# Vector Space of Cycles

arXiv: https://arxiv.org/abs/2606.08202 · HTML: https://mlsanity.com/paper/2606.08202/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 83 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 19 of 19)

- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Cosine Normalization: Using Cosine Similarity Instead of Dot Product in Neural Networks](https://mlsanity.com/paper/1702.05870/) (2017)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Stochastic Convergence of Persistence Landscapes and Silhouettes](https://mlsanity.com/paper/1312.0308/) (2013)
- [Causal Discovery from Temporal Data: An Overview and New Perspectives](https://mlsanity.com/paper/2303.10112/) (2023)
- [From Ordinary Differential Equations to Structural Causal Models: the deterministic case](https://mlsanity.com/paper/1408.2063/) (2014)
- [Constraint-based Causal Discovery for Non-Linear Structural Causal Models with Cycles and Latent Confounders](https://mlsanity.com/paper/1807.03024/) (2018)
- [A Discovery Algorithm for Directed Cyclic Graphs](https://mlsanity.com/paper/1302.3599/) (2013)
- [Statistical Model for Dynamically-Changing Correlation Matrices with Application to Brain Connectivity](https://mlsanity.com/paper/1812.10050/) (2018)
- [Measuring Information Transfer Between Nodes in a Brain Network through Spectral Transfer Entropy](https://mlsanity.com/paper/2303.06384/) (2023)
- [Counterfactual Analysis of Brain Network Dynamics](https://mlsanity.com/paper/2603.29843/) (2026)
- [Flexible Bivariate INGARCH Process With a Broad Range of Contemporaneous Correlation](https://mlsanity.com/paper/2011.08799/) (2020)
- [Hodge-Laplacian of Brain Networks](https://mlsanity.com/paper/2110.14599/) (2021)
- [Topological Analysis of Seizure-Induced Changes in Brain Hierarchy Through Effective Connectivity](https://mlsanity.com/paper/2407.13514/) (2024)

## Abstract (arXiv)

Most statistical and machine learning methods for directed interactions focus on pairwise effects among variables. Even existing cyclic models represent feedback primarily through node-level dependencies, making large-scale recurrent organization difficult to estimate and compare. This limitation is particularly acute in biological and neural systems, where interactions are highly recurrent and involve many overlapping cycles. We introduce a variational framework for statistical inference on cyclic interactions. Directed interactions are represented as edge flows on a simplicial complex and evolved under an energy-minimizing dynamical system. The resulting dynamics separate transient interaction components from persistent harmonic flows, yielding a low-dimensional cycle space that captures stable recurrent organization. Rather than enumerating individual cycles, the proposed framework represents cyclic interactions as elements of a Hilbert space, enabling projection, averaging, comparison, and population-level statistical inference. We establish theoretical properties of the harmonic projection, including characterization of the cycle space, variance reduction, and population inference. Simulations demonstrate substantially improved recovery of cyclic structure in dense recurrent systems compared with existing directed-interaction methods. Applied to resting-state fMRI from 400 human subjects, the framework reveals reproducible large-scale cyclic organization that is not detectable through edgewise averaging. These results provide a scalable statistical framework for studying recurrent interactions in high-dimensional dynamical systems.