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Background Neural Networks Computing device composed of processing elements called neurons Processing power comes from interconnection between neurons Various models are Hopfield, Back propagation, Perceptron, Kohonen Net etc

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Neural Network to solve Traveling Salesman Problem

Amit Goyal 01005009

Koustubh Vachhani 01005021

Ankur Jain 01D05007

Roadmap

Hopfield Neural Network

Solving TSP using Hopfield Network

Modification of Hopfield Neural Network

Solving TSP using Concurrent Neural Network

Comparison between Neural Network and SOM for solving TSP

Background

Neural NetworksComputing device composed of processing elements called neuronsProcessing power comes from interconnection between neuronsVarious models are Hopfield, Back propagation, Perceptron, Kohonen Net etc

Associative memory

Associative memoryProduces for any input pattern a similar stored patternRetrieval by part of dataNoisy input can be also recognized

Original

Degraded

Reconstruction

Hopfield Network

Recurrent network Feedback from output to input

Fully connectedEvery neuron connected to every other neuron

Hopfield Network

Symmetric connectionsConnection weights from unit i to unit j and from unit j to unit i are identical for all i and jNo self connection, so weight matrix is 0-diagonal and symmetricLogic levels are +1 and -1

Computation

For any neuron i, at an instant t input is

j = 1 to n, ji wij j(t)

j(t) is the activation of the jth neuron

Threshold function = 0 Activation i(t+1)=sgn(j=1 to n, ji wijj(t))

where

Sgn(x) = +1 x>0

Sgn(x) = -1 x

Modes of operation

SynchronousAll neurons are updated simultaneously

AsynchronousSimple : Only one unit is randomly selected at each step

General : Neurons update themselves independently and randomly based on probability distribution over time.

Stability

Issue of stability arises since there is a feedback in Hopfield networkMay lead to fixed point, limit cycle or chaosFixed point : unique point attractorLimit cycles : state space repeats itself in periodic cyclesChaotic : aperiodic strange attractor

Procedure

Store and stabilize the vector which has to be part of memory.

Find the value of weight wij, for all i, j such that :

is stable in Hopfield Network of N neurons.

Weight learning

Weight learning is given bywij = 1/(N-1) i j 1/(N-1) is Normalizing factori j derives from Hebbs ruleIf two connected neurons are ON then weight of the connection is such that mutual excitation is sustained.Similarly, if two neurons inhibit each other then the connection should sustain the mutual inhibition.

Multiple Vectors

If multiple vectors need to be stored in memory like

.

Then the weight are given by:

wij = 1/(N-1) m=1 to pim jm

Energy

Energy is associated with the state of the system.

Some patterns need to be made stable this corresponds to minimum energy state of the system.

Energy function

Energy at state = E() = - i ji wij ijLet the pth neuron change its state from pinitial to pfinal so Einitial = - jp wpj pinitial j + T Efinal = - jp wpj pfinal j + TE = Efinal Einitial

T is independent of p

Continued

E = - (pfinal - pinitial ) jp wpj j

i.e. E = - p jp wpj j

Thus: E = - p x (netinputp)

If p changes from +1 to -1 then p is negative and netinputp is negative and vice versa.

So, E is always negative. Thus energy always decreases when neuron changes state.

Applications of Hopfield Nets

Hopfield nets are applied for Optimization problems.Optimization problems maximize or minimize a function.In Hopfield Network the energy gets minimized.

Traveling Salesman Problem

Given a set of cities and the distances between them, determine the shortest closed path passing through all the cities exactly once.

Traveling Salesman Problem

One of the classic and highly researched problem in the field of computer science.

Decision problem Is there a tour with length less than k" is NP - Complete

Optimization problem What is the shortest tour? is NP - Hard

Hopfield Net for TSP

N cities are represented by an N X N matrix of neuronsEach row has exactly one 1 Each column has exactly one 1Matrix has exactly N 1s

kj = 1 if city k is in position j

kj = 0 otherwise

Hopfield Net for TSP

For each element of the matrix take a neuron and fully connect the assembly with symmetric weights

Finding a suitable energy function E

Determination of Energy Function

E function for TSP has four components satisfying four constraints

Each city can have no more than one

position i.e. each row can have no more

than one activated neuron

E1= A/2 k i ji ki kj A - Constant

Energy Function (Contd..)

Each position contains no more than one city i.e. each column contains no more than one activated neuron

E2= B/2 j k rk kj rj B - constant

Energy Function (Contd..)

There are exactly N entries in the output matrix i.e. there are N 1s in the output matrix

E3= C/2 (n - ki ki)2 C - constant

Energy Function (cont..)

Fourth term incorporates the requirement of the shortest path

E4= D/2 krkj dkr kj(r(j+1) + r(j-1))

where dkr is the distance between city-k and city-r

Etotal = E1 + E2 + E3 + E4

Energy Function (cont..)

Energy equation is also given by

E= -kirj w(ki)(rj) ki rj

ki City k at position i

rj City r at position j

Output function ki

ki = ( 1 + tanh(uki/u0))

u0 is a constant

uki is the net input

Weight Value

Comparing above equations with the energy equation obtained previously

W(ki)(rj) = -A kr(1 rj) - Bij(1 kr) C Ddkr(j(i+1) + j(i-1))

Kronecker Symbol : kr

kr = 1 when k = r

kr = 0 when k r

Observation

Choice of constants A,B,C and D that provide a good solution vary betweenAlways obtain legitimate loops (D is small relative to A, B and C)

Giving heavier weights to the distances (D is large relative to A, B and C)

Observation (cont..)

Local minimaEnergy function full of dips, valleys and local minimaSpeedFast due to rapid computational capacity of network

Concurrent Neural Network

Proposed by N. Toomarian in 1988

It requires N(log(N)) neurons to compute TSP of N cities.

It also has a much higher probability to reach a valid tour.

Objective Function

Aim is to minimize the distance between city k at position i and city r at position i+1

Ei = krri kir(i+1) dkr

Where is the Kronecers Symbol

Cont

Ei = 1/N2 krri dkr i= 1 to ln(N) [1 + (2i 1) ki] [1 + (2i 1) ri]

Where (2i 1) = (2i 1) [1 j= 1 to i-1 i ]

Also to ensure that 2 cities dont occupy same position

Eerror = krr kr

Solution

Eerror will have a value 0 for any valid tour.So we have a constrained optimization problem to solve.

E = Ei + Eerror

is the Lagrange multiplier to be calculated form the solution.

Minimization of energy function

Minimizing Energy function which is in terms of kiAlgorithm is an iterative procedure which is usually used for minimization of quadratic functionsThe iteration steps are carried out in the direction of steepest decent with respect to the energy function E

Minimization of energy function

Differentiating the energy

dUki/dt = - E/ ki = - Ei/ ki - Eerror/ ki

d/dt = E/ = Eerror

ki = tanh(Uki) , const.

Implementation

Initial Input Matrix and the value of is randomly selected and specified

At each iteration, new value of ki and is calculated in the direction of steepest descent of energy function

Iterations will stop either when convergence is achieved or when the number of iterations exceeds a user specified number

Comparison Hopfield vs Concurrent NN

Converges faster than Hopfield Network

Probability to achieve valid tour is higher than Hopfield Network

Hopfield doesnt have systematic way to determine the constant terms.

Comparison SOM and Concurrent NN

Data set consists of 52 cities in Germany and its subset of 15 cities.Both algorithms were run for 80 times on 15 city data set.52 city dataset could be analyzed only using SOM while Concurrent Neural Net failed to analyze this dataset.

Result

Concurrent neural network always converged and never missed any city, where as SOM is capable of missing cities.Concurrent Neural Network is very erratic in behavior , whereas SOM has higher reliability to detect every link in smallest path.Overall Concurrent Neural Network performed poorly as compared to SOM.

Shortest path generated

Concurrent Neural Network (2127 km)

Self Organizing Maps (1311km)

Behavior in terms of probability

Concurrent Neural Network

Self Organizing Maps

Conclusion

Hopfield Network can also be used for optimization problems.Concurrent Neural Network performs better than Hopfield network and uses less neurons.Concurrent and Hopfield Neural Network are less efficient than SOM for solving TSP.

References

N. K. Bose and P. Liang, Neural Network Fundamentals with Graphs, Algorithms and Applications, Tata McGraw Hill Publication, 1996

P. D. Wasserman, Neural computing: theory and practice, Van Nostrand Reinhold Co., 1989

N. Toomarian, A Concurrent Neural Network algorithm for the Traveling Salesman Problem, ACM