see things differently

@IgorCarron blogs recent applications of compressive sensing and matrix factorisation every week.

(Compressive sensing solves underdetermined systems of equations, for example trying to fill in missing data, by L₁-norm minimisation.)

This week: reverse-engineering biochemical pathways and complex systems analysis.

The word “Evolution” nowadays suggests “evolution of organisms by natural selection as per Darwin & modern population genetics”.

But what about other kinds of evolution? Any unary endomorphism from a system onto itself, applied over and over to generate “time”, could be considered a kind of “evolution”.

Crystals and quasicrystals evolve naturally.
Caves and stalagmites evolve naturally.
History evolves artificially … although no-one knows the mapping.
Art evolves artificially … again, no-one knows the mapping. (but we know there is cross-pollination. Could we call it “Art sex”?)
Proto-biological chemical compounds, like basic amino acids, “evolved” by a method similar to natural selection.
Businesses (and entrepreneurs) that grow through trial-and-error, evolve their ideas and their business processes by artificial selection.
Romantic relationships evolve. Any human relationship evolves. Sometimes “a” relationship can evolve with a group of people as-a-unit. But here I’ve found the selection to be more exogenous than endogenous. Friendships seem to pick up exactly where they left off γᵢ(Tᵢ) rather than be pitched to the dustbin of 0. Even romantic relationships seem to hang mostly where they were, even after a breakup. Perhaps the breakup zeroes the romantic part ⌊γᵢ⌋ᵣ, but everything else—sexual chemistry, personality dynamics, humour dynamics, and history—remains stubbornly unbudged by a breakup per se.
And like I said, any endomorphism, repeatedly applied to a system, could count as “evolution”. (An endomorphism draws both the input and the output from the same domain, i.e. ƒ: X→X.) If there is a throw-away criterion (mapping ↦ 0), we could call that “selection”. Any fixed point of the mapping ƒ(p)↦p is an endpoint of evolution.

In this video, John Baez talks about how the inherent interestingness of the number 5 has made itself apparent to us through several processes:

artists (mosque designers) discovered it. God speaks to us in the language of mathematics, remember?
crystals and quasicrystals discovered 5 by evolution — but not the biological kind
soot and space dust found ⬟, also by natural non-bio evolution
BTW, unrelated but some Scots (perhaps Picts) carved some Platonic solids out of stone centuries before Plato … so perhaps they should be called Scottish solids.
the Pariacoto virus found ⬟ by biological evolution
Roger Penrose (a mathematical physicist) discovered & described 5-way symmetry in modern mathematical (group theoretic) terms

In each case, logic is the canvas. Art — nature — mathematicians are the painters.

Logical Circular Logic

Are the Israelis bellicose because countries are always antagonizing them, or are countries always antagonizing the Israelis because they’re bellicose?
Do you think that boy is cute because he likes you, and he likes you more because you like him, and where did this liking start in the first place?
If the rich get richer, how do you get rich in the first place? And if poverty leads to violence, lack of education, and ill health, which leads to more poverty, how do you stop the vicious cycle?
Do cops mess with people because there are too many criminals around, or do people become uncooperative and untrusting of cops because they’re always messing with people?
Nature or nurture?
Do Americans commute long distances to work because the country is built around roads, highways, and the assumption that everyone has a car, or is the highway infrastructure good because Americans have so many cars?
If child abuse is caused by the parent being abused as a child — if gang violence is usually retaliation for other gang violence — where did it all begin?
Does the culture generate the media, or does the media generate the culture?

So many chicken-and-egg questions. Dynamical systems theory comes to the rescue.

A dynamical system is a time-evolving list of equations that mutually affect each other. In the real world the arrow of causation often runs both ways. By thinking about chicken-and-egg questions as a dynamical system, suddenly circular causation becomes logical, even quantifiable.

fig3

More technicals on linear operators on dynamical systems, some famous examples including ecosystem modeling, mind-body dynamics, Jon Gottman, toy models, and the similarity to (I thought boring) differential equations later.

ƒ(ƒ(ƒ(ƒ(x))))

Above pictures the phase-space of some observable, x, that evolves over time x₀, x₁, x₂, x₃, …. The next time step after value x₃ has been reached is shown as the height of the graph above x₃.

(So unlike the graph of a baseball thrown from the origin to (x,y) = (1,0), this is not a time-path picture.)

Also notice that, at whatever time the process might reach, say, x₇ or x₅₅ = .34197, the same thing will always result next turn: x₈ or x₅₆ will become ~.6.

So time in this picture proceeds by jumping to parabola, diagonal, parabola, diagonal, parabola, diagonal, parabola, diagonal. When one reads the width of the parabola, or the height-or-width of the diagonal, one knows the value of the process at this particular time step.

What’s happening mathematically is that the transformation

$f(x) = 2.8 x (1 - x)$

is being applied over and over again. So ƒ(ƒ(ƒ(.5))) represents three time steps after starting in the middle. ƒ(ƒ(ƒ(ƒ(ƒ(ƒ(ƒ(0.000001))))))) represents seven time steps after starting at the left side. This is a spiffy way to describe how systems behave, even if it’s just a one-dimensional observable. Iterative mappings, or “difference equations”, are a popular way to describe real-life systems. I first saw it in economics class, in Robert Solow’s model of capital growth. It also appears in weather models, psychological models, and biological models.

Question. What happens if you do f(f(f(f(f(f( something )))))) ad infinitum? Do the answers ever settle down? This is where, if you had a button on your calculator to do “apply f”, you would hit the button and see what happens. Unless you can program, though, you’ll have to think through this one logically.

x̄̄ is a stable fixed point of this iterative mapping.

$x_{k+1} = x_k (1 - x_k) 4mu$

However consider a very similar cousin,

$x_{k+1} = x_k (1 - x_k) 4mu$ .

In the .9 case, x̄ is unstable. What’s up with that?

hi-res