Updated space formatting

Author: mrudhvika940

experiment/theory.md

@@ -35,16 +35,19 @@ There can be direct evidence for a hypothesis through an external input. The cor
 
 A constraint satisfaction model can be displayed by means of a Hinton diagram as shown in the following figure. Each larger square in the figure represents a unit. There are 40 units corresponding to 40 descriptors. Within the square of each unit a replica of all the 40 units are displayed as dots, each dot representing the unit in its relative position in the diagram. Around each dot, a white square indicates a positive weight connecting the unit representing the dot and the unit enclosing the dot. Thus for example, the white square on the second dot in the unit for 'ceiling' indicates that the 'walls' unit is connected to the ceiling unit with a positive weight. The size of the white square indicates the strength of the positive connection. Likewise in the last unit corresponding to 'oven', the small dark square around the last but one dot indicates that the units 'oven' and 'computer' are connected with a negative weight. There are many units which have no connections at all.
 
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 <img src="images/figA2.jpg"  width="500px;" height = "500px">
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 **Figure 2**: *Hinton diagram for the 'rooms' example.*
 
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 The model is allowed to relax by computing the next state for each unit selected at random, computing sum of its weighted inputs and thresholding the weighted sum using a hard-limiting output function. For a given external evidence, say like 'oven' and 'ceiling' in the following figure, the state of the network after each cycle is shown in the figure. After 17 cycles the model settles down to an equilibrium state closest to the given external evidence, and the state description gives a description of the concept of the room satisfying the external evidence, namely 'kitchen', in this case. Thus the PDP model clearly demonstrates the concepts of rooms captured by the weak constraints derived from the data given by the subjects. The model captures the concepts of the five room types at the equilibrium states corresponding to the description that best fits each room type. A goodness-of-fit function (g) is defined for each state (\(x_1,x_2, ...,x_N)\) , where \(x_i\) = 1 or 0, as $$ g = \sum\limits_{i,j=1}^{N}w_{ij} x_i x_j + \sum\limits_{i=1}^{N} e_i x_i + \sum\limits_{i=1}^{N} b_i x_i \qquad(3)$$
 
 where \(e_i\) is the external input given as output of the \(i^{th}\) unit and \(b_i\) is the bias of the unit \(i\). At each of the equilibrium states the goodness-of-fit function is maximum. The model not only captures the concepts of the room types, but it also gives an idea of their relative separation in the 40 dimensional space.
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 #### FORMULAE USED
 
@@ -56,12 +59,11 @@ activation of each descriptor is calculated for 16 cycles one after another and
 
 
 For further information, refer the references given in the references section for this experiment.
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 <img src="images/clamping.png">
 
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 **Figure 3**: *The state of the CS model after each cycle, starting with an initial state where the units 'ceiling' and 'oven' are clamped.*