# Parameter-Free Ordered Partial Match Alignment with Hidden State Time Warping
## 3 Results
### Experimental Setup--Music Synchronization  
The first task was the synchronization of different music performances of the same piece in the presence of non-matching regions. We used the Mazurka dataset [25], which consists of multiple performances of five different Chopin Mazurkas. For each performance, the dataset contains manual annotations of beat-level timestamps. We set aside two Mazurkas for training and the other three Mazurkas for testing, resulting in a total of 2514 training alignment pairs and 7069 test alignment pairs. This dataset has been used in a number of previous studies on music alignment [6; 25; 26; 27]. Table 1 shows an overview of the Mazurka dataset.  
There are a number of factors that might cause two classical music recordings of the same piece to have non-matching regions. For example, a CD audio recording might start and end at the beginning and end of the actual performance, while a Youtube recording might contain silence or applause at the beginning and/or end of the video. Non-matching regions in the middle of a recording might be due to structural differences such as repeats, silence between movements of a piece, and a different cadenza.  
We generated tampered versions of the Mazurka performances in order to study the effect of non-matching regions. The beginning, middle, and end of each original [PERSON] performance was altered in the following manner. First, a filler audio recording of a _different_ Mazurka was randomly selected from the same train-test split. This filler recording was used to replace original content with non-matching content. Second, the first \(L_{start}\) seconds of the original recording was replaced with a random interval selected from the filler recording, where \(L_{start}\) was selected uniformly in an interval of \([0,10]\) seconds. Third, the last \(L_{end}\) seconds of the original recording was likewise replaced with a random interval  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Piece** & **Files** & **Mean** & **Std** & **Min** & **Max** \\ \hline Opus 17, No. 4 & 64 & 259.7 & 32.5 & 194.4 & 409.6 \\ Opus 24, No. 2 & 64 & 137.5 & 13.9 & 109.6 & 180.0 \\ \hline Opus 30, No. 2 & 34 & 85.0 & 9.2 & 68.0 & 99.0 \\ Opus 63, No. 3 & 88 & 129.0 & 13.4 & 96.2 & 162.9 \\ Opus 68, No. 3 & 51 & 101.1 & 19.4 & 71.8 & 164.8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Overview of the Chopin Mazurka data used in the music alignment task. All durations are in seconds. The top two pieces were used for validation (i.e., setting hyperparameters), and the bottom three pieces were used for testing. Tampered versions of the performances were generated to simulate non-matching regions at the beginning, middle, and end of the recording.  
in the filler recording, where \(L_{end}\) was selected uniformly from \([0,10]\) seconds. Finally, a random \(L\) second interval in the middle of the original recording was replaced with content from the filler recording. We ensured that the tampered middle region was at least 10 s away from the tampered regions at the beginning and end, so as to avoid creating very short matching fragments. Each tampered recording was aligned against all other (untampered) original recordings of the same [PERSON].  
We evaluated system performance based on how well it classified each audio frame in the query (i.e., the tampered recording) as matching or non-matching. Note that determining whether or not an audio frame was matching or non-matching requires solving the alignment problem accurately. (We found that all systems had roughly the same alignment accuracy, so we only focus on reporting classification performance.) We characterized performance using a standard receiver operating characteristic (ROC) curve, which shows the tradeoff between false positives and false negatives. When evaluating classification performance, we excluded frames that lie within a \(\pm\Delta=100\) ms scoring collar around tampering locations.