs come from naturally occurring generative processes, e.g., successive video frames. We show that existing methods struggle to flexibly capture this conditional uncertainty. As a remedy, we introduce a latent variable to account for this uncertainty and derive a variational lower bound on the mutual information between paired embeddings. Our derivation yields a simple regularization term for standard SSL objectives. The resulting method, which we call AdaSSL, applies to both contrastive and distillation-based SSL objectives, and we empirically show its versatility in causal representation learning, fine-grained image understanding, and world modeling on videos.

Online handwritten character recognition leverages stroke order and dynamic features, which generally provide higher accuracy and robustness compared with offline recognition. However, in practical applications, rotational deformations can disrupt the spatial layout of strokes, substantially reducing recognition accuracy. Extracting rotation-invariant features therefore remains a challenging open problem. In this work, we employ the Sliding Window Path Signature (SW-PS) to capture local structural features of characters, and introduce the lightweight Linear Recurrent Units (LRU) as the classifier. The LRU combine the fast incremental processing capability of recurrent neural networks (RNN) with the efficient parallel training of state space models (SSM), while reliably modelling dynamic stroke characteristics. We conducted recognition experiments with random rotation angle up to $\pm 180^{\circ}$ on three subsets of the CASIA-OLHWDB1.1 dataset: digits, English upper letters, and Chinese radicals. The accuracies achieved after ensemble learning were $99.62\%$, $96.67\%$, and $94.33\%$, respectively. Experimental results demonstrate that the proposed SW-PS+LRU framework consistently surpasses competing models in both convergence speed and test accuracy.

Machine Learning (ML) has deeply changed some fields recently, like Language and Vision and we may expect it to be relevant also to the analysis of of complex systems. Here we want to tackle the question of how and to which extent can one regress scale-free processes, i.e. processes displaying power law behavior, like earthquakes or avalanches? We are interested in predicting the large ones, i.e. rare events in the training set which therefore require extrapolation capabilities of the model. For this we consider two paradigmatic problems that are statistically self-similar. The first one is a 2-dimensional fractional Gaussian field obeying linear dynamics, self-similar by construction and amenable to exact analysis. The second one is the Abelian sandpile model, exhibiting self-organized criticality. The emerging paradigm of Geometric Deep Learning shows that including known symmetries into the model's architecture is key to success. Here one may hope to extrapolate only by leveraging scale invariance. This is however a peculiar symmetry, as it involves possibly non-trivial coarse-graining operations and anomalous scaling. We perform experiments on various existing architectures like U-net, Riesz network (scale invariant by construction), or our own proposals: a wavelet-decomposition based Graph Neural Network (with discrete scale symmetry), a Fourier embedding layer and a Fourier-Mellin Neural Operator. Based on these experiments and a complete characterization of the linear case, we identify the main issues relative to spectral biases and coarse-grained representations, and discuss how to alleviate them with the relevant inductive biases.

The core of video-based visible-infrared person re-identification (VVI-ReID) lies in learning sequence-level modal-invariant representations across different modalities. Recent research tends to use modality-shared language prompts generated by CLIP to guide the learning of modal-invariant representations. Despite achieving optimal performance, such methods still face limitations in efficient spatial-temporal modeling, sufficient cross-modal interaction, and explicit modality-level loss guidance. To address these issues, we propose the language-driven sequence-level modal-invariant representation learning (LSMRL) method, which includes spatial-temporal feature learning (STFL) module, semantic diffusion (SD) module and cross-modal interaction (CMI) module. To enable parameter- and computation-efficient spatial-temporal modeling, the STFL module is built upon CLIP with minimal modifications. To achieve sufficient cross-modal interaction and enhance the learning of modal-invariant features, the SD module is proposed to diffuse modality-shared language prompts into visible and infrared features to establish preliminary modal consistency. The CMI module is further developed to leverage bidirectional cross-modal self-attention to eliminate residual modality gaps and refine modal-invariant representations. To explicitly enhance the learning of modal-invariant representations, two modality-level losses are introduced to improve the features' discriminative ability and their generalization to unseen categories. Extensive experiments on large-scale VVI-ReID datasets demonstrate the superiority of LSMRL over AOTA methods.

PCA can be used for rotation invariant features, describing a shape with its $p_{ab}=E[(x_i-E[x_a])(x_b-E[x_b])]$ covariance matrix approximating shape by ellipsoid, allowing for rotation invariants like its traces of powers. However, real shapes are usually much more complicated, hence there is proposed its extension to e.g. $p_{abc}=E[(x_a-E[x_a])(x_b-E[x_b])(x_c-E[x_c])]$ order-3 or higher tensors describing central moments, or polynomial times Gaussian allowing decodable shape descriptors of arbitrarily high accuracy, and their analogous rotation invariants. Its practical applications could be rotation-invariant features to include shape modulo rotation e.g. for molecular shape descriptors, or for up to rotation object recognition in 2D images/3D scans maybe also for 3D scene understanding, or shape similarity metric allowing inexpensive comparison of objects modulo rotation avoiding costly optimization over rotations.

Many learning problems are organized by group symmetries. While invariance is often imposed through architectures or group averaging, we ask when it can emerge from training on a finite random subset of an orbit. We study this question in classical Hopfield networks, where strict memorization can be expressed as a linear margin problem. Reparameterizing minimization of energy flow (MEF) as an exponential loss connects gradient descent to the corresponding minimum-norm hard-margin memorizer. Our main result shows that, for independent uniform samples from any finite permutation orbit, the exact sample hard-margin support vector machine (HSVM) concentrates exponentially around the invariant full-orbit HSVM. Consequently, an orbit-size-independent polynomial number of samples suffices both for approximate parameter invariance and for simultaneous memorization of every orbit element; directional convergence transfers this conclusion asymptotically to MEF gradient descent. For graph-isomorphism orbits, we characterize the invariant parameters as a three-dimensional subspace and show that every such orbit is memorizable. For cliques of fixed linear density, additional symmetry sharpens the uniform memorization bound to $O(v^4\log(1/δ))$, exponentially smaller than the orbit size. Together with experiments across several learning rules, these results give a finite-sample account of how optimization bias can recover symmetry from partial group-structured data.

Invariant learning on graphs aims to build predictors that rely on causal substructures rather than on environment-specific shortcuts. Current methods extract either a single invariant subgraph (CIGA) or, more recently, a set of diverse invariant subgraphs (SuGAr). We observe that the second half of the multi-subgraph pipeline, "how the extracted subgraphs are combined" has gone essentially unexamined; existing methods average branch predictions uniformly or select one greedily. We show that this is precisely where accuracy is lost. A minimum-variance combination argument establishes that uniform averaging is optimal only when every branch carries the same total error covariance with the ensemble---a knife edge balance that second order exchangeability guarantees and that the diversity regularizer used to obtain multiple subgraphs is explicitly designed to break. We then show that invariance is closed under post composition; because the branches are frozen and read the graph only through their invariant subgraphs, any measurable aggregator of their outputs remains invariant, and the aggregate's environment gap is bounded by the branches' residual invariance violation uniformly over aggregators. Learning the combination rule by plain empirical risk minimization is therefore safe, and aggregator capacity trades off against estimation error rather than robustness. PISA instantiates this with a lightweight instance-conditional MLP over stacked branch logits, trained on frozen branches at negligible cost. Across 15 synthetic and real benchmarks, PISA improves the average over the strongest prior method by 2.96 points on synthetic shifts and 1.74 points on realistic shifts, with per-dataset gains up to 5.90 points, and nearly doubles SuGAr's improvement over ERM on the six DrugOOD tasks (from +2.76 to +5.07 ROC-AUC).

The class imbalance problem refers to the disproportionate distribution of samples across different classes within a dataset, where the minority classes are significantly underrepresented. This issue is also prevalent in graph-structured data. Most graph neural networks (GNNs) implicitly assume a balanced class distribution and therefore often fail to account for the challenges introduced by class imbalance, which can lead to biased learning and degraded performance on minority classes. We identify a quality inconsistency problem in synthesized nodes, which leads to suboptimal performance under graph imbalance conditions. To mitigate this issue, we propose GraphIFE (Graph Invariant Feature Extraction), a novel framework designed to mitigate quality inconsistency in synthesized nodes. Our approach incorporates two key concepts from graph invariant learning and introduces strategies to strengthen the embedding space representation, thereby enhancing the model's ability to identify invariant features. Extensive experiments demonstrate the framework's efficiency and robust generalization, as GraphIFE consistently outperforms various baselines across multiple datasets. The code is publicly available at https://github.com/flzeng1/GraphIFE.

EEG foundation models are pretrained on heterogeneous corpora, but two challenges remain underexplored: (1) experimental protocols provide mental-process cues that are rarely used in masked pretraining, and (2) electrode montages vary across datasets, making spatial representations montage-dependent. Existing methods typically learn a single general-purpose representation and model spatial structure implicitly, limiting their ability to exploit process-related information under montage heterogeneity. We introduce BrainPro, a process-conditioned self-supervised pretraining framework that couples topographic retrieval with shared and process-associated representation learning. BrainPro maps dataset-specific montages to a universal channel-region template and retrieves channel- and region-level spatial filters to form a topographically aligned spatial basis. Over this basis, BrainPro learns a shared encoder for general EEG structure and additional affective, motor-related, and auxiliary residual branches for process-associated variation. Protocol-derived mental-process cues condition branch activation and region-weighted masked reconstruction, incorporating topographic spatial priors and process information. Across nine public BCI benchmarks, BrainPro achieves strong performance among evaluated baselines. Ablations, channel-drop analysis, encoder-configuration studies, and spatial-filter visualizations suggest that topographic retrieval and process-conditioned representation learning jointly improve EEG decoding. Our code is available at https://github.com/yi-ding-cs/BrainPro.

Predictors often exploit spurious correlations that fail under distribution shift, undermining out-of-distribution (OOD) robustness. Invariant pairs, such as the same object photographed against different backgrounds, directly indicate which variation should not affect prediction. Yet real pair collections are finite and noisy, so they can miss relevant variation or lead a learner to discard predictive information. To learn from such collections, we propose Geometric Robustness Invariant Training (GRIT), which estimates an unstable subspace from the covariance of pair differences in fixed pretrained features and projects it out before fitting a predictor. Because pair processing is separate from predictor fitting, the same projection can accompany ERM or other robustness objectives. Unlike quadratic prediction consistency, which penalizes variation along every direction with nonzero pair covariance, including variation introduced by pair noise, GRIT enforces exact invariance only to the estimated subspace. We bound GRIT's target risk in terms of pair count, noise relative to the weakest clean signal, and selected rank, assuming that the label distribution given a linear projection of the features is invariant across environments. Empirically, GRIT improves the mean OOD accuracy of each of eight learning objectives on ColoredMNIST-INV and Waterbirds-INV and has higher mean accuracy than a prediction-consistency penalty using the same pairs in 15 of 16 settings. On UrbanCars, combining pairs that vary the background with pairs that vary a co-occurring object raises worst-group accuracy above that of either pair type alone.

This paper studies continual test-time adaptation (CTTA), the task of adapting a model to constantly changing unseen domains in testing while preserving previously learned knowledge. Existing CTTA methods mostly focus on adaptation to the current test domain only, overlooking generalization to arbitrary test domains a model may face in the future. To tackle this limitation, we present a novel online test-time domain generalization framework for CTTA, dubbed TestDG. TestDG aims to learn features invariant to both current and previous test domains on the fly during testing, improving the potential for effective generalization to future domains. To this end, we propose a new model architecture and a test-time adaptation strategy dedicated to learning domain-invariant features, along with prototype selection and update mechanisms for effectively managing information from previous test domains. TestDG achieved state of the art on three of four public CTTA benchmarks and performed on par with the best prior method on ImageNet-C. Moreover, it showed superior generalization to unseen test domains.

Machine learning models are inherently bound to the distribution of the training data, often exploiting non-causal shortcuts. As a result, achieving robustness to spurious correlations remains a challenge. While existing approaches rely on data manipulation or re-weighting strategies to achieve robustness, they typically require dense group labels, multiple training domains, or specialized pre-processing. We propose Invariance Pair Guidance (IPG), a method to mitigate reliance on spurious correlations using a sparse set of counterfactual pairs. Unlike other methods demanding extensive supervision, IPG utilizes a novel dual-update mechanism to dynamically correct the optimization trajectory. We generate input pairs that isolate the spurious attribute to define the invariance, a characteristic that should not affect the outcome of the model. Based on these pairs, we define a corrective gradient that complements the traditional gradient descent approach. The correction adapts via a predefined invariance condition. Experiments on ColoredMNIST, Waterbirds-100, and CelebA datasets demonstrate the effectiveness of our approach and its robustness to group shifts, supported by a theoretical convergence analysis. IPG offers a data-efficient and theoretically grounded path to robustness.

Most self-supervised learning objectives defend against collapse but leave the target representation law unspecified. We formulate representation learning as Distribution Matching (DM), learning an augmentation-invariant encoder whose push-forward law matches an explicit reference law. The reference law specifies what the learned representation distribution should look like, while the matching mechanism determines how this target is matched. Within DM, generative learning and representation learning can be viewed as directionally inverse distributional processes: generative learning maps a tractable reference to data, whereas representation learning maps data to a designed reference law. For the Mallows-distance instantiation studied here, we connect the population objective to class-centre separation and classification error and derive a finite-sample bound on downstream classification error. Simulations and image benchmarks show manifold rectification, fine-grained structure and transferability.

Learning-based Android malware detectors degrade over time due to natural distribution drift caused by malware variants and new families. This paper systematically investigates the challenges classifiers trained with empirical risk minimization (ERM) face against such distribution shifts and attributes their shortcomings to their inability to learn stable discriminative features. Invariant learning theory offers a promising solution by encouraging models to generate stable representations across environments that expose the instability of the training set. However, the lack of prior environment labels, the diversity of drift factors, and low-quality representations caused by diverse families make this task challenging. To address these issues, we propose TIF, the first temporal invariant training framework for malware detection, which aims to enhance the ability of detectors to learn stable representations across time. TIF organizes environments based on application observation dates to reveal temporal drift, integrating specialized multi-proxy contrastive learning and invariant gradient alignment to generate and align environments with high-quality, stable representations. TIF can be seamlessly integrated into any learning-based detector. Experiments on a decade-long dataset show that TIF excels, particularly in early deployment stages, addressing real-world needs and outperforming state-of-the-art methods.

Continuous-Time Dynamic Graphs (CTDGs) enable fine-grained modeling of evolving relational systems. However, most existing CTDG representation learning methods are tailored to in-distribution settings and exhibit limited robustness under out-of-distribution (OOD) shifts. Although recent causal approaches learn invariant representations via interventions, they are primarily designed for static or discrete-time graphs and become computationally prohibitive for CTDGs due to the combinatorial explosion of structural and temporal variations. To address these challenges, we propose CIR, a framework grounded in a novel structural causal model termed the ICCM. To avoid exhaustive interventions, we leverage the Normalized Weighted Geometric Mean (NWGM) to efficiently approximate interventional predictions. We further instantiate ICCM within a practical deep learning architecture that jointly captures invariant structural and temporal patterns through dedicated subgraph extractors, and maintains an environment memory bank to model distributional shifts across evolving contexts. Extensive experiments demonstrate that CIR consistently outperforms existing methods under diverse OOD scenarios.

Density based representations of atomic environments that are invariant under Euclidean symmetries have become a widely used tool in the machine learning of interatomic potentials, broader data-driven atomistic modelling and the visualisation and analysis of materials datasets.The standard mechanism used to incorporate chemical element information is to create separate densities for each element and form tensor products between them. This leads to a steep scaling in the size of the representation as the number of elements increases. Graph neural networks, which do not explicitly use density representations, escape this scaling by mapping the chemical element information into a fixed dimensional space in a learnable way. We recast this approach as tensor factorisation by exploiting the tensor structure of standard neighbour density based descriptors. In doing so, we form compact tensor-reduced representations whose size does not depend on the number of chemical elements, but remain systematically convergeable and are therefore applicable to a wide range of data analysis and regression tasks.