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**An Illustrative Example**

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Apple/Banana Sorter

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Prototype Vectors

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**McCulloch-Pitts Perceptron**

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Perceptron Training How can we train a perceptron for a classification task? We try to find suitable values for the weights in such a way that the training examples are correctly classified. Geometrically, we try to find a hyper-plane that separates the examples of the two classes.

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**Perceptron Geometric View**

The equation below describes a (hyper-)plane in the input space consisting of real valued m-dimensional vectors. The plane splits the input space into two regions, each of them describing one class. decision region for C1 x2 w1p1 + w2p2 + b >= 0 decision boundary C1 x1 C2 w1p1 + w2p2 + b = 0

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Two-Input Case

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Apple/Banana Example

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Testing the Network

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**XOR problem A typical example of non-linealy separable function is**

the XOR. This function takes two input arguments with values in {-1,1} and returns one output in {-1,1}, as specified in the following table: If we think at -1 and 1 as encoding of the truth values false and true, respectively, then XOR computes the logical exclusive or, which yields true if and only if the two inputs have different truth values.

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XOR problem In this graph of the XOR, input pairs giving output equal to 1 and -1 are depicted with green and red circles, respectively. These two classes (green and red) cannot be separated using a line. We have to use two lines, like those depicted in blue. The following NN with two hidden nodes realizes this non-linear separation, where each hidden node describes one of the two blue lines. 1 -1 x2 x1

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Multilayer Network

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Abbreviated Notation

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Recurrent Network

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Hamming Network

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Feedforward Layer

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Recurrent Layer

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Hamming Operation

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Hamming Operation

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Hopfield Network

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Apple/Banana Problem

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**Summary Perceptron Hamming Network Hopfield Network**

Feedforward Network Linear Decision Boundary One Neuron for Each Decision Hamming Network Competitive Network First Layer – Pattern Matching (Inner Product) Second Layer – Competition (Winner-Take-All) # Neurons = # Prototype Patterns Hopfield Network Dynamic Associative Memory Network Network Output Converges to a Prototype Pattern # Neurons = # Elements in each Prototype Pattern

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**Learning Rules Network is provided with a set of examples**

• Supervised Learning Network is provided with a set of examples of proper network behavior (inputs/targets) • Reinforcement Learning Network is only provided with a grade, or score, which indicates network performance • Unsupervised Learning Only network inputs are available to the learning algorithm. Network learns to categorize (cluster) the inputs.

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Early Learning Rules These learning rules are designed for single layer neural networks They are generally more limited in their applicability. Some of the early algorithms are: Perceptron learning LMS learning Grossberg learning

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**Perceptron Architecture AGAIN!!!!**

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**Single-Neuron Perceptron**

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Decision Boundary

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Example - OR

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OR Solution

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**Multiple-Neuron Perceptron**

Each neuron will have its own decision boundary. A single neuron can classify input vectors into two categories. A multi-neuron perceptron can classify input vectors into 2S categories.

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**Learning Rule Test Problem**

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Starting Point

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**Tentative Learning Rule**

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Second Input Vector

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Third Input Vector

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Unified Learning Rule

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**Multiple-Neuron Perceptrons**

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Apple/Banana Example

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Second Iteration

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Check

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**Perceptron Rule Capability**

The perceptron rule will always converge to weights which accomplish the desired classification, assuming that such weights exist.

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**Rosenblatt’s single layer perceptron is trained as follow:**

Randomly initialize all the networks weights. Apply inputs and find outputs ( feedforward). compute the errors. Update each weight as Repeat steps 2 to 4 until the errors reach the satisfictory level.

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**What is this: Name : Learning rate.**

Where is living: usually between 0 and 1. It can change it’s value during learning. Can define separately for each parameters.

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**Perceptron Limitations**

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**You Can Find Your First Homework here**

NEXT WEEK

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