Skip to main content
Department of Mathematics College of Liberal Arts and Sciences

My Work

My thesis work, “A Schauder Basis for Multiparameter Persistence” (joint with  Peter Bubenik), develops a new method for mapping signed multiparameter persistence barcodes into Banach or Hilbert space. The main idea is to view these diagrams as elements of the dual space of compactly supported Lipschitz functionals on a polyhedral pair \((X,A)\), where \(X \) is a polyhedron in Euclidean space, and \(A\) is a subspace representing ephemeral features. This duality is illustrated in recent work by Peter Bubenik and Alex Elchesen, 2024. An example of such a pairing is the space containing persistence diagrams from 1-parameter persistence, \((\mathbb{R}^2_\leq, \Delta)\), where \(\mathbb{R}^2_\leq = \{(x,y) \in \mathbb{R}^2 \ | \ x \leq y\}\) and \( \Delta = \{(x,x) \ | \ x \in \mathbb{R}\). We construct a Schauder basis for the normed vector space of compactly-supported, Lipschitz functionals on \(X\) that are \(0\) on \(A\). We begin by generating iteratively refined triangulations of the pair \((X,A)\). These triangulations may be chosen arbitrarily, or by taking into account the mass of a specified collection of signed barcodes.

Example:\(X= \mathbb{R}^4_\leq = \{(x_1,y_1, x_2 ,y_2) \ | \ x_1 \leq y_1 \ \& \ x_2 \leq y_2)\}\) and \(A = (\mathbb{R}^2_\leq \times \Delta) \cup (\Delta \times \mathbb{R}^2_\leq)\).This space contains an embedding of signed rectangle barcodes on 2-parameter persistence modules such as those byBotnan, Opperman, & Oudot, 2024. In this case, the set \(A\) corresponds to features lived instantaneously in either of the two parameters (flat rectangles).


We may then define a list of piece-wise linear functionals on the pair \((X,A)\) by the nested sequence of triangulations. The collection we define of these functionals form a Schauder Basis of the vector space of compactly supported Lipschitz functionals on the pair \((X,A)\). That is, every compactly supported Lipschitz functional on \((X,A)\) can be approximated to arbitrary accuracy by a finite sum of scalar multiples of Schauder basis functionals.

Evaluation of a persistence diagram against these basis elements produces a sequence of real numbers in \(\ell_1\),  which serves as our vector representation of the persistence diagram.

We prove:

  1. Injectivity:  the vectorization uniquely determines the persistence diagram
  2. Lipschitz Stability: the \(\ell_1\)-distance between vectorizations is bounded above by a constant times the 1-Wasserstein distance between diagrams
  3. Minimality: in the broader setting of relative Radon measures on the pair (X,A), the constructed basis is minimal among all template systems that produce an embedding.

This work connects the algebraic nature of multiparameter persistence modules with analytic tools from functional analysis. Our vectorization method is general enough to handle:

  • -Traditional 1-parameter diagrams (as a special case),
  • -Multiparameter diagrams from bifiltrations or d-filtrations,
  • -Variants of multiparameter persistence such as mixup barcodes (Wagner, et. al. , 2024) used to compare multiple classes of datasets.

Example: Consider a set of points sampled from an annulus in \(\mathbb{R}^2\), with noise added. We then construct the signed rank barcode (Botnan, Opperman, Oudot) for the denisty-rips filtration on this data, with positive and negative rectangles plotted by their diagonals. We embed these rectangles in \(\mathbb{R}^4\), as a signed persistence diagram of polyhedral pair, \(X, A\), where \(X= \mathbb{R}^4_\leq = \{(x_1,y_1, x_2 ,y_2) \ | \ x_1 \leq y_1 \ \& \ x_2 \leq y_2)\}\) and \(A = (\mathbb{R}^2_\leq \times \Delta) \cup (\Delta \times \mathbb{R}^2_\leq)\). Vectorizing this diagram using this method yields a vector in \(\ell_1\). To visualize this, we decompose this vector into its components on individual rectangles (points) of the signed persistence diagram. Each of these vectors is visualized as a stack sheets on each line segment of the plotted signed barcode (above the xy-plane if the corresponding rectangle is positive, and below if the rectangle is negative).

Github:

Script for these project may be found on my github, under the repository name MPHvect.

Future Work

Below I’ve listed some possible future ideas for expanding on my current research.

  1. Algorithmic optimization– Implementing efficient algorithms for computing the Schauder basis elements, suitable for high-dimensional parameter spaces, and evaluating them on encoded multiparameter persistence barcodes.
  2. Generalized indexing categories -Extending the theory of Schauder basis vectorization to signed diagrams in more general metric geodesic metric spaces.
  3. Statistical theory– Studying limit theorems and hypothesis testing in the \(\ell_1\) embedding space.
  4. Undergraduate Research Project– Applying algorithms of this method to data sets coming from real applications.

My long-term research goal is to further flush out the multiparameter setting of TDA, bringing together the categorical, analytic, and statistical aspects of TDA so that the rich structure of multiparameter persistence can be effectively leveraged in data-driven applications.