The Persistent Homology of Distance Functions

under Random Projection

Don Sheehy University of Connecticut

Unions of Balls

Unions of Balls

Finite Point Set

Unions of Balls

Finite Point Set Union of Balls

Unions of Balls

Finite Point Set Union of Balls

Topologically uninteresting Potentially Interesting

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

Unions of balls are sublevels of the distance.

Unions of balls are sublevels of the distance.

P ⇢ RdInput:

Unions of balls are sublevels of the distance.

P

↵ =[

p2P

ball(p,↵) = {x 2 Rd | d(x, P ) ↵}

P ⇢ RdInput:

Unions of balls are sublevels of the distance.

P

↵ =[

p2P

ball(p,↵) = {x 2 Rd | d(x, P ) ↵}

Persistent Homology was invented to track changesin the homology of P↵ as ↵ ranges from 0 to 1.

P ⇢ RdInput:

Unions of balls are sublevels of the distance.

P

↵ =[

p2P

ball(p,↵) = {x 2 Rd | d(x, P ) ↵}

Persistent Homology was invented to track changesin the homology of P↵ as ↵ ranges from 0 to 1.

Pers({P↵})

P ⇢ RdInput:

Unions of balls are sublevels of the distance.

P

↵ =[

p2P

ball(p,↵) = {x 2 Rd | d(x, P ) ↵}

Persistent Homology was invented to track changesin the homology of P↵ as ↵ ranges from 0 to 1.

Pers({P↵})

P ⇢ RdInput:

Filtered Simplicial Complexes

Filtered Simplicial Complexes

For � ✓ P ,

rad(�) = radius of the min. encl. ball of �.diam(�) = max

p,q2Pkp� qk2.

Filtered Simplicial Complexes

For � ✓ P ,

rad(�) = radius of the min. encl. ball of �.diam(�) = max

p,q2Pkp� qk2.

ˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}

Filtered Simplicial Complexes

For � ✓ P ,

rad(�) = radius of the min. encl. ball of �.diam(�) = max

p,q2Pkp� qk2.

ˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}ˇ

Cech Filtration: {CP (↵)}↵�0

Filtered Simplicial Complexes

For � ✓ P ,

rad(�) = radius of the min. encl. ball of �.diam(�) = max

p,q2Pkp� qk2.

ˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}

Rips Complex: RP (↵) = {� ✓ P | diam(�) ↵}

ˇ

Cech Filtration: {CP (↵)}↵�0

Filtered Simplicial Complexes

For � ✓ P ,

rad(�) = radius of the min. encl. ball of �.diam(�) = max

p,q2Pkp� qk2.

ˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}

Rips Complex: RP (↵) = {� ✓ P | diam(�) ↵}

ˇ

Cech Filtration: {CP (↵)}↵�0

Rips Filtration: {RP (↵)}↵�0

Filtered Simplicial Complexes

For � ✓ P ,

rad(�) = radius of the min. encl. ball of �.diam(�) = max

p,q2Pkp� qk2.

ˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}

Rips Complex: RP (↵) = {� ✓ P | diam(�) ↵}

ˇ

Cech Filtration: {CP (↵)}↵�0

Rips Filtration: {RP (↵)}↵�0

CP (↵) ✓ RP (↵) ✓ CP (p2↵)

Filtered Simplicial Complexes

For � ✓ P ,

rad(�) = radius of the min. encl. ball of �.diam(�) = max

p,q2Pkp� qk2.

ˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}

Rips Complex: RP (↵) = {� ✓ P | diam(�) ↵}

ˇ

Cech Filtration: {CP (↵)}↵�0

Rips Filtration: {RP (↵)}↵�0

CP (↵) ✓ RP (↵) ✓ CP (p2↵)

Pers({RP (↵)}) is ap2-approximation to Pers({CP (↵)}).

Representing sublevels of distances

Representing sublevels of distances

ˇ

Cech Complex: size O(nd+1).

↵-complex (a.k.a. Delaunay Filtration): size O(ndd/2e).

Quality Meshes: size 2

(d2)n.(Sparse

ˇ

Cech Complex: 2

(d2)n).*

Representing sublevels of distances

ˇ

Cech Complex: size O(nd+1).

↵-complex (a.k.a. Delaunay Filtration): size O(ndd/2e).

Quality Meshes: size 2

(d2)n.(Sparse

ˇ

Cech Complex: 2

(d2)n).*

Representing sublevels of distances

ˇ

Cech Complex: size O(nd+1).

↵-complex (a.k.a. Delaunay Filtration): size O(ndd/2e).

Quality Meshes: size 2

(d2)n.(Sparse

ˇ

Cech Complex: 2

(d2)n).*

Key Point: Ambient Dimension Matters!

Johnson Lindenstrauss Projection

Johnson Lindenstrauss Projection

Idea: Project to lower dimensions. Preserve pairwise distances.

Johnson Lindenstrauss Projection

Idea: Project to lower dimensions. Preserve pairwise distances.

Let f : RD ! Rdbe a linear map where d = O(log n/"2) such that:

Johnson Lindenstrauss Projection

Idea: Project to lower dimensions. Preserve pairwise distances.

(1� ")ka� bk2 kf(a)� f(b)k2 (1 + ")ka� bk2Squared distances preserved up to multiplicative factor.1

Let f : RD ! Rdbe a linear map where d = O(log n/"2) such that:

Johnson Lindenstrauss Projection

Idea: Project to lower dimensions. Preserve pairwise distances.

(1� ")ka� bk2 kf(a)� f(b)k2 (1 + ")ka� bk2Squared 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 ! Rdbe a linear map where d = O(log n/"2) such that:

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� bk2Squared 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 ! Rdbe a linear map where d = O(log n/"2) such that:

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

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

Yes, for Rips filtrations, but not a tight approximation.

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

Yes, for Rips filtrations, but not a tight approximation.Distance function itself is not preserved.

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.

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.

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.

Cech Filtration, MEBs, and Approximation

Cech Filtration, MEBs, and Approximationˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}ˇ

Cech Filtration: {CP (↵)}↵�0

Cech Filtration, MEBs, and Approximationˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}ˇ

Cech Filtration: {CP (↵)}↵�0

Let P ⇢ RDand let f be any map from RD

to Rd.

Cech Filtration, MEBs, and Approximationˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}ˇ

Cech Filtration: {CP (↵)}↵�0

Let P ⇢ RDand 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.

Cech Filtration, MEBs, and Approximationˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}ˇ

Cech Filtration: {CP (↵)}↵�0

Let P ⇢ RDand 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.

Cech Filtration, MEBs, and Approximationˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}ˇ

Cech Filtration: {CP (↵)}↵�0

Let P ⇢ RDand 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.

Cech Filtration, MEBs, and Approximationˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}ˇ

Cech Filtration: {CP (↵)}↵�0

Let P ⇢ RDand 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 (p1� 4") ✓ Cf(P )(↵) ✓ CP (

p1� 4")

Cech Filtration, MEBs, and Approximationˇ

Cech Complex: CP (↵) = {� ✓ P | rad(�) 2↵}ˇ

Cech Filtration: {CP (↵)}↵�0

Let P ⇢ RDand 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 (p1� 4") ✓ Cf(P )(↵) ✓ CP (

p1� 4")

So, Pers(d(·, f(P ))) is a (1 +O("))-approximation

to Pers(d(·, P )).

MEBs under JL projection

MEBs under JL projection

Let S = {p1, . . . , pr} and let x 2 conv(S).

MEBs under JL projection

x =rX

i=1

�ipi, whererX

i=1

�i = 1.

Let S = {p1, . . . , pr} and let x 2 conv(S).

MEBs under JL projection

x =rX

i=1

�ipi, whererX

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).

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, whererX

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).

MEBs under JL projection

MEBs under JL projectionTheorem: Let P be a set of points in RD

and let f : RD ! Rdbe 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.

MEBs under JL projectionTheorem: Let P be a set of points in RD

and let f : RD ! Rdbe 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).

MEBs under JL projectionTheorem: Let P be a set of points in RD

and let f : RD ! Rdbe 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).

MEBs under JL projectionTheorem: Let P be a set of points in RD

and let f : RD ! Rdbe 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).

MEBs under JL projectionTheorem: Let P be a set of points in RD

and let f : RD ! Rdbe 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).

MEBs under JL projectionTheorem: Let P be a set of points in RD

and let f : RD ! Rdbe 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) andkf(q)� center(f(S))k � kf(q)� f(x)k.

MEBs under JL projectionTheorem: Let P be a set of points in RD

and let f : RD ! Rdbe 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) andkf(q)� center(f(S))k � kf(q)� f(x)k.

MEBs under JL projectionTheorem: Let P be a set of points in RD

and let f : RD ! Rdbe 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) andkf(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.

Extension to k-NN distances.

Extension to k-NN distances.

Extension to k-NN distances.

d

kP (x) = distance from x to k points of P .

Extension to k-NN distances.

d

kP (x) = distance from x to k points of P .

Corollary: If f is an "-JL projection then for all k,Pers(d

kf(P )) is a 1 +O(") approximation to Pers(d

kP ).

Extension to k-NN distances.

d

kP (x) = distance from x to k points of P .

Corollary: If f is an "-JL projection then for all k,Pers(d

kf(P )) is a 1 +O(") approximation to Pers(d

kP ).

Bonus: Also works for weighted points.

Going forward…

Going forward…

• Eliminate inner product condition.

Going forward…

• Eliminate inner product condition.• Eliminate constant factor (4)

Going forward…

• Eliminate inner product condition.• Eliminate constant factor (4)• Eliminate linearity condition.

Going forward…

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

Going forward…

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

Thank you.