# Sliding window persistence of quasiperiodic functions

Hitesh Gakhar  Department of Mathematics, University of Oklahoma Norman, OK,
USA<EMAIL_ADDRESS>and  Jose A. Perea  Department of Computational
Mathematics, Science & Engineering Department of Mathematics, Michigan State
University East Lansing, MI, USA<EMAIL_ADDRESS>
###### Abstract.

A function is called quasiperiodic if its fundamental frequencies are linearly
independent over the rationals. With appropriate parameters, the sliding
window point clouds of such functions can be shown to be dense in tori with
dimension equal to the number of independent frequencies. In this paper, we
develop theoretical and computational techniques to study the persistent
homology of such sets. Specifically, we provide parameter optimization schemes
for sliding windows of quasiperiodic functions, and present theoretical lower
bounds on their Rips persistent homology. The latter leverages a recent
persistent Künneth formula. The theory is illustrated via computational
examples and an application to dissonance detection in music audio samples.

###### Key words and phrases:

Topological data analysis, persistent homology, dynamical systems, sliding
window embeddings, quasiperiodicity, time series analysis

###### 2020 Mathematics Subject Classification:

Primary 55N31, 37M10; Secondary 68W05

This work was partially supported by the National Science Foundation through
grants DMS-1622301, CCF-2006661, and CAREER award DMS-1943758.
Preliminary versions of these results appeared in the first author’s PhD
thesis [14].

##  1\. Introduction

Recurrent behavior—both in time and space—is ubiquitous in nature.
_Periodicity_ and _quasiperiodicity_ are two prominent examples, characterized
by a vector of underlying non-zero frequencies: If all pairwise ratios are
rational, then the recurrence is periodic, while quasiperiodicity, on the
other hand, occurs if there are at least two frequencies whose quotient is
irrational. Quasiperiodic recurrence is at the heart of KAM (Kolmogorov-
Arnold-Moser) theory [7], it appears as a signature of biphonation (i.e., the
voicing of two simultaneous pitches) in mammalian vocalization [42], in
climate change patterns on Mars [27], in the oscillatory movement of the star
TT Arietis [18], and in the brain functioning in mice as reported by fMRI
scans [5]. The list goes on.

Quasiperiodicity in dynamical systems is typically studied with numerical
methods including Birkhoff averages [11], periodic approximations [31, 32],
estimation of Lyapunov exponents [41], power spectra [43], and recurrence
quantification analysis [40, 45]. New techniques from applied topology have
emerged recently as complements to these traditional approaches in the task of
recurrence detection—specifically for periodicity and quasiperiodicity
quantification—in time series data [26, 23, 38]. This novel framework combines
two key ingredients: _sliding window embeddings_ and _persistent homology_.

Sliding window (also known as time-delay) embeddings provide a framework to
reconstruct the topology of state-space attractors in dynamical systems, given
observed time series data. Indeed, given parameters
$d\in\mathbb{N}=\\{0,1,\ldots\\}$ (controlling the embedding dimension $d+1$)
and $\tau\in\\{x\in\mathbb{R}\mid x>0\\}$ (the time delay) the sliding window
embedding of $f:\mathbb{R}\longrightarrow\mathbb{C}$ at $t\in\mathbb{R}$ is
the vector

(1) $\displaystyle SW_{d,\tau}f(t):=\begin{bmatrix}f(t)\\\ f(t+\tau)\\\
\vdots\\\ f(t+d\tau)\end{bmatrix}\in\mathbb{C}^{d+1}.$

The motivation behind this construction is Takens’ embedding theorem [35],
which asserts that if $f$ is the result of observing the evolution of a
(potentially unknown) dynamical system, then the underlying topology of the
sliding window point cloud
$\mathbb{SW}_{d,\tau}f:=SW_{d,\tau}f(\mathbb{R})$—generically in $f$ and for
appropriate parameters $d,\tau$—recovers that of the traversed portion of the
state space. In particular, this is how attractors can be reconstructed from
observed time series data.

The topology of attractors constrains many properties of the underlying
dynamical system (e.g., periodic orbits, chaos, etc) and detecting these
features in practice is where persistent homology has come into play [29].
Persistent homology is a tool from Topological Data Analysis widely used to
quantify multiscale homological features of shapes. Its typical input is a
collection $\mathcal{K}=\\{K_{\epsilon}\\}_{\epsilon\geq 0}$ of spaces with
$K_{\epsilon}\subset K_{\epsilon^{\prime}}$ continuous for all
$\epsilon\leq\epsilon^{\prime}$. This is called a _filtration_. The output in
each dimension $j\in\mathbb{N}$ is a multiset

$\mathsf{dgm}_{j}(\mathcal{K})\subset\\{(x,y)\in[0,\infty]\times[0,\infty]\mid
0\leq x<y\\}$

called the $j$-th _persistence diagram_ of $\mathcal{K}$, where each pair
$(a,b)\in\mathsf{dgm}_{j}(\mathcal{K})$ encodes a $j$-dimensional topological
feature (like a connected component, a hole, or a void) which appears at
$K_{a}$ and disappears entering $K_{b}$. The quantity $b-a$ is the
_persistence_ of the feature, and typically measures significance across the
filtration.

In data analysis applications the input to persistent homology is often a
metric space $(X,\mathbf{d}_{X})$—e.g., a sliding window point cloud
$\mathbb{SW}_{d,\tau}f$—from which the _Rips (simplicial) complex_

(2) $R_{\epsilon}(X,\mathbf{d}_{X}):=\left\\{\\{x_{0},\ldots,x_{n}\\}\subset
X\mid\max_{0\leq j,k\leq
n}\mathbf{d}_{X}(x_{j},x_{k})<\epsilon\;\;,\;\;n\in\mathbb{N}\right\\}$

is computed at each scale $\epsilon\geq 0$, producing the _Rips filtration_

(3)
$\mathcal{R}(X,\mathbf{d}_{X}):=\\{R_{\epsilon}(X,\mathbf{d}_{X})\\}_{\epsilon\geq
0}.$

Points in the Rips persistence diagrams
$\mathsf{dgm}_{j}^{\mathcal{R}}(X):=\mathsf{dgm}_{j}\big{(}\mathcal{R}(X,\mathbf{d}_{X})\big{)}$
quantify the underlying topology of $X$ in that pairs $(a,b)$ with large
persistence $b-a$ represent likely topological features of a continuous space
around which $X$ accumulates.

The diagrams $\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{SW}_{d,\tau}f)$ have
shown to be rich signatures for recurrence detection in time series, with
applications including: periodicity quantification in gene expression data
[25], (quasi)periodicity detection in videos [38], synthesis of slow-motion
videos from repetitive movements [37], wheezing detection [12], and chatter
prediction [19]. See [24] for a recent survey. One of the main challenges in
these applications is the validation of empirical results, which stems, in
part, from the current limited theoretical understanding of how
$\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{SW}_{d,\tau}f)$ depends on $f,d,\tau$
and $T$. That said, there are recent explicit conditions on $f$ for
$\mathbb{SW}_{d,\tau}f$ to provide appropriate reconstructions [44], as well
as analyses of sliding window persistence for periodic functions [26], and
quasiperiodic functions of the form [23]

(4) $f(t)=c_{1}e^{it\omega_{1}}+\cdots+c_{N}e^{it\omega_{N}}.$

In Eq. (4) the $\omega_{n}>0$ are $\mathbb{Q}$-linearly independent (i.e.,
incommensurate), and the coefficients $c_{n}\in\mathbb{C}$ are nonzero. Our
goal in this paper is to extend [26] and [23] to general quasiperiodic
functions; i.e., those beyond Eq. (4).

###  1.1. Contributions

The first contribution of this paper is methodological: we develop techniques
to study the persistent homology of sliding window point clouds from general
quasiperiodic functions. Specifically, we show that if
$f:\mathbb{R}\longrightarrow\mathbb{C}$ is quasiperiodic with incommensurate
frequencies $\omega=(\omega_{1},\ldots,\omega_{N})$ (Definition 2.4), and if
for $\mathbf{k}\in\mathbb{Z}^{N}$, $K\in\mathbb{N}$, we let

$\widehat{F}(\mathbf{k})=\lim_{\lambda\to
0}\frac{1}{\lambda}\int_{0}^{\lambda}f(t)e^{-i\langle\mathbf{k},t\omega\rangle}dt\;\;\;\;\;,\;\;\;\;\;S_{K}f(t)=\sum_{\|\mathbf{k}\|_{\infty}\leq
K}\widehat{F}(\mathbf{k})e^{i\langle\mathbf{k},t\omega\rangle}$

then the Rips persistence diagrams
$\mathsf{dgm}^{\mathcal{R}}_{j}(\mathbb{SW}_{d,\tau}f)$, $j\in\mathbb{N}$, can
be approximated in bottleneck distance by
$\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{SW}_{d,\tau}S_{K}f)$ as $K\to\infty$.
The diagrams of $\mathbb{SW}_{d,\tau}S_{K}f$ are then studied directly with
methods extending those of [23, 26]; the approximation to
$\mathsf{dgm}^{\mathcal{R}}_{j}(\mathbb{SW}_{d,\tau}f)$ is of order
$O\left(K^{\frac{N}{2}-r}\right)$ when
$|\widehat{F}(\mathbf{k})|=O\left(\|\mathbf{k}\|_{2}^{-r}\right)$ and $r>N/2$
(Corollary 3.2).

This approximation strategy leads to our second contribution: computational
schemes for optimizing the choice of parameters $d\in\mathbb{N}$ and $\tau>0$,
so that the geometry (and hence the Rips persistent homology) of the sliding
window point cloud $\mathbb{SW}_{d,\tau}f=SW_{d,\tau}f(\mathbb{R})$ robustly
reflects that of an $N$-torus. Explicitly:

(1) Given $f$, estimate the coefficients $\widehat{F}(\mathbf{k})$ and their
frequency locations $\langle\mathbf{k},\omega\rangle$. This can be done
numerically with methods leveraging the Discrete Fourier Transform [16, 20] or
Wavelet analysis [39]. (2) Let $K\in\mathbb{N}$ be the smallest integer so
that
$\mathsf{supp}(\widehat{F}_{K}):=\left\\{\mathbf{k}\in\mathbb{Z}^{N}\mid\widehat{F}(\mathbf{k})\neq
0\mbox{ and }\|\mathbf{k}\|_{\infty}\leq K\right\\}$ spans an $N$-dimensional
$\mathbb{Q}$-vector space. This is possible provided $f$ is smooth enough
(Lemma 4.1). (3) Let $d$ be the cardinality of
$\mathsf{supp}(\widehat{F}_{K})$, or alternatively, the number of prominent
peaks in the spectrum of $f$. This choice is so that $\mathbb{SW}_{d,\tau}f$
has the right toroidal dimension (Theorem 4.2). (4) Let $\tau>0$ be a
minimizer over $[0,\tau_{\mathsf{max}}]$ of the scalar function
$\Gamma(x)=\sum_{\mathbf{k}\neq\mathbf{k}^{\prime}}\left|1+e^{ix\langle\mathbf{k}-\mathbf{k}^{\prime},\omega\rangle}+\cdots+e^{ix\langle\mathbf{k}-\mathbf{k}^{\prime},\omega\rangle
d}\right|^{2}$ where the sum runs over
$\mathbf{k},\mathbf{k}^{\prime}\in\mathsf{supp}(\widehat{F}_{K})$. This choice
is meant to amplify the toroidal features in
$\mathsf{dgm}^{\mathcal{R}}_{j}(\mathbb{SW}_{d,\tau}f)$—see Figures 4 and
5—and can be implemented via simple minimization algorithms.

Our third contribution leverages the aforementioned approximation strategy,
parameters choices, and a recent persistent Künneth formula [15], to establish
bounds for the cardinality and persistence of strong toroidal features in
$\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{SW}_{d,\tau}f)$, for $1\leq j\leq N$.
We prove the following (Section 6):

######  Theorem.

With $K,d\in\mathbb{N}$ and $\tau>0$ as before, let $\sigma_{\mathsf{min}}>0$
be the smallest singular value of the $(d+1)\times d$ Vandermonde matrix
$\left[e^{i\langle\mathbf{k},\omega\rangle\tau
j}\right]_{j=0,\ldots,d,\;\mathbf{k}\in\mathsf{supp}(\widehat{F}_{K})}$.
Moreover, let
$\mathbf{k}_{1},\ldots,\mathbf{k}_{N}\in\mathsf{supp}(\widehat{F}_{K})$ be
$\mathbb{Q}$-linearly independent with

$|\widehat{F}(\mathbf{k}_{1})|\geq|\widehat{F}(\mathbf{k}_{2})|\geq\cdots\geq|\widehat{F}(\mathbf{k}_{N})|>0.$

For $1\leq n\leq N$, let $1\leq n_{1}<\cdots<n_{\ell}\leq N$ be the longest
sequence (i.e., largest $\ell$) for which

$|\widehat{F}(\mathbf{k}_{n})|=|\widehat{F}(\mathbf{k}_{n_{1}})|=\cdots=|\widehat{F}(\mathbf{k}_{n_{\ell}})|$

and for $1\leq j\leq N$, let

$\mu_{j}(n):={n_{1}-1\choose j-1}+\cdots+{n_{\ell}-1\choose j-1}.$

Then, there are at least $\mu_{j}(1)+\cdots+\mu_{j}(n)$ toroidal features
$(a,b)\in\mathsf{dgm}^{\mathcal{R}}_{j}\left(\mathbb{SW}_{d,\tau}f\right)$,
counted with multiplicity, and with persistence

(5)
$b-a\;\;\geq\;\;\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|\sigma_{\mathsf{min}}\;-\;4\sqrt{d+1}\|f-S_{K}f\|_{\infty}.$

We envision for these kinds of theorems to be useful in separating noise from
features in applications of quasiperiodicity detection/quantification with
sliding windows and persistence. As an illustration, consider the
quasiperiodic signal

$f(t)=2\sin(t)+1.8\sin\left(\sqrt{3}t\right)$

shown in Figure 1 below (left), together with
$\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{SW}_{d,\tau}f)$ (right) in dimensions
$j=1$ (blue) and $j=2$ (orange), computed with appropriate parameters
$d,\tau$. The corresponding theoretical lower bounds on persistence from Eq.
(5) are depicted with dashed lines (see Section 7.1 for computational
details).

Figure 1. Left: the function $f(t)=2\sin(t)+1.8\sin\left(\sqrt{3}t\right)$.
Right: The persistence diagrams in dimensions $j=1$ (blue) and $j=2$ (orange)
of the sliding window point cloud $\mathbb{SW}_{d,\tau}f$, together with the
lower bounds on persistence (dashed lines) from Eq. (5).

Finally, as an application of the theory established in this paper, we
illustrate how sliding windows and persistence can be used to identify the
presence of dissonance in music audio recordings. Indeed, dissonant intervals
in music (like _the tritone_) lead to quasiperiodic recurrence in the recorded
audio waves, which we show can be effectively detected with the methods
developed here.

###  1.2. Organization

Section 2 presents the necessary mathematical background on the Fourier theory
of quasiperiodic functions and persistent homology. Section 3 provides
Fourier-type approximation theorems with explicit rates, at the level of
sliding window point clouds and Rips persistent homology. Section 4 is devoted
to studying the geometric structure of the sliding window point cloud
$\mathbb{SW}_{d,\tau}S_{K}f$ as a function of the parameters involved. In
Section 5, we establish a computational framework for the optimization of $d$
and $\tau$. Section 6 establishes the advertised theoretical lower bounds on
$\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{SW}_{d,\tau}f)$, and we end in Section
7 with computational examples and applications.

###  1.3. Notation

Let $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\cong S^{1}$, that is, $[0,2\pi]$
with the endpoints identified. Similarly for $N\in\mathbb{N}$, let
$\mathbb{T}^{N}=\left(\mathbb{R}/2\pi\mathbb{Z}\right)^{N}\cong
S^{1}\times\cdots\times S^{1}$. We will abuse notation and regard any
$F:\mathbb{T}^{N}\longrightarrow\mathbb{C}$ as both a function of a variable
$\mathbf{t}\in\mathbb{R}^{N}$ where each coordinate $t_{n}$ is
$2\pi$-periodic, and also as a function on the quotient $\mathbb{T}^{N}$.

##  2\. Preliminaries

In this section we establish the necessary background for later parts of the
paper. In particular, we provide a short review of Kronecker’s
multidimensional approximation theorem in Section 2.1, as well as of
quasiperiodic functions and their Fourier theory in Section 2.2. The theory of
persistent homology is briefly discussed in Section 2.3.

###  2.1. Kronecker’s theorem

If we regard $\mathbb{R}$ as a vector space over $\mathbb{Q}$, then a finite
set $\\{\beta_{1},\dots,\beta_{N}\\}\subset\mathbb{R}$ is called
incommensurate if $\beta_{1},\dots,\beta_{N}$ are linearly independent over
$\mathbb{Q}$. That is, for $c_{1},\dots,c_{N}\in\mathbb{Q}$

$c_{1}\beta_{1}+\cdots+c_{N}\beta_{N}=0\;\;\;\mbox{ if and only if
}\;\;\;c_{1}=\cdots=c_{N}=0.$

Kronecker’s theorem is concerned with simultaneous diophantine approximations
to real numbers, and incommensurability turns out to be the main condition.
Later on we will use this theorem to see why sliding window point clouds from
functions with incommensurate frequencies (i.e., quasiperiodic) are dense in
high-dimensional tori. For now, the theorem can be stated as follows [2,
Chapter 7].

######  Theorem 2.1 (Kronecker).

$\\{\beta_{1},\dots,\beta_{N}\\}\subset\mathbb{R}$ is incommensurate if and
only if for every $r_{1},\dots,r_{N}\in\mathbb{R}$ and every $\epsilon>0$,
there exist $t\in\mathbb{R}$ and $k_{1},\dots,k_{N}\in\mathbb{Z}$ so that

(6) $|t\beta_{n}-r_{n}-2\pi k_{n}|<\epsilon\hskip 28.45274pt\mbox{ for all
}\hskip 28.45274pt1\leq n\leq N.$

As a consequence, the entries of
$\beta=(\beta_{1},\dots,\beta_{N})\in\mathbb{R}^{N}$ are incommensurate if and
only if
$\mathbb{R}\beta/2\pi\mathbb{Z}^{N}:=\big{\\{}t(\beta_{1},\ldots,\beta_{N})\mod
2\pi\mid t\in\mathbb{R}\big{\\}}$ is dense in $\mathbb{T}^{N}$.

######  Remark 2.2.

If one further assumes that $\\{\pi,\beta_{1},\ldots,\beta_{N}\\}$ is
incommensurate, then Eq. (6) holds with $t\in\mathbb{Z}$.

Here is a useful consequence of Kronecker’s theorem:

######  Corollary 2.3.

Let $\beta=(\beta_{1},\dots,\beta_{\alpha})\in\mathbb{R}^{\alpha}$, for
$\alpha\in\mathbb{N}$. Then
$\mathsf{span}_{\mathbb{Q}}\\{\beta_{1},\ldots,\beta_{\alpha}\\}$ has
dimension $N$ over $\mathbb{Q}$, if and only if
$\mathbb{R}\beta/2\pi\mathbb{Z}^{\alpha}\subset\mathbb{T}^{\alpha}$ is dense
in an $N$-torus embedded in $\mathbb{T}^{\alpha}$.

Before presenting the proof, and as an illustration of this Corollary, Figure
2 shows what $T\beta/2\pi\mathbb{Z}^{3}$ looks like inside $\mathbb{T}^{3}$
for $T:=\left[-10^{4},10^{4}\right]\cap\mathbb{Z}$ and

$\beta=\left(\sqrt{2},\sqrt{3},0\right),\;\left(\sqrt{2},\sqrt{3},\sqrt{2}+\sqrt{3}\right),\;\left(\sqrt{2},\sqrt{3},3\sqrt{2}+2\sqrt{3}\right),\;\left(\sqrt{2},\sqrt{3},\pi^{2}+1\right).$

Figure 2. The set $T\beta/2\pi\mathbb{Z}^{3}\subset\mathbb{T}^{3}$ for
$\beta=(\sqrt{2},\sqrt{3},0)$ (left, a 2-torus),
$\beta=(\sqrt{2},\sqrt{3},\sqrt{2}+\sqrt{3})$ (second, a 2-torus),
$\beta=(\sqrt{2},\sqrt{3},3\sqrt{2}+2\sqrt{3})$ (third, a 2-torus),
$\beta=(\sqrt{2},\sqrt{3},\pi^{2}+1)$ (right, a 3-torus) and
$T=\left[-10^{4},10^{4}\right]\cap\mathbb{Z}$.

Notice that in each case, $T\beta/2\pi\mathbb{Z}^{3}$ traces a torus of the
appropriate dimension, but embedded in ways governed by the linear relations
between the entries of $\beta$.

###### Proof (of Corollary 2.3).

Let $\\{\beta_{1},\dots,\beta_{\alpha}\\}\subset\mathbb{R}$ span an
$N$-dimensional vector space over $\mathbb{Q}$, and without loss of
generality, assume that $\beta_{1},\dots,\beta_{N}$ form a basis, $N<\alpha$.
For $1\leq j\leq\alpha-N$ let $k_{j},k_{j,n}\in\mathbb{Z}$ be so that

(7) $k_{j}\beta_{N+j}=\sum_{n=1}^{N}k_{j,n}\beta_{n}$

If $N=1$ we further require that $\mathsf{gcd}(k_{j},k_{j,1})=1$, and if
$k=\mathsf{lcm}(k_{1},\ldots,k_{\alpha-1})$, then it readily follows that

$\begin{array}[]{cccc}\mathbb{T}^{1}&\longrightarrow&\mathbb{T}^{\alpha}\\\
t&\mapsto&\frac{kt}{\beta_{1}}\beta\end{array}$

is an embedding of $\mathbb{T}^{1}$ into $\mathbb{T}^{\alpha}$ with the
closure
$\overline{\mathbb{R}\beta/2\pi\mathbb{Z}^{\alpha}}=\mathbb{R}\beta/2\pi\mathbb{Z}^{\alpha}$
as its image.

Now, for $N\geq 2$ define
$\phi:\mathbb{R}^{N}\longrightarrow\mathbb{R}^{\alpha}$ and
$\psi:\mathbb{R}^{\alpha}\longrightarrow\mathbb{R}^{\alpha}$ as

(8) $\displaystyle\phi(t_{1},\ldots,t_{N})$ $\displaystyle=$
$\displaystyle\left(t_{1},\ldots,t_{N},\sum\limits_{n=1}^{N}k_{1,n}t_{n}\,,\ldots,\sum\limits_{n=1}^{N}k_{\alpha-N,n}t_{n}\right)$
(9) $\displaystyle\psi(t_{1},\ldots,t_{\alpha})$ $\displaystyle=$
$\displaystyle\big{(}t_{1},\ldots,t_{N},\,k_{1}t_{N+1}\,,\ldots,\,k_{\alpha-N}t_{\alpha}\big{)}$

Note that both $\phi$ and $\psi$ preserve classes modulo $2\pi$, and therefore
descend to continuous maps
$\phi:\mathbb{T}^{N}\longrightarrow\mathbb{T}^{\alpha}$,
$\psi:\mathbb{T}^{\alpha}\longrightarrow\mathbb{T}^{\alpha}$. Moreover, since
$\phi:\mathbb{T}^{N}\longrightarrow\mathbb{T}^{\alpha}$ is injective and tori
are compact and Hausdorff, then $\phi$ is a homeomorphism onto its image. We
claim that $\psi:\mathbb{T}^{\alpha}\longrightarrow\mathbb{T}^{\alpha}$ is
injective when restricted to $\mathbb{R}\beta/2\pi\mathbb{Z}^{\alpha}$.
Indeed, if $t\neq t^{\prime}\in\mathbb{R}$ are so that
$\psi(t\beta)=\psi(t^{\prime}\beta)$ in $\mathbb{T}^{\alpha}$, then there
exist $p,q\in\mathbb{Z}$ with

$\displaystyle(t-t^{\prime})\beta_{1}$ $\displaystyle=$ $\displaystyle 2\pi p$
$\displaystyle(t-t^{\prime})\beta_{2}$ $\displaystyle=$ $\displaystyle 2\pi q$

which implies $q\beta_{1}=p\beta_{2}$, contradicting the $\mathbb{Q}$-linear
independence of $\beta_{1},\ldots,\beta_{N}$. It follows that $\psi$ is an
injective continuous map on
$\overline{\mathbb{R}\beta/2\pi\mathbb{Z}^{\alpha}}$, and thus induces a
homeomorphism
$\overline{\mathbb{R}\beta/2\pi\mathbb{Z}^{\alpha}}\cong\psi\left(\overline{\mathbb{R}\beta/2\pi\mathbb{Z}^{\alpha}}\right)$.
Finally, since Eq. (7) implies that
$\psi(t\beta)=\phi\big{(}t(\beta_{1},\ldots,\beta_{N})\big{)}$ for every
$t\in\mathbb{R}$, and
$\overline{\mathbb{R}(\beta_{1},\ldots,\beta_{N})/2\pi\mathbb{Z}^{N}}\cong\mathbb{T}^{N}$
by Kronecker’s theorem, then

$\overline{\mathbb{R}\beta/2\pi\mathbb{Z}^{\alpha}}\cong\psi\left(\overline{\mathbb{R}\beta/2\pi\mathbb{Z}^{\alpha}}\right)\cong\phi\left(\overline{\mathbb{R}(\beta_{1},\ldots,\beta_{N})/2\pi\mathbb{Z}^{N}}\right)=\phi\left(\mathbb{T}^{N}\right)\cong\mathbb{T}^{N}.$

The only if direction follows from the fact that two tori of different
dimensions cannot be homeomorphic. ∎

###  2.2. Quasiperiodic functions

As we alluded to in the introduction, quasiperiodic functions are
superpositions of periodic oscillators with incommensurate frequencies. They
arise in dynamical systems as observation functions on invariant toroidal
submanifolds [30]. More specifically,

######  Definition 2.4.

Let $\omega_{1},\ldots,\omega_{N}>0$ be incommensurate. A function
$f:\mathbb{R}\longrightarrow\mathbb{C}$ is said to be _quasiperiodic_ with
frequency vector $\omega=(\omega_{1},\ldots,\omega_{N})$ if

$f(t)=F(\omega_{1}t,\ldots,\omega_{N}t)$

for all $t\in\mathbb{R}$ and a continuous function
$F:\mathbb{T}^{N}\longrightarrow\mathbb{C}$. That is $F\in C(\mathbb{T}^{N})$,
which we will call a _parent function_ for $f$.

######  Remark 2.5.

The case $N=1$ recovers the family of complex-valued periodic functions, and
thus the results presented here generalize those of [26].

######  Remark 2.6 (Important).

We will require throughout that the dimension $N$ of the frequency vector
$\omega\in\mathbb{R}^{N}$ for $f$ quasiperiodic, be _minimal_. The reason
being that if not, then a function like $f(t)=e^{i(1+\pi)t}$ can be regarded
as being quasiperiodic with frequency vector $\omega=1+\pi$ and parent
function $F(t)=e^{it}$, or as having $\omega=(1,\pi)$ for frequency vector and
$F(t_{1},t_{2})=e^{i(t_{1}+t_{2})}$ for parent function. Requiring that $N$ be
minimal, and showing that for a given $\omega$ the parent function is unique
(we will do so in Theorem 2.7 below), eliminates this ambiguity.

It turns out that the traditional approximation theory via Fourier series on
$L^{2}(\mathbb{T}^{N})$ and $C(\mathbb{T}^{N})$ can be leveraged to obtain
similar insights for quasiperiodic functions. We describe how in what follows
(Theorem 2.11), though the interested reader should also consult [2, 17, 30].
Indeed, let $\|(k_{1},\ldots,k_{N})\|_{\infty}=\max\limits_{1\leq n\leq
N}|k_{n}|$, and for $K\in\mathbb{N}$ let
$I^{N}_{K}=\big{\\{}\mathbf{k}\in\mathbb{Z}^{N}\mid\|\mathbf{k}\|_{\infty}\leq
K\big{\\}}$ be the integral square box of side $2K$. The $K$-truncated Fourier
polynomial of $F\in L^{2}\left(\mathbb{T}^{N}\right)$ is the function

(10) $S_{K}F(\mathbf{t})=\sum_{\mathbf{k}\in
I^{N}_{K}}\widehat{F}(\mathbf{k})e^{i\langle\mathbf{k},\mathbf{t}\rangle}$

where $\mathbf{t}=(t_{1},\dots,t_{N})\in\mathbb{R}^{N}$,
$\langle\cdot,\cdot\rangle$ is the standard inner product in $\mathbb{R}^{N}$,
and

(11)
$\widehat{F}(\mathbf{k})\;\;=\;\;\frac{1}{(2\pi)^{N}}\int\limits^{2\pi}_{0}\cdots\int\limits^{2\pi}_{0}F(t_{1},\dots,t_{N})e^{-i\langle\mathbf{k},\mathbf{t}\rangle}dt_{1}\cdots
dt_{N}\;\;=\;\;\left\langle
F,e^{i\langle\mathbf{k},\cdot\rangle}\right\rangle_{L^{2}}$

is the $\mathbf{k}$-Fourier coefficient of $F$, for
$\mathbf{k}\in\mathbb{Z}^{N}$. As it is well-known [17, Proposition 3.2.7],
the sequence $\\{S_{K}F\\}_{K\in\mathbb{N}}$ converges to $F$ in
$L^{2}(\mathbb{T}^{N})$ as $K\to\infty$. That is,

$\lim_{K\to\infty}\|F-S_{K}F\|_{L^{2}}=0.$

It is not the case, however, that one has pointwise convergence
$S_{K}F(\mathbf{t})\to F(\mathbf{t})$, $\mathbf{t}\in\mathbb{T}^{N}$, even for
$F\in C(\mathbb{T}^{N})$ (see [17, Proposition 3.4.6.] for negative results,
and Theorem 2.9 below for appropriate conditions).

One can address these difficulties with _approximate identities_ , and in
particular using the _square Cesàro (or Fejér) mean_

(12) $C_{K}F(\mathbf{t})=\sum_{\mathbf{k}\in
I_{K}^{N}}\left(1-\frac{|k_{1}|}{K+1}\right)\cdots\left(1-\frac{|k_{N}|}{K+1}\right)\widehat{F}(\mathbf{k})e^{i\langle\mathbf{k},\mathbf{t}\rangle}$

which for $F\in C(\mathbb{T}^{N})$ satisfies

(13) $\displaystyle\lim_{K\to\infty}\|F-C_{K}F\|_{\infty}=0$

(this is Fejér’s theorem) for $\|\cdot\|_{\infty}$ the sup-norm of uniform
convergence in $C(\mathbb{T}^{N})$. We now state the first main result on the
Fourier theory of quasiperiodic functions.

######  Theorem 2.7.

If $f:\mathbb{R}\longrightarrow\mathbb{C}$ is quasiperiodic with frequency
vector $\omega\in\mathbb{R}^{N}$, then $f$ has a unique parent function $F\in
C(\mathbb{T}^{N})$ with Fourier coefficients

(14)
$\widehat{F}(\mathbf{k})=\lim_{\lambda\to\infty}\frac{1}{\lambda}\int_{0}^{\lambda}f(t)e^{-i\langle\mathbf{k},t\omega\rangle}dt.$

###### Proof.

Write $f(t)=F(\omega t)$ for $F\in C(\mathbb{T}^{N})$. Since
$\lim\limits_{K\to\infty}\|F-C_{K}F\|_{\infty}=0$ (from Eq. (13)), then
$C_{K}f(t):=C_{K}F(\omega t)$ converges to $f(t)$ uniformly in
$t\in\mathbb{R}$. We claim that

$\frac{1}{\lambda}\int_{0}^{\lambda}C_{K}f(t)e^{-i\langle\mathbf{k},t\omega\rangle}dt\;\;\;\to\;\;\;\frac{1}{\lambda}\int_{0}^{\lambda}f(t)e^{-i\langle\mathbf{k},t\omega\rangle}dt$

uniformly in $\lambda>0$ as $K\to\infty$. Indeed,

$\left|\,\frac{1}{\lambda}\int_{0}^{\lambda}\big{(}C_{K}f(t)-f(t)\big{)}e^{-i\langle\mathbf{k},t\omega\rangle}dt\,\right|\;\leq\;\|C_{K}f-f\|_{\infty}$

where the right hand side goes to zero as $K\to\infty$ independent of
$\lambda$. Therefore, by the Moore-Osgood theorem, we can exchange the order
of limits as

(15)
$\lim_{\lambda\to\infty}\frac{1}{\lambda}\int_{0}^{\lambda}f(t)e^{-i\langle\mathbf{k},t\omega\rangle}dt=\lim_{K\to\infty}\lim_{\lambda\to\infty}\frac{1}{\lambda}\int_{0}^{\lambda}C_{K}f(t)e^{-i\langle\mathbf{k},t\omega\rangle}dt$

and if

$\lambda_{\mathbf{k}}=\left(1-\frac{|k_{1}|}{K+1}\right)\cdots\left(1-\frac{|k_{N}|}{K+1}\right)\widehat{F}(\mathbf{k})$

are the coefficients of $C_{K}F$ (defined in Eq. (12)), then evaluating the
right hand side of Eq. (15) yields

$\displaystyle\lim_{\lambda\to\infty}\frac{1}{\lambda}\int_{0}^{\lambda}f(t)e^{-i\langle\mathbf{k},t\omega\rangle}dt$
$\displaystyle=$
$\displaystyle\lim_{K\to\infty}\lim_{\lambda\to\infty}\left(\lambda_{\mathbf{k}}+\sum_{\mathbf{k}^{\prime}\in
I_{K}^{N}\smallsetminus\\{\mathbf{k}\\}}\lambda_{\mathbf{k}^{\prime}}\frac{e^{i\langle\mathbf{k}^{\prime}-\mathbf{k},\lambda\omega\rangle}-1}{i\langle\mathbf{k}^{\prime}-\mathbf{k},\omega\rangle\lambda}\right)$
$\displaystyle=$
$\displaystyle\lim_{K\to\infty}\left(1-\frac{|k_{1}|}{K+1}\right)\cdots\left(1-\frac{|k_{N}|}{K+1}\right)\widehat{F}(\mathbf{k})$
$\displaystyle=$ $\displaystyle\widehat{F}(\mathbf{k}).$

If there were another parent function $G\in C(\mathbb{T}^{N})$—i.e., with
$f(t)=G(t\omega)$—then the above calculation shows that
$\widehat{G}(\mathbf{k})=\widehat{F}(\mathbf{k})$ for every
$\mathbf{k}\in\mathbb{Z}^{N}$. Since functions in $L^{2}(\mathbb{T}^{N})$ with
the same Fourier coefficients are equal almost everywhere [17, Proposition
3.2.7], then continuity improves this to functional equality $G=F$. ∎

We now move onto providing conditions for the uniform convergence of

(16) $S_{K}f(t):=\sum_{\mathbf{k}\in
I_{K}^{N}}\widehat{F}(\mathbf{k})e^{i\langle\mathbf{k},t\omega\rangle}\;\;\;,\;\;\;t\in\mathbb{R}\;\;\;,\;\;\;K\in\mathbb{N}$

as $K\to\infty$. Here the $\widehat{F}(\mathbf{k})$ can be seen equivalently
as the Fourier coefficients of the parent function $F$, or as the result of
evaluating the right hand side of Eq. (14). The latter is what we expect to
have access to in practice. We start with an upper bound on the size of the
coefficients $\widehat{F}(\mathbf{k})$ [17, Theorem 3.3.9].

######  Proposition 2.8.

Let $r\in\mathbb{N}$ and suppose that the partial derivatives
$\partial^{\mathbf{l}}F$ exist and are continuous for all
$\|\mathbf{l}\|_{1}=|l_{1}|+\cdots+|l_{N}|\leq r$. That is, $F\in
C^{r}(\mathbb{T}^{N})$. Then

$\left|\widehat{F}(\mathbf{k})\right|\leq\frac{\sqrt{N}^{r}}{\|\mathbf{k}\|_{2}^{r}}\left|\widehat{\partial^{r}_{n}F}(\mathbf{k})\right|$

where $n=n(\mathbf{k})$ satisfies $|k_{n}|=\|\mathbf{k}\|_{\infty}$ and
${\partial^{r}_{n}F}$ is the $r$-th partial derivative of $F$ with respect to
$t_{n}$.

These types of inequalities can be used to estimate the degree $r$ of
regularity of the parent function $F$, by inspecting the rate of decay of the
coefficients $\left|\widehat{F}(\mathbf{k})\right|$. Proposition 2.8 yields
the following estimate for uniform approximation error.

######  Theorem 2.9.

If $F\in C^{r}(\mathbb{T}^{N})$ for $\frac{N}{2}<r\in\mathbb{N}$, then

(17) $\sum_{\mathbf{k}\notin
I^{N}_{K}}\left|\widehat{F}(\mathbf{k})\right|\leq\left(\frac{Area(\mathbb{S}^{N-1})N^{r}}{K^{2r-N}(2r-N)}\sum_{n=1}^{N}\big{\|}\partial^{r}_{n}F-S_{K}\partial^{r}_{n}F\big{\|}^{2}_{L^{2}}\right)^{1/2}$

where ${\partial^{r}_{n}F}$ is the $r$-th partial derivative of $F$ with
respect to $t_{n}$. As a result, the sequence of Fourier coefficients
$\widehat{F}(\mathbf{k})$ is absolutely summable, i.e.

$\sum_{\mathbf{k}\in\mathbb{Z}^{N}}\big{|}\widehat{F}(\mathbf{k})\big{|}<\infty.$

###### Proof.

From Proposition 2.8 we have that

(18) $\sum_{\mathbf{k}\notin
I^{N}_{K}}\big{|}\widehat{F}(\mathbf{k})\big{|}\leq\sum_{\mathbf{k}\notin
I^{N}_{K}}\frac{\sqrt{N}^{r}}{\|\mathbf{k}\|_{2}^{r}}\left|\widehat{\partial^{r}_{n}F}(\mathbf{k})\right|$

for any fixed $K\in\mathbb{N}$. Note that
$n=\operatorname*{\arg\\!\max\;}\limits_{1\leq j\leq N}|k_{j}|$ depends on
$\mathbf{k}$, so we will write it as $n(\mathbf{k})$, and the right hand side
of Eq. (18) can be bounded using Cauchy-Schwarz as

$\sum_{\mathbf{k}\notin
I^{N}_{K}}\frac{\sqrt{N}^{r}}{\|\mathbf{k}\|^{r}_{2}}\left|\widehat{\partial^{r}_{n(\mathbf{k})}F}(\mathbf{k})\right|\leq\sqrt{N}^{r}\left(\sum_{\mathbf{k}\notin
I^{N}_{K}}\frac{1}{\|\mathbf{k}\|_{2}^{2r}}\right)^{1/2}\left(\sum_{\mathbf{k}\notin
I^{N}_{K}}\left|\widehat{\partial^{r}_{n(\mathbf{k})}F}(\mathbf{k})\right|^{2}\right)^{1/2}.$

Moreover, since $\partial^{r}_{n}F$ is continuous and thus square integrable
for each $n\in\mathbb{N}$, then its Fourier coefficients are square summable:

(19)
$\sum_{\mathbf{k}\in\mathbb{Z}^{N}}\left|\widehat{\partial^{r}_{n}F}(\mathbf{k})\right|^{2}=\left\|\partial^{r}_{n}F\right\|_{L^{2}}^{2}<\infty$

by Parseval’s theorem. Hence, summing over $n$ and rearranging terms we get

$\sum_{\mathbf{k}\notin
I^{N}_{K}}\left|\widehat{\partial^{r}_{n(\mathbf{k})}F}(\mathbf{k})\right|^{2}\;\;\leq\;\;\sum_{n=1}^{N}\sum_{\mathbf{k}\notin
I^{N}_{K}}\left|\widehat{\partial^{r}_{n}F}(\mathbf{k})\right|^{2}\;\;=\;\;\sum_{n=1}^{N}\big{\|}\partial^{r}_{n}F-S_{K}\partial^{r}_{n}F\big{\|}^{2}_{L^{2}}$

which goes to zero as $K\to\infty$. In order to bound the remaining sum of
fractions, let
$J^{N}_{K}=\left\\{\mathbf{y}\in\mathbb{R}^{N}\;\mid\;\|\mathbf{y}\|_{\infty}\leq
K\right\\}$ and let
$B^{N}_{K}=\left\\{\mathbf{y}\in\mathbb{R}^{N}\;\mid\;\|\mathbf{y}\|_{2}\leq
K\right\\}$. Observe that

$\displaystyle\sum_{\mathbf{k}\notin
I^{N}_{K}}\frac{1}{\|\mathbf{k}\|_{2}^{2r}}$
$\displaystyle\leq\int\limits_{\mathbf{x}\notin
J^{N}_{K}}\frac{1}{\left(y_{1}^{2}+\cdots+y_{N}^{2}\right)^{r}}dy_{1}\cdots
dy_{N}$ $\displaystyle\leq\int\limits_{\mathbf{x}\notin
B^{N}_{K}}\frac{1}{\left(y_{1}^{2}+\cdots+y_{N}^{2}\right)^{r}}dy_{1}\cdots
dy_{N}$

which in higher dimensional spherical coordinates can be written as

$\int\limits_{\mathbb{S}^{N-1}}\int\limits_{K}^{\infty}\frac{1}{\rho^{2r-N+1}}d\rho
d\sigma$

for $d\sigma$ the differential of surface area on the unit sphere
$\mathbb{S}^{N-1}$ (the differential solid angle) and $\rho$ the distance from
a point in $\mathbb{R}^{N}$ to the origin. The integral satisfies

$\displaystyle\int\limits_{\mathbb{S}^{N-1}}\int\limits_{K}^{\infty}\frac{1}{\rho^{2r-N+1}}d\rho
d\sigma$ $\displaystyle=$ $\displaystyle
Area\left(\mathbb{S}^{N-1}\right)\frac{\rho^{N-2r}}{N-2r}\Bigg{|}^{\infty}_{K}$
$\displaystyle=$
$\displaystyle\frac{Area\left(\mathbb{S}^{N-1}\right)}{K^{2r-N}(2r-N)}$

and thus

$\sum_{\mathbf{k}\notin
I^{N}_{K}}\left|\widehat{F}(\mathbf{k})\right|\leq\left(\frac{Area(\mathbb{S}^{N-1})N^{r}}{K^{2r-N}(2r-N)}\sum_{n=1}^{N}\big{\|}\partial^{r}_{n}F-S_{K}\partial^{r}_{n}F\big{\|}^{2}_{L^{2}}\right)^{1/2}$

where the right hand side goes to zero as $K\rightarrow\infty$. ∎

######  Remark 2.10.

See [33, Chapter VII, Corollary 1.9] for a result akin to Theorem 2.9. While
both have similar hypotheses and deal with absolute Fourier convergence,
Theorem 2.9 above gives explicit bounds for the size of the error term
$\sum\limits_{\mathbf{k}\notin
I^{N}_{Z}}\left|\widehat{F}(\mathbf{k})\right|$. We will need such explicit
estimates when discussing Fourier approximations to persistent homology of
sliding window point clouds in Section 3.

Now, absolute summability of the Fourier coefficients
$\widehat{F}(\mathbf{k})$ implies uniform convergence $S_{K}F\rightarrow F$,
[33, Chapter VII, Corollary 1.8], since

$\left|F(\mathbf{t})-S_{K}F(\mathbf{t})\right|\leq\sum_{\mathbf{k}\notin
I_{K}^{N}}\left|\widehat{F}(\mathbf{k})\right|$

for all $\mathbf{t}\in\mathbb{T}^{N}$. Combining this fact with Eq. (17),
yields the following Fourier series approximation theorem for quasiperiodic
functions:

######  Theorem 2.11.

Let $f$ be quasiperiodic with parent function $F\in C^{r}(\mathbb{T}^{N})$,
$r>\frac{N}{2}$. If $S_{K}f$ is defined as in Eq. (16), then

$\|f-S_{K}f\|_{\infty}\leq\left(\frac{Area(\mathbb{S}^{N-1})N^{r}}{K^{2r-N}(2r-N)}\sum_{n=1}^{N}\big{\|}\partial^{r}_{n}F-S_{K}\partial^{r}_{n}F\big{\|}^{2}_{L^{2}}\right)^{1/2}$

which goes to zero faster than $\frac{1}{K^{r-\frac{N}{2}}}$ as $K\to\infty$.

###  2.3. Persistent Homology

As we mentioned in Section 1, persistent homology is a tool from Topological
Data Analysis used to study the evolution of topological features in filtered
spaces. Indeed, for any filtration
$\mathcal{K}=\\{K_{\epsilon}\\}_{\epsilon\geq 0}$, taking homology in
dimension $j\in\mathbb{N}$ and coefficients in a field $\mathbb{F}$ yields a
family

$H_{j}(\mathcal{K};\mathbb{F})=\left\\{\;T_{\epsilon,\epsilon^{\prime}}:H_{j}(K_{\epsilon};\mathbb{F})\longrightarrow
H_{j}(K_{\epsilon^{\prime}};\mathbb{F})\;\;,\;\;\epsilon\leq\epsilon^{\prime}\;\right\\}$

of $\mathbb{F}$-vector spaces and linear transformations
$T_{\epsilon,\epsilon^{\prime}}$ induced by the inclusion maps
$K_{\epsilon}\hookrightarrow K_{\epsilon^{\prime}}$, for
$\epsilon\leq\epsilon^{\prime}$. The $j$-th _persistent homology groups_ are

(20)
$H^{\epsilon,\epsilon^{\prime}}_{j}(\mathcal{K};\mathbb{F}):=\mathsf{Img}\left(T_{\epsilon,\epsilon^{\prime}}\right)$

and their dimension over $\mathbb{F}$ are the _persistent Betti numbers_

(21)
$\beta_{j}^{\epsilon,\epsilon^{\prime}}(\mathcal{K}):=\mathsf{rank}(T_{\epsilon,\epsilon^{\prime}})=\mathsf{dim}_{\mathbb{F}}\left(H_{j}^{\epsilon,\epsilon^{\prime}}(\mathcal{K};\mathbb{F})\right).$

If $\beta_{j}^{\epsilon,\epsilon}(\mathcal{K})<\infty$ for every
$\epsilon$—i.e., if $H_{j}(\mathcal{K};\mathbb{F})$ is _pointwise-finite_
—then a theorem of Crawley-Boevey [10] contends that the isomorphism type of
$H_{j}(\mathcal{K};\mathbb{F})$ is uniquely determined by a multiset of
intervals $I\subset[0,\infty]$, called the _barcode_ of
$H_{j}(\mathcal{K};\mathbb{F})$, and denoted $\mathsf{bcd}_{j}(\mathcal{K})$.
The (undecorated) _persistence diagram_ $\mathsf{dgm}_{j}(\mathcal{K})$, on
the other hand, is the multiset of pairs $(a,b)$ resulting from taking the
endpoints $a\leq b$ of the intervals in $\mathsf{bcd}_{j}(\mathcal{K})$. In
terms of persistent Betti numbers, one can check that

(22)
$\beta_{j}^{\epsilon,\epsilon^{\prime}}(\mathcal{K})=\\#\Big{\\{}I\in\mathsf{bcd}_{j}(\mathcal{K})\;\mid\;[\epsilon,\epsilon^{\prime}]\subset
I\Big{\\}}$

where cardinality ($\\#$) on the right hand side is that of multisets. If all
intervals in $\mathsf{bcd}_{j}(\mathcal{K})$ are of the same type (i.e., all
open, closed, right/left open), then

(23)
$\beta_{j}^{\epsilon,\epsilon^{\prime}}(\mathcal{K})=\\#\Big{\\{}(a,b)\in\mathsf{dgm}_{j}(\mathcal{K})\;\mid\;a\prec_{\ell}\epsilon\leq\epsilon^{\prime}\prec_{r}b\Big{\\}}$

where $\prec_{\ell}$ and $\prec_{r}$ are chosen to coincide with the interval
type of $\mathsf{bcd}_{j}(\mathcal{K})$.

The pointwise-finite hypothesis on $H_{j}(\mathcal{K};\mathbb{F})$ can be
relaxed to $\beta_{j}^{\epsilon,\epsilon^{\prime}}(\mathcal{K})<\infty$ for
all $\epsilon<\epsilon^{\prime}$; this is called being $\mathsf{q}$-_tame_ ,
and is a condition satisfied by the persistent homology of the Rips filtration
(defined in Eq. (3)) of any totally bounded metric space [9, Proposition 5.1].
It is known that barcodes and persistence diagrams can be defined in the
$\mathsf{q}$-tame case in such a way that Eq. (22) (and also Eq. (23) if all
intervals are of the same type) is still valid [8, Corollary 3.8, Theorem
3.9]. As a result, and when $(X,\mathbf{d}_{X})$ is totally bounded, we have
well-defined Rips persistence diagrams

$\mathsf{dgm}_{j}^{\mathcal{R}}(X,\mathbf{d}_{X}):=\mathsf{dgm}_{j}(\mathcal{R}(X,\mathbf{d}_{X}))$

for every $j\in\mathbb{N}$.

A bit more is true: these diagrams are well-behaved in the sense that they are
stable under Gromov-Hausdorff perturbations on $(X,\mathbf{d}_{X})$. Here is
what this means. The _Hausdorff_ distance in a metric space
$(\mathbb{M},\mathbf{d})$ between two bounded and non-empty subsets
$X,Y\subset\mathbb{M}$ is defined as

$\mathbf{d}_{H}^{\mathbb{M}}(X,Y):=\inf\left\\{\delta>0\mid X\subset
Y^{(\delta)}\mbox{ and }Y\subset X^{(\delta)}\right\\}.$

Here $X^{(\delta)}$ (resp. $Y^{(\delta)}$) is the union of open balls in
$\mathbb{M}$ of radius $\delta>0$ centered at points in $X$ (resp. $Y$). Also,
when the ambient metric space is clear, the notation
$\mathbf{d}_{H}^{\mathbb{M}}(X,Y)$ is simplified to $\mathbf{d}_{H}(X,Y)$. The
_Gromov-Hausdorff distance_ , on the other hand, is defined for bounded metric
spaces $(X,\mathbf{d}_{X})$, $(Y,\mathbf{d}_{Y})$ as

$\mathbf{d}_{GH}\left((X,\mathbf{d}_{X}),(Y,\mathbf{d}_{Y})\right):=\inf_{\mathbb{M},\varphi,\psi}\mathbf{d}^{\mathbb{M}}_{H}\big{(}\varphi(X)\;,\;\psi(Y)\big{)}$

where the infimum runs over all metric spaces $(\mathbb{M},\mathbf{d})$, and
all isometric embeddings
$\varphi:(X,\mathbf{d}_{X})\hookrightarrow(\mathbb{M},\mathbf{d})$,
$\psi:(Y,\mathbf{d}_{Y})\hookrightarrow(\mathbb{M},\mathbf{d})$. In
particular, if $X,Y\subset(\mathbb{M},\mathbf{d})$, then

(24)
$\mathbf{d}_{GH}((X,\mathbf{d}|_{X}),(Y,\mathbf{d}|_{Y}))\leq\mathbf{d}_{H}^{\mathbb{M}}(X,Y).$

The Gromov-Hausdorff distance is a measure of similarity between bounded
metric spaces; in fact it is a pseudometric, which is zero if and only if the
completions of the metric spaces involved are isometric. The stability of Rips
persistence diagrams, on the other hand, is an inequality comparing the
Gromov-Hausdorff distance between the input metric spaces, and a notion of
distance between their persistence diagrams called the _bottleneck distance_.
This distance is defined as follows: two persistence diagrams $\mathsf{dgm}$
and $\mathsf{dgm}^{\prime}$ are said to be $\delta$-matched, $\delta>0$, if
there is a bijection $\mu:A\longrightarrow A^{\prime}$ of multisets
$A\subset\mathsf{dgm}$ and $A^{\prime}\subset\mathsf{dgm}^{\prime}$ for which:

  1. (1)

If $(x,y)\in A$ and $(x^{\prime},y^{\prime})=\mu(x,y)$, then
$\max\left\\{|x-x^{\prime}|,|y-y^{\prime}|\right\\}<\delta$

  2. (2)

If $(x,y)\in(\mathsf{dgm}\smallsetminus
A)\cup(\mathsf{dgm}^{\prime}\smallsetminus A^{\prime})$ then $y-x<2\delta$

The bottleneck distance between $\mathsf{dgm}$ and $\mathsf{dgm}^{\prime}$ is

(25)
$\mathbf{d}_{B}\big{(}\mathsf{dgm},\mathsf{dgm}^{\prime}):=\inf\,\left(\big{\\{}\delta>0\mid\mbox{
$\mathsf{dgm}$ and $\mathsf{dgm}^{\prime}$ are $\delta$-matched
}\big{\\}}\cup\\{\infty\\}\right).$

Finally, the stability of Rips persistent homology [9, Theorem 5.2] contends
that

(26)
$\mathbf{d}_{B}\left(\mathsf{dgm}_{j}^{\mathcal{R}}(X,\mathbf{d}_{X}),\mathsf{dgm}_{j}^{\mathcal{R}}(Y,\mathbf{d}_{Y})\right)\;\leq\;2\mathbf{d}_{GH}\big{(}(X,\mathbf{d}_{X}),(Y,\mathbf{d}_{Y})\big{)}$

for $(X,\mathbf{d}_{X})$ and $(Y,\mathbf{d}_{Y})$ totally bounded.

##  3\. Fourier Approximations of Sliding Window Persistence

With the preliminaries out of the way, we now move onto studying the Rips
persistent homology of sliding window point clouds from quasiperiodic
functions. Thus far we have that if $f$ is quasiperiodic and its parent
function $F$ has enough regularity, then $f$ can be uniformly approximated by
the truncated series $S_{K}f$. This is the content of Theorem 2.11, and in
particular says that the higher the smoothness of $F$, then the faster the
degree of approximation $S_{K}f\rightarrow f$. We will see next that these
results can be readily bootstrapped to the level of sliding window point
clouds, and hence to statements about Rips persistence diagrams.

######  Theorem 3.1.

Let $f$ be quasiperiodic with parent function $F\in C^{r}(\mathbb{T}^{N})$,
$r>\frac{N}{2}$. If

$\mathbb{SW}_{d,\tau}f=SW_{d,\tau}f(T)\mbox{\;\;\; and
\;\;\;}\mathbb{SW}_{d,\tau}S_{K}f=SW_{d,\tau}S_{K}f(T)\;\;\;,\;\;\;T\subset\mathbb{R}$

are the sliding window point clouds of $f$ and $S_{K}f$, respectively, then
the Hausdorff distance between them satisfies

$\displaystyle\mathbf{d}_{H}\big{(}\mathbb{SW}_{d,\tau}f\,,\,\mathbb{SW}_{d,\tau}S_{K}f\big{)}$
$\displaystyle\leq$ $\displaystyle\sqrt{d+1}\|f-S_{K}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\left(\frac{Area(\mathbb{S}^{N-1})(d+1)N^{r}}{K^{2r-N}(2r-N)}\sum_{n=1}^{N}\big{\|}\partial^{r}_{n}F-S_{K}\partial^{r}_{n}F\big{\|}^{2}_{L^{2}}\right)^{1/2}$

which goes to zero faster than $\frac{1}{K^{r-\frac{N}{2}}}$ as $K\to\infty$.

###### Proof.

Let $\epsilon>\sqrt{d+1}\|f-S_{K}f\|_{\infty}$, and let $t\in T$. Then

$\|SW_{d,\tau}f(t)-SW_{d,\tau}S_{K}f(t)\|_{2}\;\;\leq\;\sqrt{d+1}\|f-S_{K}f\|_{\infty}\;<\;\epsilon$

which implies that $\epsilon$ satisfies both

(27)
$\mathbb{SW}_{d,\tau}f\subset\big{(}\mathbb{SW}_{d,\tau}S_{K}f\big{)}^{(\epsilon)}\;\;\;\;\mbox{and}\;\;\;\;\mathbb{SW}_{d,\tau}S_{K}f\subset\big{(}\mathbb{SW}_{d,\tau}f\big{)}^{(\epsilon)}.$

Since the Hausdorff distance in $\mathbb{C}^{d+1}$ between
$\mathbb{SW}_{d,\tau}f$ and $\mathbb{SW}_{d,\tau}S_{K}f$ is the infimum over
all $\delta>0$ satisfying Eq. (27), then we have that

$\mathbf{d}_{H}\big{(}\mathbb{SW}_{d,\tau}f\,,\,\mathbb{SW}_{d,\tau}S_{K}f\big{)}\;\leq\;\epsilon.$

Because this is true for any $\epsilon>\sqrt{d+1}\|f-S_{K}f\|_{\infty}$, then

$\mathbf{d}_{H}\big{(}\mathbb{SW}_{d,\tau}f\,,\,\mathbb{SW}_{d,\tau}S_{K}f\big{)}\leq\sqrt{d+1}\|f-S_{K}f\|_{\infty}$

and the bound from Theorem 2.11 finishes the proof. ∎

Using the stability of Rips persistent homology (Eq. (26)), we can readily
bound the bottleneck distance between the corresponding persistence diagrams:

######  Corollary 3.2.

With the same hypotheses of Theorem 3.1, and for all $j\in\mathbb{N}$,

$\displaystyle\mathbf{d}_{B}\left(\mathsf{dgm}^{\mathcal{R}}_{j}(\mathbb{SW}_{d,\tau}f),\mathsf{dgm}^{\mathcal{R}}_{j}(\mathbb{SW}_{d,\tau}S_{K}f)\right)$
$\displaystyle\leq\hskip 5.69046pt2\sqrt{d+1}\|f-S_{K}f\|_{\infty}$
$\displaystyle\leq\hskip
5.69046pt2\left(\frac{Area(\mathbb{S}^{N-1})(d+1)N^{r}}{K^{2r-N}(2r-N)}\sum_{n=1}^{N}\big{\|}\partial^{r}_{n}F-S_{K}\partial^{r}_{n}F\big{\|}^{2}_{L^{2}}\right)^{1/2}$

and thus goes to zero faster than $\frac{1}{K^{r-\frac{N}{2}}}$ as
$K\to\infty$.

The main point of these approximation results is that studying
$\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{SW}_{d,\tau}f)$ can be reduced to
understanding $\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{SW}_{d,\tau}S_{K}f)$ and
its asymptotes as $K\to\infty$. This is a vastly more accessible
simplification, as we will see shortly.

##  4\. The geometric structure of $\mathbb{SW}_{d,\tau}S_{K}f$

Our next goal is to show that for suitable choices of $d,K\in\mathbb{N}$,
$T\subset\mathbb{R}$, and $\tau>0$, the closure of the sliding window point
cloud $\mathbb{SW}_{d,\tau}S_{K}f=SW_{d,\tau}S_{K}f(T)$ in $\mathbb{C}^{d+1}$,
is homeomorphic to an $N$-torus. Indeed, for $F\in C^{r}(\mathbb{T}^{N})$ and
$K\in\mathbb{N}$, let

$\mathsf{supp}\left(\widehat{F}_{K}\right)=\left\\{\mathbf{k}\in
I_{K}^{N}\mid\widehat{F}(\mathbf{k})\neq 0\right\\}$

denote the support of the Fourier transform $\widehat{F}$ restricted to the
square box $I^{N}_{K}$.

######  Lemma 4.1.

Let $f(t)=F(\omega t)$ be quasiperiodic with frequency vector
$\omega\in\mathbb{R}^{N}$, and parent function $F\in C^{r}(\mathbb{T}^{N})$,
$r>\frac{N}{2}$. Then, for all large enough $K\in\mathbb{N}$,
$\mathsf{supp}\left(\widehat{F}_{K}\right)$ spans an $N$-dimensional
$\mathbb{Q}$-vector space.

###### Proof.

The first thing to note is that since

$\mathsf{supp}\left(\widehat{F}\right)=\bigcup_{K\in\mathbb{N}}\mathsf{supp}\left(\widehat{F}_{K}\right)\subset\mathbb{Z}^{N},$

then
$V=\mathsf{span}_{\mathbb{R}}\left(\mathsf{supp}\left(\widehat{F}\right)\right)$
is an $\mathbb{R}$-linear subspace of $\mathbb{R}^{N}$. It follows that

$L=\mathsf{span}_{\mathbb{Z}}\left(\mathsf{supp}\left(\widehat{F}\right)\right)$

is an additive discrete subgroup of $V$, and therefore a lattice [34, Theorem
6.1] of dimension $n\leq\mathsf{dim}_{\mathbb{R}}(V)\leq N$. It suffices to
show that $n=N$.

Let $\mathbf{z}_{1},\ldots,\mathbf{z}_{n}\in L$ be so that
$L=\mathsf{span}_{\mathbb{Z}}\\{\mathbf{z}_{1},\ldots,\mathbf{z}_{n}\\}$.
Incommensurability of $\omega$ implies that
$\tilde{\omega}_{j}=\langle\mathbf{z}_{j},\omega\rangle$, $j=1,\ldots,n$, are
$\mathbb{Q}$-linearly independent, and we can assume without loss of
generality that $\tilde{\omega}_{j}>0$; otherwise replace $\mathbf{z}_{j}$ by
$-\mathbf{z}_{j}$ as a basis element for $L$. Hence,
$\tilde{\omega}=\left(\tilde{\omega}_{1},\ldots,\tilde{\omega}_{n}\right)\in\mathbb{R}^{n}$
is a vector of incommensurate frequencies.

For $\mathbf{t}\in\mathbb{T}^{n}$ let

$G(\mathbf{t}):=\sum_{\mathbf{a}\in\mathbb{Z}^{n}}\widehat{F}(a_{1}\mathbf{z}_{1}+\cdots+a_{n}\mathbf{z}_{n})e^{i\langle\mathbf{a},\mathbf{t}\rangle}$

which converges uniformly in $\mathbf{t}\in\mathbb{T}^{n}$ since the Fourier
coefficients $\widehat{F}(\mathbf{k})$ are absolutely summable (Theorem 2.9).
Therefore $G\in C(\mathbb{T}^{n})$, and thus

$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{k}\in\mathbb{Z}^{N}}\widehat{F}(\mathbf{k})e^{i\langle\mathbf{k},\omega\rangle
t}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{a}\in\mathbb{Z}^{n}}\widehat{F}(a_{1}\mathbf{z}_{1}+\cdots+a_{n}\mathbf{z}_{n})e^{i\langle\mathbf{a},\tilde{\omega}\rangle
t}$ $\displaystyle=$ $\displaystyle G\left(\tilde{\omega}t\right)$

which shows that $G$ is also a parent function for $f$, with $\tilde{\omega}$
as the corresponding frequency vector. Since the dimension $N$ of the
frequency vector for $f$ is assumed to be minimal (Remark 2.6), then $n=N$,
completing the proof. ∎

Now, if we write
$\mathsf{supp}\left(\widehat{F}_{K}\right)=\left\\{\mathbf{k}_{1},\ldots,\mathbf{k}_{\alpha}\right\\}$,
for $1\leq\alpha\leq(1+2K)^{N}$, then

(28) $\displaystyle SW_{d,\tau}S_{K}f(t)=\Omega_{K,f}\cdot x_{K,f}(t)$

where

(29) $\displaystyle
x_{K,f}(t)=\begin{bmatrix}\widehat{F}(\mathbf{k}_{1})e^{i\langle\mathbf{k}_{1},\omega\rangle
t}\\\ \vdots\\\
\widehat{F}(\mathbf{k}_{\alpha})e^{i\langle\mathbf{k}_{\alpha},\omega\rangle
t}\end{bmatrix}\in\mathbb{C}^{\alpha}$

and $\Omega_{K,f}$ is the Vandermonde $(d+1)\times\alpha$ matrix

(30) $\displaystyle\Omega_{K,f}=\begin{bmatrix}1&\cdots&1\\\
e^{i\langle\mathbf{k}_{1},\omega\rangle\tau}&\cdots&e^{i\langle\mathbf{k}_{\alpha},\omega\rangle\tau}\\\
\vdots&&\vdots\\\ e^{i\langle\mathbf{k}_{1},\omega\rangle\tau
d}&\cdots&e^{i\langle\mathbf{k}_{\alpha},\omega\rangle\tau d}\end{bmatrix}$

with nodes
$e^{i\langle\mathbf{k}_{1},\omega\rangle\tau},\ldots,e^{i\langle\mathbf{k}_{\alpha},\omega\rangle\tau}\in
S^{1}\subset\mathbb{C}$ [3]. This decomposition together with Lemma 4.1 yields
conditions on the parameters $K,d,\tau$ under which the sliding window point
cloud $\mathbb{SW}_{d,\tau}S_{K}f$ is dense in a torus of the appropriate
dimension. Indeed,

######  Theorem 4.2.

Let $f(t)=F(t\omega)$ be quasiperiodic with frequency vector
$\omega\in\mathbb{R}^{N}$ and parent function $F\in C^{r}(\mathbb{T}^{N})$,
$r>\frac{N}{2}$. Let

$\mathsf{supp}(\widehat{F}_{K})=\\{\mathbf{k}_{1},\ldots,\mathbf{k}_{\alpha}\\}$

and assume that $\tau>0$ is not an integer multiple of
$\frac{2\pi}{\langle\mathbf{k}_{n}-\mathbf{k}_{m},\omega\rangle}$ for $1\leq
n<m\leq\alpha$. If $K\in\mathbb{N}$ is large enough so that
$\mathsf{supp}(\widehat{F}_{K})$ spans an $N$-dimensional $\mathbb{Q}$-vector
space, and $d\geq\alpha-1$, then the sliding window point cloud

$\mathbb{SW}_{d,\tau}S_{K}f=SW_{d,\tau}S_{K}f(\mathbb{R})$

is dense in an $N$-torus embedded in $\mathbb{C}^{d+1}$.

###### Proof.

The first thing to note is that since $\tau>0$ is not an integer multiple of
any $\frac{2\pi}{\langle\mathbf{k}_{n}-\mathbf{k}_{m},\omega\rangle}$, $1\leq
n<m\leq\alpha$, then the points
$e^{i\langle\mathbf{k}_{1},\omega\rangle\tau},\ldots,e^{i\langle\mathbf{k}_{\alpha},\omega\rangle\tau}\in
S^{1}$ are all distinct. Thus, the Vandermonde matrix $\Omega_{K,f}$ is full
rank. This can be checked via induction on $\alpha$, by showing that the
determinant of an $\alpha\times\alpha$ Vandermonde matrix with nodes
$\zeta_{1},\ldots,\zeta_{\alpha}$ is $\prod\limits_{1\leq
n<m\leq\alpha}(\zeta_{m}-\zeta_{n})$. Combining this observation with
$d+1\geq\alpha$, implies that
$\Omega_{K,f}:\mathbb{C}^{\alpha}\longrightarrow\mathbb{C}^{d+1}$ is injective
as a linear transformation.

Now, Corollary 2.3 with
$\beta_{1}=\langle\mathbf{k}_{1},\omega\rangle,\ldots,\beta_{\alpha}=\langle\mathbf{k}_{\alpha},\omega\rangle$,
together with Lemma 4.1, imply that for all large enough $K\in\mathbb{N}$ the
point cloud

(31) $\mathbb{X}_{K,f}=\left\\{x_{K,f}(t)\mid
t\in\mathbb{R}\right\\}\;\;\;,\;\;\;x_{K,f}(t)=\begin{bmatrix}\widehat{F}(\mathbf{k}_{1})e^{i\langle\mathbf{k}_{1},\omega\rangle
t}\\\ \vdots\\\
\widehat{F}(\mathbf{k}_{\alpha})e^{i\langle\mathbf{k}_{\alpha},\omega\rangle
t}\end{bmatrix}\in\mathbb{C}^{\alpha}$

is dense in an $N$-torus embedded in $\mathbb{C}^{\alpha}$. The result follows
from $\Omega_{K,f}$ being a linear homeomorphism onto its image. ∎

######  Corollary 4.3.

With the same hypotheses of Theorem 4.2, and if
$\\{\pi,\omega_{1},\ldots,\omega_{N}\\}$ is incommensurate, then the sliding
window point cloud

$\mathbb{SW}_{d,\tau}S_{K}f=SW_{d,\tau}S_{K}f(\mathbb{Z})$

is dense in an $N$-torus embedded in $\mathbb{C}^{d+1}$.

###### Proof.

If $\mathbf{k}_{1},\ldots,\mathbf{k}_{N}\in\mathbb{Z}^{N}$ are
$\mathbb{Q}$-linearly independent, then incommensurability of
$\\{\pi,\omega_{1},\ldots,\omega_{N}\\}$ implies incommensurability of
$\\{\pi,\langle\mathbf{k}_{1},\omega\rangle,\ldots,\langle\mathbf{k}_{N},\omega\rangle\\}$.
The result follows in the same way as Theorem 4.2, but using the integer
version of Kronecker’s theorem as starting point (see Remark 2.2). ∎

We would like to emphasize that the condition on $\tau$ in Theorem 4.2 only
guarantees the topology of an $N$-torus. Preserving the geometric structure as
much as possible when going from $\mathbb{X}_{K,f}$ to
$\mathbb{SW}_{d,\tau}S_{K}f$, and consequently amplifying the toroidal
features in
$\mathsf{dgm}_{j}^{\mathcal{R}}\left(\mathbb{SW}_{d,\tau}S_{K}f\right)$,
requires specific optimizations on $\tau$.

##  5\. Parameter selection: how to optimize $d$ and $\tau$?

###  5.1. The embedding dimension

In practice, the diagrams

$\mathsf{dgm}_{j}^{\mathcal{R}}\left(SW_{d,\tau}f(T)\right)\;\;\;\;,\;\;\;\;T\subset\mathbb{R}\;\;\mbox{finite}$

are computed as approximations to those of $SW_{d,\tau}f(\mathbb{R})$; the
latter set is relatively compact, and hence the stability theorem (see Eq.
(26)) implies that finite samples provide arbitrarily good approximations. The
difficulty lies in that as $d\to\infty$, it becomes necessary for $T$ to also
grow in order to overcome the curse of (ambient space) dimensionality, and
provide appropriate geometric recovery [28]. This is problematic since the
Rips filtration grows exponentially in the number of points, and the matrix
reduction algorithm for computing persistent homology is in the worst case
cubic in the number of simplices [22]. It is thus desirable for $d$ to be as
small as possible. With this and Theorem 4.2 in mind, we propose the following
procedure for choosing $d$: Let $K$ be the smallest integer so that
$\mathsf{supp}(\widehat{F}_{K})$ spans an $N$-dimensional vector space over
$\mathbb{Q}$, and let $d$ be the cardinality ($\alpha$) of
$\mathsf{supp}(\widehat{F}_{K})$. When $f$ is given numerically as a
potentially noisy time series sampled at finitely many evenly spaced time
points, then $d$ can be estimated as the number of prominent peaks in the
spectrum of $f$.

######  Remark 5.1.

The structure theorems for both periodic functions [26, Theorem 5.6] and
quasiperiodic functions (Theorem 4.2) only require $d\geq\alpha-1$. While the
choice $d=\alpha-1$ works for periodic signals in practice, we will
demonstrate in Example 5.2 that $d=\alpha$ is preferable in the quasiperiodic
case. This discrepancy arises in the computation of the time delay $\tau$.
Indeed, while for periodic functions there is a clear closed-form choice of
$\tau$, it turns out that this is typically not possible in the quasiperiodic
case. We will investigate how in what follows.

###  5.2. The time delay

One way in which $\tau$ controls the shape of $\mathbb{SW}_{d,\tau}S_{K}f$ is
via the condition number (i.e., the largest singular value divided by the
smallest singular value) of the Vandermonde matrix $\Omega_{K,f}$ (defined in
Eq. (30)). Indeed, when this number is much larger than 1 and the singular
subspaces from the smallest singular values of $\Omega_{K,f}$ intersect
$\overline{\mathbb{X}_{K,f}}$ transversally, then the persistence of the
toroidal features of $\mathbb{SW}_{d,\tau}S_{K}f$ localized along these
directions can be greatly diminished. One can avoid this problem by selecting
a $\tau>0$ promoting orthogonality between the columns
$\mathbf{u}_{1},\ldots,\mathbf{u}_{\alpha}$ of $\Omega_{K,f}$. Indeed, mutual
orthogonality together with
$\|\mathbf{u}_{1}\|=\cdots=\|\mathbf{u}_{\alpha}\|=\sqrt{d+1}$ would imply
that $\Omega_{K,f}$ is $\sqrt{d+1}$ times a linear isometry. Such a
transformation would have condition number equal to 1, and would preserve the
persistent features of $\mathbb{X}_{K,f}$ (these are described in Theorem
6.6). That said, exact mutual orthogonally of the $\mathbf{u}_{j}$’s is not
possible in general, for if $N\geq 3$, then
$\langle\mathbf{u}_{1},\mathbf{u}_{2}\rangle=\langle\mathbf{u}_{1},\mathbf{u}_{3}\rangle=0$
implies that there exist $m,m^{\prime}\in\mathbb{Z}$ satisfying

$\displaystyle\langle\mathbf{k}_{1}-\mathbf{k}_{2},\omega\rangle\tau(d+1)$
$\displaystyle=$ $\displaystyle 2\pi m$
$\displaystyle\langle\mathbf{k}_{1}-\mathbf{k}_{3},\omega\rangle\tau(d+1)$
$\displaystyle=$ $\displaystyle 2\pi m^{\prime}$

which in turn would imply

$m^{\prime}\langle\mathbf{k}_{1}-\mathbf{k}_{2},\omega\rangle=m\langle\mathbf{k}_{1}-\mathbf{k}_{3},\omega\rangle$

contradicting either the linear independence of
$\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}$, or the incommensurability of
$\omega$. We will settle for the next best option: to let $\tau$ be so that
the $\mathbf{u}_{j}$’s are, in average, as orthogonal as possible. In other
words, we will choose $\tau$ as a minimizer over $[0,\tau_{\mathsf{max}}]$ of
the scalar function

(32) $\Gamma(x):=\sum_{1\leq
j<\ell\leq\alpha}\left|1+e^{i\langle\mathbf{k}_{j}-\mathbf{k}_{\ell},\omega\rangle
x}+\cdots+e^{i\langle\mathbf{k}_{j}-\mathbf{k}_{\ell},\omega\rangle
xd}\right|^{2}$

which is exactly the sum of squared magnitudes of the inner inner products
$\langle\mathbf{u}_{j},\mathbf{u}_{\ell}\rangle$ between the columns of the
Vandermonde matrix $\Omega_{K,f}$. The thing to note is that when $f$ is given
as a noisy finite sample, then the inner products
$\langle\mathbf{k}_{j},\omega\rangle$ can be estimated as the frequency
locations of the prominent peaks in the spectrum of $f$.

######  Example 5.2.

As an illustration of our parameter selection procedure, let

$f(t)=e^{it}+e^{i\sqrt{2}t}+e^{i\sqrt{3}t}\;\;\;,\;\;0\leq t\leq 400.$

The real and imaginary part of this function are shown in Figure 3 below.

(a)

(b)

Figure 3. Real (top) and imaginary (bottom) part of the function
$f(t)=e^{it}+e^{i\sqrt{2}t}+e^{i\sqrt{3}t}$, $0\leq t\leq 400$.

It can be readily checked that $\omega=\left(1,\sqrt{2},\sqrt{3}\right)$ is
the frequency vector for $f$, and

$\mathsf{supp}(\widehat{F}_{K})=\big{\\{}(1,0,0),(0,1,0),(0,0,1)\big{\\}}\;\;\;,\;\;K\geq
1.$

Following the discussion from Section 5.1 we let $d=3$ (the cardinality of
$\mathsf{supp}(\widehat{F}_{K})$) and compute
$\mathsf{dgm}_{j}^{\mathcal{R}}\left(SW_{d,\tau}f(T)\right)$ for
$T\subset[0,400]$ in dimensions $j=1,2$ as follows. We begin by evaluating
$SW_{d,\tau}f(t)$ at 2,000 evenly spaced points in $[0,400]$, and then further
subsample this point cloud by selecting 800 points via `maxmin` sampling. That
is, we pick $t_{1}\in\widetilde{T}=\left\\{\frac{n}{5}\mid
n=0,\ldots,2000\right\\}$ uniformly at random, and if
$t_{1},\ldots,t_{\ell}\in\widetilde{T}$ have been selected, then we let

$t_{\ell+1}=\operatorname*{\arg\\!\max\;}_{t\in\widetilde{T}}\min\Big{\\{}\|SW_{d,\tau}f(t)-SW_{d,\tau}f(t_{0})\|,\ldots,\|SW_{d,\tau}f(t)-SW_{d,\tau}f(t_{\ell})\|\Big{\\}}.$

This inductive process continues until the sampling set
$T=\\{t_{1},\ldots,t_{800}\\}\subset\widetilde{T}$ is constructed, and then we
compute the Rips persistence diagrams of $SW_{d,\tau}f(T)$ in dimensions
$j=1,2$, coefficients in $\mathbb{Z}_{2}$, and two choices of time delay:
$\tau=16.458$ and $\tau=49.325$. The resulting persistence diagrams are shown
in Figure 4 below.

Figure 4. Rips persistence diagrams of $SW_{d,\tau}f(T)$ in dimensions $j=1$
(blue), $j=2$ (orange), $\mathbb{F}=\mathbb{Z}_{2}$, $d=3$, $\tau=16.458$
(left) and $\tau=49.325$ (right). Computations performed with Ripser.py [36].

For this particular example we expect persistence diagrams consistent with a
3-torus—i.e., 3 strong classes in dimension 1, and 3 strong classes in
dimension 2— since there are three linearly independent frequencies:
$1,\sqrt{2}$ and $\sqrt{3}$. That said, and as Figure 4 shows, a poor choice
of time delay (e.g., $\tau=16.458$) can completely obscure these toroidal
features with sampling artifacts (points near the diagonal). This stresses how
important the need for delay optimization can be.

A broader picture of how the persistence of the top 3 features in each
dimension varies with $\tau$ is shown in Figure 5.

(a)

(b)

Figure 5. Persistence of top 3 features in dimension 1 (top) and dimension 2
(bottom) as a function of $\tau$, for $SW_{d,\tau}f(T)$ and $d=3$.

The value $\tau=49.325$ is optimal in the sense that it jointly maximizes the
persistence of the top 3 features in both dimensions. More importantly, it is
also optimal in that it is a global minimizer over $[0,100]$ for the function
$\Gamma(x)$ (defined in Eq. (32)) as described in Section 5.2. We reiterate
that the values $\langle\mathbf{k},\omega\rangle$,
$\mathbf{k}\in\mathsf{supp}(\widehat{F}_{K})$, needed to compute $\tau$ as the
minimizer of $\Gamma(x)$ can be estimated numerically as the frequency
locations of the $d$ most prominent peaks in the spectrum of $f$. Indeed,
Figure 6 shows the result of computing the Discrete Fourier Transform
$\widehat{f}(\xi)$ of $f$ sampled at $t\in\widetilde{T}$, as well as the
locations of the most prominent peaks in amplitude.

Figure 6. Modulus of the Discrete Fourier Transform for $f$ sampled at
$t\in\widetilde{T}$. The locations of the most prominent peaks approximate the
inner products $\langle\mathbf{k},\omega\rangle$.

An important thing to note is that the Discrete Fourier Transform (DFT) by
itself is known to provide only very rough approximations to the frequency
locations of quasiperiodic functions. This can have deleterious effects in the
appropriate estimation of $\tau$ via minimization of $\Gamma(x)$. One solution
is to use methods like [16, 20], which leverage the DFT to produce high-
accuracy frequency estimates.

Finally, to illustrate the difference between the choices $d=\alpha$ and
$d=\alpha-1$ outlined in Remark 5.1, we repeat the same process above with
$d=2$. The persistence of the top 3 features in dimensions 1 and 2, as a
function of $\tau$, is shown in Figure 7 below.

(a)

(b)

Figure 7. Persistence of top 3 features in dimension 1 (top) and dimension 2
(bottom) as a function of $\tau$, for $SW_{d,\tau}f(T)$ and $d=2$.

As Figure 7 shows, the global minimizer $\tau=65.731$ of $\Gamma(x)$, $0\leq
x\leq 100$, jointly maximizes the top 3 persistence values in both dimensions.
In particular, the underlying 3-torus topology is clearly captured by this
choice of time delay. One thing to note when comparing Figure 5 ($d=\alpha=3$)
and Figure 7 ($d=\alpha-1=2$) is the number and nature of local maxima in
persistence (specially in dimension 2) as a function of $\tau$. Indeed, $d=3$
yields a larger number of stable local maxima; by stable we mean that the
values of persistence remain large in a neighborhood of a local maximizer.
This suggests that while $d=2$ still captures the right underlying topology,
as Theorem 4.2 guarantees, the embedding in $\mathbb{C}^{3}$ of the sliding
window point cloud is nonlinear enough that strong features in persistence
(with the ambient Euclidean distance) occur for only very specific time
delays.

##  6\. The Rips Persistent Homology of $\mathbb{SW}_{d,\tau}S_{K}f$ and
$\mathbb{SW}_{d,\tau}f$

We now turn our attention to the persistent homology of the sliding window
point clouds $\mathbb{SW}_{d,\tau}S_{K}f$ and $\mathbb{SW}_{d,\tau}f$, as well
as their dependence on both the Fourier coefficients
$\widehat{F}(\mathbf{k})$, and the parameters $K,d,\tau$. Our aim is to
establish bounds on the cardinality and persistence of strong toroidal
features in $\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{SW}_{d,\tau}f)$. To that
end, let $K\in\mathbb{N}$ be so that

$\mathsf{supp}\left(\widehat{F}_{K}\right)=\\{\mathbf{k}_{1},\ldots,\mathbf{k}_{\alpha}\\}\;\;\;\;\;\;,\;\;\;\;\;\;1\leq\alpha\leq(1+2K)^{N}$

spans a $\mathbb{Q}$-vector space of dimension $N$ (Lemma 4.1). We will
further assume, after re-indexing if necessary, that
$\mathbf{k}_{1},\ldots,\mathbf{k}_{N}$ are $\mathbb{Q}$-linearly independent
and that

$|\widehat{F}(\mathbf{k}_{1})|\geq|\widehat{F}(\mathbf{k}_{2})|\geq\cdots\geq|\widehat{F}(\mathbf{k}_{N})|>0.$

With this convention, let

(33)
$\mathbb{T}^{N}_{\widehat{F}}:=\left\\{\,\mathbf{z}\in\mathbb{C}^{N}\;:\;|z_{1}|=|\widehat{F}(\mathbf{k}_{1})|,\ldots,|z_{N}|=|\widehat{F}(\mathbf{k}_{N})|\,\right\\}$

regarded as a metric space with the $\|\cdot\|_{\infty}$ distance:

$\|\mathbf{z}-\mathbf{z}^{\prime}\|_{\infty}=\max\\{|z_{1}-z_{1}^{\prime}|,\ldots,|z_{N}-z_{N}^{\prime}|\\}.$

In order to understand the Rips persistent homology of
$\mathbb{SW}_{d,\tau}f$, the first step is to clarify that of
$\mathbb{T}^{N}_{\widehat{F}}$. This involves two theorems that we now
describe. The first is a result by Adams and Adamaszek [1, Theorem 7.4]
computing the homotopy type of $R_{\epsilon}(S^{1})$ at each scale
$\epsilon>0$.

######  Proposition 6.1 ([1]).

The Rips complex ${R}_{\epsilon}(S^{1}_{r})$ of a circle
$S^{1}_{r}\subset\mathbb{C}$ of radius $r$ (equipped with the Euclidean
metric) is homotopy equivalent to $S^{2\ell+1}$ if and only if

$\displaystyle
r_{\ell}=2r\sin\left(\pi\frac{\ell}{2\ell+1}\right)\;<\;\epsilon\;\leq\;2r\sin\left(\pi\frac{\ell+1}{2\ell+3}\right)=r_{\ell+1}\;\;\;,\;\;\;\ell\in\mathbb{N}.$

Moreover, for all $\ell\in\mathbb{N}$ and
$r_{\ell}<\epsilon\leq\epsilon^{\prime}\leq r_{\ell+1}$, the inclusion
$R_{\epsilon}(S^{1}_{r})\hookrightarrow R_{\epsilon^{\prime}}(S^{1}_{r})$ is a
homotopy equivalence.

As a consequence, the Rips persistent homology of $(S^{1}_{r},|\cdot|)$ is
pointwise-finite—hence $\mathsf{q}$-tame—the resulting barcodes (and hence the
persistence diagrams) are singleton multisets in odd dimensions, and empty in
positive even dimensions:

(34)
$\displaystyle\mathsf{bcd}^{\mathcal{R}}_{j}\left(S^{1}_{r},|\cdot|\right)=\begin{cases}\left\\{\left(2r\sin\left(\pi\frac{\ell}{2\ell+1}\right),2r\sin\left(\pi\frac{\ell+1}{2\ell+3}\right)\right]\right\\}&\textrm{
if }j=2\ell+1\\\\[5.69046pt] \left\\{\left(0,\infty\right)\right\\}&\textrm{
if }j=0\\\\[5.69046pt] \emptyset&\textrm{ if }j=2\ell+2\end{cases}$

The second result needed to describe
$\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})$
is a Künneth formula for Rips persistent homology and the maximum metric [15,
Corollary 4.6].

######  Proposition 6.2 ([15]).

Let $(X_{1},\mathbf{d}_{1}),\dots,(X_{N},\mathbf{d}_{N})$ be metric spaces
with pointwise-finite Rips persistent homology. Then, for all $j\in\mathbb{N}$

$\mathsf{bcd}_{j}^{\mathcal{R}}(X_{1}\times\cdots\times
X_{N},\mathbf{d}_{\mathsf{max}})=\left\\{\bigcap_{n=1}^{N}I_{j_{n}}\;\Big{|}\;I_{j_{n}}\in\mathsf{bcd}^{\mathcal{R}}_{j_{n}}(X_{n},\mathbf{d}_{n})\;\;,\;\sum\limits_{n=1}^{N}j_{n}=j\;\right\\}$

and thus

$\displaystyle\mathsf{dgm}_{j}^{\mathcal{R}}$
$\displaystyle(X_{1}\times\cdots\times
X_{N},\mathbf{d}_{\mathsf{max}})=\;\;\;$ $\displaystyle\hskip
56.9055pt\left\\{\left(\max_{n}a_{n},\min_{n}b_{n}\right)\;\Big{|}\;(a_{n},b_{n})\in\mathsf{dgm}^{\mathcal{R}}_{j_{n}}(X_{n},\mathbf{d}_{n})\;\;,\;\sum\limits_{n=1}^{N}j_{n}=j\;\right\\}$

where
$\mathbf{d}_{\mathsf{max}}(\mathbf{x},\mathbf{x}^{\prime}):=\max\limits_{1\leq
n\leq N}\mathbf{d}_{n}(x_{n},x^{\prime}_{n})$ is the maximum metric.

These results combined yield the following:

######  Lemma 6.3.

If $a<\sqrt{3}|\widehat{F}(\mathbf{k}_{N})|$, then

$(a,b)\in\mathsf{dgm}^{\mathcal{R}}_{j}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})\;\;\;\;,\;\;\;\;1\leq
j\leq N$

if and only if $a=0$ and $b=\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|$ for some
$1\leq n\leq N$.

Moreover, if $1\leq n\leq N$, and $1\leq n_{1}<\cdots<n_{\ell}\leq N$ is the
longest sequence (i.e., largest $\ell$) for which

$|\widehat{F}(\mathbf{k}_{n})|=|\widehat{F}(\mathbf{k}_{n_{1}})|=\cdots=|\widehat{F}(\mathbf{k}_{n_{\ell}})|$

then $\left(0,\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|\right)$ appears in
$\mathsf{dgm}^{\mathcal{R}}_{j}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})$
with multiplicity

$\mu_{j}(n):={n_{1}-1\choose j-1}+\cdots+{n_{\ell}-1\choose j-1}.$

###### Proof.

By Proposition 6.2, we have that
$(a,b)\in\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})$,
$1\leq j\leq N$, if and only if there exist integers $0\leq
j_{1},\ldots,j_{N}\leq j$ and

$(a_{n},b_{n})\in\mathsf{dgm}_{j_{n}}^{\mathcal{R}}(S^{1}_{|\widehat{F}(\mathbf{k}_{n})|}\,,\,|\cdot|\,)\;\;\;,\;\;\;1\leq
n\leq N$

so that $j_{1}+\cdots+j_{N}=j$, $a=\max\\{a_{1},\ldots,a_{N}\\}$ and
$b=\min\\{b_{1},\ldots b_{N}\\}$. Assume without loss of generality that
$a=a_{1}$.

If

$a_{1}<\sqrt{3}|\widehat{F}(\mathbf{k}_{N})|\leq\sqrt{3}|\widehat{F}(\mathbf{k}_{1})|$

then Eq. (34) implies that $a_{1}=0$ (hence $a_{1}=\cdots=a_{N}=0$) and
therefore

$b_{n}=\left\\{\begin{array}[]{ccl}\infty&\hbox{if}&j_{n}=0\\\\[5.69046pt]
\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|&\hbox{if}&j_{n}=1\end{array}\right.\;\;\;,\;\;\;\mbox{for
all}\;\;\;1\leq n\leq N.$

The first part of the lemma readily follows from this and $j\geq 1$.

Let us now address the multiplicity computation. We will do so by counting the
number of distinct copies of
$\left(0,\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|\right)$ contributed to
$\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})$
by each index $1\leq n_{1}<\cdots<n_{\ell}\leq N$. Indeed, start with $n_{1}$
and assume $1\leq j\leq n_{1}$. Then, each choice of $j-1$ integers $1\leq
m_{1}<\cdots<m_{j-1}<n_{1}$ yields a set of indices

$\mathcal{M}(n_{1})=\\{m_{1},\ldots,m_{j-1},n_{1}\\}$

parametrizing a unique way of writing
$\left(0,\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|\right)$ as
$\left(\max\limits_{m}a_{m}\;,\;\min\limits_{m}b_{m}\right)$ for

$(a_{m},b_{m})=\left\\{\begin{array}[]{ccl}\left(0,\sqrt{3}|\widehat{F}(\mathbf{k}_{m})|\right)&\hbox{if}&m\in\mathcal{M}(n_{1})\\\\[5.69046pt]
(0,\infty)&\hbox{if}&m\in\\{1,\ldots,N\\}\smallsetminus\mathcal{M}(n_{1}).\end{array}\right.$

Since there are ${n_{1}-1\choose j-1}$ ways of choosing $\mathcal{M}(n_{1})$,
the sets $\mathcal{M}(n_{1}),\ldots,\mathcal{M}(n_{\ell})$ are all distinct,
and this computation accounts for all copies, then this completes the proof. ∎

######  Corollary 6.4.

If $0<\delta\leq\epsilon<\sqrt{3}|\widehat{F}(\mathbf{k}_{N})|$ and $1\leq
j\leq N$, then the homomorphism

$\iota_{*}:H_{j}\left(R_{\delta}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty});\mathbb{F}\right)\longrightarrow
H_{j}\left(R_{\epsilon}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty});\mathbb{F}\right)$

induced by the inclusion
$\iota:R_{\delta}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})\hookrightarrow
R_{\epsilon}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})$, is surjective.

Let

$P:\mathbb{C}^{\alpha}\longrightarrow\mathbb{C}^{N}$

be the projection onto the first $N$-coordinates. The first thing to note is
that $P(\mathbb{X}_{K,f})\subset\mathbb{T}^{N}_{\widehat{F}}$ (see Eq. (31)),
and since

$\|P(\mathbf{z})-P(\mathbf{z}^{\prime})\|_{\infty}\;\leq\;\|P(\mathbf{z})-P(\mathbf{z}^{\prime})\|_{2}\;\leq\;\|\mathbf{z}-\mathbf{z}^{\prime}\|_{2}$

for every $\mathbf{z},\mathbf{z}^{\prime}\in\mathbb{C}^{\alpha}$, then $P$
induces simplicial maps at the level of Rips complexes

$\begin{array}[]{rccl}R_{\epsilon}(P):&R_{\epsilon}(\mathbb{X}_{K,f},\|\cdot\|_{2})&\longrightarrow&R_{\epsilon}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})\\\
&\sigma&\mapsto&P(\sigma)\end{array}$

for every $\epsilon>0$. The idea now is to use
$\mathcal{R}(P)=\\{R_{\epsilon}(P)\mid\epsilon>0\\}$ in order to derive
insights about the persistent homology of
$\mathcal{R}(\mathbb{X}_{K,f},\|\cdot\|_{2})$ from that of
$\mathcal{R}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})$.

We have the following,

######  Lemma 6.5.

For all $0<\epsilon<\sqrt{3}|\widehat{F}(\mathbf{k}_{N})|$ and $1\leq j\leq
N$, the homomorphism

$R_{\epsilon}(P)_{*}:H_{j}(R_{\epsilon}(\mathbb{X}_{K,f},\|\cdot\|_{2});\mathbb{F})\longrightarrow
H_{j}(R_{\epsilon}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty});\mathbb{F})$

is surjective.

###### Proof.

The case $N=1$ is essentially Theorem 6.8 in [26], so assume $N\geq 2$.

Our first claim is that the projection
$P:(\mathbb{C}^{\alpha},\|\cdot\|_{2})\longrightarrow(\mathbb{C}^{N},\|\cdot\|_{\infty})$
restricts to a homeomorphism

$P\,:\,\,\overline{\mathbb{X}_{K,f}}\,\xrightarrow{\hskip 7.11317pt\cong\hskip
7.11317pt}\,\mathbb{T}^{N}_{\widehat{F}}$

Indeed, surjectivity is the content of Kronecker’s Theorem (2.1), so in order
to check injectivity, assume that
$\mathbf{x},\mathbf{x}^{\prime}\in\mathbb{X}_{K,f}$ have
$P(\mathbf{x})=P(\mathbf{x}^{\prime})$, and write

$\mathbf{x}=\begin{bmatrix}\widehat{F}(\mathbf{k}_{1})e^{i\langle\mathbf{k}_{1},\omega\rangle
t}\\\ \vdots\\\
\widehat{F}(\mathbf{k}_{\alpha})e^{i\langle\mathbf{k}_{\alpha},\omega\rangle
t}\end{bmatrix}\;\;\;,\;\;\;\mathbf{x}^{\prime}=\begin{bmatrix}\widehat{F}(\mathbf{k}_{1})e^{i\langle\mathbf{k}_{1},\omega\rangle
t^{\prime}}\\\ \vdots\\\
\widehat{F}(\mathbf{k}_{\alpha})e^{i\langle\mathbf{k}_{\alpha},\omega\rangle
t^{\prime}}\end{bmatrix}$

for $t,t^{\prime}\in\mathbb{R}$. Since $P(\mathbf{x})=P(\mathbf{x}^{\prime})$
and $\widehat{F}(\mathbf{k}_{r})\neq 0$ for $r=1,\ldots,\alpha$, then there
exist $m_{1},m_{2}\in\mathbb{Z}$ for which

$\displaystyle\langle\mathbf{k}_{1},\omega\rangle(t-t^{\prime})$
$\displaystyle=$ $\displaystyle 2\pi m_{1}$
$\displaystyle\langle\mathbf{k}_{2},\omega\rangle(t-t^{\prime})$
$\displaystyle=$ $\displaystyle 2\pi m_{2}$

If $t\neq t^{\prime}$, then we would have that

$\langle m_{2}\mathbf{k}_{1}-m_{1}\mathbf{k}_{2},\omega\rangle=0$

contradicting either the incommensurability of $\omega$, or the
$\mathbb{Q}$-linear independence of the vectors
$\mathbf{k}_{1},\mathbf{k}_{2},\ldots,\mathbf{k}_{N}$. Thus $t=t^{\prime}$,
showing that $P$ is injective on $\mathbb{X}_{K,f}$, and continuity plus
Hausdorffness improves this to injectivity on $\overline{\mathbb{X}_{K,f}}$.
Finally, since $P$ provides a continuous bijection between
$\overline{\mathbb{X}_{K,f}}$ and $\mathbb{T}^{N}_{\widehat{F}}$, and the
former is compact (since it is closed and bounded), then $P$ yields the
desired homeomorphism.

Now, given $\epsilon>0$, let $0<\delta_{\epsilon}\leq\epsilon$ be so that

(35)
$\|P(\mathbf{x})-P(\mathbf{x}^{\prime})\|_{\infty}<\delta_{\epsilon}\;\;\;\mbox{
always implies }\;\;\;\|\mathbf{x}-\mathbf{x}^{\prime}\|_{2}<\epsilon$

for $\mathbf{x},\mathbf{x}^{\prime}\in\mathbb{X}_{K,f}$. The existence of
$\delta_{\epsilon}>0$ follows from the uniform continuity of
$P^{-1}:\mathbb{T}^{N}_{\widehat{F}}\longrightarrow\overline{\mathbb{X}_{K,f}}$,
and replacing $\delta_{\epsilon}$ with $\min\\{\delta_{\epsilon},\epsilon\\}$
if necessary. Density of $\mathbb{X}_{K,f}$ in $\overline{\mathbb{X}_{K,f}}$
implies that for each $\mathbf{z}\in\mathbb{T}^{N}_{\widehat{F}}$ there is
$\mathbf{x}_{\mathbf{z}}\in\mathbb{X}_{K,f}$ so that
$\|P(\mathbf{x}_{\mathbf{z}})-\mathbf{z}\|_{\infty}<\frac{\delta_{\epsilon}}{4}$.
Fixing a choice of $\mathbf{x}_{\mathbf{z}}$ for each $\mathbf{z}$ defines a
function

$\begin{array}[]{rccl}\nu:&\mathbb{T}^{N}_{\widehat{F}}&\longrightarrow&\mathbb{X}_{K,f}\\\
&\mathbf{z}&\mapsto&\mathbf{x}_{\mathbf{z}}\end{array}$

satisfying

$\|P\circ\nu(\mathbf{z})-\mathbf{z}\|_{\infty}<\frac{\delta_{\epsilon}}{4}$

for every $\mathbf{z}\in\mathbb{T}^{N}_{\widehat{F}}$. Therefore, if
$\mathbf{z},\mathbf{z}^{\prime}\in\mathbb{T}^{N}_{\widehat{F}}$ are so that
$\|\mathbf{z}-\mathbf{z}^{\prime}\|_{\infty}<\frac{\delta_{\epsilon}}{2}$,
then

$\displaystyle\|P\circ\nu(\mathbf{z})-P\circ\nu(\mathbf{z}^{\prime})\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\|P\circ\nu(\mathbf{z})-\mathbf{z}\|_{\infty}+\|\mathbf{z}-\mathbf{z}^{\prime}\|_{\infty}+\|\mathbf{z}^{\prime}-P\circ\nu(\mathbf{z}^{\prime})\|_{\infty}$
$\displaystyle<$
$\displaystyle\frac{\delta_{\epsilon}}{4}+\frac{\delta_{\epsilon}}{2}+\frac{\delta_{\epsilon}}{4}$
$\displaystyle=$ $\displaystyle\delta_{\epsilon}$

which implies $\|\nu(\mathbf{z})-\nu(\mathbf{z}^{\prime})\|_{2}<\epsilon$ (by
Eq. (35)), and $\nu$ extends to a simplicial map

$\begin{array}[]{rccl}R(\nu):&R_{\frac{\delta_{\epsilon}}{2}}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})&\longrightarrow&R_{\epsilon}(\mathbb{X}_{K,f},\|\cdot\|_{2})\\\
&\sigma&\mapsto&\nu(\sigma)\end{array}$

at the level of Rips complexes.

We claim that $R_{\epsilon}(P)\circ R(\nu)$ is contiguous to the inclusion

$\iota:R_{\frac{\delta_{\epsilon}}{2}}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})\hookrightarrow
R_{\epsilon}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})$

Indeed, if $\mathbf{z},\mathbf{z}^{\prime}\in\mathbb{T}^{N}_{\widehat{F}}$ are
so that
$\|\mathbf{z}-\mathbf{z}^{\prime}\|_{\infty}<\frac{\delta_{\epsilon}}{2}$,
then

$\displaystyle\|\mathbf{z}^{\prime}-P\circ\nu(\mathbf{z})\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\|\mathbf{z}^{\prime}-\mathbf{z}\|_{\infty}+\|P\circ\nu(\mathbf{z})-\mathbf{z}\|_{\infty}$
$\displaystyle<$
$\displaystyle\frac{\delta_{\epsilon}}{2}+\frac{\delta_{\epsilon}}{4}$
$\displaystyle<$ $\displaystyle\delta_{\epsilon}$ $\displaystyle\leq$
$\displaystyle\epsilon$

showing that the set-theoretic union

$\iota(\sigma)\;\cup\;\big{(}R_{\epsilon}(P)\circ R(\nu)\big{)}(\sigma)$

is an element of
$R_{\epsilon}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})$ for every
$\sigma\in
R_{\frac{\delta_{\epsilon}}{2}}(\mathbb{T}^{N}_{\widehat{F}},\|\cdot\|_{\infty})$.

Contiguity at the level of simplicial maps implies that
$\iota_{*}=R_{\epsilon}(P)_{*}\circ R(\nu)_{*}$ in homology, and since
$\iota_{*}$ is surjective (Corollary 6.4), then it follows that
$R_{\epsilon}(P)_{*}$ is also surjective. ∎

The next thing to note is that Lemma 6.3 together with Lemma 6.5 yields the
following estimate for the number of toroidal persistent features in
$\mathcal{R}(\mathbb{X}_{K,f},\|\cdot\|_{2})$:

######  Theorem 6.6.

Fix $1\leq n\leq N$, and let $1\leq n_{1}<\cdots<n_{\ell}\leq N$ be the
longest sequence for which

$|\widehat{F}(\mathbf{k}_{n})|=|\widehat{F}(\mathbf{k}_{n_{1}})|=\cdots=|\widehat{F}(\mathbf{k}_{n_{\ell}})|.$

Then, for each $1\leq j\leq N$, the multiset cardinality of

$\left\\{(0,b)\in\mathsf{dgm}^{\mathcal{R}}_{j}\left(\mathbb{X}_{K,f}\right)\;\Big{|}\;b\geq\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|\right\\}$

is greater than or equal to $\mu_{j}(1)+\cdots+\mu_{j}(n)$, for

$\mu_{j}(n)={n_{1}-1\choose j-1}+\cdots+{n_{\ell}-1\choose j-1}.$

In order to make statements on
$\mathsf{dgm}_{j}^{\mathcal{R}}(\mathbb{SW}_{d,\tau}S_{K}f)$, we will leverage
the diagram

(36)
${\left(\mathbb{SW}_{d,\tau}S_{K}f,\|\cdot\|_{2}\right)}$${\left(\mathbb{X}_{K,f},\|\cdot\|_{2}\right)}$$\scriptstyle{\Omega^{+}_{K,f}}$$\scriptstyle{\Omega_{K,f}}$

and the estimates in Rips persistence that it implies. Here $\Omega_{K,f}$ is
the Vandermonde matrix defined in Eq. (30), and $\Omega^{+}_{K,f}$ is its
Moore-Penrose pseudoinverse (see [6, III.3.4]).

Let $0<\sigma_{\mathsf{min}}\leq\sigma_{\mathsf{max}}$ be the smallest and
largest singular values of $\Omega_{K,f}$, respectively. Standard singular
value decomposition arguments show that

$\left\|\Omega_{K,f}\mathbf{u}\right\|_{2}\leq\sigma_{\mathsf{max}}\|\mathbf{u}\|_{2}\;\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\left\|\Omega^{+}_{K,f}\mathbf{v}\right\|_{2}\leq\frac{1}{\sigma_{\mathsf{min}}}\|\mathbf{v}\|_{2}$

for every $\mathbf{u}\in\mathbb{C}^{\alpha}$ and
$\mathbf{v}\in\mathbb{C}^{d+1}$, and thus we have induced simplicial maps

$\Omega:R_{\epsilon}(\mathbb{X}_{K,f})\longrightarrow
R_{\epsilon\sigma_{\mathsf{max}}}(\mathbb{SW}_{d,\tau}S_{K}f)$

$\Omega^{+}:R_{\delta\sigma_{\mathsf{min}}}(\mathbb{SW}_{d,\tau}S_{K}f)\longrightarrow
R_{\delta}(\mathbb{X}_{K,f})$

at the level of Rips complexes. Let

$\kappa(\Omega_{K,f})=\frac{\sigma_{\mathsf{max}}}{\sigma_{\mathsf{min}}}$

denote the condition number of $\Omega_{K,f}$. Then for every
$\epsilon\leq\epsilon^{\prime}$, the diagram

(37)
${R_{\epsilon\sigma_{\mathsf{min}}}(\mathbb{SW}_{d,\tau}S_{K}f)}$${R_{\epsilon^{\prime}\sigma_{\mathsf{min}}}(\mathbb{SW}_{d,\tau}S_{K}f)}$${R_{\epsilon/\kappa\left(\Omega_{K,f}\right)}(\mathbb{X}_{K,f})}$${R_{\epsilon}(\mathbb{X}_{K,f})}$${R_{\epsilon^{\prime}}(\mathbb{X}_{K,f})}$$\scriptstyle{\Omega^{+}}$$\scriptstyle{\Omega^{+}}$$\scriptstyle{\Omega}$

commutes. The horizontal maps are inclusions, and commutativity follows from
noting that $\Omega^{+}_{K,f}$ is a left inverse of $\Omega_{K,f}$. Indeed,
the latter (tall skinny) matrix is full-rank with our choice of $d$ and
$\tau$. Taking homology in dimension $j\in\mathbb{N}$ we get the induced
homomorphisms

${H_{j}^{\epsilon\sigma_{\mathsf{min}},\;\epsilon^{\prime}\sigma_{\mathsf{min}}}(\mathcal{R}(\mathbb{SW}_{d,\tau}S_{K}f);\mathbb{F})}$${H_{j}^{\epsilon/\kappa\left(\Omega_{K,f}\right),\;\epsilon^{\prime}}(\mathcal{R}(\mathbb{X}_{K,f});\mathbb{F})}$${H_{j}^{\epsilon,\,\epsilon^{\prime}}(\mathcal{R}(\mathbb{X}_{K,f});\mathbb{F})}$$\scriptstyle{\Omega^{+}_{*}}$

at the level of persistent homology groups (See Eq. (20)), where the
horizontal map is an inclusion as linear spaces. Commutativity of the diagram
in Eq. (37) implies that

$H_{j}^{\epsilon/\kappa\left(\Omega_{K,f}\right),\;\epsilon^{\prime}}(\mathcal{R}(\mathbb{X}_{K,f});\mathbb{F})\subset\mathsf{Img}\left(\Omega_{*}^{+}\right)$

which, after taking dimensions, yields the following inequality of persistent
Betti numbers (see Eq. (21)):

$\beta_{j}^{\epsilon\sigma_{\mathsf{min}},\;\epsilon^{\prime}\sigma_{\mathsf{min}}}(\mathcal{R}(\mathbb{SW}_{d,\tau}S_{K}f))\;\;\geq\;\;\mathsf{rank}\left(\Omega_{*}^{+}\right)\;\;\geq\;\;\beta_{j}^{\epsilon/\kappa\left(\Omega_{K,f}\right),\;\epsilon^{\prime}}(\mathcal{R}(\mathbb{X}_{K,f}))$

Letting $\epsilon\rightarrow 0$ and using Theorem 6.6, we get the following:

######  Theorem 6.7.

Let $f(t)=F(t\omega)$ be quasiperiodic with frequency vector
$\omega\in\mathbb{R}^{N}$ and parent function $F\in C^{r}(\mathbb{T}^{N})$,
$r>\frac{N}{2}$. Fix parameters $K,d,\tau$ as before.

Let $\sigma_{\mathsf{min}}>0$ be the smallest singular value of the
Vandermonde matrix $\Omega_{K,f}$ (see Eq. (30)), and for $1\leq n\leq N$, let
$1\leq n_{1}<\cdots<n_{\ell}\leq N$ be the longest sequence for which

$|\widehat{F}(\mathbf{k}_{n})|=|\widehat{F}(\mathbf{k}_{n_{1}})|=\cdots=|\widehat{F}(\mathbf{k}_{n_{\ell}})|.$

Then, for each $1\leq j\leq N$, the multiset cardinality of

$\left\\{(0,b)\in\mathsf{dgm}^{\mathcal{R}}_{j}\left(\mathbb{SW}_{d,\tau}S_{K}f\right)\;\;\Big{|}\;\;b\geq\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|\sigma_{\mathsf{min}}\right\\}$

is greater than or equal to $\mu_{j}(1)+\cdots+\mu_{j}(n)$ for

$\mu_{j}(n)={n_{1}-1\choose j-1}+\cdots+{n_{\ell}-1\choose j-1}.$

The Stability Theorem for Rips persistence (Eq. (26)), together with Theorem
6.7 and Corollary 3.2 yield the main result of this section.

######  Theorem 6.8.

With the same hypotheses of Theorem 6.7, and for $1\leq j,n\leq N$, the
multiset cardinality of

$\left\\{(a,b)\in\mathsf{dgm}^{\mathcal{R}}_{j}\left(\mathbb{SW}_{d,\tau}f\right)\;\;\Big{|}\;\;b-a\;\geq\;\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|\sigma_{\mathsf{min}}-4\sqrt{d+1}\|f-S_{K}f\|_{\infty}\right\\}$

is greater than or equal to $\mu_{j}(1)+\cdots+\mu_{j}(n)$.

The extremal singular values of Vandermonde matrices with nodes in the unit
circle haven been extensively studied in the harmonic analysis and
computational mathematics literature [3, 21, 13]. In particular, the lower
bound on $\sigma_{\mathsf{min}}$ from [3, Eq. (55)] implies the following.

######  Corollary 6.9.

With the same hypotheses of Theorem 6.7, and if

$d\;>\;\frac{1}{\delta_{\omega}}-\frac{3}{2}\;\;\;\;\;\;\;\;,\;\;\;\;\;\;\delta_{\omega}:=\min_{1\leq\ell<m\leq\alpha}\,\frac{1}{\pi}\arcsin\left(\frac{\left|e^{i\langle\mathbf{k}_{\ell},\omega\rangle\tau}-e^{i\langle\mathbf{k}_{m},\omega\rangle\tau}\right|}{2}\right)$

then, for each $1\leq j,n\leq N$, the multiset cardinality of

$\left\\{(a,b)\in\mathsf{dgm}^{\mathcal{R}}_{j}\left(\mathbb{SW}_{d,\tau}f\right)\Big{|}b-a\geq\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|\sqrt{d+\frac{3}{2}-\frac{1}{\delta_{\omega}}}-4\sqrt{d+1}\|f-S_{K}f\|_{\infty}\right\\}$

is always greater than or equal to $\mu_{j}(1)+\cdots+\mu_{j}(n)$.

This brings us to the end of the theoretical quasiperiodicity analysis in this
paper. In the next section, we focus on examples and applications.

##  7\. Experiments and Applications

This section has two goals: first, to illustrate the pipeline developed in
this paper for the analysis of quasiperiodic time series data. Indeed, we will
utilize a synthetic example to review the optimization of $d$ and $\tau$, as
well as the evaluation of our theoretical lower bounds on persistence. The
second goal is to provide an example of how quasiperiodicity can arise in
naturally-occurring time series data. Specifically, we will study a sound
recording of _dissonance_ , and illustrate how quasiperiodicity emerges
through the lens of sliding windows and persistence.

###  7.1. Computational pipeline and valuation of theoretical lower bounds

Let

$f(t)=2\sin(t)+1.8\sin\left(\sqrt{3}t\right)\;\;\;\mbox{ for }\;\;\;0\leq
t\leq 60\pi$

with graph shown in Figure 8 below.

Figure 8. $f(t)=2\sin(t)+1.8\sin\left(\sqrt{3}t\right)$ for $0\leq t\leq
60\pi$

We sample $f$ at 10,000 evenly spaced points; that is, at each

$t\in T=\left\\{\frac{3n\pi}{500}\mid n=0,\ldots,\,9,999\right\\}$

producing a discrete time series for which the Discrete Fourier Transform is
computed (see Figure 9 below). We note that since $f$ is real-valued, then
$|\widehat{f}(\xi)|$ is symmetric with respect to the origin,
$\omega=\left(1,\sqrt{3}\right)$, $\mathbf{k}_{1}=(1,0)$,
$\mathbf{k}_{2}=(0,1)$, $\mathbf{k}_{3}=(-1,0)$, $\mathbf{k}_{4}=(0,-1)$, and
that $|\widehat{F}(\mathbf{k}_{1})|=|\widehat{F}(\mathbf{k}_{3})|\approx 1$,
$|\widehat{F}(\mathbf{k}_{2})|=|\widehat{F}(\mathbf{k}_{4})|\approx 0.9$.

Figure 9. Modulus of the DFT for $f(T)$

The number of prominent peaks in $|\widehat{f}(\xi)|$ is used—as described in
Section 5.1—to select $d=4$, while the peak locations
$\langle\mathbf{k},\omega\rangle$ define the function $\Gamma(x)$ (see Eq.
(32)) whose minimizer over $\left[0,\tau_{\mathsf{max}}\right]$, for
$\tau_{\mathsf{max}}=\frac{3}{4}\frac{60\pi}{d}$, yields the choice
$\tau\approx 11.9577$ as described in Section 5.2. The value of
$\tau_{\mathsf{max}}$ is selected to guarantee that the window size $d\tau$ is
less than that of the domain $T$ over which $f$ is evaluated.

The number of points in $T$ is already large enough that computing the Rips
persistent homology of $SW_{d,\tau}f(T)$, using standard software [4], is
algorithmically intensive. Thus, we take a maxmin subsample
$SW_{d,\tau}f(\widetilde{T})$ (see Example 5.2) by selecting
$\widetilde{T}\subset T$ with 1,000 points, and compute
$\mathsf{dgm}_{j}^{\mathcal{R}}\left(SW_{d,\tau}f(\widetilde{T})\right)$ in
dimensions $j=1,2$ and coefficients in $\mathbb{F}=\mathbb{Z}_{2}$.

Since $\widetilde{T}\neq\mathbb{R}$, then the lower bounds on persistence from
Theorem 6.7 do not readily apply to the diagrams
$\mathsf{dgm}_{j}^{\mathcal{R}}(SW_{d,\tau}f(\widetilde{T}))$. That said, the
stability theorem implies that the inequality can be corrected to

(38)
$b-a\;\geq\;\sqrt{3}|\widehat{F}(\mathbf{k}_{n})|\sigma_{\mathsf{min}}-4\sqrt{d+1}\|f-S_{K}f\|_{\infty}-4d_{H}\left(SW_{d,\tau}f(\widetilde{T}),\mathbb{SW}_{d,\tau}f\right)$

where, for this example, the Hausdorff distance term was estimated as

$d_{H}\left(SW_{d,\tau}f(\widetilde{T}),\mathbb{SW}_{d,\tau}f\right)\approx
0.54292.$

Figure 10 below shows the Rips persistence diagrams
$\mathsf{dgm}_{j}^{\mathcal{R}}(SW_{d,\tau}f(\widetilde{T}))$, as well as the
estimated lower bounds in persistence with and without the correction term on
Hausdorff distance.

Figure 10. Rips persistence diagrams of $SW_{d,\tau}f(\widetilde{T})$ in
dimensions $j=1$ (blue) and $j=2$ (orange), and coefficients in
$\mathbb{F}=\mathbb{Z}_{2}$. Lower bounds on persistence are shown with dashed
lines. Left: no Hausdorff correction, and Right: Hausdorff correction

###  7.2. Application: Dissonance Detection in Music

In music theory, consonance and dissonance are classifications of multiple
simultaneous tones. While the former is associated with pleasantness, the
latter creates tension as experienced by the listener. Perfect dissonance
occurs when the audio frequencies are irrational with respect to each other.
One such instance is the _tritone_ , which is a musical interval that is
halfway between two octaves. Mathematically, for a base frequency
$\omega_{1}$, its tritone is $\sqrt{2}\omega_{1}$. We will use the theory of
sliding window embedding to quantify quasiperiodicity from a dissonant sample.
For the purpose of this application, we use a $5$-second audio recording of a
brass horn playing the tritone111generously provided by Adam Huston.. The
signal was read using wavfile.read() and the resulting amplitude plot is shown
in Figure 11 (Top). Like before, in order to perform sliding window analysis,
we need to choose appropriate parameters $d$ and $\tau$. We proceed exactly as
before with the spectral analysis shown in Figure 11 (Bottom).

(a)

(b)

Figure 11. (Left) The plot of a dissonant music sample created on a brass
horn. (Right) The amplitude-frequency spectrum of the music sample plot using
Fast Fourier Transform.

We then find peaks with height at least $0.04$ and at least $100$ Hz apart to
detect prominent frequencies which we will use for estimation of the embedding
parameters. See Table 1.

Angular Frequencies | 1384.93 | 1957.83 | 2769.86 | 3911.93
---|---|---|---|---
Frequencies (Hz) | 220.41 | 311.59 | 440.83 | 622.60
Proportion | 1 | 1.4137 $\approx\sqrt{2}$ | 2 | 2.8246 $\approx 2\sqrt{2}$
Table 1. List of frequencies in the positive side of the (symmetric)
amplitude-frequency spectrum: First row: list of detected frequencies. Second
row: their conversion to Hertz. Third row: ratio with respect to the first
row.

The resulting embedding parameters are $d=8$ and $\tau=0.0285736$. We use
cubic splines to compute the sliding window vectors and present the PCA
representation of the point cloud, along with the persistence diagrams
computed for $2200$ landmarks in Figure 12.

Figure 12. (Left) PCA representation of the sliding window point cloud.
(Right) Persistence Diagrams in homological dimensions $0$, $1$ and $2$.

## Acknowledgments

The authors of this paper would like to thank Adam Huston for the audio
recording of the brass horn. The first author would like to thank Tyler
Bongers for discussions on some of the Harmonic Analysis aspects of this
paper.

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