The geo-mapping problem in Geometric Data Science

Professor Vitaliy Kurlin is a mathematician by training and leads the Data Science Theory and
Applications group in the Materials Innovation Factory at the University of Liverpool, UK. The group
develops geometric methods for applications in crystallography, chemistry, and structural biology. The
recent funding includes the Royal Academy of Engineering Industry Fellowship and the Royal Society
APEX Fellowship in the UK.

The talk is based on the papers extending Topological Data Analysis (TDA) to a wider area of
Geometric Data Science, which aims to continuously parametrise moduli spaces of real objects under
practical equivalences. The key example is a cloud A of unordered points under isometry in ℝ.
Standard filtrations (Vietoris-Rips, Cech, Delaunay) of complexes on A are invariant under isometry
(any distance-preserving transformation). Hence, persistent homology of these filtrations can be
considered a partial solution to the following geo-mapping problem: design an invariant I of clouds of
m unordered points satisfying the conditions below.

(a) Completeness: any clouds A,B ∈ ℝⁿ are related by rigid motion if and only if I(A)=I(B);

(b) Realisability: the invariant space {I(A) for all clouds A in ℝⁿ } is explicitly parameterised so that any
sampled value I(A) can be realised by a cloud A, uniquely under motion in ℝⁿ ;

(c) Bi-continuity: the bijection from the space of clouds to the space of complete invariants is Lipschitz
continuous in both directions in a suitable metric d on the invariant space;
(d) Computability: the invariant I, the metric d, and a reconstruction of A in ℝⁿ from I(A) can be
obtained in polynomial time in the size of A, for a fixed dimension n.

The talk will outline a full solution to this problem, which remains open for other data (embedded
graphs, meshes, or complexes) and relations (dilation, affine, or projective maps).