Signal classification with a point process distance on the space of persistence diagrams

Abstract

In this paper, we consider the problem of signal classification. First, the signal is translated into a persistence diagram through the use of delay-embedding and persistent homology. Endowing the data space of persistence diagrams with a metric from point processes, we show that it admits statistical structure in the form of Fréchet means and variances and a classification scheme is established. In contrast with the Wasserstein distance, this metric accounts for changes in small persistence and changes in cardinality. The classification results using this distance are benchmarked on both synthetic data and real acoustic signals and it is demonstrated that this classifier outperforms current signal classification techniques.

Appendices

Appendix A

Proof that $d^c_p$ is a metric We adapt the proof from Schuhmacher et al. (2008) to the space $P_W$. According to Definition 5, it is clear we have that $d^c_p \ge 0$ and that $d^c_p$ is symmetric and satisfies the identity. It remains to show the triangle inequality. We consider three persistence diagrams $\mathbb {D}_1 = (t_1,\ldots ,t_\ell ), \mathbb {D}_2 = (u_1,\ldots ,u_n), \mathbb {D}_3 = (v_1,\ldots ,v_m)$. Assume that $\ell \le n$ and that at most one of the cardinalities is zero. Since W is a closed and bounded subset of $\mathbb {R}^2$, we consider some dummy points $(a_i)_{i \in \mathbb {N}}$ and $(b_i)_{i \in \mathbb {N}}$ at least distance c from W and each other. The two cases we must consider are $\ell \le n \le m$ and $\ell ,m \le n$.

We first treat the case when $\ell \le n \le m$. Extend the persistence diagram $\mathbb {D}_1$ with points $t_{\ell + j} = a_j$ for $1 \le j \le m - \ell$ and similarly for $\mathbb {D}_2$ with $u_{n + j} = b_j$ for $1 \le j \le m-n$. This way the cardinality difference is equal to zero in Eq. (2). Moreover, after the dummy points have been added in, let $\eta$ and $\nu$ be the minimum permutations from $\mathbb {D}_1$ to $\mathbb {D}_3$ and from $\mathbb {D}_3$ to $\mathbb {D}_2$ respectively. Then, according to Eq. (2) and $a \le c^pm$ implying $\frac{a}{m} \le \frac{a + c^p(n-m)}{n}$, we have

$$\begin{aligned} d^c_p(\mathbb {D}_1,\mathbb {D}_2)= & {} \left( \frac{1}{n} \min _{\pi \in \varPi _n} \sum _{i=1}^n \min (c,||t_i - u_{\pi (i)}||_\infty )^p\right) ^\frac{1}{p}\nonumber \\\le & {} \left( \frac{1}{m} \min _{\pi \in \varPi _m} \sum _{i=1}^m \min (c,||t_i - u_{\pi (i)}||_\infty )^p\right) ^\frac{1}{p} \end{aligned}$$

(12)

The right hand side of Eq. (12) can further be bounded by

$$\begin{aligned}&\left( \frac{1}{m} \sum _{i=1}^m \min (c,||t_i - v_{\eta (i)}||_\infty )^p + \min (c,||v_i - u_{\nu (i)}||_\infty )^p\right) ^\frac{1}{p}\nonumber \\&\quad \le \left( \frac{1}{m} \sum _{i=1}^m \min (c,||t_i - v_{\eta (i)}||_\infty )^p\right) ^\frac{1}{p} + \left( \frac{1}{m} \sum _{i=1}^m \min (c,||v_i - u_{\nu (i)}||_\infty )^p\right) ^\frac{1}{p}\nonumber \\&\quad = d^c_p(\mathbb {D}_1,\mathbb {D}_3) + d^c_p(\mathbb {D}_3,\mathbb {D}_2) \end{aligned}$$

(13)

Note that in Eq. (13), we are mapping from $\mathbb {D}_1$ to $\mathbb {D}_3$ in the most optimal way (via permutation $\eta$) and then from $\mathbb {D}_3$ to $\mathbb {D}_2$ in the most optimal way (via permutation $\nu$).

The second case is when $\ell ,m \le n$. Take $\eta$ and $\nu$ to be the minimum permutations from $\mathbb {D}_1$ to $\mathbb {D}_3$ and from $\mathbb {D}_3$ to $\mathbb {D}_2$ respectively as above. Then, similarly, we have that

$$\begin{aligned} d^c_p(\mathbb {D}_1,\mathbb {D}_2)= & {} \left( \frac{1}{n} \min _{\pi \in \varPi _n} \sum _{i=1}^n \min (c,||t_i - u_{\pi (i)}||_\infty )^p\right) ^\frac{1}{p}\nonumber \\\le & {} \left( \frac{1}{m} \sum _{i=1}^m \min (c,||t_i - v_{\eta (i)}||_\infty )^p + \min (c,||v_i - u_{\nu (i)}||_\infty )^p\right) ^\frac{1}{p}\nonumber \\\le & {} \left( \frac{1}{m} \sum _{i=1}^m \min (c,||t_i - v_{\eta (i)}||_\infty )^p\right) ^\frac{1}{p}\nonumber \\&+\, \left( \frac{1}{m} \sum _{i=1}^m \min (c,||v_i - u_{\nu (i)}||_\infty )^p\right) ^\frac{1}{p}\nonumber \\= & {} d^c_p(\mathbb {D}_1,\mathbb {D}_3) + d^c_p(\mathbb {D}_3,\mathbb {D}_2) \end{aligned}$$

Appendix B

Proof of Lemma 1

By “Appendix A”, it is clear that this is a metric space. We first show completeness. Let $\{\mathbb {D}_n\}_{i=1}^k$ be a Cauchy sequence of persistence diagrams. It is clear that for some $k_0$, we have that $j,l \ge k_0$ implies $|\mathbb {D}_j| = |\mathbb {D}_l| = k$, so we may assume without loss of generality that the associated cardinalities are equal. Fix an $\epsilon > 0$. Note there is N such that for $n,m > N$, $d_p^c(\mathbb {D}_n,\mathbb {D}_m) < \epsilon$. In particular, since their cardinalities are the same, we have that

$$\begin{aligned} d^c_p(\mathbb {D}_n,\mathbb {D}_m) = \left( \frac{1}{k}\min _{\pi \in \varPi _k} \sum _{i=1}^k \left\| x^n_i - x^m_{\pi (i)}\right\| _\infty ^p\right) ^\frac{1}{p} < \epsilon \end{aligned}$$

and so we have that, for a given point $x^n_i \in \mathbb {D}_n$,

$$\begin{aligned} \left\| x^n_i - x^m_{\pi (i)}\right\| _\infty < (k)^\frac{1}{p}\epsilon \end{aligned}$$

where ${\pi (i)}$ is the minimal permutation.

Thus, there is a sequence of points $x^n_i, x^{n+1}_{\pi _{n+1}(i)}, x^{n+2}_{\pi _{n+2}(i)},\ldots$ such that the distance between any two points in this sequence is less than $2(k)^\frac{1}{p}\epsilon$ via the triangle inequality, where $\pi _{n+\alpha }$ is the minimal permutation between persistence diagrams $\mathbb {D}_n$ and $\mathbb {D}_{n+\alpha }$. This is a Cauchy sequence in W under the $\inf$-norm. Since W is complete, this sequence converges to some limit $x_i \in S$. Repeating this for each element in $\mathbb {D}_n$, we generate a persistence diagram $\mathbb {D}^*$ consisting of points $(x_1,\ldots ,x_k)$ chosen as the limits above.

Therefore, for any fixed $\epsilon ^p$, since each sequence above converges to the corresponding limits, there is some N such that for $j > N$ we have $||x^{j}_{i} - x_i||_\infty < \epsilon$ This implies that

$$\begin{aligned} d_p^c(\mathbb {D}_j,\mathbb {D}^*)= & {} \left( \frac{1}{k}\min _{\pi \in \varPi _k} \sum _{i=1}^k ||x^n_i - x_{\pi (i)}||_\infty ^p\right) ^\frac{1}{p} \le \left( \frac{1}{k} \sum _{i=1}^k ||x^n_i - x_i||_\infty ^p\right) ^\frac{1}{p} \\< & {} \left( \frac{1}{k} k\epsilon ^p\right) ^\frac{1}{p} = \epsilon \end{aligned}$$

Since this sequence converges to a limit in this space, this space is complete.

Finally, it remains to show separability. Consider the space $P_{\mathbb {Q} \bigcap W,k}$ of all persistence diagrams with points in $\mathbb {Q} \bigcap W$ and cardinality less than or equal to k. Then for any persistence diagram $\mathbb {D}$, find $\mathbb {D}_q \in P_{\mathbb {Q} \bigcap W,k}$ such that $|\mathbb {D}| = |\mathbb {D}_q| = k$ and for all $x_i \in \mathbb {D}$, there is a corresponding $y_{x_i} \in \mathbb {D}_q$ such that $||x_i - y_{x_i}||_\infty ^p \le \epsilon$. Then

$$\begin{aligned} d^c_p(\mathbb {D},\mathbb {D}_q) = \frac{1}{k} \left( \min _{\pi \in \varPi _k} \sum _{i=1}^k ||x_i - y_{\pi (i)}||_\infty ^p\right) \le \frac{1}{k} \sum _{i=1}^k ||x_i - y_{x_i}||_\infty ^p) \le \frac{1}{k} \sum _{i=1}^k \epsilon = \epsilon \end{aligned}$$

$\square$