Music genre profiling based on Fisher manifolds and Probabilistic Quantum Clustering

Abstract

Probabilistic classifiers induce a similarity metric at each location in the space of the data. This is measured by the Fisher Information Matrix. Pairwise distances in this Riemannian space, calculated along geodesic paths, can be used to generate a similarity map of the data. The novelty in the paper is twofold; to improve the methodology for visualisation of data structures in low-dimensional manifolds, and to illustrate the value of inferring the structure from a probabilistic classifier by metric learning, through application to music data. This leads to the discovery of new structures and song similarities beyond the original genre classification labels. These similarities are not directly observable by measuring Euclidean distances between features of the original space, but require the correct metric to reflect similarity based on genre. The results quantify the extent to which music from bands typically associated with one particular genre can, in fact, crossover strongly to another genre.

Appendices

A Derivation of FI matrix as a function of MLP

For obtaining $\mathbf {FI} \left( \mathbf {x} \right)$ as a function of the MLP output estimators, the soft-max logarithms and their derivatives are needed:

$$\begin{aligned}&p_j = \frac{ \text {exp} \left( a_j \right) }{\sum _{k=1}^{J} \text {exp} \left( a_k \right) }\end{aligned}$$

(31)

$$\begin{aligned}&\text {log} (p_j) = a_j - \text {log} \left( \sum _{k=1}^{J} \text {exp} \left( a_k \right) \right) \end{aligned}$$

(32)

$$\begin{aligned}&\nabla \text {log} (p_j) = \nabla a_j - \sum _{k=1}^{J} p_k \nabla a_k \end{aligned}$$

(33)

where $p_j = p(c_j|\mathbf {x})$, $\nabla = \nabla _{\mathbf {x}} = \frac{d}{d{\mathbf {x}}}$ and $a_j = a_j( \mathbf {x} )$ for notation abbreviation. Now, combining Eqs. 2 and 31 and expanding the product:

$$\begin{aligned} \begin{aligned}&\mathbf {FI} \left( \mathbf {x} \right) = \sum _{j=1}^J \left( \nabla a_j - \sum _{k=1}^{J} p_k \nabla a_k \right) \left( \nabla a_j - \sum _{l=1}^J p_l \nabla a_l \right) ^T p_j \\&\quad = \sum _{j=1}^J \Biggl ( (\nabla a_j)(\nabla a_j)^T - \sum _{l=1}^J (\nabla a_j)(\nabla a_l)^T p_l \\&\qquad - \sum _{k=1}^J (\nabla a_k)(\nabla a_j)^T p_k + \sum _{k=1}^J \sum _{l=1}^J (\nabla a_k)(\nabla a_l)^T p_k p_l \Biggr ) p_j \end{aligned} \end{aligned}$$

(34)

Rearranging terms and considering that any variable t can be expressed as $\sum _i^J tp_i = t \sum _i^J p_i = t$, we get:

$$\begin{aligned} & \mathbf {FI}\left( \mathbf {x} \right) = \sum _{j=1}^J \Biggl ( \sum _{k=1}^J \sum _{l=1}^J (\nabla a_j)(\nabla a_j)^T p_k p_l \\&\quad - \sum _{k=1}^J \sum _{l=1}^J (\nabla a_j)(\nabla a_l)^T p_k p_l - \sum _{k=1}^J \sum _{l=1}^J (\nabla a_k)(\nabla a_j)^T p_k p_l \\&\quad + \sum _{k=1}^J \sum _{l=1}^J (\nabla a_k)(\nabla a_l)^T p_k p_l \Biggr ) p_j \\& = \sum _{j=1}^J \Biggl ( \sum _{k=1}^J \sum _{l=1}^J \biggl ( (\nabla a_j)(\nabla a_j)^T -(\nabla a_j)(\nabla a_l)^T \\&\quad -(\nabla a_k)(\nabla a_j)^T + (\nabla a_k)(\nabla a_l)^T \biggr ) p_k p_l \Biggr ) p_j \end{aligned}$$

(35)

After merging the summations, the final expression of the $\mathbf {FI} \left( \mathbf {x} \right)$ for the MLP is obtained:

$$\begin{aligned} \mathbf {FI} \left( \mathbf {x} \right) = \sum _{j=1}^J \sum _{k=1}^J \sum _{l=1}^J \nabla \left( a_j - a_k \right) \nabla \left( a_j - a_l \right) ^T p_j p_k p_l \end{aligned}$$

(36)

With this Eq. 36 (equivalent to Eq. 5), the metric is estimated locally, computing differential distances as:

$$\begin{aligned} d(\mathbf {x}, \, \mathbf {x}+d\mathbf {x})^2 = d\mathbf {x}^T \cdot \mathbf {FI}(\mathbf {x}) \cdot d\mathbf {x} \end{aligned}$$

(37)

B Silhouette figures

The Silhouette metric is especially adequate when the clustering method is based on minimizing pairwise distances within the cluster members. However, when the clustering also takes account of density similarity, non-spherical shapes are expected and hence, the Silhouette metric of a PQC probably might be worse than K-means Silhouette even if the PQC cluster better reflects the data profiles. In any case, the Silhouette metric is computed for all cluster solutions based in the cMDs data (Fig. 12).

Fig. 12

Silhouette figures for the different cluster solutions. The genre labels correspond with: Rap (0), Pop-Rock (1), Jazz (2)