Online TT-ALS for Streaming Tensor Decomposition with Incremental Orthogonalization

Tensor Train (TT) decomposition is a powerful technique for analyzing high-dimensional data. Existing algorithms for computing TT decompositions can be categorized into two main types: conventional batch-based approaches and recursive online methods. In the context of streaming data, batch methods typically achieve higher reconstruction accuracy but often suffer from memory exhaustion, while online methods provide greater computational efficiency. In this work, we introduce Online TT-ALS (Alternating Least Squares), an algorithm that sequentially enforces orthogonality constraints. This approach allows for efficient and exact updates of the core tensor while maintaining high reconstruction accuracy. Theoretically, we prove that enforcing these orthogonal gauge constraints guarantees monotonic decrease of the local objective function and temporal smoothness. Computationally, our deterministic single-sweep update reduces the rank dependence from quadratic to linear, achieving an overall complexity of $\mathcal{O}(I^{n-1} r)$. Experimental results demonstrate that the proposed method outperforms existing online techniques not only in terms of mathematical approximation accuracy but also in human perception-based video quality metrics. Furthermore, compared to recent deep learning-based paradigms, our algebraic approach achieves speedups of several orders of magnitude. Consequently, our method exhibits high computational efficiency and is suitable for low-latency real-time processing applications.