\section{Introduction}
\label{introduction}

Given a set of points $\Xn = \pb{\Xv_1,\Xv_2,\dots,\Xv_n}$ observed from a probability distribution $\pr$ on an input space $\X \subseteq \R^d$, understanding the shape of $\Xn$ sheds important insights on low-dimensional geometric and topological features which underlie $\pr$, and this question has received increasing attention in the past few decades. To this end, Topological Data Analysis (TDA), with a special emphasis on persistent homology \citep{edelsbrunner2000topological,zomorodian2005computing}, has become a mainstay for extracting the shape information from data. In statistics and machine-learning, persistent homology has facilitated the development of novel methodology (e.g., \citealt{chazal2013persistence,chen2019topological,bruel2018topology}), which has been widely used in a variety of applications dealing with massive, unconventional forms of data (e.g., \citealt{gameiro2015topological,bendich2016persistent,xu2019finding}).

Informally speaking, persistent homology detects the presence of topological features across a range of resolutions by examining a nested sequence of spaces, typically referred to as a \textit{filtration}. The filtration encodes the birth and death of topological features as the resolution varies, and is presented in the form of a concise representation---a persistence diagram or barcode. In the context of data-analysis, there are two different methods for obtaining filtrations. The first is computed from the pairwise Euclidean distances of $\Xn$, such as the Vietoris-Rips, \cech{}, and Alpha filtrations \cite{edelsbrunner2000topological}. The second approach is based on 

choosing a function on $\X$ that reflects the density of $\pr$ (or its approximation based on $\Xn$), and, then, constructing a filtration. While the two approaches explore the topological features governing $\pr$ in different ways, in essence, they generate equivalent insights.

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\begin{figure}[!t]
  \centering
  \begin{subfigure}[b]{0.24\linewidth}
    \includegraphics[width=\linewidth]{./plots/examples/signal.pdf}
    
  \end{subfigure}
  \begin{subfigure}[b]{0.24\linewidth}
    \includegraphics[width=\linewidth]{./plots/examples/signal-dgm.pdf}
    
  \end{subfigure}
  \begin{subfigure}[b]{0.24\linewidth}
    \includegraphics[width=\linewidth]{./plots/examples/X.pdf}
    
  \end{subfigure}
  \begin{subfigure}[b]{0.24\linewidth}
    \includegraphics[width=\linewidth]{./plots/examples/X-dgm.pdf}
    
  \end{subfigure}
  \caption{(Left) $\Xn$ is sampled from a circle with small perturbations to each point. The persistence diagram detects the presence of the loop, as guaranteed by the stability of persistence diagrams \cite{chazal2016structure,cohen2007stability}. (Right) $\Xn$ is sampled from a circle but with just a few outliers. The resulting persistence diagram changes dramatically --- the persistence of the main loop plummets, and other spurious loops appear, as elaborated in Section \ref{preliminaries}.}
  \label{fig:distfct}
  
\end{figure}

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Despite obvious advantages, the adoption of persistent homology in mainstream statistical methodology is still limited. An important limitation among others, in the statistical context, is that the resulting persistent homology is highly sensitive to outliers. While the stability results of \cite{chazal2016structure,cohen2007stability} guarantee that small perturbations on all of $\Xn$ induce only small changes in the resulting persistence diagrams, a more pathological issue arises when a small fraction of $\Xn$ is subject to very large perturbations. Figure~\ref{fig:distfct} illustrates how inference from persistence diagrams can change dramatically when $\Xn$ is contaminated with only a few outliers.  Another challenge is the mathematical difficulty in performing sensitivity analysis in a formal statistical context. Since the space of persistence diagrams have an unusual mathematical structure, it falls victim to issues such as non-uniqueness of Fr\'{e}chet means and bounded curvature of geodesics \citep{mileyko2011probability,turner2014frechet,divol2019understanding}. With this background, the \textit{central objective} of this paper is to develop outlier robust persistence diagrams, develop a framework for examining the sensitivity of the resulting persistence diagrams to noise, and establish statistical convergence guarantees. To the best of our knowledge, not much work has been carried out in this direction, except for \citet{bendich2011improving} where robust persistence diagrams are constructed from Vietoris-Rips or \cech{} filtrations on $\Xn$ by replacing the Euclidean distance with diffusion distance. However, no sensitivity analysis of the resultant diagrams are carried out in \citep{bendich2011improving} to demonstrate their robustness.


\textbf{Contributions.} The main contributions of this work are threefold. 1) We propose robust persistence diagrams constructed from filtrations induced by an RKHS-based robust KDE (kernel density estimator) \cite{kim2012robust} of the underlying density function of $\pr$ (Section~\ref{robust-dgm}). While this idea of inducing filtrations by an appropriate function---\citep{fasy2014confidence,chazal2017robust,phillips2015geometric} use KDE, distance-to-measure (DTM) and kernel distance (KDist), respectively---has already been explored, we show the corresponding persistence diagrams to be less robust compared to our proposal.

2)


In Section~\ref{robustness}, we generalize the notions of \textit{influence function} and \textit{gross error sensitivity}---which are usually defined for normed spaces---to the space of persistence diagrams, which lack the vector space structure. Using these generalized notions, we investigate the sensitivity of persistence diagrams constructed from filtrations induced by different functions (e.g., KDE, robust KDE, DTM) and demonstrate the robustness of the proposed method, both \mbox{theoretically (Remark~\ref{remark:influence})} \linebreak

and numerically (Section~\ref{experiments}). 

3) We establish the statistical consistency of the proposed robust persistence diagrams and provide uniform confidence bands by deriving exponential concentration bounds for the uniform deviation of the robust KDE (Section~\ref{consistency}).


\textbf{Definitions and Notations.} For a metric space $\pa{\X,\varrho}$, the ball of radius $r$ centered at $\xv\in \X$ is denoted by $B_{\X}(\xv,r)$. $L_p(\X,\mu)$ is the Banach space of functions of $p^{th}$-power $\mu$-integrable functions with norm $\Vert \cdot\Vert_{p}$, where $\mu$ is a Borel measure defined on $\X$. $\mathcal{P}(\R^d)$ is the set of all Borel probability measures on $\R^d$, and $\M(\R^d)$ denotes~the set of probability measures on $\R^d$ with compact support and \textit{tame}, bounded density function. For bandwidth $\sigma > 0$, $\HH$ denotes a reproducing kernel Hilbert space (RKHS) with ${\K : \R^d\times\R^d \rightarrow \R}$ as its reproducing~kernel. We assume that $\K$ is radial, i.e., $\K (\xv,\yv) = \s^{-d}\psi({\norm{\xv-\yv}_2}/{\s})$ with $\psi(\Vert\cdot\Vert_2)$ being a probability density function on $\R^d$, 
where $\Vert \xv\Vert^2_2=\sum^d_{i=1}x^2_i$ for ${\xv=(x_1,\ldots,x_d)\in\R^d}$. We denote ${\kinf \defeq \sup_{\xv,\yv\in \R^d}{\K(\xv,\yv)} = \s^{-d}\psi(0)}$. For ${\pr \in \mathcal{P}(\R^d)}$, $\mu_{\pr} \defeq \int\K(\cdot,\yv)d\pr(\yv) \in \HH$ is called the \mbox{mean embedding of $\pr$}, and ${\D \defeq \pb{\mu_{\pr} : \pr  \in \mathcal{P}(\R^d)}}$ is the space of all mean embeddings. $\delta_{\xv}$ denotes a Dirac measure \mbox{at $\xv$}.
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