PhD Candidate in Data Science and Analytics, Kennesaw State University

Advisor: Mehmet Emin Aktas

Research Interests

Mathematics:

Algebraic Topology | Harmonic Analysis | Approximation Theory | Functional Analysis | Metric Geometry

Computer Science:

Topological Data Analysis | Computational Geometry | Graph Representation Learning | Geometric Deep Learning | Computer Vision

Probability and Statistics:

Stochastic Processes | Kernel Methods

Applications:

Network Science | Computational Biology | Medical Imaging

Selected Research Projects

Reproducing Kernel Hilbert Spaces for Virtual Persistence Diagrams

Journal of Applied and Computational Topology (2026).

For a metric pair (X,d,A), the group of Virtual persistence diagrams is the free abelian metric group of persistence diagrams with possibly signed multiplicities and diagonal A. The metric on this group is the unique translation invariant extension of the 1-Wasserstein distance on the canonical inclusion of unsigned persistence diagrams.

When this group is finite-rank, it is also locally compact abelian, and the Fourier transform and Fourier–Stieltjes transform are well-defined. Its group of characters—continuous homomorphisms into the unit circle—is the torus with dimension equal to the rank of the virtual persistence diagram group.

The real part of the character map of a virtual persistence diagram.

This is equivalent to the real part of the Fourier transform of the indicator function or the Fourier–Stieltjes transform of the Dirac measure of the virtual persistence diagram.

The imaginary part of the character map of a virtual persistence diagram.

This is equivalent to the imaginary part of the Fourier transform of the indicator function or the Fourier–Stieltjes transform of the Dirac measure of the virtual persistence diagram.

Experimental Results

Digital image processing gives a natural case in which the space X of persistence intervals for a virtual persistence diagram group is finite. For a filtration on an 8-bit grayscale image, there can be no more than 2⁸ Choose 2 = 32,640 distinct persistence intervals.

We train U-Net for image segmentation on a noisy synthetic image segmentation dataset with (1) no topological regularization, (2) classical topological regularization using the Wasserstein distance, and (3) a regularization term using virtual persistence diagrams.

Reproducing Kernel Hilbert Spaces on Banach Completions of Virtual Persistence Diagram Groups

Virtual persistence diagram groups may also be locally compact abelian without having finite rank. We prove that, for a metric pair (X,d,A), the corresponding metric group of virtual persistence diagrams is locally compact abelian if and only if X/A is uniformly discrete.

But many important spaces of persistence intervals, such as classical persistence diagrams with intervals in ℝ², are not uniformly discrete.

A space of persistence intervals

This is the metric pair (ℂ,ℓ²,S¹). For the virtual persistence diagram group defined on this metric pair, ℂ represents the space of persistence intervals, and S¹ is taken to be the diagonal.

The solid red circle in the figure to the left of this paragraph is the diagonal for this space of persistence intervals.

The solid blue line is the distance between the two persistence intervals before quotienting by the diagonal.

The sum of the dashed blue lines is the distance between the two persistence intervals after quotienting by the diagonal.

A virtual persistence diagram on (ℂ,ℓ²,S¹)

In this case, the space of persistence intervals is the wedge sum of two copies of S².

This space is not uniformly discrete. We develop methods for the functional analysis of virtual persistence diagram groups for which the Fourier transform and the Fourier–Stieltjes transform are not well defined.

We first extend the group of virtual persistence diagrams on an arbitrary metric pair to the Banach completion of its real span: persistence diagrams with multiplicities in ℝ.

We next define a translation-invariant Gaussian kernel on this extension.

We then derive structural inequalities which recover information from the Gaussian kernel back to the original space, as we did when the Fourier transform on the virtual persistence diagram group was well-defined.

Higher-order Persistence Diagrams

In multiple instance learning, multiple observations are aggregated into a single observation before any machine learning takes place. Persistence diagrams have the useful property that a pair of persistence intervals from distinct persistence diagrams generated by the same filtration are interpretable in a shared geometry.

Two higher-order persistence intervals

A higher-order persistence interval is an implication: if a persistence interval is born after and dies before another persistence interval, we form a higher-order persistence interval from their pair.

We then generalize this construction to implication between pairs of persistence intervals, to implication between pairs of pairs of persistence intervals, etc.

A higher-order persistence diagram

In this example, each persistence diagram is generated from that of the previous order.

Harmonic aggregation

Computing higher-order persistence diagrams as aggregates of lower-order persistence diagrams is computationally expensive. We develop an algorithm of harmonic aggregation and show that it is more computationally efficient across several random network model families.

Education

Ph.D. in Data Science and Analytics, Kennesaw State University (Expected May 2027)
M.S. in Data Science, Lewis University (2024)
B.S. in Mathematics, Lewis University (2023)

Current Position

Graduate Research Assistant, Kennesaw State University (2024–Present)

Page updated

Google Sites

Report abuse