Dynamic Time Warping (DTW) is a widely used metric for time series matching. However, when applied to few-shot action recognition (FSAR), DTW often encounters the "identical matching" issue, where multiple frames from one video are matched to a single frame from another. To address this, we introduce FTP-FSAR, a novel metric-based FSAR approach designed to mitigate this challenge. FTP-FSAR proposes an innovative alignment metric that incorporates temporal priors, guiding the matching process by encouraging the alignment of frames with similar temporal progression, thus improving the accuracy of frame matching. Additionally, FTP-FSAR integrates a dual framework, combining a foundation model with transductive learning to optimize feature extraction. Extensive experiments across multiple datasets demonstrate that FTP-FSAR outperforms existing methods, achieving the best results in 3 out of 4 benchmarks across 1-shot, 3-shot, and 5-shot settings, with performance improvements of up to 4.5%.

This paper studies the Partial Optimal Transport (POT) problem between two unbalanced measures with at most n supports and its applications in various AI tasks such as color transfer or domain adaptation. There is hence a need for fast approximations of POT with increasingly large problem sizes in arising applications. We first theoretically and experimentally investigate the infeasibility of the state-of-the-art Sinkhorn algorithm for POT, which consequently degrades its qualitative performance in real world applications like point-cloud registration. To this end, we propose a novel rounding algorithm for POT, and then provide a feasible Sinkhorn procedure with a revised computation complexity of O(n^2/epsilon^4). Our rounding algorithm also permits the development of two first-order methods to approximate the POT problem. The first algorithm, Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD), finds an epsilon-approximate solution to the POT problem in O(n^2.5/epsilon). The second method, Dual Extrapolation, achieves the computation complexity of O(n^2/epsilon), thereby being the best in the literature. We further demonstrate the flexibility of POT compared to standard OT as well as the practicality of our algorithms on real applications where two marginal distributions are unbalanced.

We study the Unbalanced Optimal Transport (UOT) between two measures of possibly different masses with at most $n$ components, where the marginal constraints of standard Optimal Transport (OT) are relaxed via Kullback-Leibler divergence with regularization factor $\tau$. Although only Sinkhorn-based UOT solvers have been analyzed in the literature with the iteration complexity of ${O}\big(\tfrac{\tau \log(n)}{\varepsilon} \log\big(\tfrac{\log(n)}{{\varepsilon}}\big)\big)$ and per-iteration cost of $O(n^2)$ for achieving the desired error $\varepsilon$, their positively dense output transportation plans strongly hinder the practicality. On the other hand, while being vastly used as heuristics for computing UOT in modern deep learning applications and having shown success in sparse OT problem, gradient methods applied to UOT have not been formally studied. In this paper, we propose a novel algorithm based on Gradient Extrapolation Method (GEM-UOT) to find an $\varepsilon$-approximate solution to the UOT problem in $O\big( \kappa \log\big(\frac{\tau n}{\varepsilon}\big) \big)$ iterations with $\widetilde{O}(n^2)$ per-iteration cost, where $\kappa$ is the condition number depending on only the two input measures. Our proof technique is based on a novel dual formulation of the squared $\ell_2$-norm UOT objective, which fills the lack of sparse UOT literature and also leads to a new characterization of approximation error between UOT and OT. To this end, we further present a novel approach of OT retrieval from UOT, which is based on GEM-UOT with fine tuned $\tau$ and a post-process projection step. Extensive experiments on synthetic and real datasets validate our theories and demonstrate the favorable performance of our methods in practice. We showcase GEM-UOT on the task of color transfer in terms of both the quality of the transfer image and the sparsity of the transportation plan.