[Paper Review] From trees to barcodes and back again: theoretical and statistical perspectives

Lida Kanari, Adélie Garin+1 more|arXiv (Cornell University)|Oct 22, 2020

Topological and Geometric Data Analysis26 references4 citations

TL;DR

This paper proposes a theoretical and computational framework to study the inverse relationship between topological data analysis and geometric tree structures, focusing on the Topological Morphology Descriptor (TMD) and Topological Neuron Synthesis (TNS) algorithms. It establishes that TNS acts as a stable, stochastic inverse to TMD by leveraging symmetric groups to classify tree realizations, proving that barcode permutations significantly affect the number of possible geometric trees, with biological trees occupying a constrained subset of possible combinatorial types.

ABSTRACT

Methods of topological data analysis have been successfully applied in a wide range of fields to provide useful summaries of the structure of complex data sets in terms of topological descriptors, such as persistence diagrams. While there are many powerful techniques for computing topological descriptors, the inverse problem, i.e., recovering the input data from topological descriptors, has proved to be challenging. In this article we study in detail the Topological Morphology Descriptor (TMD), which assigns a persistence diagram to any tree embedded in Euclidean space, and a sort of stochastic inverse to the TMD, the Topological Neuron Synthesis (TNS) algorithm, gaining both theoretical and computational insights into the relation between the two. We propose a new approach to classify barcodes using symmetric groups, which provides a concrete language to formulate our results. We investigate to what extent the TNS recovers a geometric tree from its TMD and describe the effect of different types of noise on the process of tree generation from persistence diagrams. We prove moreover that the TNS algorithm is stable with respect to specific types of noise.

Motivation & Objective

• To investigate the extent to which the Topological Neuron Synthesis (TNS) algorithm can stochastically recover geometric trees from their TMD barcodes.
• To formalize a combinatorial classification of geometric trees using symmetric groups, linking barcode permutations to tree realization counts.
• To analyze the stability of TNS under noise, particularly bar transpositions, and quantify its robustness.
• To compare biological neuronal morphologies with random trees by analyzing their TMD-equivalence classes and realization frequencies.
• To identify structural constraints in biological trees by measuring their deviation from random barcode distributions.

Proposed method

• The TMD algorithm maps any finite binary tree embedded in R³ to a persistence barcode (a multiset of intervals), encoding topological features via birth and death times of topological invariants.
• The TNS algorithm stochastically generates geometric trees from barcodes using a parameterized stochastic process, with tree structure determined by permutation of barcode bars.
• Symmetric groups are used to classify barcodes by permutation of death times, enabling a combinatorial equivalence relation on tree realizations.
• An explicit formula is derived for the number of geometric trees realizing a given barcode, based on the permutation associated with the barcode’s death times.
• Cayley graphs of symmetric groups are used to visualize how tree-realization counts change under bar transpositions.
• Computational experiments analyze the distribution of tree-realization numbers, empirical combinatorial types, and the diversity of TMD-equivalence classes across random and biological barcodes.

Experimental results

Research questions

• RQ1 To what extent can the TNS algorithm recover the original geometric tree from its TMD barcode, and how does this depend on barcode permutation?
• RQ2 How does the number of geometric trees realizing a given barcode change when bars are permuted, and what role do symmetric groups play in this?
• RQ3 What is the stability of the TNS algorithm under small perturbations, such as transposition of two bars in the barcode?
• RQ4 Why do biological neuronal morphologies occupy only a small fraction of possible TMD-equivalence classes, and what does this imply about biological constraints?
• RQ5 How does the stochastic nature of TNS affect the variance in the combinatorial types of generated trees, especially when birth or death times are similar?

Key findings

• The number of geometric trees realizing a given barcode is explicitly determined by the permutation of death times, with the formula being n! for n+1 bars when all death times are distinct.
• The TNS algorithm is stable with respect to bar transpositions, as proven via bottleneck and transposition stability, ensuring small changes in barcodes lead to small changes in generated trees.
• Biological barcodes from neuronal reconstructions represent only a small fraction of possible TMD-equivalence classes, indicating strong biological constraints on tree morphology.
• The probability of generating a specific combinatorial tree type via TNS depends on the parameter λ, reflecting the stochastic nature of the algorithm and leading to variable output across runs.
• When barcodes contain bars with similar birth or death times, TNS exhibits oscillatory behavior between multiple equivalence classes, increasing output variance.
• The distribution of tree-realization numbers for biological apical dendrites is significantly lower than the theoretical maximum (n!), suggesting non-random, biologically constrained tree structures.