Updated theory

Author: mrudhvika940

experiment/theory.md

@@ -36,12 +36,14 @@ 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.
 
 
+
+
 <img src="images/figA2.jpg"  width="500px;" height = "500px">
-\
-\
+
+
 **Figure 2**: *Hinton diagram for the 'rooms' example.*
-\
-\
+
+
 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.
@@ -59,47 +61,65 @@ 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.
 
 
+
 <img src="images/clamping.png">
-\
-\
+
+
+
 **Figure 3**: *The state of the CS model after each cycle, starting with an initial state where the units 'ceiling' and 'oven' are clamped.*
-\
-\
+
+
+
+
 ### Illustration of Constraint Satisfaction Model
 
 In this experiment, we use the PDP model called the Constraint Satisfaction Model, to illustrate how we as humans, attempt building of concepts as well as to arrive at conclusions given weak evidences. We start with a set of 40 descriptors used to characterize different room types. These descriptors can also be referred as hypotheses . Also present with the model is the relationship between these descriptors, acting as constraints, which can either be weak or strong. So eventually given a set of such hypotheses and weak or strong constraints between these, we arrive at some knowledge towards a particular room type in this example. The experiment first shows all the 40 descriptors involved in capturing knowledge for a given room type.
-\
-\
+
+
+
 <img src="images/CSNN_1.png">
-\
-\
+
+
+
 **Figure 1**: *Experiment window showing descriptors and other buttons.*
-\
-\
+
+
+
+
 The constraint satisfaction model which is illustrated in this experiment is already trained. We can view the model using Hinton Diagram by clicking on the "CLICK" button on top. A new panel showing the Hinton Diagram for all the descriptors appears where 40 hypotheses have their corresponding squares with constraints between them and other hypotheses being displayed within the larger square. Moving the mouse over the squares presents their zoomed picture at the bottom of the page.
-\
-\
+
+
+
+
 <img src="images/CSNN_2.png">
-\
-\
+
+
+
+
 **Figure 2**: *Experiment panel showing the Hinton diagram for the descriptors.*
-\
-\
+
+
+
 In the above diagram we notice that each large square 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 smaller squares or dots, each dot representing the unit in its relative position in the diagram. Around each dot, weak constraints are depicted by smaller squares in lighter gray shade. Similarly strong constraints is represented by darker gray or black color with a square of larger size. There are many units which have no connections at all.
 
 The constraint satisfaction model is used to illustrate the ability of human beings to build concepts or arrive at conclusion based on partial and sometime erroneous knowledge. The key idea of the model is that a large number of weak constraints when put together, evolve into a definitive conclusion. To test the model, some units can be clamped or say provided a constant external bias. As shown in the figure below, we can click on the units and have these clamped.
-\
-\
+
+
+
 <img src="images/CSNN_3.png">
-\
-\
+
+
+
 **Figure 3**: *Testing of the model by clamping some units.*
-\
-\
+
+
+
 After clamping the model is allowed to relax by computing the next state for each unit selected at random. The state computation is done by computing sum of the weighted inputs and thresholding the weighted sum using hard-limiting output function. After we click the test network button, we find that after running through some iterations the network settles to a state where no further change in the state of descriptors is visible. And then the state description gives the description of the concept of the room satisfying the external evidence or clamping
-\
-\
+
+
+
 <img src="images/CSNN_4.png">
-\
-\
-Figure 4: Figure showing the states of descriptors after reaching equilibrium.
+
+
+
+**Figure 4**: *Figure showing the states of descriptors after reaching equilibrium.*