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4.4.3. Fuzzy Hebbian learning and associative memory as a realization of rule-based systems

Let us remind that in any associative memory we are concerned with the mechanisms of an efficient storage and recall. The data (items) to be handled are organized as the finite family of input-output pairs (associations),

Here we confine ourselves to the pairs xi, yi lying in the unit hypercubes, [0,1]n and [0,1]m, respectively. Those are nothing but fuzzy sets in the condition and conclusion parts of the individual rules. The basic formulation of the associative memory problem is as follows: store the associations and propose a recall mechanism capable of retrieving the data without any corruption (error).

Let be an association architecture. The recall scheme is called error-free when acting upon satisfies the recall formula,

for any x = x1, x2, ... , xn thus y becomes y1, y2,...,yn, respectively. Note that this nothing but the testing condition promoted during the design of the rule-based system at its linguistic (conceptual) level.

In what follows we outline a generic form of the transformation scheme. The functional concept of the rule-based system splits into two phases:

1  - summarization (storing): at this stage the data (associations) are aggregated (summarized) into the format of a single relation (rather than function),
2  - recall: here for any new data (input variables) the associated output (control) becomes recalled.

The above general scheme can be realized in various ways (by coming up with the different realizations of the summarization and association steps). The standard way is the one exploiting fuzzy Hebbian learning and recall (this becomes an essence of the processing module).

Before getting to that, let us summarize some of the basic facts about associative memories, cf. (Kosko, 1992). In this setting the association structure is very often achieved as a correlational convolution of the data items to be stored producing a correlation matrix R,

This form of the aggregation is known as Hebbian learning.1 The recall scheme is also of a linear nature producing the recall vector,


1The essence of the Hebbian learning is that the strength of each synapse is changed proportionally to the product of the activations of the pre - and post-synaptic neurons.

for a given item x. The fundamental issue of error-free recall can be addressed by analyzing the character of the associations to be stored in the memory. It is easy to verify that if the inputs of the associations are mutually orthonormal, viz.

then the recall is error-free. To validate this claim, simply put ; the detailed computations reveal that

as the first term of the expression vanishes because of the orthonormality condition. Note also that if the orthonormality requirement does not hold, then yyk0; the first term forms a crosstalk component

that is responsible for any distortions in the recall process. Two observations arise:

1  - the correlational associative memory is a conceptually simple and quite attractive construct noting that there is no extensive learning involved; essentially all the associations are put together in the form of a single correlational matrix built within a single pass through the data set,
2  - the conceptual and computational simplicity is offset by a limited capacity (again one can easily figure out that only N different associations can be stored without error). There is also no special mechanism to treat some of the associations more selectively even knowing that they might be more reliable than the others.

We have concentrated on this scheme of the associative memory as the fuzzy controller forms a straightforward analog of this construct; the only technical difference lies in the different operations being used in the associative memory and fuzzy controller. Similarly, the strengths and weaknesses of the discussed structure can be also found in the fuzzy controller.

Let us focus on the input-output associations (conditions - conclusions), with Ak and Bk viewed as fuzzy sets defined in the corresponding finite universes of discourse, namely Ak ∈ [0, 1]n, Bk ∈ [0, 1]m.

The crosstalk between the rules occurring due to Hebbian learning and the introduced architecture can be well illustrated in Fig. 4.6 where for clarity of presentation the rules include overlapping sets (as opposed to fuzzy sets). This figure illustrates that in the regions of their overlap the result of the mapping generates a sum of B1 and B2.


Figure 4.6  Crosstalk between control rules: X and its recalled item; note that it subsumes B1 as well as B2

The following are the fundamental information processing properties of the Hebbian realization of the rule-based system:

boundary conditions

if X is an empty set, X = ∅, so is B, B = ∅,
X = X implies that the fuzzy set of control is given as which constitutes a union of all the control actions forming the control protocol.

monotonicity

if X ∴ X′ then B ∴ B′ so B′ is less specific than that produced by B. Especially, the second boundary condition outlined above becomes a clear-cut illustration of the monotonicity aspect.

crosstalk

This phenomenon leads to the containment rather than equality condition between the recalled items, namely,


This relationship is a manifestation of the crosstalk phenomenon; refer to Fig. 4.6.


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