V arious dedicated algorithms have been recently proposed and studied by the machine-learning community. In contrast, active bipartite ranking rule is poorly documented in the literature. Due to its global nature, a strategy for labeling sequentially data points that are difficult to rank w.r.t. to the others is

The primal and dual are related by the Lagrangian L (,',), L ( ',,)= E We proceed to proofs of the theorems stated in Section 4. Assumption NTK, the regularization parameter, and it may also depend indirectly on the bound R . Theorem 4.2 follows immediately from Lemmas B.1 and B.2. Theorem The following result follows from Proposition E.4 and E.5 of of Luise et al. Interestingly, the rate of estimation of the Sinkhorn plan breaks the curse of dimensionality. B.2 Log-concavity of Sinkhorn Factor The optimal entropy regularized Sinkhorn plan is given by The optimal potentials satisfy fixed point equations. Using this result, one can prove the following lemma.

Thus, rank( P) = 1 followed by the definition of the tensor rank. The following proposition is related with the second paragraph in Section 3.4. Next, we show the opposite direction. The following proposition is related to the second paragraph in Section 3.4. A.5 Proof of Proposition 5 The following discussion is related to the third paragraph in Section 3.4.

Many problems in science and engineering involve time-dependent, high dimensional datasets arising from complex physical processes, which are costly to simulate. In this work, we propose WeldNet: Windowed Encoders for Learning Dynamics, a data-driven nonlinear model reduction framework to build a low-dimensional surrogate model for complex evolution systems. Given time-dependent training data, we split the time domain into multiple overlapping windows, within which nonlinear dimension reduction is performed by auto-encoders to capture latent codes. Once a low-dimensional representation of the data is learned, a propagator network is trained to capture the evolution of the latent codes in each window, and a transcoder is trained to connect the latent codes between adjacent windows. The proposed windowed decomposition significantly simplifies propagator training by breaking long-horizon dynamics into multiple short, manageable segments, while the transcoders ensure consistency across windows. In addition to the algorithmic framework, we develop a mathematical theory establishing the representation power of WeldNet under the manifold hypothesis, justifying the success of nonlinear model reduction via deep autoencoder-based architectures. Our numerical experiments on various differential equations indicate that WeldNet can capture nonlinear latent structures and their underlying dynamics, outperforming both traditional projection-based approaches and recently developed nonlinear model reduction methods.