# A Robust Cover Song Identification System with Two-Level Similarity Fusion and Post-Processing
## 2 Proposed Model
### First-Level Fusion  
For each pair of tracks (\(\mathbf{v}_{q}\) and \(\mathbf{v}_{p}\)), the \(j\)-th kind of similarity function is performed on their \(i\)-th descriptors to obtain the similarity score \(s_{j}^{(i)}(q,p)\):  
\[s_{j}^{(i)}(q,p)=s_{j}(f_{i}(\mathbf{v}_{q}),f_{i}(\mathbf{v}_{p})),i=1, \cdots,M,j=1,\cdots,R. \tag{1}\]  
Figure 1: Block diagram and illustrative example of the proposed model, taking part results on Covers40 as an example. (**a**) Extract the harmonic pitch class profile (HPCP) descriptor and main melody (MLD) descriptor from each track in the music collection. (**b**) A track-by-track similarity graph is constructed based on each descriptor and corresponding similarity function. The similarity graphs based on the same descriptor and different similarity functions are fused with similarity network fusion (SNF). (**c**) The first-level fused similarity graphs for each descriptor are integrated with the mixture Markov model (MMM) technique to obtain a second-level fused similarity graph. (**d**) Post-processing. First, diffusion is performed on the second-level fused similarity graph to take advantage of the structure of the underlying track manifold contained within it to reduce noise and enhance retrieval accuracy, then mutual proximity (MP) is adopted to modify the diffused similarity to reduce the “hubness” phenomenon.  
Thus, the track-by-track similarity matrix obtained based on the \(i\)-th descriptor and \(j\)-th similarity function can be represented as a graph, denoted as \(G_{j}^{(i)}\left\{\mathbf{V},\mathbf{E},\mathbf{s}_{j}^{(i)}\right\}\), where the vertices \(\mathbf{V}\) correspond to the tracks in the collection, and the edges \(\mathbf{E}\) are weighted by the corresponding similarity scores \(\mathbf{s}_{j}^{(i)}\).  
To take advantage of the complementarity between the Qmax and Dmax similarity functions in representing the similarity between cover versions, the similarity graphs based on the same descriptor (HPCP or MLD) and two different similarity functions (Qmax [4] and Dmax [5]) are fused with the SNF technique [13]. The specific details of the SNF technique can be found in [13] and [7]. The first-level fused similarity graph for the \(i\)-th descriptor can be denoted as \(G^{(i)}(\mathbf{V},\mathbf{E},\mathbf{A}^{(i)})\), \(i=1,2\), which is obtained with Equation (2):  
\[G^{(i)}\left(\mathbf{V},\mathbf{E},\mathbf{A}^{(i)}\right)=SNF\left(s_{j}^{(i) }\right),i=1,\cdots,M,j=1,\cdots,R. \tag{2}\]  
To test the validity of the first-level fusion, three cover sets shown in Table 1 are studied here. The six tracks were used both as the queries and the targets. The corresponding 6 \(\times\) 6 similarity matrices obtained by MLD-Qmax, MLD-Dmax, and the first-level fused version of them (denoted as SNF-MLD-QD), are shown in Figure 2a-c, respectively. The cells corresponding to the query/cover pairs are marked with white boxes. It can be seen that MLD-Qmax and MLD-Dmax did not work on the No. 1 and No. 3 cover sets, respectively. However, after first-level fusion, this problem was solved.