**DNA-like Structure of Nonlinear Functions**

Moody T. Chu

(Joint work with Zhenyue Zhang)

North Carolina State University

March 8, 2012 @ National Cheng Kung University

**Disclaimer**

I This talk is about mathematics, not biology.

I This talk is elementary, involving only fundamental calculus.

I This work is just a beginning. More need be done.

I The importance of DNA is well documented.

• Found in all living organisms.

• Supplies the information for building all cell proteins.

I Basic structure of DNA:

• Two strands coiled around to form a double helix.

• Each rung of the spiral ladder consists of a pair of chemical groups called bases (of which there are four types)

• Base pairing combines A to T and C to G, and the sequence on one strand is complementary to that on the other.

• The specific sequence of bases constitutes the genetic information.

**Take Home Message**

I There is a considerably similar structure in all nonlinear functions.

• The structure determines the properties of the underlying function?

**Outline**

**Basics**

Gradient Adaption

Singular Value Decomposition Deformation Effect

**Singular Curves**
Dynamical Systems
Examples

Critical Curves
**Local Bearing**

Curvilinear Coordinate System Generic Behaviors

**Base Pairing**

Concavity Property
Pairings and Traits
**Applications**

**Conclusion**

**Outline**

**Basics**

Gradient Adaption

Singular Value Decomposition Deformation Effect

**Singular Curves**
Dynamical Systems
Examples

Critical Curves
**Local Bearing**

Curvilinear Coordinate System Generic Behaviors

**Base Pairing**

Concavity Property
Pairings and Traits
**Applications**

**Conclusion**

**Outline**

**Basics**

Gradient Adaption

Singular Value Decomposition Deformation Effect

**Singular Curves**
Dynamical Systems
Examples

Critical Curves
**Local Bearing**

Curvilinear Coordinate System Generic Behaviors

**Base Pairing**

Concavity Property
Pairings and Traits
**Applications**

**Conclusion**

**Outline**

**Basics**

Gradient Adaption

Singular Value Decomposition Deformation Effect

**Singular Curves**
Dynamical Systems
Examples

Critical Curves
**Local Bearing**

Curvilinear Coordinate System Generic Behaviors

**Base Pairing**

Concavity Property
Pairings and Traits
**Applications**

**Conclusion**

**Outline**

**Basics**

Gradient Adaption

Singular Value Decomposition Deformation Effect

**Singular Curves**
Dynamical Systems
Examples

Critical Curves
**Local Bearing**

Curvilinear Coordinate System Generic Behaviors

**Base Pairing**

Concavity Property
Pairings and Traits
**Applications**

**Conclusion**

**Outline**

**Basics**

Gradient Adaption

Singular Value Decomposition Deformation Effect

**Singular Curves**
Dynamical Systems
Examples

Critical Curves
**Local Bearing**

Curvilinear Coordinate System Generic Behaviors

**Base Pairing**

Concavity Property
Pairings and Traits
**Applications**

**Conclusion**

**Gradient**

I Given a scalar function

η : R^{n}−→ R,
define the gradient ofη by

∇η := ∂η

∂x1

, . . . , ∂η

∂xn

.

I Significance:

• Points in the direction where the function η(x) ascends most rapidly.

• Attainable maximum rate of change is precisely k∇η(x)k.

**Gradient Adaption**

I Heat transfer by conduction.

• Opposite to the temperature gradient and is perpendicular to the equal-temperature surfaces.

I Osmosis.

• Passive transport of substances across the cell membrane down a concentration gradient without requiring energy use.

I Image gradients.

• Fundamental building blocks in image processing such as edge detection and computer vision.

**Jacobian**

I Given a vector function

f : R^{n}−→ R^{m},
define the Jacobian of f by

Jf :=

∂f_{1}

∂x1 . . . _{∂x}^{∂f}^{1}
.. n

. . .. ...

∂fm

∂x1 . . . ^{∂f}_{∂x}^{m}

n

.

• A natural generalization of the gradient.

• Both offer linear approximations.

• Does not indicate critical directions or rates of change?

**Singular Value Decomposition**

I Any given matrix A ∈ R^{m×n}enjoys a factorization of the form
A = V ΣU^{>}.

• Known as a singular value decomposition (SVD) of A.

I Singular vectors:

• V ∈ R^{m×m}, U ∈ R^{n×n}are orthogonal matrices.

I Singular values:

• Σ ∈ R^{m×n}is diagonal with nonnegative elements
σ1≥ σ2≥ . . . ≥ σκ> σκ+1= . . . =0.

• κ =rank(A).

**Applications**

I A long conceived notion popping up in various disciplines.

I Frequent appearance in a remarkably wide range of important applications.

I A few examples –

• Data analysis.

• Dimension reduction.

• Signal processing.

• Image compression.

• Principal component analysis.

• ...

**Variational Formulation**

I Many ways to characterize the SVD of a matrix A.

I Cast as an optimization problem over the unit disk:

**kxk=1**max**kAxk.**

• Unit stationary points**u**i ∈ R^{n}= Right singular vectors.

• Singular values = kAuik.

I In the neighborhood of the origin:

• Right singular vectors = Directions where the linear map A changes most critically.

• Singular values = Extent of deformation.

I Similar role by the left singular vectors by the duality theory.

**Linear Approximation**

I Nearby any given pointe**x, approximate f (x) by the affine map**
g(x) := f (e**x) + f**^{0}(e**x)(x −**e**x).**

I Under the function g,

• The unit sphere centered ate**x gets mapped into an ellipsoid**
centered at f (e**x).**

• Semi-axes are aligned with the left singular vectors of f^{0}(e**x).**

• Semi-axis lengths are precisely the singular values.

**Infinitesimal Deformation**

I Reducing the radius of the sphere,

• Downsizes the ellipsoid proportionally.

• Does not alter the directions of the semi-axes.

• g becomes a more accurate approximation of f .

I The gradually reduced ellipsoids silhouette the images of the gradually reduced spheres under f .

I The SVD information of the linear operator f^{0}(e**x) manifests the**
infinitesimal deformation property of the nonlinear map f ate**x.**

**Directional Derivatives**

I Consider the norm of the directional derivative

t→0lim

f (e**x + tu) − f (**e**x)**
t

= kf^{0}(e**x)uk.**

• **u is an arbitrary unit vector.**

I Along which direction will the norm of the directional derivative be maximized?

• The right singular vectors of f^{0}(e**x)!**

I This is the generalization of the conventional gradient to vector functions.

**Singular Vector Field**

I At every point**x ∈ R**^{n},

• Have a set of orthonormal vectors pointing in particular directions related to the variation of f .

• These orthonormal vectors form a natural frame point by point.

I Tracking down the “motion" of these frames might help to reveal some innate peculiarities of the underlying function f .

**Dynamical Systems**

I Let (σi**, u**i**, v**i)= the ith singular triplet of f^{0}(x_{i}). Interested in the
solution flows:

• **x**i(t) ∈ R^{n}defined by

**˙x**i := ±ui(xi), **x**i(0) =e**x.**

• **y**i(t) ∈ R^{m}defined by

**˙y**i:= ±σi(xi)vi(xi), **y**i(0) = f (e**x).**

I Minor notes:

• Scaling ensures**y**i(t) = f (xi(t)).

• Select the sign ± so as to avoid discontinuity jump.

• Integrate in both forward and backward time.

**Critical Points**

I The vector field may not be well defined at certain points.

• When singular values coalesce.

• f^{0}(x) has multiple singular vector

• Makes ˙**x**i(or ˙**y**i) discontinuous.

I Not an issue of the factorization.

• An analytic factorization as a whole for a function analytic in**x does**
exist.

• The continuity of a fixed order singular vectors, say,**u**1(x), may not
be maintained.

**First Singular Curve**

I Moves in the direction along which f (x) changes most rapidly, when measured in the Euclidean norm.

I Serves as the backbone in the moving frame.

I Can be demonstrated and explained in the case f : R^{2}→ R^{n}.

• Parametric surfaces.

I More need be done in higher dimensional spaces.

**Example 1**

sin (x_{1}+x_{2}) +cos (x_{2}) −1
cos (2 x_{1}) +sin (x_{2}) −1

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

### Right Singular Curves for Example 1

x1

x 2

**Example 2a**

e^{x}^{1}cos(x_{2})
20e^{x}^{1}sin(x_{1})

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

### Right Singular Curves for Example 2a

x1

x 2

**Example 2b**

e^{x}^{1}cos(x_{2})
e^{x}^{1}sin(x_{1})

x2

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

### Right Singular Curves for Example 2b

x1

x 2

**Example 3**

4 + x_{1}cos(x_{2}/2)
x_{2}
x1sin(x1x2/2)

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x1

x 2

### Right Singular Curves for Example 3

**Example 4**

e^{x}^{1}cos(20x2)
20e^{sin(x}^{2}^{)}sin(x1)

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x1

x 2

### Right Singular Curves for Example 4

**Example 5**

sin(x_{1}^{2}+x_{2}^{2})cos(x_{2})
2e^{−2x}^{2}^{2}^{x}^{1}^{2}cos(10 sin(x1))

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x1

x 2

### Right Singular Curves for Example 5

**Example 6**

−270x_{1}^{4}x_{2}^{3}− 314x1x_{2}^{4}− 689x1x_{2}^{3}+1428

36x_{1}^{7}+417x_{1}^{6}x2−422x_{1}^{5}x_{2}^{2}−270x_{1}^{4}x_{2}^{3}+1428x_{1}^{3}x_{2}^{4}−1475x_{1}^{2}x_{2}^{5}+510x1x_{2}^{6}

−200x_{1}^{6}−174x_{1}^{5}x2−966x_{1}^{4}x_{2}^{2}+529x_{1}^{3}x_{2}^{3}+269x_{1}^{2}x_{2}^{4}+49x1x_{2}^{5}−267x_{2}^{6}+529x_{1}^{4}x2

+1303x_{1}^{2}x_{2}^{3}−314x1x_{2}^{4}+262x_{2}^{5}+36x_{1}^{4}−788x_{1}^{2}x_{2}^{2}−689x1x_{2}^{3}+177x_{2}^{4}

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x1

x 2

### Right Singular Curves for Example 6

**Example 7**

x1−^{x}_{3}^{1}^{2} +x1x_{2}^{2}
x2−^{x}_{6}^{2}^{3} +x2x_{1}^{3}

x_{1}^{2}− x_{2}^{3}

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x1

x 2

### Right Singular Curves for Example 7

**Example 8**

1

2 2ρ^{2}− φ^{2}− ψ^{2}+2φψ(φ^{2}− ψ^{2}) +ψρ(ρ^{2}− ψ^{2}) +ρφ(φ^{2}− ρ^{2})

√3

2 φ^{2}− ψ^{2}+ (ψρ(ψ^{2}− ρ^{2}) +ρφ(φ^{2}− ρ^{2}))
(ρ + φ + ψ) (ρ + φ + ψ)^{3}+4(φ − ρ)(ψ − φ)(ρ − ψ)

with

ρ = cos(x1)sin(x2) φ = sin(x1)sin(x2) ψ = cos(x2)

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x1

x 2

### Right Singular Curves for Example 8

## Why?

**A Closer Look**

I Write

f^{0}(x) =

**a**_{1}(x), a_{2}(x) .

I Define scalar functions

( n(x) := **ka**_{2}(x)k^{2}**− ka**_{1}(x)k^{2},
o(x) := 2a_{1}(x)^{>}**a**_{2}(x).

• n(x) measures the disparity of lengths.

• o(x) measures nearness of orthogonality.

**Critical Curves**

I Define

( N := **{x ∈ R**^{n}**| n(x) = 0} ,**
O := **{x ∈ R**^{n}**| o(x) = 0} .**

I Each forms generically a 1-dimensional manifold in R^{2}.

• Possibly composed of multiple curves or loops.

• Will play the role of “polynucleotide" connecting a string of interesting points.

**First Right Singular Pair**

I The first singular value of f^{0}(x):

σ1(x) := 1 2

**ka**_{1}(x)k^{2}+ ka2(x)k^{2}+
q

o(x)^{2}+n(x)^{2}

^{1/2}

I The first right singular vector:

**u**_{1}(x) := ±1
**p1 + ω(x)**^{2}

"

**ω(x)**
1

# .

with

**ω(x) :=**

o(x) n(x)+√

o(x)^{2}+n(x)^{2}, if n(x)> 0,

**−n(x)+**√

o(x)^{2}+n(x)^{2}

o(x) **, if n(x) < 0.**

• Take the limit if ω(x) becomes infinity.

**Crossings**

I When singular curves coming across critical curves, their tangent vectors point in specific directions.

I Orientations of tangent vectors:

• At N − O, are parallel to either [1, 1]^{>}or [1, −1]^{>}, depending on
whether o(x) is positive or negative.

• At O − N , are parallel to [0, 1]^{>}or [1, 0]^{>}, depending on whether
n(x) is positive or negative.

**Singular Points**

I NT O = singular points.

I At singular points,

• Singular values coalesce.

• The (right) singular vectors become ambiguous.

• Singular curves are“terminated" or “reborn".

I The angles cut by N and O at the singular point affects the intriguing dynamics observed.

• The 1-dimensional manifolds N and O string singular points together along their strands.

x1

x 2

### Critical Curves and Singular Curves for Example 1

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

ridges of near singular values

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

Right Singular Curves for Example 2a

x_{1}

x 2

x_{1}

x 2

Critical Curves for Example 2a

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

Right Singular Curves for Example 2b

x_{1}

x 2

x_{1}

x 2

Critical Curves for Example 2b

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x_{1}

x 2

Right Singular Curves for Example 3

x_{1}

x 2

Critical Curves for Example 3

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x1 x2

### Right Singular Curves for Example 4

x1 x2

### Critical Curves for Example 4

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x1 x2

### Right Singular Curves for Example 5

x1 x2

### Critical Curves for Example 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x_{1}

x 2

Right Singular Curves for Example 6

x_{1}

x 2

Critical Curves for Example 6

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x_{1}

x 2

Right Singular Curves for Example 7

x_{1}

x 2

Critical Curves for Example 7

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

x_{1}

x 2

Right Singular Curves for Example 8

x_{1}

x 2

Critical Curves for Example 8

−5 −4 −3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1 0 1 2 3 4 5

**Curvilinear Coordinate System**

I Denote theα-halves portions of N and O by by nαand oα, where

• The crossing singular vectors are parallel to the unit vectors
**u**n_{α} :=^{√}^{1}

2[1, 1]^{>}and**u**o_{α}:= [0, 1]^{>}.

I Flag the sides of nαand oαby arrows .

• Naturally divides the neighborhood of**x**0into “quadrants"

distinguished by the signs (sgn(n(x)), sgn(o(x)).

I When the “orientation" is changed, the nearby dynamical behavior might also change its topology.

**A Scenario of Propellant**

N O x0

(+,+) (+,−) (−,+)

(−,−) N

O x0

(+,+) (+,−) (−,+)

(−,−)

1

I Red segments = tangent vectors crossing the critical curves.

• Take into account the signs of o(x) and n(x).

I Invariant on each half of the critical curves.

I Flows of singular curves near**x**_{0}should move away from**x**_{0}as a
repellant.

**A Scenario of Roundabout**

N

O x0

(+,+)

(+,−) (−,+)

(−,−)

N

O x0

(+,+)

(+,−) (−,+)

(−,−)

1

**Generic Behaviors**

I Divide the plane into eight sectors with a central angle ^{π}_{4}.

I Relative position of nαand oαwith respect to these sectors is critical for deciding the local behavior.

**Regular** **Cases**

Aa Aa Ac Ac Bb Bb Bd Bd

Dd Dd Ca Ca Cc Cc Db Db

Db Db Cc Cc Ca Ca Dd Dd

Bb

Bb Bd Bd Aa Aa Ac Ac

Bb

Bb Bd Bd Aa Aa Ac Ac

Ca Ca Dd Dd Db Db Cc Cc

Cc Cc

Db Db

Dd Dd

Ca Ca

Aa

Aa Ac Ac Bb Bb Bd Bd

**Mutative Cases**

Bb Ca Dd Ac Bb Ca Dd Ac

Aa Db Cc Bd Aa Db Cc Bd

**Second Derivative**

I Express**ω(x) as**

ω(x) :=

sgn (o(x))−^{n(x)}_{o(x)}+sgn(o(x))n(x)^{2}
2o(x)^{2} +O

n(x)^{3}

, near n(x) = 0,

o(x)

2n(x)−_{8n(x)}^{o(x)}^{3}3+_{16n(x)}^{o(x)}^{5}_{5}+O
o(x)^{7}

, near o(x) = 0 and if n(x) > 0,

−1 o(x)

2n(x)−^{o(x)3}
8n(x)3+^{o(x)5}

16n(x)5+O(^{o(x)}^{7}), near o(x) = 0 and if n(x) < 0.

• The first derivative of**x**1(t) is related to ω(x1(t)).

• The first term of ω(x) estimates the the second derivative of x1(t).

• Can characterize the concavity property observed.

**Variation near N**

1 -1

n(x) > 0

n(x) < 0 n(x) < 0

o(x) > 0 o(x) < 0

Increase ω(x), decrease the slope

Slope =±1 when n(x) = 0 A typical point on the critical curveN

ω(x)

1

I In the direction**u**_{n}_{α},**ω(x(t)) must be increased if x(t) moves to**
the side where n(x)< 0.

• The slope of**u**1(x(t)) must be less than 1.

I Only four basic ways to cross N .

**Four Bases along N**

(A)

(C)

n(x) > 0 n(x) > 0 n(x) < 0

n(x) < 0

x0

x0

τn

τn

(B)

(D)

n(x) > 0

n(x) > 0

n(x) < 0

n(x) < 0 x0

x0

τn

τn

1

**Variation near O**

o(x) n(x)> 0

o(x) n(x)> 0

o(x) n(x)< 0

o(x)

n(x)< 0 Increase ω(x), decrease the slope Slope =∞ or 0 when o(x) = 0

A typical point on the critical curveO ω(x)

1 ω(x)

1

**Four Bases along O**

(a)

(c)

o(x) n(x)> 0

o(x) n(x)> 0

o(x) n(x)< 0

o(x) n(x)< 0

x0

x0

τn

τn

τo

τo

(b)

(d)

o(x) n(x)> 0

o(x) n(x)> 0

o(x) n(x)< 0

o(x) n(x)< 0

x0

x0

τn

τn

τo

τo

1

**Pairing**

I Entire dynamics can be classified into 8 categories.

• These base parings are Aa, Ac, Bb, Bd, Ca, Cc, Db, and Dd only, with no other possible combinations.

I Each base pairing results in 8 dynamics in the regular cases and 2 in the mutative cases.

• Distinctive by their characteristic traits.

• Fascinating, but no time in this talk.

I Identify each dynamics by two letters of base paring at the upper left corner.

**Trait Characterization**

I Base pairings characterize dynamical details.

I Can also characterize the general behavior by a single quantity.

• Define θ(nα,oα)= Angles measured clockwise from τnand to τo.

• Assume the generic condition that τnis not forming an angle^{π}_{4}
with the north.

• Singular point**x**0is

• A repeller, if 0 < θ(nα,oα) < π.

• A roundabout, if θ(nα,oα) > π.

I Crossovers/hybrids are possible.

I ...

I Too detailed to include here.

**Making Mosaics**

I Classify of all possible local behaviors.

• A simplistic collection of “tiles" for the delicate and complex

“mosaics".

I Inherent characteristics of the underlying function arrange these local pieces together along the strands of N and O to form the various patterns.

**A Comparison**

I Consider examples 2a and 2b.

• Easy critical curves.

• **O forms horizontal lines with alternating o(x) in between.**

• N forms closed loops.

• One additional vertical, continuous, ogee N curve in Example 2b.

• n(x) > 0 inside the loops and to the left of the ogee curve.

I Similar, but different dynamics.

**Zoom In**

x1

x 2

Curves near Singular Points for Example 2a

R S R S

−1 −0.5 0 0.5 1 1.5 2 2.5 3

0.5 1 1.5 2 2.5

x1

x 2

Curves near Singular Points for Example 2b

T

U

V

−1.50 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

0.5 1 1.5 2 2.5 3

**A Jigsaw Puzzle**

x_{1}

x 2

### α −Halves and Base Pairings for Example 1

−2 −1 0 1 2 3 4

−2

−1 0 1 2 3

4 Dd/Db

Ca/Cc

Cc/Ca

Db/Dd Aa/s

Bd

Bd

Ac

Aa

Ac

Bb

Bb

Ac Ac

Bd Bd

Aa/s

Bb

Ac Bb

1

**Conclusion**

I Gradient adaption is an important mechanism occurring in nature.

• Its generalization to the Jacobian does not “discriminate" directions per se.

I Adaption information is coded in the singular curves.

• Forms a natural moving frame telling intrinsic properties per the given function.

• Results in intricate and complicated patterns.

I Global behavior in general and interpretation in specific are not conclusively understood yet.

• Two stands joined by singular points with one of eight distinct base pairings make up the underlying function.

• Amazingly analogous to the DNA structure essential for all known forms of life.

I Are the patterns discovered “the trace of DNA" within an abstract, “inorganic" function?