nhaliday + acm + dimensionality   27

ON THE GEOMETRY OF NASH EQUILIBRIA AND CORRELATED EQUILIBRIA
Abstract: It is well known that the set of correlated equilibrium distributions of an n-player noncooperative game is a convex polytope that includes all the Nash equilibrium distributions. We demonstrate an elementary yet surprising result: the Nash equilibria all lie on the boundary of the polytope.
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may 2019 by nhaliday
teaching - Intuitive explanation for dividing by \$n-1\$ when calculating standard deviation? - Cross Validated
The standard deviation calculated with a divisor of n-1 is a standard deviation calculated from the sample as an estimate of the standard deviation of the population from which the sample was drawn. Because the observed values fall, on average, closer to the sample mean than to the population mean, the standard deviation which is calculated using deviations from the sample mean underestimates the desired standard deviation of the population. Using n-1 instead of n as the divisor corrects for that by making the result a little bit bigger.

Note that the correction has a larger proportional effect when n is small than when it is large, which is what we want because when n is larger the sample mean is likely to be a good estimator of the population mean.

...

A common one is that the definition of variance (of a distribution) is the second moment recentered around a known, definite mean, whereas the estimator uses an estimated mean. This loss of a degree of freedom (given the mean, you can reconstitute the dataset with knowledge of just n−1 of the data values) requires the use of n−1 rather than nn to "adjust" the result.
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january 2017 by nhaliday
gt.geometric topology - Intuitive crutches for higher dimensional thinking - MathOverflow
Terry Tao:
I can't help you much with high-dimensional topology - it's not my field, and I've not picked up the various tricks topologists use to get a grip on the subject - but when dealing with the geometry of high-dimensional (or infinite-dimensional) vector spaces such as R^n, there are plenty of ways to conceptualise these spaces that do not require visualising more than three dimensions directly.

For instance, one can view a high-dimensional vector space as a state space for a system with many degrees of freedom. A megapixel image, for instance, is a point in a million-dimensional vector space; by varying the image, one can explore the space, and various subsets of this space correspond to various classes of images.

One can similarly interpret sound waves, a box of gases, an ecosystem, a voting population, a stream of digital data, trials of random variables, the results of a statistical survey, a probabilistic strategy in a two-player game, and many other concrete objects as states in a high-dimensional vector space, and various basic concepts such as convexity, distance, linearity, change of variables, orthogonality, or inner product can have very natural meanings in some of these models (though not in all).

It can take a bit of both theory and practice to merge one's intuition for these things with one's spatial intuition for vectors and vector spaces, but it can be done eventually (much as after one has enough exposure to measure theory, one can start merging one's intuition regarding cardinality, mass, length, volume, probability, cost, charge, and any number of other "real-life" measures).

For instance, the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be interpreted as a manifestation of the law of large numbers, using the interpretation of a high-dimensional vector space as the state space for a large number of trials of a random variable.

More generally, many facts about low-dimensional projections or slices of high-dimensional objects can be viewed from a probabilistic, statistical, or signal processing perspective.

Scott Aaronson:
Here are some of the crutches I've relied on. (Admittedly, my crutches are probably much more useful for theoretical computer science, combinatorics, and probability than they are for geometry, topology, or physics. On a related note, I personally have a much easier time thinking about R^n than about, say, R^4 or R^5!)

1. If you're trying to visualize some 4D phenomenon P, first think of a related 3D phenomenon P', and then imagine yourself as a 2D being who's trying to visualize P'. The advantage is that, unlike with the 4D vs. 3D case, you yourself can easily switch between the 3D and 2D perspectives, and can therefore get a sense of exactly what information is being lost when you drop a dimension. (You could call this the "Flatland trick," after the most famous literary work to rely on it.)
2. As someone else mentioned, discretize! Instead of thinking about R^n, think about the Boolean hypercube {0,1}^n, which is finite and usually easier to get intuition about. (When working on problems, I often find myself drawing {0,1}^4 on a sheet of paper by drawing two copies of {0,1}^3 and then connecting the corresponding vertices.)
3. Instead of thinking about a subset S⊆R^n, think about its characteristic function f:R^n→{0,1}. I don't know why that trivial perspective switch makes such a big difference, but it does ... maybe because it shifts your attention to the process of computing f, and makes you forget about the hopeless task of visualizing S!
4. One of the central facts about R^n is that, while it has "room" for only n orthogonal vectors, it has room for exp⁡(n) almost-orthogonal vectors. Internalize that one fact, and so many other properties of R^n (for example, that the n-sphere resembles a "ball with spikes sticking out," as someone mentioned before) will suddenly seem non-mysterious. In turn, one way to internalize the fact that R^n has so many almost-orthogonal vectors is to internalize Shannon's theorem that there exist good error-correcting codes.
5. To get a feel for some high-dimensional object, ask questions about the behavior of a process that takes place on that object. For example: if I drop a ball here, which local minimum will it settle into? How long does this random walk on {0,1}^n take to mix?

Gil Kalai:
This is a slightly different point, but Vitali Milman, who works in high-dimensional convexity, likes to draw high-dimensional convex bodies in a non-convex way. This is to convey the point that if you take the convex hull of a few points on the unit sphere of R^n, then for large n very little of the measure of the convex body is anywhere near the corners, so in a certain sense the body is a bit like a small sphere with long thin "spikes".
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december 2016 by nhaliday
CS229T/STATS231: Statistical Learning Theory
Course by Percy Liang covers a mix of statistics, computational learning theory, and some online learning. Also surveys the state-of-the-art in theoretical understanding of deep learning (not much to cover unfortunately).
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june 2016 by nhaliday