## 1 Introduction  
Video, especially short video, has become the mainstream information medium on the Internet. Facing the huge daily output and segmented production of short videos, however, review and recommendation of short video still highly dependent on tag-based and manual recognition. In order to achieve more intelligent and efficient content audit, short video recognition algorithms are in great demand. Since the first step of recognition is feature extraction, many feature extraction methods based on static image have been established. However, their application to video data still faces great challenges. Meanwhile, Riemannian manifolds have been proven to be robust in extracting video features under different imaging conditions and therefore have been successfully employed in many branches of video recognition, including face recognition and action recognition. In particular, symmetric positive definite (SPD) matrices are widely used in video representation because of second-order statistical information provided. Moreover, the space of all SPD matrices possessing the Riemannian metric is called SPD manifold [1, 2]. In general, existing SPD-based methods for video recognition construct an SPD matrix to represent each video and then take the resulting SPD manifold into account for recognition; for example, modeling video as covariance matrix [3], kernel matrix [4, 5], and Gaussian model [6, 7]. Moreover, dimensionality reduction (DR) techniques can effectively extract valid information from the original SPD matrix and reduce the calculation cost drastically with regard to the dimension of SPD matrix. One of the DR approaches directly extracts feature vector, which is in the Euclidean space, from the original SPD matrix, taking the notions of Riemannian geometry [8]. Or, to maintain SPD geometry, the bilinear DR [9, 10] aims to learn a mapping that transforms the original manifold into a new manifold. This new manifold possesses a lower dimension and a more discriminative metric. Metric learning on SPD manifolds learns a more discriminative function that narrows the similarity between SPD matrices from different categories while expanding the similarity between SPD matrices from the same category. Metric learning on SPD manifold usually projects the original SPD manifolds to their tangent space [11; 12], or embeds the SPD manifolds to a subspace of reproducing the kernel Hilbert space (RKHS) [13; 14; 15]. The tangent space and subspace of RKHS are isomorphic with Euclidean spaces that possess the same dimension. It is obvious that the SPD-based methods introduced above regard the whole video as an image-set-based SPD matrix but without considering temporal correlation between frame images. Moreover, the experiments of these methods are based on databases composed of segmented video clips. These video clips are actually short videos. Through the benefit of the few transitions in short videos, tracing the trajectories of regions of interest (ROIs) through video is available. Moreover, in order to extract both spatial and temporal features from video, the authors of [16; 17] adopted space-time geometric representations of human landmark configurations. Inspired by the work in [16; 17], we propose a framework for short video recognition that models the spatiotemporal trajectories of ROIs as a family of feature curves on SPD manifold in this paper. The main steps of our short video recognition framework are as follows:  
1. According to the practical application, a key frame is extracted from the short video to determine the spatial feature blocks (i.e., ROIs) of the target. 2. Using motion estimation [18] in video encoding, each ROI in the key frame is traced forward, backward, or two-way to string time series of ROIs through the short video. The resulting family of time series of ROIs represent the temporal and spatial feature of short video. 3. Each ROI is transformed into an SPD matrix by regional covariance descriptor (RCD).