Fig 1.
Accuracy comparison between DAO-CP (proposed) and its competitors on Airport Hall (upper) and Sample Video (lower) datasets.
DAO-CP automatically detects a change of theme (for example, an object starts moving or a scene changes) and re-decomposes the data depending on the degree of changes. Note that DAO-CP results in much more clear images than the competitors with little sacrifice in speed.
Table 1.
Table of symbols.
Fig 2.
In this figure, the length of time factor (with horizontal axis) becomes larger for each time step. Contrary to static decomposition methods, OnlineCP reduces computational cost using approximation with an additional constraint: it updates only the non-temporal mode, reusing the same temporal factor for all time steps. However, this leads to a substantial loss of accuracy if an incoming tensor has a different theme compared to previous tensors (e.g., theme changes A → B, B′ → C, or C → D). Thus, OnlineCP cannot achieve an accurate decomposition due to the lack of consideration on the change of themes in data.
Fig 3.
Visualization of split and refinement processes of DAO-CP.
When the z-score exceeds the split threshold Ls (e.g., change from theme A to B), DAO-CP re-initializes the new tensor slice using the static CP decomposition. When the z-score is between Lr and Ls (e.g, change from theme B to B′), the refinement process determines how much information from the previous factors should be retained. Consequently, DAO-CP provides both fast and accurate decomposition for tensor streams even with inconsistent temporal patterns.
Table 2.
Execution criteria for split and refinement processes.
Ls is the threshold of splitting and initializing the decomposition, and Lr is the threshold of refining the previous decomposition. By changing the two hyperparameters, we can fine-tune the re-decomposition process to balance the trade-off between accuracy and speed.
Fig 4.
Reconstruction errors: Global (upper) and local (lower) fitness.
Since Full–CP is not an online method, we evaluate its fitness whenever a new slice is added. Detecting the change points of theme, DAO-CP successfully increases the accuracy of decomposition, which is even higher than that of Full–CP.
Table 3.
Comparison of existing tensor decomposition methods (Full–CP refers to the static CP decomposition method).
Table 4.
Summary of datasets.
Fig 5.
Time cost: Average of local running time.
Since Full–CP is not an online method, we evaluate its fitness whenever a new slice is added. Note that DAO-CP results in a promising speed comparable to DTD and OnlineCP with much more accurate decomposition, and significantly faster than Full–CP.
Table 5.
Effect of thresholds Lr and Ls.
The memory usage means the summation of byte allocation to store intermediate data to calculate next decomposition results (e.g. auxiliary matrices G and H). We use Korea Air Quality dataset with rank 20, and change Lr and Ls to investigate the effect of refinement and split processes. Note that the lower the thresholds is set, the more frequently the re-decomposition processes are executed. Thus, one can benefit from this observation when there is a particular importance in one of accuracy, speed, or memory usage depending on target tasks.
Fig 6.
Refinement and split processes: Effects of split (solid line) and refinement (dashed line) processes in terms of local error norm (upper) and running time (lower).
Each re-decomposition process (at split point) significantly reduces the local error norm with only a modest sacrifice of running time (e.g., vertical line connecting Pprev, Pnext, Qprev, and Qnext). Note that DAO-CP runs slower than the other dynamic methods (OnlineCP and DTD) only when one of split or refinement processes is performed to increase the accuracy (horizontal line R: average running time of competitor methods).