arXiv:2606.26900v2 Announce Type: replace Abstract: To mitigate the computational burden due to the high dimensionality of linear kinetic equations, low-rank solvers and data-driven reduced order models (ROMs) have been developed. Low-rank solvers based on matrix or tensor compression exploit phase-space low-rank structures within individual solutions, while data-driven ROMs exploit low-rank structures of the solution manifold across parameters of parametric multi-query applications such as sensitivity analysis and uncertainty quantification. It is attractive to combine them to exploit their complementary strengths. A key challenge is to extract a reduced basis without reconstructing full solution or prescribing the ranks. To address this challenge, we develop a deterministic incremental tensor-train (TT) compression algorithm that operates directly on streaming TT data and adapts the ranks to a prescribed approximation tolerance. Given a new TT tensor, the proposed method updates an accumulated TT representation via core-wise projection, residual orthogonalization, and adaptive enrichment, retaining only the complementary information that cannot be represented within a prescribed tolerance. By operating entirely at the core level, the algorithm avoids reconstructing either the incoming tensor or the accumulated full tensor. We establish approximation error bounds for the proposed approach. Moreover, we show that the accumulated TT representation provides a compressed analogue of proper orthogonal decomposition for full-order snapshot data. Using this connection, we construct reduced order models for linear kinetic equations directly from streaming low-rank solution data. Numerical experiments on parametric radiative transfer equations demonstrate that the proposed method achieves comparable reconstruction accuracy with substantially reduced wall time and yields efficient and accurate ROMs directly from compressed low-rank data.