# Parameter-Free Ordered Partial Match Alignment with Hidden State Time Warping
## 3 Results
### Music Synchronization Results  
For the music task, we compared the performance of six different systems. The first two systems (\"DTW-233\" and \"DTW-111\") were standard DTW with transitions \(\{(1,1),(1,2),(2,1)\}\) and corresponding transition weights \(\{2,3,3\}\) or \(\{1,1,1\}\). Scores were imputed to each frame in the query (i.e., the tampered recording) and thresholded, as before. The third system (\"NWTW-threshold\") was proposed in [20] as a way to handle time warping as well as insertions and deletions. It allowed for \(\{(1,1),(1,2),(2,1)\}\) transitions to handle time warping, as well as \((0,1)\) and \((1,0)\) skip transitions with a fixed penalty \(\gamma\). Scores were imputed to the query frames and thresholded, as above. The fourth system (\"NWTW-skip\") was identical to the third system but simply classified insertion and deletion transitions as non-matches and all other regular transitions as matches. The fifth system (\"HSTW-threshold\") used our proposed algorithm to estimate an optimal global alignment path and then imputed scores to each query frame and thresholds, as before. The sixth system (\"HSTW-plane\") estimated the alignment with our proposed algorithm and simply classified path elements in the visible plane as matches and path elements in the hidden plane as non-matches. For each system above, hyperparameters were tuned on the training set.  
Figure 4 shows the results on the music synchronization task with \(L=5,10,20,30\) s tampering durations. We can see that HSTW-plane again had the best performance, but that the music task was much more challenging than the speech task based on the equal error rates. This is because the speech task involves comparing two exactly matching recordings (with possible tampering operations), whereas the music task involves comparing two different performances of a piece (with additional tampering operations). Again, we can see that HSTW-plane outperformed HSTW-threshold despite the fact that both shared the same predicted alignment path. This suggests that the hidden and visible planes are capturing important and relevant information that the 2D alignment path is not capturing.  
Figure 4: Results on the music synchronization task with non-matching regions. Bars indicate the equal error rate for classifying audio frames as matching vs. non-matching. For systems that are not threshold-based (indicated with \({}^{*}\)), bars indicate the mean values of the false positive and false negative rates.  
Figure 5 shows two example HSTW alignments from the music task that illustrate when HSTW performs well (top) and when HSTW performs poorly (bottom). The green and red regions indicate ground truth matching and non-matching regions in time (i.e., the red regions correspond to tampered operations), the blue line shows which plane (hidden or visible) each element of the optimal HSTW path is in, and the black line shows the corresponding costs in the pairwise cost matrix. We can see that HSTW correctly identified the tampered regions: it transitions from the visible plane to the hidden plane when there is tampering. In the bottom example, we can see that the matching and non-matching regions were difficult to differentiate based on pairwise cost values, which led to higher error rates compared to the speech task. Looking at the pairwise costs, the HSTW predictions are reasonable though incorrect.