Hodge Laplacians on Sequences and Dimensionality Reduction

Open Access
- Author:
- Santa Cruz, Hannah Rocio
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 26, 2024
- Committee Members:
- Pierre-Emmanuel Jabin, Program Head/Chair
Dezhe Jin, Outside Unit & Field Member
Jeffrey Case, Major Field Member
Carina Curto, Major Field Member
Vladimir Itskov, Chair & Dissertation Advisor - Keywords:
- Laplacian
Sequential data
Embedding
feature space
Hodge Laplacian
Simplicial complexes - Abstract:
- An important goal in theoretical neuroscience is to understand the organizing principles underlying the brain’s representation of the physical world. Entangled with this goal is the problem of recovering the underlying manifold the stimuli are mapped onto within the feature space describing neural activity. In this dissertation, we propose a low dimensional, Hodge Laplacian eigen-embedding for probability distributions on sequences. We study such distributions in Chapter 2 and give some background on Hodge Laplacians in Chapter 3. In Chapter 4, we demonstrate this Hodge Laplacian operator has desirable properties with respect to natural null-models, where the underlying features are modeled as independent variables, both in the cases of sequential data and more conventional simplicial data. In particular, we show that for the case of simplicial complexes, with vertices acting independently, the appropriate Laplacians are multiples of the identity and thus have no meaningful Fourier modes. For the null model over sequences with independent vertices, we show the corresponding Hodge Laplacian has an integer spectrum with high multiplicities and describe its eigenspaces. In Chapter 5, we describe a Hodge Laplacian eigen-embedding method for sequences, and illustrate how the method reveals the underlying geometry of the dataset. We introduce a "sweep complex" of sequences induced by hyperplanes sweeping a collection of points, and show the eigen embedding reveals the arrangement of the hyperplanes. Additionally, we apply the developed machinery to spike train recordings from songbirds, to study the different topologies over two brain regions.