Higher Order

We study high-order safety-critical control of quadrotor teams under bounded inputs and pairwise collision-avoidance constraints. Squared-distance barriers may lose thrust effectiveness when the relative displacement is orthogonal to the available thrust directions, while nonsingular constraints may still be jointly infeasible under shared bounds. We characterize both phenomena through pairwise effectiveness and aggregate feasibility measures. A torque-aware dynamic extension exposes attitude torques in a fourth-order barrier and prevents the extended-input row from vanishing under positive thrust. Gaussian processes directly learn the fourth-order HOCBF residual, providing robust margins without differentiating unknown perturbations. Under residual-bound and persistent-feasibility assumptions, the resulting QP guarantees collision avoidance and recovers the nominal input whenever it satisfies the robust safety and actuator constraints.

We provide an expository introduction on the importance of higher-arity tensor operations to deep learning. Then, we conduct a novel empirical investigation of higher-arity phenomenon in trained neural networks, introduce a hypergraphical generalization of the multilayer perceptron, and explore connections to evolutionary algorithms. We conclude with a discussion of promising directions for future research.

Topological neural networks (TNNs) enable leveraging high-order structures on graphs (e.g., cycles and cliques) to boost the expressive power of message-passing neural networks. In turn, however, these structures are typically identified a priori through an unsupervised graph lifting operation. Notwithstanding, this choice is crucial and may have a drastic impact on a TNN's performance on downstream tasks. To circumvent this issue, we propose $\partial$lift (DiffLift), a general framework for learning graph liftings to hypergraphs and cellular- and simplicial complexes in an end-to-end fashion. In particular, our approach leverages learned vertex-level latent representations to identify and parameterize distributions over candidate higher-order cells for inclusion. This results in a scalable model which can be readily integrated into any TNN. Our experiments show that $\partial$lift outperforms existing lifting methods on multiple benchmarks for graph and node classification across different TNN architectures. Notably, our approach leads to gains of up to 45% over static liftings, including both connectivity- and feature-based ones.

Tensor operations represent a cornerstone of modern scientific computing. However, the Numpy-like notation adopted by predominant tensor frameworks is often difficult to read and write and prone to so-called shape errors, i.a., due to following inconsistent rules across a large, complex collection of operations. Alternatives like einsum and einops have gained popularity, but are inherently restricted to few operations and lack the generality required for a universal model of tensor programming. To derive a better paradigm, we revisit vectorization as a function for transforming tensor operations, and use it to both lift lower-order operations to higher-order operations, and conceptually decompose higher-order operations to lower-order operations and their vectorization. Building on the universal nature of vectorization, we introduce einx, a universal notation for tensor operations. It uses declarative, pointful expressions that are defined by analogy with loop notation and represent the vectorization of tensor operations. The notation reduces the large APIs of existing frameworks to a small set of elementary operations, applies consistent rules across all operations, and enables a clean, readable and writable representation in code. We provide an implementation of einx that is embedded in Python and integrates seamlessly with existing tensor frameworks: https://github.com/fferflo/einx

The problem of learning the graphical Markov blanket (MB) of a variable from data has applications in many areas such as structure learning for Bayesian networks and Markov random fields, causal discovery, and feature selection. However, a common assumption most methods make is that the conditional independencies in the distribution imply the same separation in the graphical structure -- also known as the faithfulness assumption. Unfortunately, this assumption can be violated by higher-order dependencies such as XOR and parity-type relations, and -- on finite samples -- by empirical violations that, in extreme cases, even induce spurious dependencies absent from the true distribution. Therefore, in this paper we propose a "k-order" relaxation of the faithfulness assumption that captures parity type relationships between k+2 variables. We then propose a proof of concept algorithm called k-order Markov blanket (kOMB) that uses this relaxation for MB discovery. Finally, we empirically show how kOMB can recover the MB of a variable under both true and empirical violations of faithfulness. Code available at: https://github.com/lklee9/k-order-Markov-blanket

Non-binary bottom-up constituency parsing is usually taken to require arity actions: reductions such as $\textsc{Reduce-}X\#k$ specify both the mother label and the number of children to be composed. We show that this arity parameter is not a necessary transition primitive. Our parser introduces constituent labels separately and recovers reduction spans from delimiter-bounded stack configurations. In a well-formed reduction configuration, arity is uniquely determined by the active delimiter and the label marker, making it a derived property of parser state rather than an action label. This factorization removes label--arity-specific reduce actions while preserving direct construction of original non-binary trees. Experiments on PTB and CTB show that the delimiter-guided parser remains competitive with an arity-specific bottom-up baseline under the same implementation framework, with substantially smaller action inventories. Analyses further show that its predicted arity profile remains close to the gold treebanks and that high-arity constituents do not collapse when arity actions are removed.

Tool utilization enables Large Language Model (LLM) agents to interact with the real world and resolve complex tasks. However, existing agent frameworks predominantly rely on static toolsets composed of granular atomic actions (e.g., basic file I/O or single-turn search), which forces agents to reinvent low-level logic for every recurring workflow, leading to increased reasoning overhead and failure rates. In this study, we propose that agents can achieve self-evolution by synthesizing these atomic actions into reusable Standard Operating Procedures (SOPs), which function as callable higher-order tools that encapsulate multi-step logic. We further introduce EvoSOP, a framework that empowers agents to extract SOPs from execution trajectories and iteratively optimize the toolset through a systematic lifecycle of construction, merging, evaluation, and pruning. Extensive experiments demonstrate that EvoSOP significantly boosts task success rates while substantially reducing the number of interaction rounds compared to baselines. Our analysis also reveals that iterative tool optimization fosters reliable and efficient tool-use patterns, providing a scalable pathway for the development of self-evolving agents.

Event-based object Detection (EvDet), as a biologically inspired visual perception paradigm, demonstrates superior performance in scenarios demanding high temporal resolution and a wide dynamic range. Nevertheless, the inherent sparse representations and inadequate visual semantics of event data result in a considerable performance disparity between EvDet and frame-based object detection. Previous works attempt to alleviate this cross-modal discrepancy through knowledge distillation, yet they only focus on spatial visual semantics or pair-wise relational information, thus limiting performance in more complex scenarios. To address this challenge, this paper proposes M^2C-EvDet, a Multi-domain and Multi-order Cross-modal knowledge distillation framework for EvDet. Built upon frequency learning and hypergraph computation, M^2C-EvDet integrates two specialized modules: Adaptive Frequency-Decoupled Feature Distillation (AF^2D^2) and Multi-Order Relational Distillation (MORD).

Higher-order structures are powerful relational modeling tools, yet existing spectral operators decompose the topology into separate ranks, leaving practitioners to fuse the information back to vertices through ad hoc choices. We introduce Collapsed Effective Operators, which condense higher-order degrees of freedom into a single vertex-level operator via Schur complementation of a graded Laplacian. This yields a (generally dense) operator that encodes long-range interactions mediated by topology and is applicable to arbitrary higher-order constructs. We show it preserves positive semi-definiteness with a spectral upper bound relative to the rank-0 Hodge Laplacian, effectively lowering system energy under higher-order connectivity. Empirically, our operator improves spectral clustering, signal smoothing, and enables the inclusion of topological features in neural network architectures via positional encoding. The project page can be found http://circle-group.github.io/research/CollapsedEffectiveOperators

Graph neural networks (GNNs) are limited to modeling pairwise interactions, while higher-order models based on cell complexes achieve greater expressivity but often suffer from poor scalability. We introduce simplified and factored cellular Weisfeiler Leman tests (sCWL and fCWL), which preserve the expressivity of the CWL test while improving computational efficiency. We further introduce the maximal clique complex, enabling scalable CWNs with reduced time and memory complexity while retaining strong empirical performance. To avoid explicit clique enumeration, we propose CliqueWalk, a biased random walk that samples maximal cliques and scales linearly with graph size. These contributions yield a scalable topological learning framework for higher-order graph representation.

Sensor networks increasingly govern modern infrastructure, yet the data they lose are rarely missing in the uniform-random patterns assumed by standard imputation benchmarks. Loop detectors go offline during calibration, roadside cabinets silence clusters of nearby sensors, and newly installed instruments provide no history. Such failures create structured absences whose values are constrained by higher-order relations among groups of sensors, not merely by pairwise proximity. Existing low-rank and graph-based methods often miss this collective structure and can fail when missingness becomes coherent. We introduce Multi-Scale Hypergraph Laplacians (MSHL), a two-stage framework for learning higher-order structure from incomplete spatiotemporal observations. The Discovery stage builds a multi-scale hypergraph from complementary topology and residual-correlation evidence, with an observation-only selector that adapts to the supported interaction scale. The Refinement stage adds a small hypergraph-conditioned residual network that is safe by construction: it learns nonlinear corrections where informative residual features exist and defers to the linear estimate where they do not. We prove that MSHL represents group-conservation patterns inaccessible to pairwise graph priors, adapts to the best fixed scale up to a logarithmic factor, transfers this advantage to held-out imputation error, and admits a one-sided refinement guarantee. On two real traffic networks evaluated across scattered cell missingness, contiguous block outages, and whole-sensor blackouts at five rates, MSHL improves over a pairwise-graph baseline whenever higher-order structure is identifiable and otherwise matches it within sampling noise. The results point to a broader principle for reliable infrastructure learning: missing data should be treated not as isolated entries to fill, but as evidence of structure to discover.

Hypergraphs model higher-order relations that drive real-world decisions, from drug prescriptions to recommendations. A central structural signal in such data, beyond what pairwise relations can express, is interaction compositionality: whether a higher-order relation is compositional, emergent, or inhibitory with respect to its observed or unobserved sets. In polypharmacy, the regime decides whether a drug should be dropped, kept, or excluded: a compositional drug triple can be safely simplified, an emergent triple requires all drugs jointly, and an inhibitory triple flags a drug that disrupts an existing interaction. However, existing hypergraph learning methods, which merely propagate messages over observed hyperedges, leave this compositional signal unmodeled, allowing dangerous drug combinations to slip through and be misclassified. To this end, we propose the Hypergraph Pattern Machine (HGPM), shifting the paradigm from message passing to learning the compositional pattern of subsets. It tokenizes compositional subsets, organizes them in an inclusion DAG, and trains an inclusion-aware Transformer under masked reconstruction. On ten hypergraph benchmarks, HGPM matches or exceeds state-of-the-art methods. Notably, in a real adverse-event prediction case, HGPM correctly identifies the drug addition that inhibits the side effect among feature-identical candidates, a discrimination existing methods cannot make. The code and data are in https://github.com/KryieZhao/HGPM.git.

We study language generation in the limit under a global preference ordering on strings, as introduced by Kleinberg and Wei. As is done in previous work, we aim for breadth, but impose an additional requirement of timeliness: higher-ranked strings should be generated earlier. A string is then only credited if it is generated before a deadline, where its deadline is defined by a function that maps a string's rank in the target language to the time by which it must be produced. This is in keeping with a central consideration in machine learning, where inductive bias favors simpler'' or more plausible'' outputs, all else being equal. We show that timely generation is impossible in a strong sense for eventually consistent generators -- the protagonists of most prior related work. Under what is perhaps the mildest natural relaxation of consistency, a hallucination rate that vanishes over time, we show that we can circumvent our impossibility result. In particular, we can achieve optimal density with respect to any superlinear deadline function. We also show this is tight by ruling out timely generation with linear deadlines and vanishing hallucination rate.

After a somewhat rocky start, geometry and topology have established a foothold in machine learning. Message passing, either on graphs or higher-order complexes, is one of the main drivers of geometric deep learning, and paradigms that were once considered to be firmly in the realm of the abstract-like sheaves-have been "tamed" to serve as novel inductive biases for model architectures in topological deep learning. The veritable diversity of models, however, is in stark contrast to the scarcity of suitable benchmark datasets. As a result, researchers often resort to lifting existing graph datasets to include higher-order information. In this opinion paper, I want to encourage the community to also source new datasets, which may be used to prop up the foundations of our research field.

In this paper, we describe procedures for computing higher-order time derivatives of the Lie-group Newton-Euler, Articulated-Body Inertia, and hybrid dynamics algorithms for floating-base trees, where the base configuration evolves on SE(3) and the attached mechanism is an open kinematic tree with configuration on the (n1+n2)-dimensional manifold T^{n1} \times R^{n2}, using spatial representation of twists. After presenting the algorithms, we collect the resulting recursions into closed-form equations of motion, identifying an admissible Coriolis matrix satisfying the passivity property, and showing that the articulated inertia tensor remains unchanged across all time derivatives. We then apply the developed methods to a 12-DoF aerial manipulator to derive analytical expressions for its geometric forward and inverse dynamics along with their first time derivatives whereas the numerical simulations successfully evaluate these dynamics up to fifth order. Finally, to demonstrate their practical utility, we benchmark the proposed extensions and show that, in the considered tests, their computational cost scales quadratically with the derivative order, whereas the automatic-differentiation baseline exhibits exponential scaling.

This paper investigates singular configurations of planar 3-RPR parallel manipulators, which result from applying the averaging technique to solution pairs of their direct kinematic problem. Without computing the zeros of the corresponding degree 6 polynomial we parametrize the input pairs and determine their relative orientation in a way that the flexion order of the averaged configurations increases. Moreover, the obtained results are visualized for concrete examples. The presented methodology can also be used for studying the spherical and spatial analogues of planar 3-RPR parallel manipulators.

Graph neural networks (GNNs) are the standard for learning on graphs, yet they have limited expressive power, often expressed in terms of the Weisfeiler-Leman (WL) hierarchy or within the framework of first-order logic. In this context, topological neural networks (TNNs) have recently emerged as a promising alternative for graph representation learning. By incorporating higher-order relational structures into message-passing schemes, TNNs offer higher representational power than traditional GNNs. However, a fundamental question remains open: what is the logical expressiveness of TNNs? Answering this allows us to characterize precisely which binary classifiers TNNs can represent. In this paper, we address this question by analyzing isomorphism tests derived from the underlying mechanisms of general TNNs. We introduce and investigate the power of higher-order variants of WL-based tests for combinatorial complexes, called $k$-CCWL test. In addition, we introduce the topological counting logic (TC$_k$), an extension of standard counting logic featuring a novel pairwise counting quantifier $\exists^{N}(x_i,x_j)\, \varphi(x_i,x_j),$ which explicitly quantifies pairs $(x_i, x_j)$ satisfying property $\varphi$. We rigorously prove the exact equivalence: $\text{k-CCWL} \equiv \text{TC}_{k{+}2} \equiv \text{Topological }(k{+}2)\text{-pebble game}.$ These results establish a logical expressiveness theory for TNNs.

Higher-order learning is fundamentally rooted in exploiting compositional features. It clearly hinges on enriching the representation by more elaborate interactions of the data which, in turn, tends to increase the model complexity of conventional large-scale deep learning models. In this paper, a kernelized Volterra Neural Network (kVNN) is proposed. The key to the achieved efficiency lies in using a learnable multi-kernel representation, where different interaction orders are modeled by distinct polynomial-kernel components with compact, learnable centers, yielding an order-adaptive parameterization. Features are learned by the composition of layers, each of which consists of parallel branches of different polynomial orders, enabling kVNN filters to directly replace standard convolutional kernels within existing architectures. The theoretical results are substantiated by experiments on two representative tasks: video action recognition and image denoising. The results demonstrate favorable performance-efficiency trade-offs: kVNN consistently yields reduced model (parameters) and computational (GFLOPs) complexity with competitive and often improved performance. These results are maintained even when trained from scratch without large-scale pretraining. In summary, we substantiate that structured kernelized higher-order layers offer a practical path to balancing expressivity and computational cost in modern deep networks.

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.