data_mining:neural_network:perceptron [AE Wiki]

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====== Perceptron ======
  * Popularized by Frank Rosenblatt (1960s)
  * Used for tasks with very big vectors of features

Decision Unit: Binary trheshold neuron.

Bias can be learned like weights, it's weight with value 1.

Perceptron convergence
  * If output correct => no weight changes
  * If output unit incorrectly outputs 0 => add input vector to weight vector.
  * If output unit incorrectly outputs 1 => substract input vector from the weight vector.

This generates set of weights that gets the right answer for all training cases, if such a set exists.
=> Deciding the features is the important distinction

====== Geometrical Interpretation ======

===== Weight-Space view =====

  * 1 dimension for each weight
  * Point represents a setting of all weights
  * Leaving the threshold out, each **training case** can be represented as a **hyperplane** through the **origin**. Inputs represent planes (or Constraints)
     * For a particular training case: Weights must lie on one side of this hyper-plane to get the answer correct.



Plane goes through **origin**, is perpendicular to the input vector (with correct answer = 1 (or 0)). Good weight vector needs to be on the same side of the hyperplane.
Scalar product of wight vector and input vector positiv (angle < 90°).

**Cone of feasable solutions**

Need to find a point on right side of all the planes (training cases): Might not exist.
If there are weight vectors that get the right side for all cases, they lie in a hyper-cone, with apex in origin.
Average of two good wight vectors is a good weight vector => Convex problem.


==== Why learning works ====

Proof that squared distance between feasable and current weight vector get's smaller. Hopeful claim: Every time perceptron makes mistakes, the current weight vector gets closer to all feasble wight vectors. Not true.

"generously feasable" weight vector, that lie within a feasible region by a **margin** at least as great as the **length of the input vector** that defines **each constraint plane**.

Every time perceptron makes mistake, squared distance to all of the **generously feasible** weight vectors is always decreased by at least the **squared length** of the **update vector**.

After finite number of mistakes, the weight vector must lie in the **feasible** region, if the region exists.

==== What perceptrons can't do ====

Separate feature unit for each of the many binary vectors => any possible discrimination. Exponential many features are necessary. 

This type of table look-up won't generalize. (guess due to the binary encoding).

Binary threshold output unit can't tell if two single bit features are the same.
=> contradicting constraints.

=== Data-Space ===
- Each input vector is point in space
- Weight vector defines plane
- Weight plane is perpendicular to the weight vector and misses the origin by a distance equal to the threshold.

Pos and neg cases can not be seperated by a plane => not linearly separable.

=== Discriminate patterns under translation with wrap-around ===

Binary threshold neuron can't discriminate between different patterns that have same number of on pixels (if pattern can translate with wrap-around).

Proof:

Pattern A on all possible translations (4 on pixels). 
Total Input will be 4x sum of all the weights.

Pattern B on all possible translations (4 on pixels)
Total Input will be 4x sum of all the weights.

But every single case of pattern A must provide more import than all single cases of pattern B.

Case:

3 Patterns with 2 Classes.
First class contains patterns with 4 pixels on.
Second class contains pattern with either 1 or 3 pixels on.

Weight from each pixel = 1.
Bias = -3.5

Any example with 3 pixels on: activation = -0.5

Any example with 1 pixel on: activation = -2.5

Any example with 4 pixel on: acitvation = 0.5

=> correctly classified.

Whole point of pattern recognition is to recognize patterns despite transformations like translation. Minksy and Paper ("Group Invariance Theorem"): Part of perceptron that **learns** cannot learn to do this if the transformation form a **group**:
- Translation with **wrap-around** form a group.
To deal with such transf. tricky part of recognition must be solved by hand-coded **feature detectors** (not learning procedure).

Conclusion (after 20 yrs): Neuronal Network has to learn the feature detectors (not only weights).
  * More layers of linear units do not help (still linear).
  * Fixed output non-linearities are not enough.

=> Need multiple layers of **adaptive**, **non-linear** hidden units.

  * Efficient way of adapting all the weights.
  * Learning weights going into hidden units is equivalent to learning features.
  * No one is telling us what the feature vectors should be.