Persistent homology

Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters. Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters. To find the persistent homology of a space, the space must first be represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested sequence of increasing subsets. Formally, consider a real-valued function on a simplicial complex f : K → R {displaystyle f:K ightarrow mathbb {R} } that is non-decreasing on increasing sequences of faces, so f ( σ ) ≤ f ( τ ) {displaystyle f(sigma )leq f( au )} whenever σ {displaystyle sigma } is a face of τ {displaystyle au } in K {displaystyle K} . Then for every a ∈ R {displaystyle ain mathbb {R} } the sublevel set K ( a ) = f − 1 ( − ∞ , a ] {displaystyle K(a)=f^{-1}(-infty ,a]} is a subcomplex of K, and the ordering of the values of f {displaystyle f} on the simplices in K {displaystyle K} (which is in practice always finite) induces an ordering on the sublevel complexes that defines a filtration When 0 ≤ i ≤ j ≤ n {displaystyle 0leq ileq jleq n} , the inclusion K i ↪ K j {displaystyle K_{i}hookrightarrow K_{j}} induces a homomorphism f p i , j : H p ( K i ) → H p ( K j ) {displaystyle f_{p}^{i,j}:H_{p}(K_{i}) ightarrow H_{p}(K_{j})} on the simplicial homology groups for each dimension p {displaystyle p} . The p th {displaystyle p^{ ext{th}}} persistent homology groups are the images of these homomorphisms, and the p th {displaystyle p^{ ext{th}}} persistent Betti numbers β p i , j {displaystyle eta _{p}^{i,j}} are the ranks of those groups. Persistent Betti numbers for p = 0 {displaystyle p=0} coincide withthe size function, a predecessor of persistent homology. Any filtered complex over a field F {displaystyle F} can be brought by a linear transformation preserving the filtration to so called canonical form, a canonically defined direct sum of filtered complexes of two types: one-dimensional complexes with trivial differential d ( e t i ) = 0 {displaystyle d(e_{t_{i}})=0} and two-dimensional complexes with trivial homology d ( e s j + r j ) = e r j {displaystyle d(e_{s_{j}+r_{j}})=e_{r_{j}}} . A persistence module over a partially ordered set P {displaystyle P} is a set of vector spaces U t {displaystyle U_{t}} indexed by P {displaystyle P} , with a linear map u t s : U s → U t {displaystyle u_{t}^{s}:U_{s} o U_{t}} whenever s ≤ t {displaystyle sleq t} , with u t t {displaystyle u_{t}^{t}} equal to the identity and u t s ∘ u s r = u t r {displaystyle u_{t}^{s}circ u_{s}^{r}=u_{t}^{r}} for r ≤ s ≤ t {displaystyle rleq sleq t} . Equivalently, we may consider it as a functor from P {displaystyle P} considered as a category to the category of vector spaces (or R {displaystyle R} -modules). There is a classification of persistence modules over a field F {displaystyle F} indexed by N {displaystyle mathbb {N} } : Each of these two theorems allows us to uniquely represent the persistent homology of a filtered simplicial complex with a barcode or persistence diagram. A barcode represents each persistent generator with a horizontal line beginning at the first filtration level where it appears, and ending at the filtration level where it disappears, while a persistence diagram plots a point for each generator with its x-coordinate the birth time and its y-coordinate the death time.Equivalently the same data is represented by Barannikov's canonical form, where each generator is represented by a segment connecting the birth and the death values plotted on separate lines for each p {displaystyle p} .