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February 11 2013

The Mathematical Journey | Life Skills | National Numeracy

Numeracy is a key skill necessary to allow each of us to make informed choices and decisions in all aspects of everyday life.

 

See it on Scoop.it, via oAnth's day by day interests - via its scoop.it contacts
Reposted from02mysoup-aa 02mysoup-aa

June 20 2012

Alan Turing: the short, brilliant life and tragic death of an enigma

Codebreaker and mathematician Alan Turing's legacy comes to life in a Science Museum exhibition

A German Enigma coding machine on loan from Mick Jagger and a 1950 computer with less calculating power than a smartphone but which was once the fastest in the world, are among the star objects in a new exhibition at the Science Museum devoted to the short, brilliant life and tragic death of the scientist Alan Turing.

"We are in geek heaven," his nephew Sir John Turing said, surrounded by pieces of computing history which are sacred relics to Turing's admirers, including a computer-controlled tortoise that had enchanted the scientist when he saw it at the museum in the 1951 Festival of Britain. "This exhibition is a great tribute to a very remarkable man," Turing said.

"My father was in awe of him, the word genius was often used in speaking of him in the family," he said, "but he also spoke of his eccentricity, of how he cycled to work at Bletchley wearing a gas mask to control his hayfever so the local people he passed dreaded that a gas attack was imminent."

The exhibition, marking the centenary of Turing's birth, tackles both the traumatic personal life and the brilliant science of the man who was a key member of the codebreaking team at Bletchley Park, and devised the Turing Test which is still the measure of artificial intelligence.

Turing was gay, and in 1952 while working at Manchester University, where he had a relationship with a technician called Arnold Murray, he was arrested and charged with gross indecency. He escaped prison only by agreeing to chemical castration through a year's doses of oestrogen – which curator David Rooney said had a devastating effect on him, mentally and physically. In 1954 he was found dead in his bed, a half eaten apple on the table beside him, according to legend laced with the cyanide which killed him.

His mother insisted that his death was accidental, part of an experiment to silver plate a spoon – he had previously gold plated another piece of cutlery by stripping the gold from a pocket watch – with the chemicals found in a pot on the stove. However the coroner's report, also on display, is unequivocal: Turing had consumed the equivalent of a wine glass of poison and the form records bleakly "the brain smelled of bitter almonds".

The death is wreathed with conspiracy theories, but Rooney's explanation for the apple is pragmatic: not an obsession with the poisoned apple in the Disney film of Snow White, as some have claimed, but a very intelligent man who had it ready to bite into to counteract the appalling taste of the cyanide.

His nephew said both the prosecution and death were devastating for the family, but they were delighted by the formal public apology offered in 2009 by then prime minister Gordon Brown.

The campaign for a posthumous pardon is more problematic he said, speaking as a senior partner at the law firm Clifford Chance.

"So many people were condemned properly under the then law for offences which we now see entirely differently. One would not wish to think that Turing won a pardon merely because he is famous, that might be just a step too far. But the suggestion that there might be some reparation by having him appear on the back of a bank note – that might indeed be good."

The exhibition includes the only surviving parts of one of the 200 bombe machines which ran day and night decoding German messages at sites around the country, each weighing a ton and all broken up for scrap after the war. The components were borrowed from the government intelligence centre at GCHQ after tortuous negotiations. Although visitors will not realise it, a short interview filmed at GCHQ is even more exceptional, the only film for public viewing ever permitted inside the Cheltenham complex.

By 1950 when the Pilot Ace computer, on which Turing did key development work, was finally running at the National Physical Laboratory, he had moved to Manchester, impatient at the slow pace of work in the postwar public sector. It is displayed beside a panel of tattered metal, part of a Comet, the first civilian passenger jet, which exploded over the Mediterranean killing all on board: the computer ran the millions of calculations to work out why.

Rooney says the exhibition is also intended to destroy the impression of Turing as a solitary boffin: it includes many of the people he worked with, who regarded him with awe and affection. When he came to see the computer tortoises in 1951 – they responded to light and scuttled back home when the bulb was switched on in their hutches – he also managed to break a game playing computer by recognising the work of a protege and cracking the algorithm on the spot: the computer flashed both "you've won" and "you've lost" messages at him, and then shut itself down in a sulk.

In an interview filmed for the exhibition his last researcher, Professor Bernard Richards of Manchester University, the man he was due to meet on the day of his death, says: "Turing struck me as a genius. He was on a higher plane."

Codebreaker – Alan Turing's life and legacy, free at the Science Museum, London, until June 2013.


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January 18 2012

The Man of Numbers: How Fibonacci Changed the World | brainpickings.org

What Medieval mathematics have to do with remix culture, publishing entrepreneurship, and gamification.

 

----------------------------

 // oAnth

 "The change in society brought about by the teaching of modern arithmetic was so pervasive and all-powerful that within a few generations people simply took it for granted. There was no longer any recognition of the magnitude of the revolution that took the subject from an obscure object of scholarly interest to an everyday mental tool. Compared with Copernicus’s conclusions about the position of Earth in the solar system and Galileo’s discovery of the pendulum as a basis for telling time, Leonardo’s showing people how to multiply 193 by 27 simply lacks drama.” ~ Keith Devlin

Original URL -


November 16 2011

Talk Nondually & Mathematics Without Appearing Ludicrous


----------------------

oAnth:

recommendation - talk ~35 min - math as a factor of cultures and socialisation.
Reposted fromsigaloninspired sigaloninspired

October 26 2011

02mydafsoup-01

September 06 2011

Philosophia Scientiæ - Travaux d'histoire et de philosophie des sciences

Philosophia Scientiæ est une revue scientifique à comité de lecture qui publie des travaux en épistémologie, en histoire et en philosophie des sciences. Elle accueille notamment des études traitant des mathématiques, de la physique et de la logique, mais elle est ouverte aux travaux portant sur les autres disciplines scientifiques.

--------------------------

// oAnth

préface est librement disponible

http://philosophiascientiae.revues.org/354



Reposted from02mysoup-aa 02mysoup-aa

June 29 2011

02mydafsoup-01

April 13 2011

CAPTCHA chaos by Marianne Freiberger | plus.maths.org

[...]

The new method, proposed by T.V. Laptyeva and S. Flach from the Max-Planck-Institute for the Physics of Complex Systems in Germany and K. Kladko from Axioma Research in the US, also uses standard encryption algorithms, like AES, but it works some magic with the password. It uses a long password consisting of two parts. One part is an easily memorable short password (SP) and the other is longer and known as the strong key. The confidential text is encrypted using a standard algorithm like AES and the combination of short password and strong key as the password.

Image embedding

The strong key — in this example the word "chaos" — is embedded in an image.

The strong key doesn't have to be memorised by the user. Instead the strong key is then embedded in an image in a similar way as for the familiar CAPTCHA test, which shows the user a distorted image of a word which computers used to find difficult to identify (though good image recognition software can these days do the trick).

But here's the trick: the image can also be interpreted as a snapshot in time of a dynamical system — a system that evolves over time according to a set of mathematical equations. The state of the system at each individual time step can be described using numbers and represented by an image.

The particular dynamical system used in this case comes from physics, where it is used to model the behaviour of ferroelectric materials, such as barium titanate. These are akin to magnetic materials, but rather than responding to magnetic fields, they respond to electric fields, becoming electrically polarised when an electric field is applied.

Ferroelectric materials lose their ferrorelectric property at a certain temperature. Such a sudden change in a material's porperties is called a phase transition. Typically a system undergoing a phase transition moves from an ordered state — electric polarisation in the case of ferroelectric materials — to a disordered state. The dynamical system the researchers chose for their encryption method is a simple model of the behaviour of ferroelectric materials close to the phase transition. One of the properties the system exhibits near this order-disorder transition point is chaos: parameters describing the system fluctuate wildly and the system is so sensitive to small changes that it is impossible to predict exactly what state it will be in after a number of time steps.

Image evolution

The image containing the strong key (called the ISK) is evolved in time (downward direction). Even the smallest change in the resulting image will cause you to miss the original when you go back in time.

Now suppose you have arrested this dynamical system at a particular moment in time, say at time $t_0$. The encryption method works by embedding the strong key in the corresponding image using something akin to the CAPTCHA method. The system is then allowed to evolve for a certain number of time steps, until it reaches time $t_ n$. The state of the system at time $t_ n$ will have similar overall features for whatever initial state $t_0$ we start with, but due to the chaotic dynamics the precise nature of these features is impossible to predict and they appear random. The numerical information which encodes the state of the system at time $t_ n$ can therefore be split into two parts. One part describes the regular features of the system — this can be stored in plain text. The other describes the random-looking detail and is encoded using the short password.

Now imagine that you're the legitimate user in possesion of the correct SP. You enter the SP, which decrypts the random-looking part of the information. This is then combined with the information that encodes the regularities of the dynamical system, to give you all the information about the system at time $t_ n$. The system is now evolved backwards in time until it gets back to time $t_0$. In theory, such a backward evolution does not always get you back to the exact state you started from: because the system is chaotic, the smallest inaccuracy in your information (coming for example from rounding off numbers) can amplify as you move backwards in time and cause you to miss the original image. But by making sure you don't use too many time steps (i.e. that the number $n$ isn't too large) you can guarantee that you end up with something nearly identical to the original image. Then, instead of having to remember it, you can just read the strong key off that image. It is now combined with the short password to give you the overall password which decrypts the original data.

But what if you're a hostile attacker systematically trying out a large number of short passwords? Suppose that you have access to the plain-text information describing the regular features of the dynamical system at time $t_ n$ and to the encrypted data representing the random-looking part. First you use a particular short password guess to get a candidate decryption of the random-looking part. Usually the candidate decryption would be searched for patterns and correlations which indicate if it's the true text. But in this case the true text itself looks random, so any such search is futile. You just have to plough on: you evolve the system backwards until time $t_0,$ hoping that the resulting image contains the correct strong key.

[...]

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April 11 2011

02mydafsoup-01
Play fullscreen
ran-dom – Shell sort algorithm performed by a hungarian folk...
Reposted fromyetzt yetzt viasofias sofias

March 30 2011

02mydafsoup-01
For Deleuze, the distinction between problematics and axiomatics is reflected in the distinction between two different conceptions of the multiple: differential of continuous multiplicities (problematics) and extensional or discrete sets (axiomatics). As we have seen, royal or ‘major’ mathematics is defined by the perpetual translation or conversion of the latter into the former. But it would be erroneous to characterize differential multiplicities as ‘merely’ intuitive and operative, and extensional sets as conceptual and formalizable. ‘The fact is’ writes Deleuze, ’ that the two kinds of science have different modes of formalization […]. What we have are two formally different conceptions of science, and ontologically, a single field of interaction in which royal science [axiomatics] continually appropriates the contents of vague or nomad science [problematics], while nomad science continually cuts the contents of royal science loose.’ One of Badiou’s most insistent claims is that Deleuze’s theory of multiplicities is drawn from a ‘vitalist’ paradigm, and not a mathematical one. But in fact, Deleuze’s theory of multiplicity is drawn exclusively from mathematics - but from its problematic pole. Badiou implicitly admits this when he complains that Deleuze’s ‘experimental construction of multiplicities is anachronistic because it is pre-Cantorian. More accurately, however, one should say that Deleuze’s theory of multiplicities is non-Cantorian.’ Cantor’s set theory represents the crowning moment of the tendency toward ‘discretization’ in mathematics; Deleuze’s project, by contrast, is to formalize the conception of ‘continuous’ multiplicities that corresponds to the problematic pole of mathematics. Problematics, no less than axiomatics, is the object of pure mathematics; just as Weierstrass, Dedekind and Cantor are the great names in the discretization programme, and Hilbert, Zermelo, Frankel, Godel and Cohen the great names in the movement toward formalization and axiomatization, it is Abel, Galois, Riemann and Poincaré who appear among the great names in the history of problematics. ”
— Blog: concrete rules and abstract machines 2011-03-29 | For Deleuze, the distinction between problematics... from -  Peter Hallward: Think Again. Alain Badiou and the Future of Philosophy.

November 14 2010

My bright idea: Innovation is born when art meets science

The technology and design guru argues that for invention to occur, scientists must embrace the art world

A graphic designer and computer scientist, known for his work on the online computer game Second Life, as well as the author of bestselling self-help book The Laws of Simplicity, John Maeda has made great use of dual educations at the Massachusetts Institute of Technology and art school. Drawing from his experiences in these two disciplines, the 44-year-old has come to believe that too stark a distinction is drawn between science and the arts. It is Maeda's conviction that scientists need art and artists in their professional lives in order to invent and innovate successfully, and with a particular focus on education he has toured the world to promote the idea that government-approved "Stem" subjects (science, technology, engineering and maths) should be widened to include art; "turning Stem into Steam," as he puts it. This week Maeda, who is president of the Rhode Island School of Design, will expound on these ideas at an experimental installation at London's Riflemaker gallery, where he will "dispense wisdom from a sandpit". See Riflemaker.org for more on this eccentric project.

Why does science need artists?

We seem to forget that innovation doesn't just come from equations or new kinds of chemicals, it comes from a human place. Innovation in the sciences is always linked in some way, either directly or indirectly, to a human experience. And human experiences happen through engaging with the arts – listening to music, say, or seeing a piece of art.

So to help them become more humanist, you'd parachute artists and musicians into laboratories?

Which already happens to some degree with artist-in-residence programmes in scientific labs. They're usually very small, but these programmes are seen as quite desirable by scientists. Because all scientists are humans, and they are humanists inside, and by bringing that part out, innovation happens more naturally.

Can you think of an example where an injection of the arts has helped the sciences?

I recently saw something in Time magazine, a famous Nobel laureate chemist making molecular models out of clay. It shows how these more fluid, abstract materials traditionally belonging to the artist lend themselves better to ways of thinking about the world, as opposed to some kind of ball-and-stick model that shows a constrained view. Art helps you see things in a less constrained space. Our economy is built upon convergent thinkers, people that execute things, get them done. But artists and designers are divergent thinkers: they expand the horizon of possibilities. Superior innovation comes from bringing divergents (the artists and designers) and convergents (science and engineering) together.

Such as?

Look at Apple's iPod. A perfect example of technology – an MP3 player – that existed for a long time but that nobody ever wanted, until design made it something desirable, useful, integrated into your lifestyle. Look at the success of Mint.com [a colourful money-management website] which has recently been sold. It's an app in which 80% of the experience is what you see, how you touch it. Not the technology. I'm also interested in how art and design links into leadership. Because leaders now are facing a very chaotic landscape, things are no longer black and white, things are harder to predict. What better mindset to adopt than the artist's, who is very used to living in an ambiguous space? Real innovation doesn't just come from technology, it comes from places like art and design.

George Osborne recently announced protection in the higher-education cuts for the so-called Stem subjects, but not the arts. Is this blinkered?

You know, it's easy for politicians to look at the measurability of a science and maths education. I mean, fill out 100 questions, you get 100 right or 50 right or zero right, it's easy to measure. There's no test that can give you a score from zero to 100 on the question, "Is this student a good writer?" And society's so focused on measurement. It's awkward and sad. Singapore or Japan are highly known test-taking countries focused on science and engineering, yet are desperate to find innovation. And where are they looking? They're looking to the west for new ideas. It's kind of like a dog chasing after its tail a little bit – this weeding out of the idea that expression, something that exists in the intuition space, can matter. I mean, it's ironic that the people who talk about these kind of things [cuts to the arts] are all counting on things to carry their message – like images, the written word – as givens.

Do you think that scientists tend to lack humanity?

Scientists would say otherwise. But scientists strive to be pure, to live in what's called a "concept space". And by doing so they tend to move away from the core humanist principles that actually put those two arms and legs on them in the first place. The best scientists that I've met are those that are humanists and scientists at the same time.


guardian.co.uk © Guardian News & Media Limited 2010 | Use of this content is subject to our Terms & Conditions | More Feeds


October 19 2010

The art of geometry

The sphere of maths has borne few as provocative as the man whose 'fractals' demonstrated the universe's playful irregularity

Few recent thinkers have woven such a beautiful braid of art and science as Benoît B Mandelbrot, who has died aged 85 in Cambridge, Massachusetts. (The B apparently doesn't stand for anything. He just felt like adding it.) Mandelbrot was a provocative mathematician, a subversive geometer. He left a beautiful legacy in visual art, for Mandelbrot was the man who named and explained fractals – those complex, apparently chaotic yet geometrically ordered shapes that delight the eye and fascinate the mind. They are icons of modern understanding of the universe's complexity.

The Mandelbrot set, one of the most famous fractal designs, is named after him. With its fizzing fringe of crystal-like microforms blossoming out of a conjunction of black circles, this fractal pattern looks crazy but is the outcome of geometrical calculations.

Geometry, said Mandelbrot, is seen as "dry" because it can only explain regular shapes like the square, the cylinder and the cone. Such shapes have been analysed mathematically since the time of the ancient Greeks, which is why traditional geometry is known as Euclidean geometry. But in the 19th and 20th centuries, physicists and mathematicians started to think beyond Euclid and his regular universe. Mandelbrot was not the first, but with his startling fractals concept he created a visual manifesto for a non-Euclidean universe.

Fractals – and I'd be delighted if mathematicians can give a better explanation below– are shapes that are irregular but repeat themselves at every scale: they contain themselves in themselves. Mandelbrot used the example of a cauliflower which, like a fern, is a fractal found in nature; if you look at the smallest sections of these vegetable forms, you see them mirroring the whole.

Mandelbrot, who worked at IBM before becoming a professor at Yale, started thinking about irregular shapes by looking at maps of Great Britain. The squiggly shape of the UK mainland fascinated him and he wondered whether it was possible to make a mathematical model of its perimeter. Can you measure the British coastline? He discovered that you can at a distance, but that then the closer you look, the more you find. In a sense, the British coastline is "infinite".

Artists have been fascinated by geometry for as long as mathematicians have. The studies of Euclid are reflected in the regularities of classical and Renaissance architecture, from the Pantheon in Rome to the duomo in Florence. But artists and architects were also thinking centuries ago about non-regular, curving geometries. You could argue that fractals give us the mathematics of the Baroque – they were anticipated by Borromini and Bach. I have a facsimile, given away by an Italian newspaper, of part of Leonardo da Vinci's Atlantic Codex, which contains page after page of his attempts to analyse the geometry of twisted, curving shapes.

Mandelbrot was a modern Leonardo, a man who showed the beauty in nature. He was a prophet of the curving universe and gave us, in the endlessly playful geometry of fractals, a visual lexicon for our complex world.


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April 21 2010

02mydafsoup-01
Computer, sagt der amerikanische Mathematiker Steve Strogatz, berechnen mittlerweile Dinge, die auch die brillantesten Mathematiker nicht mehr überprüfen können.
Macht der Simulation: Plötzlich sind wir alle Zuschauer - Digitales Denken - Feuilleton - FAZ.NET

February 10 2010

Mathematics and Art

Nikki Graziano's intriguing integration of mathematical curves into her photography sparked a Radar discussion about the relationship between mathematics and the real world. Does her work give insight into the nature of mathematics? Or into the nature of the world? And if so, what kind of insight?


Mathematically, matching one curve to another isn't a big deal. Given N points, it's trivial to write an N+1 degree equation that passes through all of them. There are many more subtle ways of solving the same problem, with more aesthetically pleasing results: you can use sine functions, wavelets, square waves, whatever you want. Take out a ruler, measure some points, plug them into Mathematica, and in seconds you can generate as many curves as you like. So finding an equation that matches the curve of an artfully trimmed hedge is easy. The question is whether that curve tells us anything, or whether it's just another stupid math trick.


It's not a simple question. A few weeks ago, I came across a brilliant essay from 1960, The Unreasonable Effectiveness of Mathematics in the Natural Sciences. The point of the essay is that while mathematics is an incredibly abstract discipline, with no apparent contact to the real world, against all reasonable expectation mathematical results have proven to be invaluable in studying the physical world. Why should purely mathematical results be relevant in everything from population studies to quantum physics? Why do the Fibonacci series, fractals, and other abstract mathematical concepts show up in nature?


My favorite example is the square root of -1. In grade school, we're first told that there's no such thing. Then, sometime in high school, we're told "well we lied, but the square root of -1 is imaginary, call it i". Except it isn't imaginary. The square root of -1 makes a profound connection between the world of exponentials and the world of circles. And the world of circles is the basis for every every wave phenomenon we know about. Much of physics, and just about all of electrical engineering wouldn't exist without i--or would be so hampered by grossly complex mathematics that we couldn't explore it. But even though it's what makes the world go 'round, you can never point me to the square root of -1. It has extraordinary power, both mathematically and physically, but it's "imaginary."


So--back to the connection between art and math. Is she showing a deep connection between the objects she photographs and the mathematical world, or is it just a math trick? And is that deep connection necessary? A number of years ago, I went to a concert of the works of György Ligeti. Ligeti gave a talk in which he said the works were inspired by Benoit Mandelbrot's work--though he had not attempted to replicate Mandelbrot curves in the music itself. You didn't need to understand fractals to appreciate the music, and upon hearing the music you wouldn't say "damn, is that a fractal I just heard?" The music didn't attempt to represent the math, and the math doesn't define the music.


In Graziano's photography, as presented in Wired, I find the relationship more troubling. The math is imposed on (both in the sense of "superimposed" and in the sense of a power relationship) the natural scenes. Does the math inform the natural world? Or does it usurp it? I think this ambiguity is both healthy and intentional. But what's less healthy is that I don't find the math that convincing. In the first of the Wired pictures, the mathematical curve is (log(x)/2)^sin(x). (Or so I think. The equation is typeset poorly; this interpretation comes closes to matching the picture). It's difficult to believe that this log^sin relationship corresponds to nature. And it blows up outside of the domain used in the photograph--an almost certain sign of ham-fisted curve fitting run amok. The second photo, some clouds that match a Gaussian bell curve, is much more convincing: I can believe that there's a deep relationship between a Gaussian curve and cloud formation. I find the fifth and sixth photos particularly interesting. The fifth shows a canopy of vines, with a hyperbolic paraboloid mapped on top of it (z = y^2 -x^2). Wikipedia says that this curve is useful in the design of large roofs because it is inherently strong and allows the use of lightweight buildng materials. I'm more than willing to guess that a canopy of vines would take this form. The final picture in the series is similar: another vine or bush (right now, I wish I were better at horticulture), overlayed with a surface that looks like a paraboloid, but is really sin(3*y*x^2/4). An interesting choice, but it's hard for me to believe that there's any intrinsic relationship between this surface and the way plants grow. Ligeti's music was "inspired" by Mandelbrot; Mandelbrot didn't force his equations into an analysis of the score. In Graziano's case, the math frequently appears to be "inspired" by the photography in a way that feels false to me.


But ambiguities abound. Math has been "unreasonably effective" throughout history; could that be happening here? The first picture doesn't look like a wild scene; I think it's a garden, and that the plants were possibly pruned by a gardener. So this odd log/sine combination might not be a statement about how plants grow, but a statement about human aesthetics (like the Golden Ratio). And the final photograph--while I don't believe the sin(xy^2) relationship says much about how a tangle of vines hangs together, plant growth depends strongly on sunlight. The sun travels in a more-or-less circular path, and in a large, overgrown tangle, it's at least plausible (though I'm not convinced) that this curve says something about exposure to the sun, given the sun's path through the sky and the effects of shadows.


Are mathematical complex effects hiding in plain sight? In the curve trimmed by a careful gardener, or in the seemingly random growth of a tangled vine? Absolutely. I'm not convinced that these effects are always the ones Graziano is showing us. Is Graziano's math arising out of the natural world, or is it imposed upon it? I don't know whether she's doing something profound, or just being clever, and this ambiguity bothers me. But it has made me think, and that's certainly the function of art.

January 17 2010

Manifold Learning, Calculus & Friendship, and Other Math Links

One of the largest gatherings of mathematicians, the joint meetings of the AMS/MAA/SIAM, took place last week in San Francisco. Knowing that there were going to be over 6,000 pure and applied mathematicians at Moscone West, I took some time off from work and attended several sessions. Below are a few (somewhat technical) highlights. (It's the only conference I've attended where the person managing the press room, was also working on some equations in-between helping the media.)


The Machine-Learning Bubble in Computational Medicine (Challenges in Computational Medicine and Biology)


Donald Geman gave a nice survey of the problems and mathematical techniques frequently used in computational biology. He also raised something that struck a chord with me. While computational biology has things in common with other fields ("small n, large d problem": small samples, relative to the number of dimensions), techniques that work in fields like computer vision don't automatically translate to biology. First, the size of samples in biology and medicine are orders of magnitude smaller compared to other fields. Secondly, while black boxes (think SVM's or neural nets) are acceptable in other fields, biologists want accurate predictions and explanations for why/how algorithms work. Finally, it isn't clear if there are underlying low-dimensional structures in biological data. Taken together, Geman wonders if machine-learning's possible role in biology and medicine has been overhyped.


Using Unlabeled Data To Identify Optimal Classifiers (A Geometric Perspective on Learning Theory and Algorithms)


Revisiting, the "small n, large d" problem, Partha Niyogi gave an overview of recent geometric approaches to machine-learning. In order to mitigate the curse of dimensionality, Niyogi and his fellow researchers exploit the tendency of (natural) data be be non-uniformly distributed. In particular, they use the shape of the data to determine optimal machine-learning classifiers. In their version of manifold learning, they assume that the space of target functions (e.g. all possible classifiers), consists of functions supported on a submanifold†† of the original high-dimensional euclidean/feature space. One of the most interesting features of their geometric approach, is their use of both labeled and unlabeled data††† to identify optimal classifiers. The traditional approaches to training classifiers require labeled data. So while one can use mechanical turks to increase the amount of labeled data for learning purposes, the geometric techniques outlined by Dr. Niyogi actually take advantage of any unlabeled data you may already have. Lest you think that these are purely academic/theoretical techniques, Dr. Niyogi cites a company that uses these algorithms to analyze and classify child speech patterns. With so much Data Exhaust available, I can't help but think that techniques that can leverage unlabeled data will prove useful in many domains. (Niyogi and his collaborators have many papers on Manifold Learning, including one that describes the algorithms, and another that provides the theoretical foundations.)


pathint


The Calculus of Friendship


Mathematician Stephen Strogatz is known to many Radar readers for his work in network theory ("small-world networks") with his student Duncan Watts. I went to his talk thinking it would cover recent developments in random graphs. The talk turned out to be about his recent book chronicling his long friendship with his high school math teacher. What started out with letters that talked only about calculus and math problems, evolved into a deeper relationship over the last decade. The letters ranged from humorous calculus problems, to moving personal correspondence. For a preview of his book, listen to this recent Radiolab segment featuring Dr. Strogatz and his teacher:





[What made his teacher into a great instructor/mentor? Dr. Strogatz mentioned a few characteristics, many of which could be be re-purposed into advice for business leaders and managers. Yet another reason to read his book.]


Geomathematics (Mathematics and the Geological Sciences)


Another highlight of the conference was a symposium devoted to the emerging field of geomathematics. Given that the geological sciences routinely deal with Big Data sets, developments in geomathematics are worth paying attention to.

(†) To illustrate how geometric these techniques are, Niyogi outlined versions of the Laplace-Beltrami operator, the Heat Kernel, and Homology in his short talk. I went to another interesting talk on geometric structures and discrete graphs, but from what I could gather, it was mostly theoretical in nature.
(††) Niyogi and his fellow researchers assert that "... for almost any imaginable source of meaningful high-dimensional data, the space of possible configurations occupies only a tiny portion of the total volume available. One therefore suspects that a non-linear low-dimensional manifold may yield a useful approximation to this structure."
(†††) In classification problems, labeled data are ordered pairs of feature vectors and their corresponding class labels. In the geometric approach to learning classifiers, unlabeled data can be used to recover the "intrinsic geometric structure" of marginal probability density functions.

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