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间密度生成建模中面临的计算困难，我们提出了一种计算成本低廉的替代方案，该方案需要极少的超参数调整，并能较好地扩展到高维问题。具体而言，我们采用基于投影的最优传输求解器（Meng等，2019）连接连续样本，随后利用传输样条（Chewi等，2020）对演化的密度进行插值。当采样频率足够高时，最优映射接近恒等映射，从而能够高效计算。此外，由于所有最优映射相互独立，可同时学习，使得训练过程高度可并行化。最后，该方法完全基于数值线性代数而非最小化非凸目标函数，便于我们轻松分析和控制算法。我们通过合成数据集和真实数据集上的多个数值实验展示了该方法的效率。特别地，这些实验表明，在多种维度条件下，所提方法与基于时间条件的最先进归一化流模型相比具有高度竞争力。