Understanding and Compressing Music with Maximal Transformable Patterns

We present a polynomial-time algorithm that discovers all maximal patterns in a point set, $D\subset\mathbb{R}^k$, that are related by transformations in a user-specified class, $F$, of bijections over $\mathbb{R}^k$. We also present a second algorithm that discovers the set of occurrences for each of these maximal patterns and then uses compact encodings of these occurrence sets to compute a losslessly compressed encoding of the input point set. This encoding takes the form of a set of pairs, $E=\left\lbrace\left\langle P_1, T_1\right\rangle,\left\langle P_2, T_2\right\rangle,\ldots\left\langle P_{\ell}, T_{\ell}\right\rangle\right\rbrace$, where each $\langle P_i,T_i\rangle$ consists of a maximal pattern, $P_i\subseteq D$, and a set, $T_i\subset F$, of transformations that map $P_i$ onto other subsets of $D$. Each transformation is encoded by a vector of real values that uniquely identifies it within $F$ and the length of this vector is used as a measure of the complexity of $F$. We evaluate the new compression algorithm with three transformation classes of differing complexity, on the task of classifying folk-song melodies into tune families. The most complex of the classes tested includes all combinations of the musical transformations of transposition, inversion, retrograde, augmentation and diminution. We found that broadening the transformation class improved performance on this task. However, it did not, on average, improve compression factor, which may be due to the datasets (in this case, folk-song melodies) being too short and simple to benefit from the potentially greater number of pattern relationships that are discoverable with larger transformation classes.

RECURSIA-RRT: Recursive translatable point-set pattern discovery with removal of redundant translators

Two algorithms, RECURSIA and RRT, are presented, designed to increase the compression factor achieved using SIATEC-based point-set cover algorithms. RECURSIA recursively applies a TEC cover algorithm to the patterns in the TECs that it discovers. RRT attempts to remove translators from each TEC without reducing its covered set. When evaluated with COSIATEC, SIATECCompress and Forth's algorithm on the JKU Patterns Development Database, using RECURSIA with or without RRT increased compression factor and recall but reduced precision. Using RRT alone increased compression factor and reduced recall and precision, but had a smaller effect than RECURSIA.