Add Isotropic Gaussian Kernels at Own RiskMore and More Resilient Modes in Higher Dimensions

Herbert Edelsbrunner, BRITTANY TERESE FASY, and Gunter Rote

Symposium on Computational Geometry 2012Chapel Hill, North Carolina

18 June 2012

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 1 / 31

Introduction Counting Modes

Counting Modes and Critical Points

Definition

A critical point is a point with a zero gradient.

Definition

A mode is a local maximum.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 2 / 31

Introduction Counting Modes

Counting Modes and Critical Points

Definition

A critical point is a point with a zero gradient.

Definition

A mode is a local maximum.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 2 / 31

Introduction Counting Modes

Counting Modes and Critical Points

Definition

A critical point is a point with a zero gradient.

Definition

A mode is a local maximum.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 2 / 31

Introduction Counting Modes

How Many Modes (Local Maxima)?

we see

Existence proven in [M. Carreira-Perpinan and C. Williams, Scotland 2003].

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 3 / 31

Introduction Counting Modes

How Many Modes (Local Maxima)?

In the begining, we see 1 local maximum.

Existence proven in [M. Carreira-Perpinan and C. Williams, Scotland 2003].

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 3 / 31

Introduction Counting Modes

How Many Modes (Local Maxima)?

At the end, we see 3 local maxima.

Existence proven in [M. Carreira-Perpinan and C. Williams, Scotland 2003].

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 3 / 31

Introduction Counting Modes

How Many Modes (Local Maxima)?

In the middle, we see 4 local maxima.

Existence proven in [M. Carreira-Perpinan and C. Williams, Scotland 2003].

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 3 / 31

Introduction Counting Modes

How Many Modes (Local Maxima)?

In the middle, we see 4 local maxima.

Existence proven in [M. Carreira-Perpinan and C. Williams, Scotland 2003].

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 3 / 31

Introduction Counting Modes

How Many Modes (Local Maxima)?

In the middle, we see 4 local maxima.

Existence proven in [M. Carreira-Perpinan and C. Williams, Scotland 2003].

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 3 / 31

Contributions

Brief Overview

Define Gaussian kernel and mixture.

Analyze 1-dimensional mixtures.

Locate and count all critical points of an n-dimensional mixture.

Locate and count all modes of an n-dimensional mixtures.

(Describe the resilience of the ghost mode.)

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 4 / 31

Contributions

Brief Overview

Define Gaussian kernel and mixture.

Analyze 1-dimensional mixtures.

Locate and count all critical points of an n-dimensional mixture.

Locate and count all modes of an n-dimensional mixtures.

(Describe the resilience of the ghost mode.)

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 4 / 31

Contributions

Brief Overview

Define Gaussian kernel and mixture.

Analyze 1-dimensional mixtures.

Locate and count all critical points of an n-dimensional mixture.

Locate and count all modes of an n-dimensional mixtures.

(Describe the resilience of the ghost mode.)

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 4 / 31

Contributions

Brief Overview

Define Gaussian kernel and mixture.

Analyze 1-dimensional mixtures.

Locate and count all critical points of an n-dimensional mixture.

Locate and count all modes of an n-dimensional mixtures.

(Describe the resilience of the ghost mode.)

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 4 / 31

Contributions

Brief Overview

Define Gaussian kernel and mixture.

Analyze 1-dimensional mixtures.

Locate and count all critical points of an n-dimensional mixture.

Locate and count all modes of an n-dimensional mixtures.

(Describe the resilience of the ghost mode.)

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 4 / 31

Gaussian Kernel One Dimensional

Gaussian Kernel

Definition

gz(x) =1√

2πσ2e

−(x−z)2

2σ2

Center: zStandard Deviation: σHeight: 1√

2πσ2

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 5 / 31

Gaussian Kernel One Dimensional

Gaussian Kernel

Definition

gz(x) =1√

2πσ2e

−(x−z)2

2σ2

Center: z

Standard Deviation: σHeight: 1√

2πσ2

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 5 / 31

Gaussian Kernel One Dimensional

Gaussian Kernel

Definition

gz(x) =1√

2πσ2e

−(x−z)2

2σ2

Center: zStandard Deviation: σ

Height: 1√2πσ2

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 5 / 31

Gaussian Kernel One Dimensional

Gaussian Kernel

Definition

gz(x) =1√

2πσ2e

−(x−z)2

2σ2

Center: zStandard Deviation: σHeight: 1√

2πσ2

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 5 / 31

Gaussian Kernel One Dimensional

Standardized Gaussian Kernel

Definition

gz(x) = e−π(x−z)2

Center: zStandard Deviation: σ0 = 1√

2πHeight: 1

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 6 / 31

Gaussian Kernel One Dimensional

Standardized Gaussian Kernel

Definition

gz(x) = e−π(x−z)2

Center: z

Standard Deviation: σ0 = 1√2π

Height: 1

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 6 / 31

Gaussian Kernel One Dimensional

Standardized Gaussian Kernel

Definition

gz(x) = e−π(x−z)2

Center: zStandard Deviation: σ0 = 1√

2π

Height: 1

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 6 / 31

Gaussian Kernel One Dimensional

Standardized Gaussian Kernel

Definition

gz(x) = e−π(x−z)2

Center: zStandard Deviation: σ0 = 1√

2πHeight: 1

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 6 / 31

Gaussian Kernel n-Dimensional

n-Dimensional Isotropic Gaussian Kernel

Definition

gz(x) = e−π||x−z||2

Center: zWidth: σ0 = 1√

2πHeight: 1

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 7 / 31

Gaussian Kernel n-Dimensional

n-Dimensional Isotropic Gaussian Kernel

Definition

gz(x) = e−π||x−z||2

Center: z

Width: σ0 = 1√2π

Height: 1

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 7 / 31

Gaussian Kernel n-Dimensional

n-Dimensional Isotropic Gaussian Kernel

Definition

gz(x) = e−π||x−z||2

Center: zWidth: σ0 = 1√

2π

Height: 1

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 7 / 31

Gaussian Kernel n-Dimensional

n-Dimensional Isotropic Gaussian Kernel

Definition

gz(x) = e−π||x−z||2

Center: zWidth: σ0 = 1√

2πHeight: 1

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 7 / 31

Gaussian Kernel n-Dimensional

Separability of the Gaussian Kernel

Separability Lemma

e−π||x−z||2

= e−π||x−y ||2e−π||y−z||

2

||x − z ||2 = ||x − y ||2 + ||y − z ||2

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 8 / 31

Gaussian Kernel n-Dimensional

Separability of the Gaussian Kernel

Separability Lemma

e−π||x−z||2

= e−π||x−y ||2

e−π||y−z||2

||x − z ||2 = ||x − y ||2 + ||y − z ||2

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 8 / 31

Gaussian Kernel n-Dimensional

Separability of the Gaussian Kernel

Separability Lemma

e−π||x−z||2

= e−π||x−y ||2e−π||y−z||

2

||x − z ||2 = ||x − y ||2 + ||y − z ||2

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 8 / 31

Gaussian Kernel n-Dimensional

Separability of the Gaussian Kernel

Separability Lemma

e−π||x−z||2

= e−π||x−y ||2e−π||y−z||

2

||x − z ||2 = ||x − y ||2 + ||y − z ||2

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 8 / 31

Gaussian Kernel Restrictions

Restrictions of Kernels

Definition

A restriction of gz is theevaluation of the function on alower-dimensional plane P.

gz |P(x) = gy (x).

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 9 / 31

Gaussian Kernel Restrictions

Restrictions of Kernels

Definition

A restriction of gz is theevaluation of the function on alower-dimensional plane P.

gz |P(x) = gz(y)gy (x).

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 9 / 31

Gaussian Kernel Restrictions

Restrictions of Kernels

Definition

A restriction of gz is theevaluation of the function on alower-dimensional plane P.

gz |P(x) = czgy (x).

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 9 / 31

Gaussian Mixtures

Gaussian Mixture

A Gaussian mixture is the sum of Gaussian kernels.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 10 / 31

Gaussian Mixtures

Gaussian Mixture

A Gaussian mixture is the sum of Gaussian kernels.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 10 / 31

Gaussian Mixtures

Gaussian Mixture

A Gaussian mixture is the sum of Gaussian kernels.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 10 / 31

Gaussian Mixtures

Restrictions of Mixtures

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 11 / 31

Gaussian Mixtures

No Ghost Modes

Theorem

In R1, the number of modes is at most the number of components.

Balanced sum of two kernels: [Burke, 1956].

Weighted sum of two kernels: [Behboodian, 1970].

General sum: [M. Carreira-Perpinan and C. Williams, LNCS 2003]relies heavily on [Silverman, 1981].

Question

When is the transition between having one mode and two?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 12 / 31

Gaussian Mixtures

No Ghost Modes

Theorem

In R1, the number of modes is at most the number of components.

Balanced sum of two kernels: [Burke, 1956].

Weighted sum of two kernels: [Behboodian, 1970].

General sum: [M. Carreira-Perpinan and C. Williams, LNCS 2003]relies heavily on [Silverman, 1981].

Question

When is the transition between having one mode and two?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 12 / 31

Gaussian Mixtures

No Ghost Modes

Theorem

In R1, the number of modes is at most the number of components.

Balanced sum of two kernels: [Burke, 1956].

Weighted sum of two kernels: [Behboodian, 1970].

General sum: [M. Carreira-Perpinan and C. Williams, LNCS 2003]relies heavily on [Silverman, 1981].

Question

When is the transition between having one mode and two?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 12 / 31

Gaussian Mixtures

No Ghost Modes

Theorem

In R1, the number of modes is at most the number of components.

Balanced sum of two kernels: [Burke, 1956].

Weighted sum of two kernels: [Behboodian, 1970].

General sum: [M. Carreira-Perpinan and C. Williams, LNCS 2003]relies heavily on [Silverman, 1981].

Question

When is the transition between having one mode and two?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 12 / 31

Gaussian Mixtures

No Ghost Modes

Theorem

In R1, the number of modes is at most the number of components.

Balanced sum of two kernels: [Burke, 1956].

Weighted sum of two kernels: [Behboodian, 1970].

General sum: [M. Carreira-Perpinan and C. Williams, LNCS 2003]relies heavily on [Silverman, 1981].

Question

When is the transition between having one mode and two?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 12 / 31

Gaussian Mixtures Sums in R1

Weighted Gaussian Mixture

Gw (x) = ckg−z(x) + c`gz(x).

The Weighted Mixture

1 If z is small enough, thenGw has one critical point.

2 If z is large enough, then Gw

has three critical points.

3 Gw has exactly 2 criticalpoints when ck

c`= r(x) + 1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 13 / 31

Gaussian Mixtures Sums in R1

Weighted Gaussian Mixture

Gw (x) = ckg−z(x) + c`gz(x).

The Weighted Mixture

1 If z is small enough, thenGw has one critical point.

2 If z is large enough, then Gw

has three critical points.

3 Gw has exactly 2 criticalpoints when ck

c`= r(x) + 1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 13 / 31

Gaussian Mixtures Sums in R1

Weighted Gaussian Mixture

Gw (x) = ckg−z(x) + c`gz(x).

The Weighted Mixture

1 If z is small enough, thenGw has one critical point.

2 If z is large enough, then Gw

has three critical points.

3 Gw has exactly 2 criticalpoints when ck

c`= r(x) + 1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 13 / 31

Gaussian Mixtures Sums in R1

Weighted Gaussian Mixture

Gw (x) = ckg−z(x) + c`gz(x).

The Weighted Mixture

1 If z is small enough, thenGw has one critical point.

2 If z is large enough, then Gw

has three critical points.

3 Gw has exactly 2 criticalpoints when ck

c`= r(x) + 1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 13 / 31

Gaussian Mixtures Ghosts Exist

Counting Modes in Rn

For n ≥ 2, there can be moremodes than components of aGaussian mixture in Rn.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 14 / 31

Design n-Simplex

Standard n-Simplex, ∆n

An n-simplex is the convex hullof n + 1 vertices.

The standard n-simplex has thestandard basis elements as thevertices:

e1, e2, . . . , en+1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 15 / 31

Design n-Simplex

Standard n-Simplex, ∆n

An n-simplex is the convex hullof n + 1 vertices.The standard n-simplex has thestandard basis elements as thevertices:

e1, e2, . . . , en+1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 15 / 31

Design n-Simplex

Standard n-Simplex, ∆n

An n-simplex is the convex hullof n + 1 vertices.The standard n-simplex has thestandard basis elements as thevertices:

e1, e2, . . . , en+1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 15 / 31

R3

Design n-Simplex

Standard n-Simplex, ∆n

An n-simplex is the convex hullof n + 1 vertices.The standard n-simplex has thestandard basis elements as thevertices:

e1, e2, . . . , en+1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 15 / 31

R3

Design n-Simplex

Scaled n-Simplex, s∆n

The scaled standard n-simplex inRn+1 is defined by the n + 1standard basis elements, scaledby a factor s

se1, se2, . . . , sen+1.

The barycenter is the averagevertex position:(

s

n + 1,

s

n + 1, . . . ,

s

n + 1

).

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 16 / 31

R3

Design n-Design

Scaled n-Design

Definition

The Scaled n-Design is theGaussian mixture with centers atthe n + 1 vertices of the scaledn-simplex:

Gs(x) =n+1∑i=1

gsei (x)

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 17 / 31

R3

Design Properties

Scaled n-Simplex, s∆n

The scaled standard n-simplex inRn+1 is defined by the n + 1standard basis elements, scaledby a factor s

se1, se2, . . . , sen+1.

The barycenter is the averagevertex position:(

s

n + 1,

s

n + 1, . . . ,

s

n + 1

).

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 18 / 31

R3

Design Properties

Scaled n-Simplex, s∆n

The scaled standard n-simplex inRn+1 is defined by the n + 1standard basis elements, scaledby a factor s

se1, se2, . . . , sen+1.

The barycenter is the averagevertex position:(

s

n + 1,

s

n + 1, . . . ,

s

n + 1

).

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 18 / 31

R3

Design Properties

Complementary Faces

We partition the vertices of thescaled n-simplex into two sets:

Let k = |K | − 1 and ` = |L| − 1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 19 / 31

R3

Design Properties

Complementary Faces

We partition the vertices of thescaled n-simplex into two sets:

K = {se3},L = {se1, se2}.

Let k = |K | − 1 and ` = |L| − 1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 19 / 31

R3

Design Properties

Complementary Faces

We partition the vertices of thescaled n-simplex into two sets:

K = {se3},L = {se1, se2}.

Let k = |K | − 1 and ` = |L| − 1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 19 / 31

R3

Design Properties

Complementary Faces

We partition the vertices of thescaled n-simplex into two sets:

K = {se2, se5},L = {se1, se3, se4}.

Let k = |K | − 1 and ` = |L| − 1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 19 / 31

R5

Design Axes

Location of Critical Points

The axis AK ,L is the line definedby bK and bL.

Location of Critical Values

All critical points of the scaledn-design lie on an axis of s∆n.

Proof

Assume a critical point x is noton an axis ...

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 20 / 31

R5

Design Axes

Location of Critical Points

The axis AK ,L is the line definedby bK and bL.

Location of Critical Values

All critical points of the scaledn-design lie on an axis of s∆n.

Proof

Assume a critical point x is noton an axis ...

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 20 / 31

R5

Design Axes

Location of Critical Points

The axis AK ,L is the line definedby bK and bL.

Location of Critical Values

All critical points of the scaledn-design lie on an axis of s∆n.

Proof

Assume a critical point x is noton an axis ...

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 20 / 31

R4

Design Axes

Location of Critical Points

The axis AK ,L is the line definedby bK and bL.

Location of Critical Values

All critical points of the scaledn-design lie on an axis of s∆n.

Proof

Assume a critical point x is noton an axis ...

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 20 / 31

R3

Design Axes

Location of Critical Points

The axis AK ,L is the line definedby bK and bL.

Location of Critical Values

All critical points of the scaledn-design lie on an axis of s∆n.

Proof

Assume a critical point x is noton an axis ...

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 20 / 31

R3

Design Axes

Location of Critical Points

The axis AK ,L is the line definedby bK and bL.

Location of Critical Values

All critical points of the scaledn-design lie on an axis of s∆n.

Proof

Assume a critical point x is noton an axis ...

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 20 / 31

R3

Design Restrictions

Restriction to an Axis

Gs |A(x) = cke−πh(x) + c`e

−π(Dk,`−h(x)),

where ck = (k + 1)gsei (bL), c` = (`+ 1)gsej (bK ).

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 21 / 31

R3

Design Restrictions

Restriction to an Axis

Gs |A(x) = cke−πh(x) + c`e

−π(Dk,`−h(x)),

where ck = (k + 1)gsei (bL), c` = (`+ 1)gsej (bK ).

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 21 / 31

R3

Design Restrictions

Restriction to an Axis

Gs |A(x) = cke−πh(x) + c`e

−π(Dk,`−h(x)),

where ck = (k + 1)gsei (bL), c` = (`+ 1)gsej (bK ).

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 21 / 31

R3

Ghosts

Weighted Gaussian Mixture

Gw (x) = ckg−z(x) + c`gz(x).

The Weighted Mixture

1 If z is small enough, thenGw has one critical point.

2 If z is large enough, then Gw

has three critical points.

3 Gw has exactly 2 criticalpoints when ck

c`= r(x) + 1.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 22 / 31

Ghosts Lower Transition Scale Factor

Lower Transition Scale Factor Tk ,`

Definition

Tk,` is the scale factor for which

ckc`

= r(x) + 1.

1-Dimensional Maxima Lemma

For all s > Tk,`, the axis AK ,L

witnesses two one-dimensionalmaxima.

Lemma

If s ∈ (Tk,`,Un), then AK ,L witnesses two one-dimensional maxima,one of which is at the barycenter.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 23 / 31

Ghosts Lower Transition Scale Factor

Lower Transition Scale Factor Tk ,`

Definition

Tk,` is the scale factor for which

ckc`

= r(x) + 1.

1-Dimensional Maxima Lemma

For all s > Tk,`, the axis AK ,L

witnesses two one-dimensionalmaxima.

Lemma

If s ∈ (Tk,`,Un), then AK ,L witnesses two one-dimensional maxima,one of which is at the barycenter.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 23 / 31

Ghosts Lower Transition Scale Factor

Lower Transition Scale Factor Tk ,`

Definition

Tk,` is the scale factor for which

ckc`

= r(x) + 1.

1-Dimensional Maxima Lemma

For all s > Tk,`, the axis AK ,L

witnesses two one-dimensionalmaxima.

Lemma

If s ∈ (Tk,`,Un), then AK ,L witnesses two one-dimensional maxima,one of which is at the barycenter.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 23 / 31

Ghosts Lower Transition Scale Factor

Lower Transition Scale Factor Tk ,`

Definition

Tk,` is the scale factor for which

ckc`

= r(x) + 1.

1-Dimensional Maxima Lemma

For all s > Tk,`, the axis AK ,L

witnesses two one-dimensionalmaxima.

Lemma

If s ∈ (Tk,`,Un), then AK ,L witnesses two one-dimensional maxima,one of which is at the barycenter.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 23 / 31

Ghosts Mode at Barycenter

Upper Transition Scale Factor Un

Definition

Tk,` is the scale factor for which:

ckc`

= r(x) + 1.

1-Dimensional Maxima Lemma

For all s > Tk,`, the axis AK ,L

witnesses two one-dimensionalmaxima.

Definition

Un =

√n + 1

2π.

Barycenter Lemma

The barycenter of s∆n is a modefor s < Un, and a saddle ofindex 1 for s > Un.

Theorem

If s ∈ (Tk,`,Un), then AK ,L witnesses two one-dimensional maxima, one ofwhich is at the barycenter.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 24 / 31

Ghosts Mode at Barycenter

Upper Transition Scale Factor Un

Definition

Tk,` is the scale factor for which:

ckc`

= r(x) + 1.

1-Dimensional Maxima Lemma

For all s > Tk,`, the axis AK ,L

witnesses two one-dimensionalmaxima.

Definition

Un =

√n + 1

2π.

Barycenter Lemma

The barycenter of s∆n is a modefor s < Un, and a saddle ofindex 1 for s > Un.

Theorem

If s ∈ (Tk,`,Un), then AK ,L witnesses two one-dimensional maxima, one ofwhich is at the barycenter.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 24 / 31

Ghosts Mode at Barycenter

One-Dimensional Maxima

Definition

Tk,` is the scale factor for which:

ckc`

= r(x) + 1.

1-Dimensional Maxima Lemma

For all s > Tk,`, the axis AK ,L

witnesses two one-dimensionalmaxima.

Definition

Un =

√n + 1

2π.

Barycenter Lemma

The barycenter of s∆n is a modefor s < Un, and a saddle ofindex 1 for s > Un.

Theorem

If s ∈ (Tk,`,Un), then AK ,L witnesses two one-dimensional maxima, one ofwhich is at the barycenter.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 24 / 31

Ghosts Mode at Barycenter

Restriction to an Axis

Gs |A(x) = e−πh(x) + c`e−π(D0,n−1−h(x)),

where c` = (`+ 1)gsej (bL).

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 25 / 31

R5

Ghosts Witnessed Ghost

Witnessing the Modes

Witnessing Modes

If |K | = 1, then AK ,L witnessestwo modes for s ∈ (T0,n−1,Un).

Witnessing Critical Points

If |K | > 1, then M is a criticalpoint, not a mode.

Lemma

If s ∈ (Tk,`,Un), then AK ,L witnesses two one-dimensional maxima, one ofwhich is at the barycenter.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 26 / 31

Ghosts Witnessed Ghost

Witnessing the Modes

Witnessing Modes

If |K | = 1, then AK ,L witnessestwo modes for s ∈ (T0,n−1,Un).

Witnessing Critical Points

If |K | > 1, then M is a criticalpoint, not a mode.

Lemma

If s ∈ (Tk,`,Un), then AK ,L witnesses two one-dimensional maxima, one ofwhich is at the barycenter.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 26 / 31

Ghosts Witnessed Ghost

Witnessing the Modes

Witnessing Modes

If |K | = 1, then AK ,L witnessestwo modes for s ∈ (T0,n−1,Un).

Witnessing Critical Points

If |K | > 1, then M is a criticalpoint, not a mode.

Lemma

If s ∈ (Tk,`,Un), then AK ,L witnesses two one-dimensional maxima, one ofwhich is at the barycenter.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 26 / 31

Ghosts Axes

Many Axes

Number of Axes with k = 0:

n + 1.

The scaled design hasn + 2 modes.

Total Number of Axes:

1

2

n+1∑k=1

(n + 1

k

)= 2n − 1.

The scaled design has Θ(2n)critical points.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 27 / 31

R4

Ghosts Axes

Many Axes

Number of Axes with k = 0:

n + 1.

The scaled design hasn + 2 modes.

Total Number of Axes:

1

2

n+1∑k=1

(n + 1

k

)= 2n − 1.

The scaled design has Θ(2n)critical points.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 27 / 31

R4

Ghosts Axes

Many Axes

Number of Axes with k = 0:

n + 1.

The scaled design hasn + 2 modes.

Total Number of Axes:

1

2

n+1∑k=1

(n + 1

k

)= 2n − 1.

The scaled design has Θ(2n)critical points.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 27 / 31

R4

Ghosts Axes

Many Axes

Number of Axes with k = 0:

n + 1.

The scaled design hasn + 2 modes.

Total Number of Axes:

1

2

n+1∑k=1

(n + 1

k

)= 2n − 1.

The scaled design has Θ(2n)critical points.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 27 / 31

R4

Ghosts Axes

Many Axes

Number of Axes with k = 0:

n + 1.

The scaled design hasn + 2 modes.

Total Number of Axes:

1

2

n+1∑k=1

(n + 1

k

)= 2n − 1.

The scaled design has Θ(2n)critical points.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 27 / 31

R4

Wrapping Up Summary

Summary of Results

The n-design has:

1 at most ONE ghost mode.

2 an exponential number ofcritical points.

3 all critical points on axes.

Wednesday at 2:50 in SN011:

1 How does Un − T0,n−1 (theresilience) grow withdimension?

2 What is the persistence ofthe ghost mode?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 28 / 31

Wrapping Up Summary

Summary of Results

The n-design has:

1 at most ONE ghost mode.

2 an exponential number ofcritical points.

3 all critical points on axes.

Wednesday at 2:50 in SN011:

1 How does Un − T0,n−1 (theresilience) grow withdimension?

2 What is the persistence ofthe ghost mode?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 28 / 31

Wrapping Up Summary

Summary of Results

The n-design has:

1 at most ONE ghost mode.

2 an exponential number ofcritical points.

3 all critical points on axes.

Wednesday at 2:50 in SN011:

1 How does Un − T0,n−1 (theresilience) grow withdimension?

2 What is the persistence ofthe ghost mode?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 28 / 31

Wrapping Up Summary

Summary of Results

The n-design has:

1 at most ONE ghost mode.

2 an exponential number ofcritical points.

3 all critical points on axes.

Wednesday at 2:50 in SN011:

1 How does Un − T0,n−1 (theresilience) grow withdimension?

2 What is the persistence ofthe ghost mode?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 28 / 31

Wrapping Up Summary

Summary of Results

The n-design has:

1 at most ONE ghost mode.

2 an exponential number ofcritical points.

3 all critical points on axes.

Wednesday at 2:50 in SN011:

1 How does Un − T0,n−1 (theresilience) grow withdimension?

2 What is the persistence ofthe ghost mode?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 28 / 31

Wrapping Up Summary

Summary of Results

The n-design has:

1 at most ONE ghost mode.

2 an exponential number ofcritical points.

3 all critical points on axes.

Wednesday at 2:50 in SN011:

1 How does Un − T0,n−1 (theresilience) grow withdimension?

2 What is the persistence ofthe ghost mode?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 28 / 31

Wrapping Up Summary

Add Isotropic Gaussian Kernels at Own RiskMore and More Resilient Modes in Higher Dimensions

Herbert Edelsbrunner, BRITTANY TERESE FASY, and Gunter Rote

Symposium on Computational Geometry 2012Chapel Hill, North Carolina

18 June 2012

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 29 / 31

Wrapping Up Summary

Summary of Results

The n-design has:

1 at most ONE ghost mode.

2 an exponential number ofcritical points.

3 all critical points on axes.

Wednesday at 2:50 in SN011:

1 How does Un − T0,n−1 (theresilience) grow withdimension?

2 What is the persistence ofthe ghost mode?

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 30 / 31

Wrapping Up References

References

Behboodian, J.On the modes of a mixture of two normal distributions.Technometrics 12, 1 (1970), 131–139.

Burke, P. J.Solution of problem 4616 [1954, 718], proposed by A. C. Cohen, Jr.Amer. Math. Monthly 63, 2 (Feb. 1956), 129.

Carreira-Perpinan, M., and Williams, C.On the number of modes of a Gaussian mixture.Scale Space Methods in Computer Vision, Lecture Notes in ComputerScience 2695 (2003), 625–640.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 31 / 31

Wrapping Up References

References

Carreira-Perpinan, M., and Williams, C.An isotropic Gaussian mixture can have more modes thancomponents.Informatics Research Report EDI-INF-RR-0185, Institute for Adaptiveand Neural Computation, University of Edinbugh, Dec. 2003.

Fasy, B. T.Modes of Gaussian mixtures and an inequality for the distancebetween curves in space, June 2012.PhD Dissertation, Duke University Comput. Sci. Dept.

Silverman, B. W.Using kernel density estimates to investigate multimodality.J. R. Stat. Soc. Ser. B. Stat. Methodol. 43 (1981), 97–99.

Edelsbrunner, Fasy and Rote (SoCG 2012) Gaussian Mixtures 18 June 2012 31 / 31