Given n points P in a Euclidean space, the Johnson-Lindenstrauss lemma guarantees that the distances between pairs of points is preserved up to a small constant factor with high probability by random projection into O(log n) dimensions. In this paper, we show that the persistent homology of the distance function to P is also preserved up to a comparable constant factor. One could never hope to preserve the distance function to P pointwise, but we show that it is preserved sufficiently at the critical points of the distance function to guarantee similar persistent homology. We prove these results in the more general setting of weighted k-th nearest neighbor distances, for which k=1 and all weights equal to zero gives the usual distance to P.

1. The Persistent Homology
of Distance Functions
under Random Projection
Don Sheehy
University of Connecticut

2. Unions of Balls

3. Unions of Balls
Finite Point Set

4. Unions of Balls
Finite Point Set Union of Balls

5. Unions of Balls
Finite Point Set Union of Balls
Topologically uninteresting Potentially Interesting

6. Unions of Balls
Finite Point Set Union of Balls
Topologically uninteresting Potentially Interesting
Idea: Fill in the gaps in the ambient space.
Examples: Molecules and Manifolds

7. Unions of balls are sublevels of the distance.

8. Unions of balls are sublevels of the distance.
P ⇢ RdInput:

9. Unions of balls are sublevels of the distance.
P↵
=
[
p2P
ball(p, ↵) = {x 2 Rd
| d(x, P)  ↵}
P ⇢ RdInput:

10. Unions of balls are sublevels of the distance.
P↵
=
[
p2P
ball(p, ↵) = {x 2 Rd
| d(x, P)  ↵}
Persistent Homology was invented to track changes
in the homology of P↵
as ↵ ranges from 0 to 1.
P ⇢ RdInput:

11. Unions of balls are sublevels of the distance.
P↵
=
[
p2P
ball(p, ↵) = {x 2 Rd
| d(x, P)  ↵}
Persistent Homology was invented to track changes
in the homology of P↵
as ↵ ranges from 0 to 1.
Pers({P↵
})
P ⇢ RdInput:

12. Unions of balls are sublevels of the distance.
P↵
=
[
p2P
ball(p, ↵) = {x 2 Rd
| d(x, P)  ↵}
Persistent Homology was invented to track changes
in the homology of P↵
as ↵ ranges from 0 to 1.
Pers({P↵
})
P ⇢ RdInput:

13. Filtered Simplicial Complexes

14. Filtered Simplicial Complexes
For ✓ P,
rad( ) = radius of the min. encl. ball of .
diam( ) = max
p,q2P
kp qk2.

15. Filtered Simplicial Complexes
For ✓ P,
rad( ) = radius of the min. encl. ball of .
diam( ) = max
p,q2P
kp qk2.
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}

16. Filtered Simplicial Complexes
For ✓ P,
rad( ) = radius of the min. encl. ball of .
diam( ) = max
p,q2P
kp qk2.
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
ˇCech Filtration: {CP (↵)}↵ 0

17. Filtered Simplicial Complexes
For ✓ P,
rad( ) = radius of the min. encl. ball of .
diam( ) = max
p,q2P
kp qk2.
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
Rips Complex: RP (↵) = { ✓ P | diam( )  ↵}
ˇCech Filtration: {CP (↵)}↵ 0

18. Filtered Simplicial Complexes
For ✓ P,
rad( ) = radius of the min. encl. ball of .
diam( ) = max
p,q2P
kp qk2.
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
Rips Complex: RP (↵) = { ✓ P | diam( )  ↵}
ˇCech Filtration: {CP (↵)}↵ 0
Rips Filtration: {RP (↵)}↵ 0

19. Filtered Simplicial Complexes
For ✓ P,
rad( ) = radius of the min. encl. ball of .
diam( ) = max
p,q2P
kp qk2.
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
Rips Complex: RP (↵) = { ✓ P | diam( )  ↵}
ˇCech Filtration: {CP (↵)}↵ 0
Rips Filtration: {RP (↵)}↵ 0
CP (↵) ✓ RP (↵) ✓ CP (
p
2↵)

20. Filtered Simplicial Complexes
For ✓ P,
rad( ) = radius of the min. encl. ball of .
diam( ) = max
p,q2P
kp qk2.
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
Rips Complex: RP (↵) = { ✓ P | diam( )  ↵}
ˇCech Filtration: {CP (↵)}↵ 0
Rips Filtration: {RP (↵)}↵ 0
CP (↵) ✓ RP (↵) ✓ CP (
p
2↵)
Pers({RP (↵)}) is a
p
2-approximation to Pers({CP (↵)}).

21. Representing sublevels of distances

22. Representing sublevels of distances
ˇCech Complex: size O(nd+1
).
↵-complex (a.k.a. Delaunay Filtration): size O(ndd/2e
).
Quality Meshes: size 2(d2
)
n.
(Sparse ˇCech Complex: 2(d2
)
n).*

23. Representing sublevels of distances
ˇCech Complex: size O(nd+1
).
↵-complex (a.k.a. Delaunay Filtration): size O(ndd/2e
).
Quality Meshes: size 2(d2
)
n.
(Sparse ˇCech Complex: 2(d2
)
n).*

24. Representing sublevels of distances
ˇCech Complex: size O(nd+1
).
↵-complex (a.k.a. Delaunay Filtration): size O(ndd/2e
).
Quality Meshes: size 2(d2
)
n.
(Sparse ˇCech Complex: 2(d2
)
n).*
Key Point: Ambient Dimension Matters!

25. Johnson Lindenstrauss Projection

26. Johnson Lindenstrauss Projection
Idea: Project to lower dimensions. Preserve pairwise distances.

27. Johnson Lindenstrauss Projection
Idea: Project to lower dimensions. Preserve pairwise distances.
Let f : RD
! Rd
be a linear map where d = O(log n/"2
) such that:

28. Johnson Lindenstrauss Projection
Idea: Project to lower dimensions. Preserve pairwise distances.
(1 ")ka bk2
 kf(a) f(b)k2
 (1 + ")ka bk2
Squared distances preserved up to multiplicative factor.
1
Let f : RD
! Rd
be a linear map where d = O(log n/"2
) such that:

29. Johnson Lindenstrauss Projection
Idea: Project to lower dimensions. Preserve pairwise distances.
(1 ")ka bk2
 kf(a) f(b)k2
 (1 + ")ka bk2
Squared distances preserved up to multiplicative factor.
1
|(b a)>
(c a) (f(b) f(a))>
(f(b) f(a))|  "kb akkc ak.
Inner products preserved up to additive factor.
2
Let f : RD
! Rd
be a linear map where d = O(log n/"2
) such that:

30. Johnson Lindenstrauss Projection
Idea: Project to lower dimensions. Preserve pairwise distances.
a
b
c f(c)
f(b)
f(a)
(1 ")ka bk2
 kf(a) f(b)k2
 (1 + ")ka bk2
Squared distances preserved up to multiplicative factor.
1
|(b a)>
(c a) (f(b) f(a))>
(f(b) f(a))|  "kb akkc ak.
Inner products preserved up to additive factor.
2
Let f : RD
! Rd
be a linear map where d = O(log n/"2
) such that:

31. Can we use JL for P.H. of distances?

32. Can we use JL for P.H. of distances?
Yes, for Rips filtrations, but not a tight approximation.

33. Can we use JL for P.H. of distances?
Yes, for Rips filtrations, but not a tight approximation.
Distance function itself is not preserved.

34. Can we use JL for P.H. of distances?
Yes, for Rips filtrations, but not a tight approximation.
Distance function itself is not preserved.
Pairwise distances in sublevels are not preserved.

35. Can we use JL for P.H. of distances?
Yes, for Rips filtrations, but not a tight approximation.
Distance function itself is not preserved.
Pairwise distances in sublevels are not preserved.
Is topology preserved? Maybe yes, maybe no.

36. Can we use JL for P.H. of distances?
Yes, for Rips filtrations, but not a tight approximation.
Distance function itself is not preserved.
Pairwise distances in sublevels are not preserved.
Is topology preserved? Maybe yes, maybe no.
Is persistent homology preserved? YES.

37. Cech Filtration, MEBs, and Approximation

38. Cech Filtration, MEBs, and Approximation
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
ˇCech Filtration: {CP (↵)}↵ 0

39. Cech Filtration, MEBs, and Approximation
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
ˇCech Filtration: {CP (↵)}↵ 0
Let P ⇢ RD
and let f be any map from RD
to Rd
.

40. Cech Filtration, MEBs, and Approximation
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
ˇCech Filtration: {CP (↵)}↵ 0
Let P ⇢ RD
and let f be any map from RD
to Rd
.
Idea: If f “preserves M.E.B. radii”, then it preserves
the persistent homology of the distance function.

41. Cech Filtration, MEBs, and Approximation
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
ˇCech Filtration: {CP (↵)}↵ 0
Let P ⇢ RD
and let f be any map from RD
to Rd
.
Idea: If f “preserves M.E.B. radii”, then it preserves
the persistent homology of the distance function.

42. Cech Filtration, MEBs, and Approximation
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
ˇCech Filtration: {CP (↵)}↵ 0
Let P ⇢ RD
and let f be any map from RD
to Rd
.
Idea: If f “preserves M.E.B. radii”, then it preserves
the persistent homology of the distance function.
For S ✓ P, (1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.

43. Cech Filtration, MEBs, and Approximation
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
ˇCech Filtration: {CP (↵)}↵ 0
Let P ⇢ RD
and let f be any map from RD
to Rd
.
Idea: If f “preserves M.E.B. radii”, then it preserves
the persistent homology of the distance function.
For S ✓ P, (1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.
For all ↵ 0, CP (
p
1 4") ✓ Cf(P )(↵) ✓ CP (
p
1 4")

44. Cech Filtration, MEBs, and Approximation
ˇCech Complex: CP (↵) = { ✓ P | rad( )  2↵}
ˇCech Filtration: {CP (↵)}↵ 0
Let P ⇢ RD
and let f be any map from RD
to Rd
.
Idea: If f “preserves M.E.B. radii”, then it preserves
the persistent homology of the distance function.
For S ✓ P, (1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.
For all ↵ 0, CP (
p
1 4") ✓ Cf(P )(↵) ✓ CP (
p
1 4")
So, Pers(d(·, f(P))) is a (1 + O("))-approximation
to Pers(d(·, P)).

45. MEBs under JL projection

46. MEBs under JL projection
Let S = {p1, . . . , pr} and let x 2 conv(S).

47. MEBs under JL projection
x =
rX
i=1
ipi, where
rX
i=1
i = 1.
Let S = {p1, . . . , pr} and let x 2 conv(S).

48. MEBs under JL projection
x =
rX
i=1
ipi, where
rX
i=1
i = 1.
kx pk2
=
rX
i=1
i(pi p)
2
=
rX
i=1
rX
j=1
i j(pi p)>
(pj p).For any p 2 S,
Let S = {p1, . . . , pr} and let x 2 conv(S).

49. MEBs under JL projection
kp xk2
kf(p) f(x)k2
=
rX
i=1
rX
j=1
i j (pi p)>
(pj p) (f(pi) f(p))>
(f(pj) f(p))

rX
i=1
rX
j=1
i j"kpi pkkpj pk

rX
i=1
rX
j=1
i j4" rad(S)2
= 4" rad(S)2
.
x =
rX
i=1
ipi, where
rX
i=1
i = 1.
kx pk2
=
rX
i=1
i(pi p)
2
=
rX
i=1
rX
j=1
i j(pi p)>
(pj p).For any p 2 S,
Let S = {p1, . . . , pr} and let x 2 conv(S).

50. MEBs under JL projection

51. MEBs under JL projection
Theorem: Let P be a set of points in RD
and let f : RD
! Rd
be an "-JL
projection for P. For every subset S of P,
(1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.

52. MEBs under JL projection
Theorem: Let P be a set of points in RD
and let f : RD
! Rd
be an "-JL
projection for P. For every subset S of P,
(1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.
Let x = center(S).

53. MEBs under JL projection
Theorem: Let P be a set of points in RD
and let f : RD
! Rd
be an "-JL
projection for P. For every subset S of P,
(1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.
Upper Bound:
Let x = center(S).

54. MEBs under JL projection
Theorem: Let P be a set of points in RD
and let f : RD
! Rd
be an "-JL
projection for P. For every subset S of P,
(1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.
Upper Bound: rad(f(S))2
 max
p2P
(kx pk2
+ 4" rad(S)2
)
 max
p2P
((1 + 4")rad(S)2
)
= (1 + 4")rad(S)2
.
Let x = center(S).

55. MEBs under JL projection
Theorem: Let P be a set of points in RD
and let f : RD
! Rd
be an "-JL
projection for P. For every subset S of P,
(1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.
Upper Bound: rad(f(S))2
 max
p2P
(kx pk2
+ 4" rad(S)2
)
 max
p2P
((1 + 4")rad(S)2
)
= (1 + 4")rad(S)2
.
Lower Bound:
Let x = center(S).

56. MEBs under JL projection
Theorem: Let P be a set of points in RD
and let f : RD
! Rd
be an "-JL
projection for P. For every subset S of P,
(1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.
Upper Bound: rad(f(S))2
 max
p2P
(kx pk2
+ 4" rad(S)2
)
 max
p2P
((1 + 4")rad(S)2
)
= (1 + 4")rad(S)2
.
Lower Bound:
Let x = center(S).
Let q 2 S be such that kq xk = rad(S) and
kf(q) center(f(S))k kf(q) f(x)k.

57. MEBs under JL projection
Theorem: Let P be a set of points in RD
and let f : RD
! Rd
be an "-JL
projection for P. For every subset S of P,
(1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.
Upper Bound: rad(f(S))2
 max
p2P
(kx pk2
+ 4" rad(S)2
)
 max
p2P
((1 + 4")rad(S)2
)
= (1 + 4")rad(S)2
.
Lower Bound:
Let x = center(S).
Let q 2 S be such that kq xk = rad(S) and
kf(q) center(f(S))k kf(q) f(x)k.

58. MEBs under JL projection
Theorem: Let P be a set of points in RD
and let f : RD
! Rd
be an "-JL
projection for P. For every subset S of P,
(1 4")rad(S)2
 rad(f(S))2
 (1 + 4")rad(S)2
.
Upper Bound: rad(f(S))2
 max
p2P
(kx pk2
+ 4" rad(S)2
)
 max
p2P
((1 + 4")rad(S)2
)
= (1 + 4")rad(S)2
.
Lower Bound:
Let x = center(S).
Let q 2 S be such that kq xk = rad(S) and
kf(q) center(f(S))k kf(q) f(x)k.
rad(f(S))2
kf(q) center(f(S))k2
kf(q) f(x)k2
kq xk2
4" rad(S)2
= (1 4")rad(S)2
.

59. Extension to k-NN distances.

60. Extension to k-NN distances.

61. Extension to k-NN distances.
dk
P (x) = distance from x to k points of P.

62. Extension to k-NN distances.
dk
P (x) = distance from x to k points of P.
Corollary: If f is an "-JL projection then for all k,
Pers(dk
f(P )) is a 1 + O(") approximation to Pers(dk
P ).

63. Extension to k-NN distances.
dk
P (x) = distance from x to k points of P.
Corollary: If f is an "-JL projection then for all k,
Pers(dk
f(P )) is a 1 + O(") approximation to Pers(dk
P ).
Bonus: Also works for weighted points.

64. Going forward…

65. Going forward…
• Eliminate inner product condition.

66. Going forward…
• Eliminate inner product condition.
• Eliminate constant factor (4)

67. Going forward…
• Eliminate inner product condition.
• Eliminate constant factor (4)
• Eliminate linearity condition.

68. Going forward…
• Eliminate inner product condition.
• Eliminate constant factor (4)
• Eliminate linearity condition.
• Extend to distances to measures.

69. Going forward…
• Eliminate inner product condition.
• Eliminate constant factor (4)
• Eliminate linearity condition.
• Extend to distances to measures.
Thank you.