Discretization and tropicalization: How are they …...Discretization and tropicalization: How are...

Discretization andtropicalization:

How are they related?Christer O. Kiselman

Member of the Reference Group for Mathematics of ISP

First Network Meeting for Sida- and ISP-funded

PhD Students in Mathematics

Stockholm, Sida Headquarters

2017 March 07 15:00–15:45

sida20170307

1

Abstract

Discretization and tropicalization are two important proceduresin contemporary mathematics.

I will present them and motivate why they are important.

Discrete objects, like carpets and mosaics, have been around forthousands of years, but the advent of computers and digitalcameras has made them ubiquitous.

Tropical mathematics is by comparison a relatively new branchof mathematics.

I shall compare the two and also pose aphilosophical/mathematical problem in relation to them.

2

Abstract






3

Abstract






4

Abstract






5

Abstract






6

Contents1. Introduction2. What is a discrete set?3. To discretize a set or a function4. Are discrete objects easier than non-discrete?5. What is tropicalization?6. Comparing discretization and tropicalization7. Is there a Rosetta Stone?

7

IntroductionCarpets, embroidery work, and mosaics are examples ofartifacts that are discrete.

A carpet can consist of thousands of knots, but they are onlyfinite in number.

8

IntroductionCarpets, embroidery work, and mosaics are examples ofartifacts that are discrete.

A carpet can consist of thousands of knots, but they are onlyfinite in number.

9

Persian carpet.

10

A mosaic is made up of finitely many little stones, tessellas.

11

Ravenna mosaic.

12

Embroidery . . .

13

All three can present pictures quite well.

These objects have been around for a long time, but now, withcomputers and digital cameras, they are everywhere. A photoconsists of many pixels (picture elements) but only finitelymany.

14

All three can present pictures quite well.

These objects have been around for a long time, but now, withcomputers and digital cameras, they are everywhere. A photoconsists of many pixels (picture elements) but only finitelymany.

15

So all this makes digital geometry into a kind of geometry ofgrowing interest. Discretization of sets and function is studiedfrom many points of view.

Tropicalization is somehow similar to discretization . . .

Here I will talk about both procedures.

16




17




18

DiscretizationWhat is a discrete set?Intuitively, a discrete set is a set where every point is a bit awayfrom all the other points.

To make this precise, we need some kind of distance. So denoteby d(x ,y) the distance between two points x and y in somekind of space, like the plane or three-space.

Then a set A is said to be discrete if for every element a of A,there is a positive number r such that d(a,b)> r for allelements b 6= a of A.

19




20




21

Let us take R as an example, with d(x ,y) = |x � y |. Then forevery number a there are other numbers b that are as close aswe like to a. So R with this metric is not discrete.

The subset Z of integers is discrete, because now d(a,b)> 1

for all points b 6= a.

But beware! Every set is discrete if we define the distance asd(x ,y) = 1 when x 6= y and d(x ,x) = 0. So also R is discretewith this kind of distance.

This means that the distance between p and

3.141592653589793238462643383279502884197

is 1 with this metric, whereas it is rather small in the unusalmetric.

22






3.141592653589793238462643383279502884197


23






3.141592653589793238462643383279502884197


24






3.141592653589793238462643383279502884197


25

Mathematical models based on real or complex numbers havebeen very successful—think of celestial mechanics and thetheory of electric circuits.

However, we should not think of real numbers as being morereal than complex numbers or than discrete models. Theattribute real is misleading.

26

Mathematical models based on real or complex numbers havebeen very successful—think of celestial mechanics and thetheory of electric circuits.

However, we should not think of real numbers as being morereal than complex numbers or than discrete models. Theattribute real is misleading.

27

That time is discrete was a theory developed in Spain in theMiddle Ages:

“As regards the theoretical and philosophical analysis of time,the most important and original contribution of medievalIslamic thinkers was their theory of discontinuous, or atomistic,time.” (Whitrow 1990:79)

28

That time is discrete was a theory developed in Spain in theMiddle Ages:

“As regards the theoretical and philosophical analysis of time,the most important and original contribution of medievalIslamic thinkers was their theory of discontinuous, or atomistic,time.” (Whitrow 1990:79)

29

Moshe ben Maimun (born in 1135 or 1138 in Cordoba anddeceased in 1204 near Cairo), a Jewish philosopher working inMuslim cultural circles and better known under his Greek nameMaimonides,Maimwn–dhc, wrote:

“Time is composed of time-atoms, i.e. of many parts, which inaccount of their short durations cannot be divided. . . . An houris, e.g. divided into sixty minutes, the second into sixty partsand so on; at last after ten or more successive divisions by sixty,time-elements are obtained which are not subject to division,and in fact are indivisible.” (Whitrow 1990:79)

Often we do not go from models based on real numbers todiscrete models, but from one very fine discrete model toanother model, also discrete, but less fine.

Physicists study discrete models of spacetime . . . and not justrecently—but much later than Maimonides . . .

30





31





32





33

To discretize

To discretize a set means to map it in some way to a discrete set.

A simple example of a discretization is the mapping

P(Rn) 3 A 7! A\Zn 2 P(Zn),

mapping any subset of Rn to the set of its points with integercoordinates.

34

To discretize

To discretize a set means to map it in some way to a discrete set.

A simple example of a discretization is the mapping

P(Rn) 3 A 7! A\Zn 2 P(Zn),

mapping any subset of Rn to the set of its points with integercoordinates.

35

Since A may lack points with integer coordinates, we oftenwant to fatten the set before we intersect it with Zn:

P(Rn) 3 A 7! (A+U)\Zn 2 P(Zn),

where U can be any set; most typically a ball or cube toguarantee that the image is nonempty as soon as A is nonempty.

36

We can discretize a function f : R ! R partially or completely:

We may just take its restriction to the integers, i.e., f |Z, andkeep its values in R.

Or we may discretize also its values, e.g., by takingg(x) = bf (x)c, x 2 Z. Here btc denotes the floor function,defined by R 3 t 7! btc 2 Z and t 6 btc< t +1.

37




38



Or we may discretize also its values, e.g., by takingg(x) = bf (x)c, x 2 Z.

Here btc denotes the floor function,defined by R 3 t 7! btc 2 Z and t 6 btc< t +1.

39




40

0 2 4 6 8 10 12 14 16 18-1

0

1

2

3

4

5

6

Fonction plancherDroite euclidienne

Covering the Euclidean straight line of equation y = 1

3

x by adilation with structural element equal to the rectangle[�1

2

, 1

2

]⇥ [�5

6

, 5

6

] (courtesy Adama Arouna Kone).

41

Covering a Euclidean plane by a dilation with structuralelement equal to the box [�1

2

, 1

2

]⇥ [�1

2

, 1

2

]⇥ [�9

8

, 9

8

] (courtesyAdama Arouna Kone).

42

Small children start to count with natural numbers. Realnumbers are much more difficult to define.

So one may think that discrete sets, like the set of naturalnumbers N = {0,1,2,3, . . .}, are easier to handle than the set Rof real numbers, with numbers such as p and e etc.

However, . . .

43



However, . . .

44



However, . . .

45

In A decision method for elementary algebra and geometry,Alfred Tarski (1901–1983) showed that the first-order theory ofthe real numbers under addition and multiplication is decidable.Via Descartes (1596–1650), this applies to elementaryEuclidean geometry.

(This result was published only in 1948,but it dates back to 1930 and was mentioned in Tarski (1931).)

46

In A decision method for elementary algebra and geometry,Alfred Tarski (1901–1983) showed that the first-order theory ofthe real numbers under addition and multiplication is decidable.Via Descartes (1596–1650), this applies to elementaryEuclidean geometry. (This result was published only in 1948,but it dates back to 1930 and was mentioned in Tarski (1931).)

47

This is a most remarkable result, because Alonzo Church(1903–1995) proved in 1936 that Peano arithmetic (the theoryof natural numbers) is not decidable: there is no algorithm.

Peano arithmetic is also incomplete by Godel’s incompletenesstheorem (Kurt Godel, 1906–1978).

So, all this points to the fact that, from the point of view oflogic, natural numbers are much more difficult to treat than realnumbers.

But not only logic!

48




But not only logic!

49




But not only logic!

50




But not only logic!

51

The derivative of f (x) = x

5, x 2 R, is f

0(x) = 5x

4.

But the difference quotient is

f (x +h)� f (x)

h

= 5x

4 +10x

3

h+10x

2

h

2 +5xh

3 +h

4.

And coming to integrals we have to compareZ

a

0

f (x)dx =1

6

a

6

and1

n

an

Â

j=0

f (j/n) =1

6

a

6 +1

2

a

5

n

+5

12

a

4

n

2

� 1

12

a

2

n

4

.

This may explain some of the success of differential andintegral calculus.

52


5, x 2 R, is f

0(x) = 5x

4.


f (x +h)� f (x)

h

= 5x

4 +10x

3

h+10x

2

h

2 +5xh

3 +h

4.


a

0

f (x)dx =1

6

a

6

and1

n

an

Â

j=0

f (j/n) =1

6

a

6 +1

2

a

5

n

+5

12

a

4

n

2

� 1

12

a

2

n

4

.


53


5, x 2 R, is f

0(x) = 5x

4.


f (x +h)� f (x)

h

= 5x

4 +10x

3

h+10x

2

h

2 +5xh

3 +h

4.


a

0

f (x)dx =1

6

a

6

and1

n

an

Â

j=0

f (j/n) =1

6

a

6 +1

2

a

5

n

+5

12

a

4

n

2

� 1

12

a

2

n

4

.


54


5, x 2 R, is f

0(x) = 5x

4.


f (x +h)� f (x)

h

= 5x

4 +10x

3

h+10x

2

h

2 +5xh

3 +h

4.


a

0

f (x)dx =1

6

a

6

and1

n

an

Â

j=0

f (j/n) =1

6

a

6 +1

2

a

5

n

+5

12

a

4

n

2

� 1

12

a

2

n

4

.


55

Discretization by balayageBalayage is a method developed in classical potential theory formeasures (masses or electric charges).

The term is of French origin and means ‘sweeping (using abroom)’.

56

Discretization by balayageBalayage is a method developed in classical potential theory formeasures (masses or electric charges).

The term is of French origin and means ‘sweeping (using abroom)’.

57

In this context we may define the balayage of a function f

defined in Rn

onto a discrete set P as the function

g(p) =Â

a2V(p)

f (a), p 2 P,

where the sets V (p), p 2 P, form a tessellation of Rn, meaningthat

V (p)\V (q) = Ø when p 6= q, and[

p2P

V (p) = Rn.

58

We may for example take P = Zn ⇢ Rn andV (p) = {a 2 Rn; bac= p}, p 2 Zn,

or like modified Voronoi cells:V (p) = {a 2 Rn; ba+(1

2

, 1

2

, . . . , 1

2

)c= p}, p 2 Zn.

The Voronoi cells are named for Georgiı Feodosevic VoronoıGeorg�⇢ Feodos�⇢oviq Voroni⇢; Georgi⇢ Feodos~eviq

Vorono⇢, Georges Voronoı (1868–1908).

Given a set P of points in a space X with distance d , we definethe Voronoi cell with kernel p 2 P as the set

V (p) = {x 2 X ; d(x ,p)6 d(x ,q) for all points q 2 P}.

The Voronoi cells cover the whole space, but they are usuallynot disjoint.

59



2

, 1

2

, . . . , 1

2

)c= p}, p 2 Zn.






60



2

, 1

2

, . . . , 1

2

)c= p}, p 2 Zn.






61



2

, 1

2

, . . . , 1

2

)c= p}, p 2 Zn.






62



2

, 1

2

, . . . , 1

2

)c= p}, p 2 Zn.






63

Voronoi cells in the plane.

64

Tropicalization

Tropicalization means, roughly speaking, to replace a sum orintegral by a supremum.

So a sum x + y is replaced by the maximum of the two:x _ y = max(x ,y).

Children learn to add numbers, like 5+3 = 8 . . . but evenbefore that, they learn that 5 comes after 3. So they learn that5_3 = 5 even before they learn that 5+3 = 8. Therefore themax operation is not something new or strange.

65

Tropicalization




66

Tropicalization




67

However, an equation like a+ x = b, where a and b are knownand x unknown, has, in the set Z of integers, a unique solutionx = b�a. The equation a_ x = b sometimes has• no solution (when b < a),• sometimes a unique solution (when b > a),• and sometimes infinitely many solutions (when b = a).

68

A typical example is the tropicalization of the l

p-norm:

kxkp

=�Â

|xj

|p�

1/p is replaced by (sup |xj

|p)1/p = sup |xj

|= kxk•

.

In this case we have convergence:

kxkp

=�Â

|xj

|p�

1/p ! sup

j

|xj

|= kxk•

as p !+•, x 2 Rn.

69

More algebraically, tropicalization occurs when we replaceaddition by the operation of taking the maximum andmultiplication by addition.

Tropical lines and tropical hyperplanes, as well as tropicalpolynomials are of interest. It seems that there is not yet anaxiomatic approach to this kind of geometry.

70

More algebraically, tropicalization occurs when we replaceaddition by the operation of taking the maximum andmultiplication by addition.

Tropical lines and tropical hyperplanes, as well as tropicalpolynomials are of interest. It seems that there is not yet anaxiomatic approach to this kind of geometry.

71

We can compare this procedure with the operation of taking thelogarithm:

log(xy) = logx + logy , x ,y > 0.

log(x _ y)< log(x + y)6 log(x _ y)+ log2, x ,y > 0.

So the logarithm transforms multiplication to addition, and asum to something close to the maximum—the error is small bycomparison if x _ y is large.

72

We can compare this procedure with the operation of taking thelogarithm:

log(xy) = logx + logy , x ,y > 0.

log(x _ y)< log(x + y)6 log(x _ y)+ log2, x ,y > 0.

So the logarithm transforms multiplication to addition, and asum to something close to the maximum—the error is small bycomparison if x _ y is large.

73

Tropical straight linesA first degree polynomial P(x ,y) = ax +by + c is transformedto Ptrop(x ,y) = (a+ x)_ (b+ y)_ c.

The zeros of P correspond to the points where two terms ofPtrop are equal.

So this means that we should look ata+ x = b+ y , a+ x = c, and b+ y = c. All three are equal at(x ,y) = (c�a,c�b).

74


The zeros of P correspond to the points where two terms ofPtrop are equal. So this means that we should look ata+ x = b+ y , a+ x = c, and b+ y = c.

All three are equal at(x ,y) = (c�a,c�b).

75


The zeros of P correspond to the points where two terms ofPtrop are equal. So this means that we should look ata+ x = b+ y , a+ x = c, and b+ y = c. All three are equal at(x ,y) = (c�a,c�b).

76

A tropical straight line in the plane.

77

Two tropical straight lines in the plane always intersect.

78

Two tropical straight lines in the plane always intersect . . . in aunique point . . . in a stable unique point . . .

79

ConvolutionGiven two function f and g defined on Rn, we define theirconvolution product h = f ⇤g by

h(x) = (f ⇤g)(x) =Â

y2Rn

f (x � y)g(y), x 2 Rn.

We have to impose some condition to guarantee convergence.

80

Infimal convolutionLet us study the convolution product of two functions of theform e

�f :

e

�h

1

(x) =Â

y2Rn

e

�f (x�y)e

�g(y), x 2 Rn,

assuming that f ,g are equal to +• outside some discrete set. Iffor instance f , g have their support in Zn andf (x),g(x)> ekxk�C, we have good convergence. Thetropicalization of this convolution product is

e

�h

•

(x) = sup

y2Rn

e

�f (x�y)e

�g(y), x 2 Rn,

which can be written

h

•

(x) = inf

y2Rn

(f (x � y)+g(y)), x 2 Rn.

81

Also in this case we have a nice convergence: If we define h

p

by

e

�ph

p

(x) =Â

y2Rn

e

�pf (x�y)e

�pg(y), x 2 Rn, p > 0,

then h

p

converges to h

•

as p !+•.

82

The function h

•

is the infimal convolution of f and g, denotedby f ug. Here we of course need not assume that f and g areequal to +• outside a discrete set. More generally, we define itwhen f and g take values in [�•,+•] using upper addition:

(f ug)(x) = inf

y2Rn

(f (x � y)+· g(y)), x 2 Rn.

The function ind{0} is a neutral element for u: f u ind{0} = f

for all f .

83

The Fenchel transformationThe Fenchel transform of a function f : Rn ! [�•,+•] isdefined as

f (x) = sup

x2Rn

(x · x � f (x)), x 2 Rn.

Clearly x · x � f (x)6 f (x), which can be written as

x · x 6 f (x)+· f (x), (x,x) 2 Rn ⇥Rn,

called Fenchel’s inequality. It follows that the secondtransform ˜

f satisfies ˜f 6 f . We have equality here if and only if f

is convex, lower semicontinuous, and takes the value �• onlyif it is �• everywhere.

84

The Fenchel transformation f 7! f , named for Werner Fenchel(1905–1988), is a tropical counterpart of the Fouriertransformation. This is perhaps even more obvious if we look atthe Laplace transform of a function g:

(L g)(x) =Z

•

0

g(x)e�xx

dx .

If we replace the integral by a supremum and take thelogarithm, we get

log(Ltrop g)(x)= sup

x

(logg(x)�xx)= f (�x), f (x)=� logg(x).

85

We have(f ug) = f +· g 6 f +· g.

If j and y are convex, then j+· y is convex, but not alwaysj+· y. However, when j = f and y = g, this is true: f +· g isalways convex, and is often equal to f +· g. In fact equality holdsexcept for a few special cases.

This formula should be compared with (f ⇤g) = f g.

86




87




88

Discretization and tropicalization:Is there a Rosetta Stone?So we discretize by replacing an integral by a sum:

Z7!

Â

,

and we tropicalize by replacing an integral by a supremum:Z

7! sup .

There are certain similarities between discretization andtropicalization. This is difficult to make precise.

A first step would be to make explicit all similarities and thenfind a more general procedure, of which both discretization andtropicalization are special cases.

89


Z7!

Â

,


7! sup .



90


Z7!

Â

,


7! sup .



91

The Rosetta Stone, 196 BC March 27, found in 1799: Twolanguages (Egyptian, Greek); three writing systems

(Hieroglyphs, Demotic script, Greek alphabet).

92

Thank you!

93

Discretization and tropicalization: How are they …...Discretization and tropicalization: How are...

Documents

Transcript of Discretization and tropicalization: How are they …...Discretization and tropicalization: How are...