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

TL;DR

This paper presents RECURSIA and RRT algorithms that enhance pattern discovery and compression in point-set data by recursively applying TEC cover algorithms and removing redundant translators, improving compression and recall.

Contribution

The paper introduces recursive TEC cover algorithms and a translator removal technique to improve pattern compression in point-set pattern discovery.

Findings

Increased compression factor and recall with RECURSIA.

RRT reduces translators, increasing compression but lowering precision.

RECURSIA with RRT outperforms existing algorithms in compression.

Abstract

We introduce two algorithms, RECURSIA and RRT, designed to increase the compression factor achievable using point-set cover algorithms based on the SIA and SIATEC pattern discovery algorithms. SIA computes the maximal translatable patterns (MTPs) in a point set, while SIATEC computes the translational equivalence class (TEC) of every MTP in a point set, where the TEC of an MTP is the set of translationally invariant occurrences of that MTP in the point set. In its output, SIATEC encodes each MTP TEC as a pair, <P,V>, where P is the first occurrence of the MTP and V is the set of non-zero vectors that map P onto its other occurrences. RECURSIA recursively applies a TEC cover algorithm to the pattern P, in each TEC, <P,V>, that it discovers. RRT attempts to remove translators from V in each TEC without reducing the total set of points covered by the TEC. 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.

Equations

$$V(p,T)=\{p-q\mid p-q\in V(T)\wedge q\in P(T)\}.$$

$$⟨\{⟨1,1⟩,⟨2,2⟩,⟨3,3⟩\},\{⟨0,0⟩,⟨1,1⟩,⟨2,2⟩,⟨3,3⟩,⟨4,4⟩\}⟩$$

$$⟨⟨1,⟨1,1⟩⟩,⟨1,⟨7,7⟩⟩,⟨2,⟨2,2⟩⟩,⟨2,⟨6,6⟩⟩,⟨3,⟨3,3⟩⟩,⟨3,⟨4,4⟩⟩,⟨3,⟨5,5⟩⟩⟩.$$

$$\begin{array}{rl}⟨& ⟨⟨-1,-1⟩,⟨3,3⟩⟩,\\ & ⟨⟨0,0⟩,⟨2,2⟩⟩,⟨⟨0,0⟩,⟨3,3⟩⟩,\\ & ⟨⟨1,1⟩,⟨1,1⟩⟩,⟨⟨1,1⟩,⟨2,2⟩⟩,⟨⟨1,1⟩,⟨3,3⟩⟩,\\ & ⟨⟨2,2⟩,⟨1,1⟩⟩,⟨⟨2,2⟩,⟨2,2⟩⟩,⟨⟨2,2⟩,⟨3,3⟩⟩,\\ & ⟨⟨3,3⟩,⟨1,1⟩⟩,⟨⟨3,3⟩,⟨2,2⟩⟩,⟨⟨3,3⟩,⟨3,3⟩⟩,\\ & ⟨⟨4,4⟩,⟨1,1⟩⟩,⟨⟨4,4⟩,⟨2,2⟩⟩,\\ & ⟨⟨5,5⟩,⟨1,1⟩⟩⟩\end{array}$$

Full text

1 Introduction

The principle of parsimony posits that, when given two models that account equally accurately for a given set of observations (data), then the simpler model is less likely to be an accurate description of the data by chance. That is, the simpler model is more likely to be a faithful representation of the true process that gave rise to the data. This principle, commonly known as “Ockham’s razor”, has been formalized in various ways in recent times, including Rissanen’s minimal description length principle [17] and Kolmogorov’s structure function [18]. The principle has been one of the foundational principles of scientific enquiry since antiquity and recent results in information theory [19] have shown that data compression is almost always the best strategy both for model selection and prediction.

In recent years, we have had some success in using compression-based point-set pattern discovery algorithms, such as COSIATEC [13, 10, 14, 16], SIATECCompress [13, 11, 14] and Forth’s algorithm [4, 5], in conjunction with normalized compression distance, to carry out classification tasks such as folk song tune family detection [8, 13, 12]. Moreover, Louboutin and Meredith [8] found a highly significant correlation between compression factor and performance on the task of automatically discovering fugue subjects and countersubjects [6, 7]. This motivates us to search for ways to improve the compression factor achieved by such algorithms in the hope that improving compression factor may also result in improved performance on a variety of musicological tasks. Our research programme is driven by the hypothesis that shorter encodings of data objects represent better ways of understanding those objects. We therefore strive to devise algorithms that compute encodings of musical data objects that are as parsimonious as possible.

Let $D$ be a set of $k$–dimensional points, such that $D\subset\mathbb{R}^{k}$ and $|D|=n$. We call $D$ a dataset. For any vector, $v\in\mathbb{R}^{k}$, the maximal translatable pattern (MTP) in $D$ is defined as $\textsc{MTP}(v,D)=D\cap\left(D-v\right)$. The SIA algorithm [15] computes all the non-empty MTPs in such a dataset in $\Theta(n^{2}\log_{2}n)$ time. Two point sets, $P_{1},P_{2}$, are translationally equivalent, denoted by $P_{1}\mathbin{\equiv_{\mathrm{T}}}P_{2}$, if and only if there exists a vector, $v$, such that $P_{1}=P_{2}+v$. The translational equivalence relation partitions the powerset of $D$ exhaustively and exclusively into translational equivalence classes (TECs), such that the TEC to which a point set, $P\subseteq D$, belongs is defined to be $\textsc{TEC}(P)=\left\{Q\mid Q\subseteq D\land Q\mathbin{\equiv_{\mathrm{T}}}P\right\}$. The SIATEC algorithm [15] computes the TEC of every non-empty MTP in a dataset, $D$, in $\Theta(n^{3})$ time. A TEC, $\textsc{TEC}(P)$, can be encoded in a compressed form as a pair, $\left\langle P,V\right\rangle$, where $V$ is the set of non-zero vectors, $\left\{v\mid P+v\subseteq D\right\}$. Each TEC in the output of SIATEC is encoded in this form. Given a TEC, $T=\textsc{TEC}(P)=\left\langle P,V\right\rangle$, we define $P(T)=P$ and ${V}(T)=V$. $P(T)$ is called the TEC’s pattern and ${V}(T)$ is called the TEC’s translator set or set of translators. The covered set of a TEC, $T$, is the union of the point sets in the TEC and is given by $C(T)=P\cup\bigcup_{v\in{V}(T)}\left(P(T)+v\right)$. The compression factor of a TEC, $T=\textsc{TEC}(P)=\left\langle P,V\right\rangle$ is defined as $\textsc{CF}(T)=|C(T)|/\left(|P(T)|+|{V}(T)|\right)$. It is the ratio of $|C(T)|$, the number of points whose coordinates need to be explicitly specified if the covered set of the TEC is described in extenso, to $|P(T)|+|{V}(T)|$, the number of points and vectors whose coordinates need to be specified if the TEC is encoded as a pair, $\left\langle P,{V}\right\rangle$, as defined above.

SIATECCompress and Forth’s algorithm use SIATEC to compute the MTP TECs in a dataset, $D$, and then attempt, using a greedy strategy, to select a subset of these TECs, $E$, such that $\bigcup_{T\in E}C(T)=D$ and $\sum_{T\in E}\left(|P(T)|+|{V}(T)|\right)$ is minimized. That is, these algorithms attempt to find a minimum-length description of the dataset in terms of a cover constructed from TEC covered sets. The TEC covered sets in the covers computed by SIATECCompress and Forth’s algorithm may share points. However, the COSIATEC algorithm typically achieves better compression than these algorithms by partitioning the input dataset exhaustively and exclusively into non-intersecting TEC covered sets. It does this by incrementally constructing an encoding, $E$, by (1) running SIATEC, (2) adding the TEC with the best compression factor to $E$, (3) removing the covered set of this TEC from $D$ and then repeating this three-step process on progressively smaller, unencoded subsets of the dataset until all the points in the dataset have been covered.

In this paper, we introduce two novel techniques for improving the compression factor achieved using TEC cover algorithms. First, an algorithm, RecurSIA, is presented, that recursively applies a TEC cover algorithm to the pattern, $P$, in each TEC in the cover it generates. Second, an approximation algorithm, RRT, is presented, that aims to remove as many translators from each TEC as possible without removing points from its covered set. The two techniques are evaluated separately and in combination on the effect that they have on compression factor, recall and precision, when used with COSIATEC, SIATECCompress and Forth’s algorithm on the JKU Patterns Development Database [2].

2 The RecurSIA algorithm

Figure 2 gives pseudocode for the RecurSIA algorithm. RecurSIA has two parameters, a TEC cover algorithm, $\mathcal{A}$ (e.g., COSIATEC, SIATECCompress or Forth’s algorithm) and a dataset $D$. RecurSIA runs $\mathcal{A}$ on $D$ to obtain an encoding, $\mathbf{E}$ (line 1 in Fig. 2), which is a list of TECs, $\mathbf{E}=\langle T_{1},T_{2},\ldots,T_{|\mathbf{E}|}\rangle$. Each TEC, $T_{i}$, is encoded as a pair, $\langle P_{i},{V}_{i}\rangle$, as defined above. If the encoding, $\mathbf{E}$, contains only one TEC and the pattern for this TEC has only one occurrence, then $\mathcal{A}$ failed to find any non-trivial MTPs in $D$. In this case, $\mathcal{A}$ is not applied to the pattern in this TEC, so RecurSIA returns $\mathbf{E}$ (see line 2 in Fig. 2). If $\mathcal{A}$ finds more than one TEC or at least one TEC whose pattern has more than one occurrence, then RecurSIA is applied recursively to the pattern, $P_{i}=\mathbf{E}[i][0]$, in each TEC in $\mathbf{E}$ (Fig. 2, lines 3–4). This generates a new encoding, $\mathbf{e}_{i}$, for each pattern, $P_{i}$. If the encoding, $\mathbf{e}_{i}$, for a pattern, $P_{i}$, contains more than one TEC, or a TEC whose pattern occurs more than once, then $\mathbf{e}_{i}$ is a compressed encoding of $P_{i}$ and $\mathbf{e}_{i}$ replaces $P_{i}$ in the TEC, $\mathbf{E}[i]$ (Fig. 2, lines 5–6).