## 6 Extended Experiments
### Dependency on Query Length  
A possible explanation for the results of the previous section is that the query length of 20 s is not discriminative enough to identify all versions of the same piece. To analyze this hypothesis, we increased the query length in the following experiments. An obvious way to do this is to increase the query shingle length. However, we want to keep our shingle size fixed to keep the same database structure for different query lengths. Therefore, instead of increasing the query shingle length, we used multiple successive shingles for each query. Recall that, in our previous approach (see Section 3), we compared a query (Q) and a document (D) by performing a nearest neighbor search of a single query shingle \(\mathcal{S}^{\text{Q}}\) and all shingles from the set \(\mathcal{S}^{\text{D}}\) of document submatrices (see Equation (2)). The squared Euclidean distance to the nearest neighbor \(\delta_{\text{D}}(\mathcal{S}^{\text{Q}})\in\mathbb{R}_{\geq 0}\) is regarded as the document-wise distance between Q and D and was used for ranking all documents of the database. Here, instead of a single query shingle \(\mathcal{S}^{\text{Q}}\), we collected a set of successive shingles \(\mathcal{S}^{\text{Q}}\) from the query recording, as done for the documents (see Equation (2)). Instead of using a hop size \(H=1\) (as for the documents), we used a hop size of \(H=L/2=10\). Denoting the number of shingles per query by \(\lambda\coloneqq|\mathcal{S}^{\text{Q}}|\), a query covers \((\lambda-1)\times 10+20\) s of audio content. For each of these shingles, we computed the document-wise distance \(\delta_{\text{D}}\), as done previously (see Equation (6)). To obtain a single distance value between the query and the database document, the resulting shingle-wise distances were simply averaged:  
\[\overline{\delta_{\text{D}}}(\mathcal{S}^{\text{Q}})=\frac{1}{\lambda}\sum_{ \mathcal{S}\in\mathcal{S}^{\text{Q}}}\delta_{\text{D}}(\mathcal{S}). \tag{10}\]  
Finally, all database documents \(\text{D}\in\mathbb{D}\) were ranked by their averaged distances \(\overline{\delta_{\text{D}}}(\mathcal{S}^{\text{Q}})\in\mathbb{R}_{\geq 0}\) in ascending order. Note that our previous experiments are the special case of \(\lambda=1\) (query length: 20 s). For our experiments, we sampled 10 queries from each recording in an equidistant way, as before. Note each query now consists of multiple shingles. Figure 7 shows the MAP evaluation results for \(K=12\) using different query lengths on the extended data set. The retrieval quality considerably improves with increasing query length, e.g., the brute-force approach improves from an MAP value of 0.924 for \(\lambda=1\) to 0.976 for \(\lambda=5\). Similarly, the MAP values for the PCA- and DNN-based approaches increase strongly with the query length. Figure 7: Evaluation results with varying \(\lambda\) on the extended data set \(\mathbb{D}_{3}\). Table 9 shows the results for each of the composers of the data set for \(\lambda=5\) (query length: 60 s). For [PERSON], the DNN (MAP: 0.937) outperforms PCA (MAP: 0.813) and comes close to brute force (MAP: 0.974). For [PERSON], PCA (MAP: 0.935) is slightly better than the DNN (MAP: 0.891). For [PERSON]'s piano concerto, the DNN (MAP: 0.840) substantially outperforms PCA (MAP: 0.681). In contrast to the [PERSON] results, this cannot be explained by overfitting, because neither [PERSON] nor any piano concerto was part of the training set. Furthermore, we achieved an MAP value of 0.967 for the pieces by [PERSON] using the DNN approach. This is the best result among all composers for this approach, even better than for the composers of the training set. This shows a certain generalizability of the DNN embedding method. In summary, our experiments show that low-dimensional embeddings need a longer query length to be discriminative enough when using a larger data set. Note that a query length of 60 s is still a medium duration compared to studies for related tasks.