# A Novel Space-Time Representation on the Positive Semidefinite Con for Facial Expression Recognition

arXiv: https://arxiv.org/abs/1707.06440 · HTML: https://mlsanity.com/paper/1707.06440/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jul 2017.

## Built on by (most cited first, 6 of 6)

- [Co-Transport for Class-Incremental Learning](https://mlsanity.com/paper/2107.12654/) (2021)
- [Sparse to Dense Dynamic 3D Facial Expression Generation](https://mlsanity.com/paper/2105.07463/) (2021)
- [Generating Multiple 4D Expression Transitions by Learning Face Landmark Trajectories](https://mlsanity.com/paper/2208.00050/) (2022)
- [Face-GCN: A Graph Convolutional Network for 3D Dynamic Face Identification/Recognition](https://mlsanity.com/paper/2104.09145/) (2021)
- [RepSPD: Enhancing SPD Manifold Representation in EEGs via Dynamic Graphs](https://mlsanity.com/paper/2602.22981/) (2026)
- [Varifold Moment Invariants for Sustainable and Explainable Contour Feature Extraction](https://mlsanity.com/paper/2606.07333/) (2026)

## Abstract (arXiv)

In this paper, we study the problem of facial expression recognition using a novel space-time geometric representation. We describe the temporal evolution of facial landmarks as parametrized trajectories on the Riemannian manifold of positive semidefinite matrices of fixed-rank. Our representation has the advantage to bring naturally a second desirable quantity when comparing shapes -- the spatial covariance -- in addition to the conventional affine-shape representation. We derive then geometric and computational tools for rate-invariant analysis and adaptive re-sampling of trajectories, grounding on the Riemannian geometry of the manifold. Specifically, our approach involves three steps: 1) facial landmarks are first mapped into the Riemannian manifold of positive semidefinite matrices of rank 2, to build time-parameterized trajectories; 2) a temporal alignment is performed on the trajectories, providing a geometry-aware (dis-)similarity measure between them; 3) finally, pairwise proximity function SVM (ppfSVM) is used to classify them, incorporating the latter (dis-)similarity measure into the kernel function. We show the effectiveness of the proposed approach on four publicly available benchmarks (CK+, MMI, Oulu-CASIA, and AFEW). The results of the proposed approach are comparable to or better than the state-of-the-art methods when involving only facial landmarks.