Training of DHN Part 3

AI Summary

An AI-generated summary of this video lecture.

The video lecture covers the testing and recall process of a Discrete Hopfield Network, followed by an introduction to Self-Organizing Maps (SOM). The first half demonstrates the asynchronous update algorithm where a network recalls a stored pattern from a partial input vector. The instructor calculates net inputs for individual units, applies activation functions, and checks for convergence. The second half transitions to SOM, defining it as a competitive learning method for dimensionality reduction, explaining the 'winner-takes-all' mechanism, and illustrating the network structure with input and output grids.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The video begins with a calculation already in progress on the screen. The equation shows a scalar addition: $1 + [1 \ 0 \ 1 \ 0] imes ext{column vector} = 2$. The text below explains the logic: 'Since here $y_{in3} > 0$; applying the activation function we get the updated value of $y_3 = 1$. Thus no updation required.' This indicates the unit was already in the correct state or the net input confirms the current state. The instructor then transitions to the next step in the asynchronous update process, stating, 'Let us choose $y_2$ for updation and testing now.' The slide introduces the formula for calculating the net input for the second unit: $y_{in2} = x_2 + \sum_{j=1}^4 y_j W_{j2}$. This formula sums the external input $x_2$ and the weighted sum of inputs from other units $y_j$ connected via weights $W_{j2}$. The instructor is guiding the student through the step-by-step verification of the Hopfield network's recall capability.

2. 2:00 – 5:00 02:00-05:00

   The instructor proceeds to solve the equation for $y_{in2}$. The slide displays the calculation: $0 + [1 \ 0 \ 1 \ 0] imes ext{column vector} = 0 + 2 = 2$. The result is positive, leading to the conclusion: 'Since here $y_{in2} > 0$; applying the activation function we get the updated value of $y_2 = 1$.' The instructor explains that this value is then broadcasted to all other units, resulting in the updated vector $y_i = [1 \ 1 \ 1 \ 0]$. A key check is performed: 'Since $y_i = ( ext{original}) x_i$; Thus it has converged.' This confirms the network has successfully recalled the stored pattern from the partial input. The slide also includes a 'Points to note' section, stating the purpose of a Hopfield net is to store patterns and recall full patterns based on partial input. The instructor emphasizes that the network can store one or more patterns and recall them even when the input is corrupted.

3. 5:00 – 10:00 05:00-10:00

   The video rewinds to the beginning of the testing algorithm to show the setup. The slide displays the weight matrix $W$ initialized with zeros on the diagonal, reflecting the rule that 'we don't have self connections.' The matrix is a $4 imes 4$ grid with values like $1, -1$. The input vector is defined as $X = [0 \ 0 \ 1 \ 0]$, representing a corrupted or partial version of the stored pattern. The initial activation of the network is set equal to this external input vector: $y_i = x_i$, so $y_i = [0 \ 0 \ 1 \ 0]$. The instructor then selects a unit randomly for the first update, choosing $y_1$. This sets the stage for the asynchronous update process where units are updated one by one until convergence. The slide explicitly shows the weight matrix $W$ with the first row being $[0 \ 1 \ 1 \ -1]$.

4. 10:00 – 15:00 10:00-15:00

   The calculation for the net input of unit $y_1$ is performed. The formula is $y_{in1} = x_1 + \sum_{j=1}^4 y_j W_{j1}$. The slide shows the substitution: $0 + [0 \ 0 \ 1 \ 0] imes ext{column vector} = 0 + 1 = 1$. Since the net input $y_{in1}$ is greater than 0, the activation function updates the value of $y_1$ to 1. The instructor notes that this value is broadcasted, resulting in the updated vector $y_i = [1 \ 0 \ 1 \ 0]$. However, the slide explicitly states, 'Still it has not converged to the vector X.' This highlights that the network requires multiple iterations of updating different units to reach the stable state. The instructor then prepares to choose $y_3$ for the next updation step. The slide shows the vector $y_i$ changing from $[0 \ 0 \ 1 \ 0]$ to $[1 \ 0 \ 1 \ 0]$.

5. 15:00 – 18:25 15:00-18:25

   The lecture transitions to a new topic: Self-Organizing Maps (SOM). The slide defines SOM as a discretized representation of the input space, often two-dimensional, used for dimensionality reduction. It contrasts SOM with other neural networks by stating they apply 'competitive learning' rather than 'error-correction learning.' The text explains that in competitive learning, output neurons compete to be activated, resulting in a 'winner-takes-all' neuron. A diagram illustrates the SOM structure with input units connecting to a 2D grid of output units. The text mentions that SOM was introduced by Teuvo Kohonen in the 1980s and is sometimes called a 'Kohonen feature map.' It also notes that neighborhood constraints are imposed on output units, meaning the winning neuron and its neighbors are updated. The diagram shows 2 inputs ($x_1, x_2$) and 49 outputs arranged in a grid.

The lecture provides a comprehensive walkthrough of Hopfield Network testing, demonstrating how a network can recall a complete pattern from a partial input through asynchronous updates. The instructor meticulously calculates net inputs for each unit, applies threshold activation functions, and verifies convergence against the original stored vector. This practical example reinforces the theoretical concept of associative memory. The video then pivots to Self-Organizing Maps (SOM), introducing them as a dimensionality reduction tool based on competitive learning. Key distinctions are made between error-correction learning and competitive learning, where neurons compete to become the 'winner.' The lecture concludes by describing the SOM architecture, highlighting the 2D grid of output units and the concept of neighborhood constraints, which are essential for preserving the topological properties of the input space.