In the eye of the Beholder

Why is the picture on the right more appealing than the one on the left?

What is it that we find more interesting about the picture on the right, compared to the one on the left? The picture on the left contains more information. So we are certainly not looking for more information. One might say we don't know how to interpret the image on the left into anything familiar, but it is television static. A more precise answer is given by Jurgen Schmidhuber, who argues convincingly that:

Artists (and observers of art) get rewarded for making (and observing) novel patterns: data that is neither arbitrary (like incompressible random white noise) nor regular in an already known way, but regular in way that is new with respect to the observer's current knowledge, yet learnable (that is, after learning fewer bits are needed to encode the data).

This explains the pictures on top. The picture on the left is not compressible because it is a matrix of uniformly random 0/1 pixels. The Monet on the right evokes familiar feelings, and yet adds something new. I think what Schmidhuber is saying is that the amount of compressibility should neither be too little, nor too much. If  something is not very compressible, then it is too unfamiliar. If something is too compressible, then it is basically boring. In other words, the pleasure derived first increases and then decreases with the compressibility, not unlike this binary entropy curve.

Let us ask the same question again for the following pair of images (you have to pick one over the other):

My guess is that most people will find the image on the right more appealing (it is for me at least). Please drop me a comment with a reason if you differ. When I look at the image on the right, it feels a little more familiar, there are some experiences in my mind that I can relate to the image - for example looking straight up at the sky through a canopy of trees (white = sky, black=tree leaves), or a splatter of semisolid food in the kitchen.

In order for an object to be appealing, the beholder must have some side information, or familiarity with the object beforehand. I learnt this lesson the hard way. About 2 years ago, I gave a talk at a premier research institution in the New York area. Even though I had received complements when I'd given this talk at other venues, to my surprise, this time, audience almost slept through my talk. I learnt later that I had made the following mistake: in the abstract I'd sent to the talk's organizer, I had failed to signal that my work would likely appeal to an audience of information theorists and signal processing researchers. My audience had ended up being a bunch of systems researchers. The reason they dozed through my talk was that they had just a bit less than the required background to connect the dots I was showing them.

It is the same with cultural side information or context -- the familiar portion of the object allows the observer to latch on. The extra portion is the fun. Without the familiar, there is nothing to latch on to. The following phrases suddenly take on a precise quantifiable meaning:

  • "Beauty lies in the eyes of the beholder": the beholder carries a codebook that allows her to compress the object she is observing. Each beholder has a different codebook, and this explains 'subjective taste'.
  • "Ahead of its time": Something is ahead of its time if it is very good but does not have enough of a familiar portion to it, to be appreciated by the majority of observers.

I can think of lots of examples of art forms that deliberately incorporate partial familiarity into them -- e.g. music remixes, Bollywood story lines. Even classical music first establishes a base pattern and then builds on top of it. In this TED talk, Kirby Ferguson argues that all successful creative activity is a type of remix, meaning that it builds upon something familiar.

Takeaways:

  1. When writing a paper or giving a talk, always make sure the audience has something familiar to latch on to first. Otherwise, even a breakthrough result will appear uninteresting
  2. Ditto for telling a story or a joke, DJing music at a party, or building a consumer facing startup. Need something familiar to latch on to.
  3. In some situations it may be possible to gauge the codebook of the audience (e.g. having a dinner party conversation with a person you just met), to make sure you seem neither too familiar, nor too obscure.

Prime Factorization Trees

A positive integer can be visualized as a tree, structured by its prime factors. I wrote a little program which takes a positive integer as input, and creates a tree layout that shows its prime factors visually. Here's the tree for 70:

Screen shot 2012-10-07 at 9.55.37 AM

Screen shot 2012-10-07 at 9.55.37 AM

The tree is drawn recursively, starting with a root node in the center of a circular layout. The degree of the root node is the largest prime factor. Then, more prime factors are added recursively (including multiplicities of the same prime factor) in decreasing order of factor, by growing the tree. Finally, the leaf nodes are displayed as filled black circles. As seen from the tree above, the prime factorization of 70 is 7*5*2.

Here is the layout for 700 = 7 * 5 * 5 * 2 * 2:

Screen shot 2012-10-07 at 11.17.22 AM

Screen shot 2012-10-07 at 11.17.22 AM

This method of displaying numbers has been proposed and implemented by others on the internet, with much prettier results. I was too tempted to quickly try it for myself using a graph visualization library, because the graph drawing library can do all the heavy lifting. I used Graphviz (developed at AT&T Labs) which I use often for visualizing network data. I also wanted to see if the underlying tree structure revealed by the edges in the graph makes the prime factorization more evident.

Here are a few more trees:

243 = 3^5:

Screen shot 2012-10-07 at 11.44.01 AM

Screen shot 2012-10-07 at 11.44.01 AM

43 (a prime):

Screen shot 2012-10-07 at 10.04.39 AM

Screen shot 2012-10-07 at 10.04.39 AM

128 = 2^7:

Screen shot 2012-10-07 at 11.04.56 AM

Screen shot 2012-10-07 at 11.04.56 AM

121 = 11 * 11:

Screen shot 2012-10-07 at 10.55.17 AM

Screen shot 2012-10-07 at 10.55.17 AM

625 = 5^4:

Screen shot 2012-10-07 at 11.42.04 AM

Screen shot 2012-10-07 at 11.42.04 AM

2310 = 11 * 7 * 5 * 3 * 2:

Screen shot 2012-10-07 at 11.09.27 AM

Screen shot 2012-10-07 at 11.09.27 AM

Code gist (Without prime factorization implementation and graph library):

[sourcecode language="python"] def createnode(x): # x == 1: support node # x == 0: real node global nodenames name = random.random() while name in nodenames: name = random.random() nodenames.add(name) if x == 1: retval = '"'+str(name)+'"'+whitenode else: retval = '"'+str(name)+'"'+blacknode print retval return '"'+str(name)+'"'

def createedge(x,y,t): # x and y are nodes, t is type (0=last stage of the tree, 1=all other stages) if t == 1: print x + '--' + y + ' [color="#DDDDDD"]' if t == 0: print x + '--' + y + ' [color=grey]' return x + '--' + y

pf = factorization(n) # keys are factors, values are the multiplicity of the prime factors factors = [] for fac in pf: for multiplicity in range(0, pf[fac]): factors.append(fac) factors = sorted(factors, reverse=True) print 'graph G{' print 'overlap = false' nodenames = set() edgeset = set() whitenode = ' [fixedsize=True, width=0.1, height=0.1, color = white, fontcolor = white, shape = circle]' blacknode = ' [color = black, style = filled, fontcolor = black, shape = circle]' rootnode = createnode(1) currentset = set() currentset.add(rootnode) level = len(factors) + 1 for i in range(0, len(factors)): level -= 1 f = factors[i] nodetype = 1 if i == len(factors) - 1: nodetype = 0 newnodeset = set() for eachnode in currentset: for j in range(0, f): newnode = createnode(nodetype) newnodeset.add(newnode) edgeset.add(createedge(eachnode, newnode, level, nodetype)) currentset = newnodeset print '}' [/sourcecode]

For prime factorization, I used the this implementation.