Compatible basis, Stability of persistent homology, and Homology inference
Compatible basis
- Compatible basis at
: -
- cycles
-
a partition
- bijection
, denoted by
such that
- cycles
Properties:
1.
Consider the restriction map
cf. picture
2.
3.
Example: cf. picture
You can basically read the decomposition of the persistence module from a compatible basis.
Th:
Let
and a compatible basis , , Then the persitence module is:
cf. picture
Sketch of proof:
Consider one
(cf picture)
Iterative algorithm
Suppose we have a compatible basis for the filtration
Consider the filtration
update the compatible basis to
Similar to what we did during the previous class: creator or destructor?
-
Creator: if
in , thenConsequently:
So
is a cycle: is independent of the other ’s in the compatible basis:
- and
cannot be the boundary of another chain because has no coface.
Set
compatible basis is
-
Destructor: if
in , thensince
Pick
As
we can set
(Condition 1 of the compatible bases remains true)
Now, pair
and :(like last time, we “kill the youngest” (
))
Example: cf. picture
NB: picking
Matrix Implementation
We don’t really need to keep track of
Let
in the canonical basis
Define
If
Algorithm
for i = 1 … n:
while ∃ j < i s.t. low(j) = low(i):
col_i ← col_i + col_j (in ℤ/2) # amounts to writing ∂σ_{i+1} = ∑_{g∈G} ε_g c_g + ∑_{f∈F} ε_f c_f
if col_i ≠ 0: # you have c_f's
add (low(i), i-1) to the persistence diagram # what we called (f_0 = g, h-1) before, we pair f_0 = low(i) with index i
# Tackle the (f, n)
Add all (i, n) s.t. col_i = 0 and there is no column col_j with low(j) = i to the persistence diagram
The algorithm maintains the following property: at the end of the for
loop, the matrix of
Stability of persistent homology
Let
be two persistence diagrams.
To compare them, we will introduce a notion of “distance” between them.
- The bottleneck distance between
and : -
is defined as
Note that bijections
cf. picture
How are the original of geometry and the one of the persistence diagrams related?
Stability theorem
Th (Stability): Let
Let
where the
Define similarly
Continuous version:
Th (Stability 2): Let
then
Philosophy of the proof:
so
cf. picture
Application: Homology inference
Back to our very first motivation: we’re given a set of points, how do we unravel their underlying topology?
Reminder:
- Haussdorff distance between compact sets
: -
where
(cf. picture)
- A set of points
is an -sample of a compact set : -
if
Note that for every compact set
Level-set
Measure of “complexity” of the geometry of a compact
Let
- A critical point for
: -
is a point
with at least two nearest neighbors on . I.e. - The reach of
: -
is
Inference theorem
Let
be a “nice” compact set of reach . Let
be a -sample of , with . Then there is a one-to-one correspondence between the points of the persistence diagram of the Cech filtration on
that are -away from the diagonal and the homology features of .
cf. picture
Sketch of proof:
Claims:
-
has the same homotopy type as for all(cf. picture)
-
if
is a -sample, i.e. , then -
Stability theorem:
-
- Persistent homolgy of
will be expressed as something like
cf. picture
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