Invariance

Indoor positioning is an essential technology for a wide range of applications in GNSS-denied environments, including indoor navigation and IoT systems. Combining convolutional neural networks (CNNs) and magnetic field-based features offers a low-cost, infrastructure-free solution for precise positioning. While magnetic fingerprints are a promising approach for indoor positioning, models trained on raw 3D magnetometer data are highly sensitive to device orientation. We address this by using two rotation invariant features derived from the 3D magnetic field: the norm (Mn) and the projection onto the gravity axis (Mg). We train a lightweight 7-layer dilated CNN (MagNetS/XL) on magnetic sequences to directly regress (x, y) positions. Using the MagPie dataset (three buildings, handheld trajectories), we systematically evaluate fixed and random rotations of test and/or train data. Raw 3D inputs (Mx, My , Mz) exhibit isotropic error increases under fixed 90° rotations and further degrade with growing random rotations. In contrast, 2D (Mn, Mg) inputs maintain rotation invariant accuracy and surpass the 3D inputs once rotation exceeds building-specific thresholds for three reference buildings: 0° for Loomis (large), 5° for Talbot (medium), and 6° for CSL (small). MagNetXL achieves or exceeds state-of-the-art accuracy on the MagPie dataset, and MagNetS delivers similar performance with roughly one third of the parameters, favoring mobile deployment. These results show that the robustness gained from rotation invariant inputs outweighs the loss of input dimensionality in realistic usage, allowing mapping and localization without orientation alignment or added infrastructure.

Self-supervised learning (SSL) is a standard approach for representation learning in aerial imagery. Existing methods enforce invariance between augmented views, which works well when augmentations preserve semantic content. However, aerial images are frequently degraded by haze, motion blur, rain, and occlusion that remove critical evidence. Enforcing alignment between a clean and a severely degraded view can introduce spurious structure into the latent space. This study proposes a training strategy and architectural modification to enhance SSL robustness to such corruptions. It introduces a per-sample, per-factor trust weight into the alignment objective, combined with the base contrastive loss as an additive residual. A stop-gradient is applied to the trust weight instead of a multiplicative gate. While a multiplicative gate is a natural choice, experiments show it impairs the backbone, whereas our additive-residual approach improves it. Using a 200-epoch protocol on a 210,000-image corpus, the method achieves the highest mean linear-probe accuracy among six backbones on EuroSAT, AID, and NWPU-RESISC45 (90.20% compared to 88.46% for SimCLR and 89.82% for VICReg). It yields the largest improvements under severe information-erasing corruptions on EuroSAT (+19.9 points on haze at s=5 over SimCLR). The method also demonstrates consistent gains of +1 to +3 points in Mahalanobis AUROC on a zero-shot cross-domain stress test using BDD100K weather splits. Two ablations (scalar uncertainty and cosine gate) indicate the additive-residual formulation is the primary source of these improvements. An evidential variant using Dempster-Shafer fusion introduces interpretable signals of conflict and ignorance. These findings offer a concrete design principle for uncertainty-aware SSL. Code is publicly available at https://github.com/WadiiBoulila/trust-ssl.

Composed Image Retrieval (CIR) is a challenging image retrieval paradigm that enables to retrieve target images based on multimodal queries consisting of reference images and modification texts. Although substantial progress has been made in recent years, existing methods assume that all samples are correctly matched. However, in real-world scenarios, due to high triplet annotation costs, CIR datasets inevitably contain annotation errors, resulting in incorrectly matched triplets. To address this issue, the problem of Noisy Triplet Correspondence (NTC) has attracted growing attention. We argue that noise in CIR can be categorized into two types: cross-modal correspondence noise and modality-inherent noise. The former arises from mismatches across modalities, whereas the latter originates from intra-modal background interference or visual factors irrelevant to the coarse-grained modification annotations. However, modality-inherent noise is often overlooked, and research on cross-modal correspondence noise remains nascent. To tackle above issues, we propose the Invariance and discrimiNaTion-awarE Noise neTwork (INTENT), comprising two components: Visual Invariant Composition and Bi-Objective Discriminative Learning, specifically designed to handle the two-aspect noise. The former applies causal intervention on the visual side via Fast Fourier Transform (FFT) to generate intervened composed features, enforcing visual invariance and enabling the model to ignore modality-inherent noise during composition. The latter adopts collaborative optimization with both positive and negative samples, and constructs a scalable decision boundary that dynamically adjusts decisions based on the loyalty degree, enabling robust correspondence discrimination. Extensive experiments on two widely used benchmark datasets demonstrate the superiority and robustness of INTENT.

Autonomous agent systems are governed by enforcement mechanisms that flag hard constraint violations at runtime. The Agent Control Protocol identifies a structural limit of such systems: a correctly-functioning enforcement engine can enter a regime in which behavioral drift is invisible to it, because the enforcement signal operates below the layer where deviation is measurable. We show that enforcement-based governance is structurally unable to determine whether an agent behavior remains within the admissible behavior space A0 established at admission time. Our central result, the Non-Identifiability Theorem, proves that A0 is not in the sigma-algebra generated by the enforcement signal g under the Local Observability Assumption, which every practical enforcement system satisfies. The impossibility arises from a fundamental mismatch: g evaluates actions locally against a point-wise rule set, while A0 encodes global, trajectory-level behavioral properties set at admission time. An agent can therefore drift -- systematically shifting its behavioral distribution away from admission-time expectations -- while every individual action remains within the permitted action space. We define the Invariant Measurement Layer (IML), which bypasses this limitation by retaining direct access to the generative model of A0, restoring observability precisely in the region where enforcement is structurally blind. We prove an information-theoretic impossibility for enforcement-based monitoring and show IML detects admission-time drift with provably finite detection delay. Validated across four settings: three drift scenarios (300 and 1000 steps), a live n8n webhook pipeline, and a LangGraph StateGraph agent -- enforcement triggers zero violations while IML detects each drift type within 9-258 steps of drift onset.

Large language models exhibit systematic limitations in structured logical reasoning: they conflate hypothesis generation with verification, cannot distinguish conjecture from validated knowledge, and allow weak reasoning steps to propagate unchecked through inference chains. We present a symbolic reasoning scaffold that operationalizes Peirce's tripartite inference -- abduction, deduction, and induction -- as an explicit protocol for LLM-assisted reasoning. The framework enforces logical consistency through five algebraic invariants (the Gamma Quintet), the strongest of which -- the Weakest Link bound -- ensures that no conclusion in a reasoning chain can exceed the reliability of its least-supported premise. This principle, independently grounded as weakest link resolution in possibilistic logic and empirically validated for chain-of-thought reasoning, prevents logical inconsistencies from accumulating across multi-step inference. We verify all invariants through a property-based testing suite of 100 properties and 16 fuzz tests over 10^5+ generated cases, providing a verified reference implementation of the invariants suitable as a foundation for future reasoning benchmarks.

LLM-based coding agents can generate functionally correct GPU kernels, yet their performance remains far below hand-optimized libraries on critical computations such as matrix multiplication, attention, and Mixture-of-Experts (MoE). Peak GPU performance requires coordinated reasoning over tightly coupled optimizations, including tiling, shared-memory staging, software pipelining, and instruction scheduling, while existing agents rely on sparse pass/fail feedback, leaving them unable to diagnose global constraint violations. We present Argus, an agentic framework that addresses this through data-flow invariants: compile-time specifications encoding how data must be choreographed throughout kernel execution. Argus introduces a tile-based, Pythonic DSL exposing hardware instructions and compiler policies while hiding low-level representations. The DSL provides tag functions to propagate symbolic annotations through data and control flow, and tag assertions to enforce relational constraints at use sites. When violations occur, the compiler returns concrete counterexamples identifying the thread, data element, and program point, enabling dense, structured feedback for targeted fixes. Invariants are verified at compile time via abstract interpretation over a layout algebra and SMT solving, with zero runtime overhead. An in-context reinforcement learning planner learns to select optimizations and synthesize effective invariants, supported by a curated knowledge base of GPU optimization techniques. We evaluate Argus on the AMD MI300X GPU across GEMM, flash attention, and MoE kernels accounting for over 90% of GPU time in LLM inference. Generated kernels achieve 99-104% of state-of-the-art hand-optimized assembly throughput and are 2-1543x faster than existing agentic systems. Argus further generalizes to 200 KernelBench tasks, solving 100% of Level 1 and 90% of Level 2 problems.

Grokking occurs when a model achieves high training accuracy but generalization to unseen test points happens long after that. This phenomenon was initially observed on a class of algebraic problems, such as learning modular arithmetic (Power et al., 2022). We study grokking on algebraic tasks in a class of feature learning kernels via the Recursive Feature Machine (RFM) algorithm (Radhakrishnan et al., 2024), which iteratively updates feature matrices through the Average Gradient Outer Product (AGOP) of an estimator in order to learn task-relevant features. Our main experimental finding is that generalization occurs only when a certain symmetry in the training set is broken. Furthermore, we empirically show that RFM generalizes by recovering the underlying invariance group action inherent in the data. We find that the learned feature matrices encode specific elements of the invariance group, explaining the dependence of generalization on symmetry.

Self-supervised pre-training, where large corpora of unlabeled data are used to learn representations for downstream fine-tuning, has become a cornerstone of modern machine learning. While a growing body of theoretical work has begun to analyze this paradigm, existing bounds leave open the question of how sharp the current rates are, and whether they accurately capture the complex interaction between pre-training and fine-tuning. In this paper, we address this gap by developing an asymptotic theory of pre-training via two-stage M-estimation. A key challenge is that the pre-training estimator is often identifiable only up to a group symmetry, a feature common in representation learning that requires careful treatment. We address this issue using tools from Riemannian geometry to study the intrinsic parameters of the pre-training representation, which we link with the downstream predictor through a notion of orbit-invariance, precisely characterizing the limiting distribution of the downstream test risk. We apply our main result to several case studies, including spectral pre-training, factor models, and Gaussian mixture models, and obtain substantial improvements in problem-specific factors over prior art when applicable.

Neural operators (NOs) are powerful deep learning frameworks designed to learn solution operators of partial differential equations. This study evaluates the ability of NOs' to capture the stiff spatio-temporal dynamics of the FitzHugh-Nagumo model. A key contribution of this study is the assessment of the translation invariance using a novel training strategy. Models are trained using an applied current with varying spatial locations and intensities at a fixed time, while the test set presents a challenging out-of-distribution scenario where the current is translated in both time and space. This approach significantly reduces dataset generation costs. We benchmark seven NO architectures: Convolutional Neural Operators (CNOs), Deep Operator Networks (DeepONets), DeepONets with CNN encoders, Proper Orthogonal Decomposition DeepONets, Fourier Neural Operators (FNOs), Tucker Tensorized FNOs, and Local Neural Operators. We evaluated these models based on their accuracy, efficiency, and inference speed. These results demonstrate that CNOs generalize well to translated test dynamics, whereas other architectures do not generalize well. On the training set, all architectures achieve comparable accuracy, with FNOs achieving the highest precision. However, this higher accuracy comes at an elevated computational cost. Meanwhile, DeepONets and their variants exhibit superior training and inference efficiency. These findings highlight the capabilities and limitations of NOs in modeling complex ionic dynamics and provide a comprehensive benchmark for scenarios involving translated dynamics.

The synthesis of inductive loop invariants remains a critical bottleneck in automated program verification. While Large Language Models (LLMs) show promise in mitigating this issue, they often fail on complex programs, producing invariants that are invalid or computationally ineffective. Although fine-tuning is a natural strategy to address these limitations, obtaining high-quality training data remains an open challenge. We first formalize the properties required for a high-quality training invariant, and then present Wonda, a rigorous data curation pipeline that extracts such invariants from raw verifier output via AST-based normalization followed by LLM-driven semantic rewriting and augmentation with provable quality guarantees. Fine-tuning Small Language Models (SLMs) on Wonda-curated data yields consistent gains across the Qwen3, Llama-3.1, and Mistral families: the 4B and 8B Qwen3 models nearly double invariant correctness and double speedup rates, while Llama-3.1-8B triples both. On the challenging InvBench suite, the same 4B model outperforms an off-the-shelf model 20x its size and matches the end-to-end verification time of GPT-OSS-120B, while a 14B Qwen3 model matches that of the frontier model GPT-5.2, all without test-time compute overhead. Our code is publicly available on GitHub.

Selective conformal prediction can yield substantially tighter uncertainty sets when we can identify calibration examples that are exchangeable with the test example. In interventional settings, such as perturbation experiments in genomics, exchangeability often holds only within subsets of interventions that leave a target variable "unaffected" (e.g., non-descendants of an intervened node in a causal graph). We study the practical regime where this invariance structure is unknown and must be estimated from data. Our main result quantifies how coverage degrades when the estimated safe calibration set accidentally includes interventions that affect the target, and gives a conservative correction when an upper bound on this error is available. Rather than learning a full causal graph, we learn only the intervention-target relationships needed to choose calibration interventions. We give algorithms for this partial learning task and evaluate them on synthetic structural equation models and Replogle K562 CRISPR-interference data, where the experiments illustrate synthetic gains from selective calibration and finite-sample tradeoffs on real perturbation screens.

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.

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.

Magnitude, obtained as a special case of Euler characteristic of enriched category, represents a sense of the size of metric spaces and is related to classical notions such as cardinality, dimension, and volume. While the studies have explained the meaning of magnitude from various perspectives, continuity also gives a valuable view of magnitude. Based on established results about continuity of magnitude and maximum diversity, this article focuses on continuity of weighting, a distribution whose totality is magnitude, and its variation corresponding to maximum diversity. Meanwhile, recent studies also illuminated the connection between magnitude and data analysis by applying magnitude theory to point clouds representing the data or the set of model parameters. This article will also provide an application for time series analysis by introducing a new kind of invariants of periodic time series, where the invariance follows directly from the continuity results. As a use-case, a simple machine learning experiment is conducted with real-world data, in which the suggested invariants improved the performance.

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.

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 \emph{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.