From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices

By Mehrtash Harandi, Mathieu Salzmann, Richard I. Hartley. 2014

209 citations recorded by the source corpus.

Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices –especially of high-dimensional ones– comes at a high cost that limits the applicability of existing techniques. In this paper we introduce an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. In particular, we search for a projection that yields a low-dimensional manifold with maximum discriminative power encoded via an affinity-weighted similarity measure based on metrics on the manifold. Learning can then be expressed as an optimization problem on a Grassmann manifold. Our evaluation on several classification tasks shows that our approach leads to a significant accuracy gain over state-of-the-art methods.

Subfields: Computer Vision and Pattern Recognition

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computer Vision and Pattern Recognition

1. Luc Van Gool
2. Zhiwu Huang
3. Xiaogang Wang
4. Xilin Chen
5. G. Wang
6. Dacheng Tao
7. Yi Yang
8. Vittorio Murino

Citations in the corpus

• When Kernel Methods Meet Feature Learning: Log-Covariance Network for Action Recognition From Skeletal Data (2017, methodology): “Precisely, we define va to be the vectorization of all diagonal and lower-diagonal entries1 of logXa: as similarly done in [16, 37, 3] such intermediate representation is fully able to provide and Euclidean (vectorial) representation which keeps the powerfulness of the Riemannian encoding as SPD mat”
• Projection Metric Learning on Grassmann Manifold with Application to Video based Face Recognition (2015, background): “To our knowledge, there are only two similar works [24, 13] seeking to learn the mapping from manifold to manifold.”
• Riemannian geometry for EEG-based brain-computer interfaces; a primer and a review (2017, unknown): “A relevant work, which can be readily borrowed from the computer vision community, is [132, 133].”
• Beyond Covariance: Feature Representation with Nonlinear Kernel Matrices (2015, methodology): “We use two subjects “bd” and “mm” for training and the remaining three for test [6].”
• Riemannian Approaches in Brain-Computer Interfaces: A Review (2017, unknown): “For example, the seminal article [52] tackles the problem of metric learning for data on a manifold in the context of computer vision.”
• Elastic functional coding of human actions: From vector-fields to latent variables (2015, background): “For data already lying on a known manifold, geometry aware mapping of SPD matrices [9] constructs a lower-dimensional SPD manifold, and principal geodesic analysis (PGA) [6] identifies the primary geodesics along which there is maximum variability of data points.”
• Log-Euclidean Metric Learning on Symmetric Positive Definite Manifold with Application to Image Set Classification (2015, methodology): “While comparing with other methods, it is also observed that our LEML performs much better than the competing SPD manifold learning techniques SPDML-AIM/SPDMLStein when training.”
• Spatio-Temporal Graph Convolution for Skeleton Based Action Recognition (2018, unknown): “…Graph-Based (Wang et al. 2016a) 91.63% MIMTL (Yang et al. 2017) 95.29% P-LSTM (Shahroudy et al. 2016) 95.35% STGC K 97.67% Deep STGC K 99.07% (Harandi, Salzmann, and Hartley 2014) 40.0% - Cov-RP (Tuzel, Porikli, and Meer 2006) 58.9% - Ker-RP (Wang et al. 2015a) 66.2% - SPDNet (Huang and…”
• Building Deep Networks on Grassmann Manifolds (2016, methodology): “93 80;40 30 respectively. The training time at each of 100 epoches is around 9m on average. Table.1 gives the performances of the comparative algorithms and of the state-of-the-art methods (RSR-SPDML [21] and SPD-Net [29]) on this dataset. We can see that our GrNet outperforms the state-of-the-art G”
• Efficient Temporal Sequence Comparison and Classification Using Gram Matrix Embeddings on a Riemannian Manifold (2016, background): “Note that the AIRM, JBLD and KLDM are affine invariant [15, 16], while simple computations show that, if U is unitary, then δle(X,Y) = δle(UXU ,UYU) Thus: δ•(Xσ,Yσ) =”