Unsupervised Learning in Neurodynamics 583 
Unsupervised Learning in 
the Phase Velocity 
Neurodynamics 
Field Approach 
Using 
Michail Zak 
Nikzad Toomarian 
Center for Space Microelectronics Technology 
Jet Propulsion Laboratory 
California Institute of Technology 
Pasadena, CA 91109 
ABSTRACT 
A new concept for unsupervised learning based upon examples in- 
troduced to the neural network is proposed. Each example is con- 
sidered as an interpolation node of the velocity field in the phase 
space. The velocities at these nodes are selected such that all the 
streamlines converge to an attracting set imbedded in the subspace 
occupied by the cluster of examples. The synaptic interconnections 
are found from learning procedure providing selected field. The 
theory is illustrated by examples. 
This paper is devoted to development of a new concept for unsupervised learning 
based upon examples introduced to an artificial neural network. The neural network 
is considered as an adaptive nonlinear dissipative dynamical system described by 
the following coupled differential equations: 
N 
iti + gui = E}1g(u1) + Ii i = 1,2,... ,N (1) 
j=l 
in which u is an N-dimensional vector, function of time, representing the neuron 
activity, T is a constant matrix whose elements represent synaptic interconnections 
between the neurons, g is a monotonic nonlinear function, Ii is the constant exterior 
input to each neuron, and  is a positive constant. 
584 Zak and Toomarian 
Let us consider a pattern vector 5 represented by its end point in an n-dimensional 
phase space, and suppose that this pattern is introduced to the neural net in the 
form of a set of vectors - examples u(k),k -- 1,2...K (Fig. 1). The difference 
between these examples which represent the same pattern can be caused not only 
by noisy measurements, but also by the invariance of the pattern to some changes 
in the vector coordinates (for instance, to translations, rotations etc.). If the set 
of the points u  is sufficiently dense, it can be considered as a finite-dimensional 
approximation of some subspace O(t). 
Now the goal of this study is formulated as following: find the synaptic intercon- 
nections T/i and the input to the network Ii such that any trajectory which is 
originated inside of 0  will be entrapped there. In such a performance the sub- 
space 0 (t) practically plays the role of the basin of attraction to the original pattern 
. However, the position of the attractor itself is not known in advance: the neural 
net has to create it based upon the introduced representative examples. Moreover, 
in general the attractor is not necessarily static: it can be periodic, or even chaotic. 
The achievement of the goal formulated above would allow one to incorporate into 
a neural net a set of attractors representing the corresponding clusters of patterns, 
where each cluster is imbedded into the basin of its attractor. Any new pattern 
introduced to such a neural net will be attracted to the "closest" attractor. Hence, 
the neural net would learn by examples to perform content-addressable memory 
and pattern recognition. 
Fig. 1: Two-Dimensional Vectors as Examples, u }, and Formation of Clusters 0. 
Unsupervised Learning in Neurodynamics 585 
Our approach is based upon the utilization of the original clusters of the example 
points u (k) as interpolation nodes of the velocity field in the phase space. The 
assignment of a certain velocity to an example point imposes a corresponding con- 
straint upon the synaptic interconnections 7]j and the input Ii via Eq. (1). After 
these unknowns are found, the velocity field in the phase space is determined by Eq. 
(1). Hence, the main problem is to assign velocities at the point examples such that 
the required dynamical behavior of the trajectories formulated above is provided. 
One possibility for the velocity selection based upon the geometrical center approach 
was analyzed by M. Zak, (1989). In this paper a "gravitational attraction" approach 
to the same problem will be introduced and discussed. 
Suppose that each example-point u  is attracted to all the other points u(')(k   
k) such that its velocity is found by the same rule as a gravitational force: 
v?)= Vo 
t= [ Z-j=I \ j 
in which vo is a constant scale coefficient. 
Actual velocities at the same points are defined by Eq. (1) rearranged as: 
v i- 1,2,...,N 
?)=ET/.g(u')-Uoi)-(u? )-uoi) k= l,2,...,K (3) 
j=l 
The objective is to find synaptic interconnections T/i and center of gravity Uoi such 
that they minimize the distance between the assigned velocity (Eq. 2) and actual 
calculated velocities (Eq. 3). 
Introducing the energy: 
K N 
= 
k=l i=1 
one can find 7] 1 and Uoi from the condition: 
i.e., as the static attractor of the dynamical system: 
itoi '-- --o? OE 
Ouoi (Sa) 
oqj 
in which a is a time scale parameter for learning. By appropriate selection of this 
parameter the convergence of the dynamical system can be considerably improved 
(J. Barhen, S. Gulati, and M. Zak, 1989). 
586 Zak and Toomarian 
Obviously, the static attractor of Eqs. (5) is unique. As follows from Eq. (3) 
0?_2 ): 
Ou k) du ) ' 
(i # j) (6) 
dg (? ) 
Since g(u) is a monotonic function, sgn d-. is constant which in turn implies that 
sgn Ou  ) - const 
(i  j) (7) 
Applying this result to the boundary of the cluster one concludes that the velocity 
at the boundary is directed inside of the cluster (Fig. 2). 
For numerical illustration of the new learning concept developed above, we select 
6 points in the two dimensional space, (i.e., two neurons) which constructs two 
separated clusters (Fig. 3, points 1-3 and 16-18 (three points are the minimum 
to form a cluster in two dimensional space)). Coordinates of the points in Fig. 
3 are given in Table 1. The assigned velocity vf calculated based on Eq. 2 and 
Vo = 0.04 are shown in dotted line. For a random initialization of T/j and Uoi, 
the energy decreases sharply from an initial value of 10.608 to less than 0.04 in 
about 400 iterations and at about 2000 iterations the final value of 0.0328 has been 
achieved, (Fig. 4). To carry out numerical integration of the differential equations, 
first order Euler numerical scheme with time step of 0.01 has been used. In this 
simulation the scale parameter a 2 was kept constant and set to one. By substituting 
the calculated Tj and Uoi into Eq. (3) for point u }, (k = 1,2,3, 16, 17, 18), one 
will obtain the calculated velocities at these points (shown as dashed lines in Fig. 
3). As one may notice, the assigned and calculated velocities are not exactly the 
same. However, this small difference between the velocities are of no importance as 
long as the calculated velocities are directed toward the interior of the cluster. This 
directional difference of the velocities is one of the reasons that the energy did not 
vanish. The other reason is the difference in the value of these velocities, which is 
of no importance either, based on the concept developed. 
Fig. 2: Velocities at Boundaries are directed Toward Inside of the Cluster. 
Unsupervised Learning in Neurodynamics 587 
In order to show that for different initial conditions, Eq. 3 will converge to an 
attractor which is inside one of the two clusters, this equation was started from dif- 
ferent points (4-15, 19-29). In all points, the equation converges to either (0.709,0.0) 
or (-0.709,0.0). However, the line x = 0 in this case is the dividing line, and all the 
points on this line will converge to Uo. 
The decay coefficient c and the gain of the hyperbolic tangent were chosen to be 
1. However, during the course of this simulation it was observed that the system 
is very sensitive to these parameters as well as Vo, which calls for further study in 
this area. 
15 
29 14 
21 23 8 
19 
2O 
25 
18 26 
27 12 
? 
28 
13 
1( 2 4 
11 3 5 
22 
24 
Fig. 3:  Cluster i (1-3) and Cluster 2 (16-19). 
 Assigned Velocity (..) Calculated Velocity (--) 
 Activation Dynamics initiated at different points. 
588 Zak and Toomarian 
Table 1. - Coordinate of Points in Figure 
point X Y point X Y 
I 0.50 0.00 16 -0.50 0.00 
2 1.00 0.25 17 -1.00 0.25 
3 1.00 -0.25 18 -1.00 0.25 
4 1.25 0.25 19 -1.25 0.25 
5 1.25 -0.25 20 -1.25 -0.25 
6 1.00 0.50 21 -1.00 0.50 
7 1.00 -0.50 22 -1.00 -0.50 
8 0.75 0.50 23 -0.75 0.50 
9 0.75 -0.50 24 -0.75 -0.50 
10 0.50 0.25 25 -0.50 -0.25 
11 0.50 -0.25 26 -0.50 -0.25 
12 0.25 0.10 27 -0.25 0.10 
13 0.25 -0.10 28 -0.25 -0.10 
14 0.02 1.00 29 -0.02 1.00 
15 0.00 1.00 
0 100 200 300 
ITERATIONS 
Fig 4: Profile of Neuromorphic Energy over Time Iterations 
Acknowledgement 
This research was carried out at the Center for Space Microelectronic Technology, 
Jet Propulsion Laboratory, California Institute of Technology. Support for the work 
came from Agencies of the U.S. Department of Defense, including the Innovative 
Science and Technology Office of the Strategic Defense Initiative Organization and 
the Office of the Basic Energy Sciences of the US Dept. of Energy, through an 
agreement with the National Aeronautics and Space Administration. 
Unsupervised Learning in Neurodynamics 589 
References 
M. Zak (1989), "Unsupervised Learning in Neurondynamics Using Example Inter- 
action Approach", Appl. Math. Letters, Vol. 2, No. 3, pp. 381- 286. 
J. Barhen, S. Gulati, M. Zak (1989), "Neural Learning of Constrained nonlinear 
Transformations", IEEE Computer, Vol. 22(6), pp. 67-76. 
