Ed's Big Plans

For the final project in Neural Networks (CIS 6050 W11), I decided to cluster blog posts based on the difference between an author’s word choice and the word choice of the entire group of authors.

>>> Attached Course Paper: Kohonen Self-Organizing Maps in Clustering of Blog Authors (pdf) <<<

A self-organizing map (Kohonen SOM) strategy was used. The words chosen to compose a given blog post defined wherein the map it should be placed. The purpose of this project was to figure out what predictive power could be harnessed given the combination of the SOM and each author’s lexicon; i.e. whether or not it is possible to automatically categorize an author’s latest post without the use of any tools besides the above.

Data: Thirteen authors contributing a total of fourteen blogs participated in the study (all data was retrieved on 2011 March 8th). The below table summarizes the origin of the data.

†Daniela remarks that the spelling of Citisen is intentional.

In order to place the blog posts into a SOM, each post was converted to a bitvector. Each bit is assigned to a specific word, so that the positions of each bit consistently represents the same word from post to post. An on-bit represented the presence of a word while an off-bit represented the absence of a word. Frequently used words like “the” were omitted from the word bit-vector, and seldom used words were also omitted.

Results: The center image (in the collection to the left) is a density map where darker pixels indicates a larger number of posts — this centre map represents all of the posts made by all of the authors pooled together.

Because of the number of posts and the number of authors, I’ve exploded the single SOM image into the remaining fourteen images.

It was found that posts were most often clustered together if they were both by the same author and on the same topic. Clusters containing more than one author generally did not show much agreement about the topic.

Regions of this SOM were dominated by particular authors and topics as below.

Discussion: There are some numerical results to go along with this, but they aren’t terribly exciting — the long and the short of it is that this project should to be repeated. The present work points towards the following two needed improvements.

First, the way the bitvectors were cropped at the beginning and at the end were based on a usage heuristic that doesn’t really conform to information theory. I’d likely take a look at the positional density of all bits to select meaningful words to cluster.

Second, all posts were included — this results in the dense spot in the middle of the central map. Whether these posts are short or just misfit, many of them can probably be removed by analyzing their bit density too.

Appendix: Here are two figures that describe the distribution of the data in the word bitvectors.

When we sort the words based from a high number of occurrences down to a low number of occurrences, we get graphs that look like the above two. A rank class contains all words that have the same number of occurrences across the entire study. The impulse graph on the left shows the trend for the number of unique words in each rank class. The number of words drastically increases as the classes contain fewer words. The impulse graph on the right shows the trend for the count of uses for words in a given rank class. The number of uses decreases as words become more rare.

These graphs were made before the main body of the work to sketch out how I wanted the bitvectors to behave — they verify that there was nothing unusual about the way the words were distributed amongst the data.

I found an old project in my archives the other day. The long and the short of it is — on occasion — if we use a three-layer back-propagation artificial neural network, initializing the hidden weight layers to larger values works better than initializing them to small values. The general wisdom is that one initializes weight layers to small values just so the bias of the system is broken, and weight values can move along, away and apart to different destination values. The general wisdom is backed by the thought that larger values can cause nets to become trapped in incorrect solutions sooner — but on this occasion, it manages to allow nets to converge more frequently.

The text is a bit small, but the below explains what’s going on.

There are two neural network weight layers (2D real-valued matrices) — the first going from input-to-hidden-layer and the second going from hidden-to-output-layer. We would normally initialize these with small random values, say in the range [-0.3, 0.3]. What we’re doing here is introducing a gap in the middle of that range that expands out and forces the weights to be a bit larger. The x-axis describes the size of this gap, increasing from zero to six. The data point in the far left of the graph corresponds to a usual initialization, when the gap is zero. The gap is increased in increments of 0.05 in both the positive and negative directions for each point on the graph as we move from left to right. The next point hence corresponds to an initialization of weights in the range [-0.35, -0.05] ∨ [0.05, 0.35]. In the far right of the graph, the weights are finally randomized in the range [-6.30, -6.00] ∨ [6.00, 6.30]. The y-axis is the probability of convergence given a particular gap size. Four-hundred trials are conducted for each gap size.

The result is that the best probability of convergence was achieved when the gap size is 1.0, corresponding to weights in the random range [-1.3 -1.0] ∨ [1.0, 1.3].

What were the particular circumstances that made this work? I think this is largely dependent on the function to fit.

The Boolean function I chose was a bit of an oddball — take a look at its truth table …

This comes from the function { λ : Bit_1, Bit_2, Bit_3 → (Bit_1 ∨ Bit_2) if (Bit_3) else (Bit_1 ⊕ Bit_2) }.

This function is unbalanced in the sense that it answers true more often than it answers false given the possible combinations of its inputs. This shouldn’t be too big of a problem though — we really only need any linearly inseparable function.

This project was originally done for coursework during my master’s degree. The best size for your initial weights is highly function dependent — I later found that this is less helpful (or even harmful) to problems with a continuous domain or range. It seems to work well for binary and Boolean functions and also for architectures that require recursion (I often used a gap of at least 0.6 during my master’s thesis work with the Neural Grammar Network).

Remaining parameters: training constant = 0.4; momentum = 0.4; hidden nodes = 2; convergence RMSE = 0.05 (in retrospect, using a raw count up to complete concordance would have been nicer); maximum allowed epochs = 1E5; transfer function = logistic curve.

Update: The function being sought is better described as “continuous bit-parity” rather than “Fuzzy XOR”, the title of the post has been changed from “Fuzzy Exclusive OR (XOR)” to reflect that.

About two weeks ago, I was working on a project wherein I needed to define a continuous XOR function. The only stipulations are that (1) the function must either be binary and commutative, or it must be variadic; and (2) the function must be continuous.

In my first attempt, I used the classic arrangement of four binary NAND gates to make an XOR where each NAND gate was replaced with the expression { λ: p, q → 1.0 – pq }. The algebraic product T-norm { λ: p, q → pq } is used instead of the standard fuzzy T-norm { λ: p, q → min(p, q) } in order to keep it continuous. Unfortunately, this attempt does not preserve commutativity, so the search continued.

At this point, Dr. Kremer suggested I consider a shifted sine curve. I eventually chose the equation
{ λ: p_[1..n] → 0.5 – 0.5cos(π Σ_i=1ⁿp_i) }.
This is shown graphically in the below figure …

# gnuplot source ...
set xrange[-2*pi:2*pi]
set output "a.eps"
set terminal postscript eps size 2.0, 1.5
plot 0.5 - 0.5 * cos(pi * x)

This can be considered a variadic function because it takes the sum of all fuzzy bits p_i in a given string and treats the arguments the same no matter the number of bits n.

Whenever the sum of all bits is equal to an even number, the function returns a zero — whenever the sum is an odd number, the function returns a one. This function offers a continuous (although potentially meaningless) value between integer values of the domain and can handle bitstrings of any length.

If you’re aware of a purely binary Fuzzy XOR (instead of variadic) that is a legal extension of classic XOR, continuous, and commutative — please let me know for future reference 😀