Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

Multidimensional Persistence: Topology with Multiple Parameters

2 minute read

Published:

TL;DR: In 1D persistence, a bifiltration indexed by (scale, density) encodes richer topological information than either parameter alone. However, the 1D interval decomposition theorem fails in higher dimensions — most 2-parameter persistence modules are not decomposable into intervals. The rank invariant and fibered barcodes provide computable weaker summaries.

Why Multiple Parameters?

Single-parameter persistence is limited because real data often has multiple relevant scales:

  • Noisy point clouds: simultaneously filter by distance scale \(r\) and density threshold \(\rho\). Points in sparse regions (noise) should be down-weighted.
  • Weighted networks: filter simultaneously by edge weight and node degree.
  • Image data: filter by pixel intensity and local gradient.

The Rips density bifiltration defines \(K^{(r,\rho)} = \mathrm{Rips}(P_\rho, r)\) where \(P_\rho = \{p \in P : \rho(p) \geq \rho\}\) and \(\rho(p)\) is a local density estimate at \(p\). This is monotone in both parameters: increasing \(r\) adds simplices, decreasing \(\rho\) adds more points.

Persistence Modules over Posets

A 2-parameter persistence module assigns a vector space \(M_{(a,b)}\) to each point \((a,b) \in \mathbb{R}^2\) and a linear map \(M_{(a,b)} \to M_{(a',b')}\) whenever \((a,b) \leq (a', b')\) (componentwise). This is a functor from the poset \((\mathbb{R}^2, \leq)\) to vector spaces.

The bad news (Carlsson & Zomorodian 2009): For 2-parameter persistence modules over fields, there is generally no complete discrete invariant analogous to the barcode. The indecomposable representations of the 2-parameter grid poset are not classified by a finite set of intervals — the representation theory is “wild.”

Computable Invariants

Despite the negative result, several useful invariants exist:

Rank invariant: For \((a,b) \leq (a',b')\), define:

$$\mathrm{rank}(a,b,a',b') = \dim \mathrm{im}(M_{(a,b)} \to M_{(a',b')})$$

The rank invariant captures how many topological features persist from scale \((a,b)\) to scale \((a',b')\).

Fibered barcodes: For a line \(L\) in parameter space, the restriction of \(M\) to \(L\) is a 1-parameter persistence module with a well-defined barcode. The collection of barcodes over all lines is the fibered barcode.

RIVET: A software tool (Lesnick & Wright 2015) that computes and visualises fibered barcodes efficiently using a 2D arrangement structure.

Key Insight: The failure of a complete barcode-type invariant in 2+ parameters is fundamental — it is not a computational limitation but an algebraic one. Recent work (2020–2025) on "minimal presentations" and "stable rank invariants" is making multidimensional persistence increasingly practical. For most applications, the fibered barcode computed along a relevant family of lines gives sufficient information.

References

  • G. Carlsson & A. Zomorodian, “The Theory of Multidimensional Persistence,” Discrete & Computational Geometry, 2009.
  • M. Lesnick & M. Wright, “Interactive Visualisation of 2-D Persistence Modules,” arXiv:1512.00180.
  • RIVET: rivet.readthedocs.io

You May Also Enjoy

GAPE: Remember to Forget — Gated Adaptive Positional Encoding

4 minute read

Published:

GAPE is a drop-in RoPE augmentation that adds content-aware attention logit biases: a query-gate suppresses irrelevant distant context while a key-gate preserves salient distant tokens. Provably sharper attention and improved long-context robustness — no architecture changes needed.

PolyNSD: Polynomial Neural Sheaf Diffusion

4 minute read

Published:

PolyNSD replaces the NSD propagation operator with a degree-K Chebyshev polynomial in the normalised sheaf Laplacian, achieving SOTA on homo- and heterophilic benchmarks with only diagonal restriction maps and dramatically lower memory usage.

TDA in Materials Science: Topology of Structure and Phase

3 minute read

Published:

Persistent homology characterises the multi-scale structure of materials — from pore geometry in catalysts to glass transition in amorphous materials. H₀ captures connectivity, H₁ captures channels and rings, H₂ captures enclosed voids. These topological descriptors predict mechanical properties, adsorption capacity, and phase transitions more accurately than geometric averages.

TDA in Drug Discovery: Molecular Topology

3 minute read

Published:

Persistent homology provides multi-scale topological fingerprints of molecular structures — capturing rings, cavities, and protein pockets that traditional cheminformatics descriptors miss. Applications include binding site detection, ADMET property prediction, and protein-ligand interaction modelling.