Motivated by the computational difficulties incurred by popular deep learning algorithms for the generative modeling of temporal densities, we propose a cheap alternative which requires minimal hyperparameter tuning and scales favorably to high dimensional problems. In particular, we use a projection-based optimal transport solver [Meng et al., 2019] to join successive samples and subsequently use transport splines [Chewi et al., 2020] to interpolate the evolving density. When the sampling frequency is sufficiently high, the optimal maps are close to the identity and are thus computationally efficient to compute. Moreover, the training process is highly parallelizable as all optimal maps are independent and can thus be learned simultaneously. Finally, the approach is based solely on numerical linear algebra rather than minimizing a nonconvex objective function, allowing us to easily analyze and control the algorithm. We present several numerical experiments on both synthetic and real-world datasets to demonstrate the efficiency of our method. In particular, these experiments show that the proposed approach is highly competitive compared with state-of-the-art normalizing flows conditioned on time across a wide range of dimensionalities.

翻译：受限于流行深度学习算法在时序密度生成建模中产生的计算困难，我们提出一种低成本替代方案，该方法仅需极少的超参数调优，且能有效扩展至高维问题。具体而言，我们采用基于投影的最优输运求解器[Meng等，2019]连接连续样本，随后利用输运样条[Chewi等，2020]插值演化中的密度。当采样频率足够高时，最优映射接近恒等映射，因而计算高效。此外，由于所有最优映射相互独立，训练过程可实现高度并行化，所有映射可同时学习。最终，该方法仅依赖数值线性代数而非最小化非凸目标函数，使得算法分析与控制变得简便。我们通过合成数据集与真实数据集的数值实验验证了该方法的高效性。特别地，实验表明该方案在跨维度场景下与基于时间条件的先进归一化流方法相比具有显著竞争力。