Foundation of re-normalized synergetics - issues of computability and complexity

Foundation of re-normalized
synergetics:
issues of computability and
complexity
Milan Jovovic

 Modeling approach based on free energy and
distortion energy
 Near linear model - aims for the simplest
explanations
 Estimation of dynamical parameters of
clustering by statistical inference
 Multi-spectral decomposition, in hierarchy of
scales
 Application: scale analysis of complex systems
Analysis of signal distortion
by multi-scale decomposition

Introduction (1 of 2)
 Cluster parameters:
• Selected spatial window: Wr
• Computed cluster vector within Wr
 Statistical inference defines PDF, with the associated
distortion energies, F and V
 Energy functions are generally multi-dimensional and
non-convex
 Non-linear map defines dynamical scale-space
clustering
 Clustering is important optimization problem
c

Introduction (2 of 2)

Model of signal distortion:
- definitions
 Distortion measure:
1. d = z2 = (Cx-X)
2 + (Cy-Y)
2 eg. in TSP
2. d = z2 = eg. in 3D video communication
2][ vIIt 
 Partition functions:
,
2



rW
z
rZ

 Distortion energies -
free energy, and variance:
 
rW
PvxdV

,
 PDF:
,
2
Z
r
P
z

  ,log
1
, r ZvF

 


Scale-space computing
 Series of convex min/max of free energy F
 brings in eq. up-scale melting & down-scale cooling:
    ,
1
0




 dVF
 
 




 0
dV
rZ
 
)1(
.
,2








 
F
PIc
c
F
c
r
W




 Evolution scheme – path integrals:
 Way to move through the scale-space ?

Motion through the scale-space:
- wave equation
 The same potential level difference the equilibrium point moves by (2)
and (3)
12
 
 ,vF

Vv grad

(1)
(2)
)2(
v
Vv 



)3(



 F
  





S S
d
v
V
vd
F
dU 0
 

V
F 2
2
2





Cluster Bindings
 Motion binding:
0  
1
1
2
2
2
2
2
1
1
1
v
F
v
F
v
v
F
v
F
v




















 Determinant of the map:
  .1
2
2
2
2
2
1
1
2
2
2
2
2
2
2
1
1
2
2
v
F
v
F
v
F
v
F

















 λλD
 Criteria of splitting a cluster at the “wave collapse”:
 Spatial coherency of information:
 Information content wrt the uncertanty relation:
 Coupled domains of computation:
     
V
V
vGwhereWv
v
vdvGvO
r
r
S 2
2
,0,
2
1








  



  V
VvG 2
2
, 
 

  1,2  vCov 

Scalable coding
 Coupled data structure of the hierarchy of
binary images
 Efficient coding, control, data transfer
 Parallelization: computing and control by
parallel computing architectures

(v
4
, W
4
) (v
3
, W
3
)
(v
2
, W
2
) (v
1
, W
1
) (v
0
, W
0
)

c
3
(v
3
, W
3
)

c
2
(v
2
, W
2
)

c
0
(v
0
, W
0
)

c
1
(v
1
, W
1
)

Focus on computability and complexity –
relationship to statistical physics
o Computing paradigm assumes:
o Motion via scale-space wave information propagation, and
o Uncertainty relation wrt the information content of a cluster
o What makes it, therefore, polynomial in complexity (ref. 2)?
o Unique statistical description, although chaotic motion possible
o No strange attractors due to the conservative motion
 Within this description: multi-scale decomposition of the information
content into clusters
 Coupling of the energy exchange – synergetics
 Coupled manifolds spanning the content of the information clusters

Counting dimensions
 Bringing in resonance system of 2 clusters
(ref. 2)
System of 3 clusters is much more complex !
 3D spectral components
 3D cluster covariance
 3D coupled cluster covariance
 results in 3 coupled clusters 3D manifolds
 dynamical scale parameter β
 = 10
 operators div and rot
o System of 3 clusters at the “wave resonance”:
  








12
21
21
FF
FF
F


0  

Summary presentation of current work
 Images: multi-spectral decomposition and clusters
coupling, spectral signature recognition
 Movements: trajectory analysis, learning, coding and
control by scale-space computing
 Bio/chemical informatics: data-mining and knowledge
discovery
 Scalable data decomposition: coding, control, and
transmission
 Synchronous computing scheme: upscale melting &
downscale cooling
 Parallel computing implementation

 Scale singularity of data sets is used in detecting rain
patterns
Still images decomposition

Sequence of 2 images: 2 clusters
decomposition
 2D ball expansion expansion and diagonal

Sequences – different intervals: 2 clusters decompos.
 Vortex sequence 2. images
 sequence 4. images sequence 7. images