User doublejay - MathOverflow

Good papers/books/essays about the thought process behind mathematical research

2010-05-13T17:49:36Z

Papers in mathematics are generally written as if the major insights suddenly appeared, unbidden, in a notebook on the researcher's desk and then were fleshed out into the final paper.

While this is great for finding out about results, it's terrible for finding out about how they were arrived at.

What I want are papers, books or essays written by researchers about their work on problems, especially if they describe the evolution of their work on a specific problem (Polya's writings on problem-solving are great, but not what I'm interested in). I'd like to know how hundred-page treatises on problems unsolved for decades are born.

Is there any straightforward way to substitute for Gaussian/Brownian assumptions in financial mathematics?

2011-02-01T18:01:50Z

A huge amount of financial mathematics assumes Gaussian distributions of risks and Brownian movement of prices. What efforts have there been to replace these with heavy-tailed distributions? For example, could Black-Scholes be adjusted to assume heavy-tailed distribution of price movements, or is this too mathematically difficult?

Graphs with fractal properties?

2009-11-17T05:22:42Z

For the purposes of a research project, I am wondering if there are any resources on graphs with fractal properties, by which I mean self-similarity in particular. For instance, imagine a graph where nodes could be transformed into subgraphs that were the same as the larger graph, and their nodes could be transformed likewise, etc. I don't mean a graph that is literally exactly like that - but it should have self-similarity at different levels as if it had been created that way, with maybe a little randomness thrown in afterwards.

I was told that Expander graphs were something like what I was looking for, but from what little I understand of their definition, they seem more related to small-world theory than to what I'm looking for.

Edit: I'm because I'm trying ot figure out a way in which representations of social and geographic networks of people could be compressed, probably with significant loss but maintaining basic properties.

What are the Applications of Hypergraphs

2010-02-01T23:47:10Z

Hypergraphs are like simple graphs, except that instead of having edges that only connect 2 vertices, their edges are sets of any number of vertices. This happens to mean that all graphs are just a subset of hypergraphs.

It strikes me as odd, then, that I have never heard of any algorithms based on hypergraphs, or of any important applications, for modeling real-world phenomena, for instance. I guess that the superficial explanation is that it's a much more complex structure than a regular graph, and given this and its generality it's harder to make neat algorithms for, but I would expect there to be something!

Has anyone heard of a hypergraph-based algorithm, or application? It perplexes me that ordinary graphs can be so wonderfully useful, but their big brothers have nothing to offer.

Solving NP problems in (usually) Polynomial time?

2010-01-01T07:58:37Z

Just because a problem is NP-complete doesn't mean it can't be usually solved quickly.

The best example of this is probably the traveling salesman problem, for which extraordinarily large instances have been optimally solved using advanced heuristics, for instance sophisticated variations of branch-and-bound. The size of problems that can be solved exactly by these heuristics is mind-blowing, in comparison to the size one would naively predict from the fact that the problem is NP. For instance, a tour of all 25000 cities in Sweden has been solved, as has a VLSI of 85900 points (see here for info on both).

Now I have a few questions:

1) There special cases of reasonably small size where these heuristics either cannot find the optimal tour at all, or where they are extremely slow to do so?

2) In the average case (of uniformly distributed points, let's say), is it known whether the time to find the optimal tour using these heuristics is asymptotically exponential n, despite success in solving surprisingly large cases? Or is it asymptotically polynomial, or is such an analysis too difficult to perform?

3) Is it correct to say that the existence of an average-case polynomial, worst-case exponential time algorithm to solve NP problems has no importance for P=NP?

4) What can be said about the structure of problems that allow suprisingly large cases to be solved exactly through heuristic methods versus ones that don't?

How important are publications for undergrads?

2010-01-08T06:28:03Z

I have heard vastly conflicting statements about whether undergrads applying for PhD programs should have published already, or what level of research will be expected of them. Looking at CVs of some of my school's professors, almost none of them seem to have publications from earlier than the 2nd half of their graduate studies, meaning they spent most of their time before getting their PhD without any publications or those that they had weren't worth listing, in their eyes.

Obviously, I'm going to try to get the best experience I can as an undergrad, and I hope that means getting published research, but in every area I've dipped my toe in, from probability to dynamical systems to complexity theory, the sheer amount of additional knowledge I'd need to understand even a upper-level graduate text seems intimidating.

When did you first publish, and what sort of research experience (if it's something other than publishing an article) should an undergraduate aiming for a PhD have?

Disclaimer: I'm an undergrad in CS, pretty average or maybe above average in my progress so far, and I'd like to make a career in researching some of the theoretical (and obviously math-heavy) parts of computer science, rather than software or interface.

Statistical physics of string theory

2010-07-14T03:49:04Z

Is there any connection between statistical physics and string theory, or a statistical interpretation of string theory, perhaps? I mean, the way electromagnetic forces and thermodynamic laws are described as emerging are all describable by statistical physics formulations. String theory accounts for gravity (I don't really know where the other two fit in), but does it do so (or is there any effort to develop a formulation) in a way reminiscent of statistical physics?

If this is a silly question, please explain why.

Are there intuitive/classically algorithmic analogues to Semidefinite programs on networks?

2011-12-07T23:20:54Z

Many network optimization algorithms, including shortest path, push-relabel, augmenting path, etc, actually have an interpretation in terms of linear programming.

A famous application of semidefinite programming is the max-cut approximation. Does this optimization algorithm, or any other on networks, have a network interpretation, a la augmenting path?

L1-regularized Least Squares on a matrix with Toeplitz Blocks (not block-Toeplitz)

2011-10-24T18:58:43Z

I am trying to speed up a sparse signal recovery algorithms.

My sensing matrix is a set of Toeplitz Blocks, M = [T1,T2,T3,...,Tk]

The objective is min ||Mx - b||_2^2 + ||x||1

What I'm actually doing is trying to encode an image with multiple patch-like bases, each of which can be centered anywhere on the image.

Is there any structure of my M = [T1,...,Tk] sensing matrix that I can take advantage of? For instance, can I efficiently compute (MM')^(-1/2) or do any other useful structure I can take advantage of to speed this up beyond naive applications (adapted to us the convolution operator) of other sparse recovery algorithms?

Mathematical Programming with other Algebras than Linear

2011-10-05T06:05:18Z

Linear Programming is strongly entwined with linear algebra, as are many of its generalizations under the heading of mathematical programming / convex optimization.

What analogies are there for convex optimization techniques outside of vector spaces dealt with in linear algebra. For example, Gaussian Elimination is generalized by Buchberger's algorithm for finding Groebner bases (or so I'm told); is there any algorithm that has a relationship with Buchberger's analogous to the Simplex Method's relationship with Gaussian Elimination?

Estimating the fractal dimension of a point cloud

2011-08-29T00:36:19Z

I have finite set of geolocation point data, and I'd like to estimate the fractal dimension. I know there are several ways to do this, and some of them give different numbers. What is the most appropriate fractal dimension to look at and what method do you recommend I use to estimate it numerically?

Thanks

Probabilistic (and other mathematical) methods of physics without the physics?

2011-07-25T05:49:07Z

Many of the methods of physics are vastly more general than their use in that discipline. For example, information theory overlaps with a lot of statistical mechanics, and the latter actually developed first. ET Jaynes wrote a famous paper illustrating the connections. However, each is comprehensible without the language and intuition of the other (though I do not deny that a richer understanding comes from knowing both).

What other methods of physics (particularly those with a statistical or computational bent) have interpretations (Please mention useful introductory texts!) that are completely physics free? I understand that various field theories meet this criterion; any good non-physics introductions?

Optimizing directly on the eigenspectrum of a matrix

2011-06-16T19:01:14Z

I have an application where I want to the eigenvalues of the graph to be involved in the objective and constraints in a flexible way (moreso than just the nuclear or frobenius norm). Whats a good survey or intro source to this sort of optimization?

Is all non-convex optimization heuristic?

2010-07-19T20:42:50Z

Convex Optimization is a mathematically rigorous and well-studied field. In linear programming a whole host of tractable methods give your global optimums in lightning fast times. Quadratic programming is almost as easy, and there's a good deal of semi-definite, second-order cone and even integer programming methods that can do quite well on a lot of problems.

Non-convex optimization (and particularly weird formulations of certain integer programming and combinatorial optimization problems), however, are generally heuristics like "ant colony optimization". Essentially all generalizable non-convex optimization algorithms I've come across are some (often clever, but still) combination of gradient descent and genetic algorithms.

I can understand why this is - in non-convex surfaces local information is a lot less useful - but I would figure that there would at least be an algorithm that provably learns for a broad class of functions whether local features indicate a nearby global optimum or not. Also, perhaps, general theories of whether and how you can project a non-convex surface into higher dimensions to make it convex or almost convex.

Edit: An example. A polynomial of known degree k only needs k + 1 samples to reconstruct - does this also give you the minimum within a given range for free, or do you still need to search for it manually? For any more general class of functions, does "ability to reconstruct" carry over at all to "ability to find global optima"?

What function smoothly interpolates between the identity and exponential (or log and identity) functions?

2011-05-27T02:58:39Z

Or rather, what function can be parametrized with some value t in [0,1] such that f(x, t= 0) = x, f(x, t = 1) = e^x, and f(x, 0 < t < 1) is a principled interpolation between those two, kind of like the gamma function is a principled interpolation for the discrete factorial.

Obviously, many functions fit the bill, like f(x,t) = x^(1 + t*(x/ln(x)-1) ), but that seems kind of arbitrary.

What mathematically useful/elegant function exists?

What can be said about an infinite linear chain of conjugate prior distributions?

2011-04-30T03:59:32Z

We can sample a discrete value from the multinomial distribution.

We can also sample the parameters of the multinomial distribution from its conjugate prior the dirichlet distribution.

Since the dirichlet distribution is part of the exponential family, it too must have a conjugate prior distribution in the exponential family.

I hope you see where I'm going: what happens as this chain of priors is taken to infinity?

For a simpler example, what happens with the self-conjugate Gaussian distribution?

Best Practices for Learning Mathematics (especially in the classroom)

2011-01-21T19:14:09Z

I am an undergraduate CS major with strong interests in applied math and theoretical computer science. In the past, I've done reasonably well grade-wise in all math-related (that is, pure math, applied or theoretical CS) classes, but I feel that I still haven't taken away as much as i could have from most.

As people who have often taught math courses and had to deal with the inevitable fact that no lecture will be universally effective, what are your suggestions for how I (as a student) can best learn in these classes.

A few problems I've experienced regularly:

When professors try to present long and difficult proofs on the blackboard. I always find it ridiculously hard to understand proofs in real time or to understand verbal and visual explication of the proof simultaneously. I have to look the proof up in a textbook, and the comprehensibility of textbook proofs varies widely.

More generally, accessing the "kernel" of the proof that really makes it comprehensible is sometimes difficult, especially when it's presented more formally. I tend to think of proofs in terms of algorithms, and proofs that don't fit this well tend often evade me.

Definitions, even, (especially in pure math) tend to blend together and become obscure. I've re-learned the basic definitions of probability waaaay too many times.

How much reading do you do before you attack a problem?

2010-07-23T01:21:38Z

When going off on a tangent from your regular area, where, presumably, you have such mastery of all cutting-edge research from your routine reading that you hardly need to do any extra (if this is false, please correct me), how much do you try to familiarize yourself with that area before beginning to directly attack your problem? Do you read just a few canonical papers and surveys, look thoroughly over a dozen and glance at a couple dozen more, or do enough to write a whole survey article of your own?

Compressed Sensing with an Unusual Basis

2010-07-18T07:33:37Z

I'm wondering if compressed sensing can be applied to a problem I have in the way I describe, and also whether it should be applied to this problem (or whether it's simply the wrong tool).

I have a large network and plenty of data on each node. I want to find a set of communities that explain most data similarity and track these communities over time. In a compressed sensing formulation this amounts to the following:

-My graph's representation basis is a weighted set of communities, where each community is a subset of the set of all nodes (candidate communities can be narrowed down to a tractable number rather easily)

-Different feature measures (e.g. bigrams, topic profiles) serve as my sensing basis, with correlations between community membership and features serving as the coefficients of my measurement matrix. The big assumptions, that my feature measurements have the Restricted Isometry Property, that similarity is incoherent with community, and that similarity is a linear combination of community, are all almost certainly incorrect, however they seem plausible approximations to within (possibly significant) noise.

Ideally, I can use this strategy to describe my network as a collection of communities and to track over time the prominence of these communities. I wonder, however, if there isn't some straightforward bayesian method that I'm overlooking.

Misc Questions about Compressed Sensing:

i) If my measurements are not linear combinations of my representation basis, but at least convex, then can I still usefully use compressed sensing? Edit: In the above case, for instance, the generally accepted submodularity property of networks means that a node's membership in additional communities that correlate positively with a feature have reduced effect. In this particular case it might be best to transform everything to logs, but in general this option might not be viable.

ii) What is the meaning of the dual of basis pursuit?

iii) How does one avoid basis mismatch in general? You choose your representation basis elements beforehand, so how do you make sure they're capable of representation?

Edit iv) If your measurements are naturally represented as vectors, rather than scalars, is there any way to represent this other than counting each component of the vector as a separate measurement (though I suppose this works fine in general, if you have enough information about each component and everything is linear).

Answer by DoubleJay for How to generate a net on a 8-dimensional sphere

2010-07-19T17:39:56Z

In general, for generating extra-regular but not-too-regular distributions of points (in a technical sense, "low discrepancy", meaning that the variance in the length of gaps between points is smaller than a uniform distribution), you can use a class of methods called quasi monte carlo methods. There are libraries in MATLAB.

http://en.wikipedia.org/wiki/Quasi-Monte_Carlo_method

http://www.mathworks.com/matlabcentral/fileexchange/17457-quasi-montecarlo-halton-sequence-generator

Though if you want a totally uniform set of points, these won't help you.

Good sources for linear algebra for convex optimization and graph analysis?

2010-07-11T11:17:22Z

What are some good sources for linear algebra for convex optimization and graph analysis? In Particular, is Gilbert Strang's MIT course suitable, or some other online course? I prefer online courses (video or lecture notes) to books because they're usually much better organized.

I'm at the undergrad level, but interested in doing research in machine learning and network theory, which use these two things respectively. I've taken one (basic, computationally-oriented: i.e. we didn't learn anything about basis, vector space or the meanings of linear mappings, but instead learned a lot about iterative methods Householder reflections) course on linear algebra, and all the time I find myself stymied by matrix formulations of optimization problems and assertions about eigenvalues of graphs.

I want to become fluent in it for these purposes - I don't plan to go further in the pure math direction, just engineering, so keep that in mind (though a few proofs and abstractions won't kill me, and are indeed welcome.

Side Question: Can you have a useful matrix of quaternions (since you can have complex as well as real matrices) or is that just a silly idea, for whatever reason?

Shortest Paths on fractals

2010-05-25T05:59:46Z

How can one find shortest paths between 2 specified points on fractals, or (since I'm pretty sure this is quite complicated) make useful generalizations about them?

Since the above question is broad, how about this one: What is the general formulation (in a direct equation, recursive formulation, or other form) for distance between 2 points on the sierpinski carpet?

Obviously for some fractals all points are infinite. Identifying these is often easy, but are there any edge cases where it's hard to decide whether all paths are infinite length? And if so, how does one decide?

Edit: This question was inspired, by the way, by this thread on a different website (where it became clear that it was beyond the average math knowledge there). http://echochamber.me/viewtopic.php?f=3&t=40348#p1618494 That particular post shows paths(whose presence is recursive) in the carpet.

Visual representation of mathematical research interrelationships

2009-10-24T07:51:38Z

I remember seeing a visualization in the form of a 2d (nodal) graph of all areas of academia, with math, physics and engineering over in one section, connecting in an arc to the central area of economics and statistics which was intermediate between it and the humanities. Psychology and sociology were in their own cluster between the humanities and econ/stats, while biology and medicine formed a huge cloud that branched towards almost everything, though it was further from the humanities than the sciences.

I'm afraid I don't have a link, which would illustrate it much better than I can.

Anyways, I wonder if there's something similar to this for mathematics alone, based on the specialities of co-authors, the references in research papers, common sense, etc. I'd be very interested to see what it looks like.

Algorithm for generating a size k error-correcting code on n bits

2010-07-04T18:18:29Z

I want to generate a code on n bits for k different inputs that I want to classify. The main requirement of this code is the error-correcting criteria: that the minimum pairwise distance between any two encodings of different inputs is maximized. I don't need it to be exact - approximate will do, and ease of use and speed of computational implementation is a priority too.

In general, n will be in the hundreds, k in the dozens.

Also, is there a reasonably tight bound on the minimum hamming distance between k different n-bit binary encodings?

Does the Axiom of Choice (or any other "optional" set theory axiom) have real-world consequences?

2010-06-08T04:31:05Z

Or another way to put it: Could the axiom of choice, or any other set-theoretic axiom/formulation which we normally think of as undecidable, be somehow empirically testable? If you have a particular scheme for testing it, that's great, but even the existence or non-existence of a proof regarding potential testability is wonderful.

How about something a little simpler: can we even test the Peano axioms? Are there experiments that can empirically verify theorems not provable by them?

This is a slightly fuzzy question, so to clarify what I mean, consider this: the parallel postulate produces good, useful geometry, yet beyond its inapplicability to the sphere, there's evidence to suggest that the universe is actually hyperbolic - this can be considered an experimental evidence "against" the parallel postulate in our universe.

Edit: Thanks to all the people who answered - I understand the concerns of those who don't like this questions, and I appreciate all those who answered a more modest interpretation that I should, in retrospect, have stated. That is, "Is the axiom of choice agreeable with testable theories of mathematical physics, is it completely and forever irrelevant, or is it conceivably relevant but in a way not yet known," to which I got several compelling answers indicating the former.

Cluster-preserving and distance-maximizing embedding into Hamming Space?

2010-06-22T07:40:50Z

I have a set of data, each instance in the real $[0,1]^{d}$. However, it's actually all in a relatively small range around 0.5, clustered into classes in even smaller ranges. The actual origin of the data is the output of an untrained neural network, but don't worry about that.

The distribution is correlated between variables in an unknown but not ridiculously ill-conditioned way, and I have no guarantee that the intra-cluster distance is dramatically smaller than the inter-cluster distance (it may be of the same order of magnitude).

I would like a method for embedding into d-dimensional Hamming Space that:

1) Preserves clusters as best as possible

2) Maximizes inter-cluster distance

3) Maintains relative inter-cluster distances

4) Minimizes intra-cluster distance

In that order.

The obvious solution is some sort of machine learning method, but since i'm actually trying to apply this to improve a different machine learning method, I want a method that's rather quick and simple instead. What I was doing was just rounding, but that failed #2 spectacularly. Rounding based on centroids or mediods of the data instead would be a little more sophisticated, but still wouldn't do a great job of #2, #3, and sometimes #1.

Mathematics of the Anthropic Principle

2010-06-20T16:39:44Z

A form of the anthropic principle is as follows: "We can observe the universe only because we can exist within it in some way such that we can observe it, and it exists such that we can observe it."

What mathematical consequence does this have? I know it's broadly a problem of Bayesian probability, and we must consider all that we see from the perspective P(A|B), A = some aspect of observed reality, B = we think, therefore we are.

Can this be formulated in some useful and general way to answer questions about the universe, existential, cosmological or otherwise, or do the mathematics here give us little information?

NOTE: I know that the anthropic principle is often stated in a much more specific way and looked at from the perspective of cosmology, but that's not what I'm looking for here.

Edit: To clarify the mathematical content of this question I'll give two examples (one from a comment below).

1) I've seen claims like "the anthropic principle indicates that we most likely live at a time such that half of all people that have ever been born have been born". I want to know if a statement like this is at all reasonable or not.

2) Consider it in these (not entirely sufficient) terms: You have a vague outline of a set of prior distributions in addition to some error-prone observations whose errors depend on the prior distribution. How can you glean information about the prior distribution.

Is perfect play possible in continuous rock-paper-scissors? game "step size" vs. "acceleration"

2010-06-04T15:21:25Z

The first part of my question is simple: Is every game continuous in time and strategy-space also a game of perfect information with a good equilibrium? For example, consider rock-paper-scissors. The discrete version has no nash equilibrium - a perfectly uniform random mixed strategy is the best option.

Continuous rock-paper-scissors, by contrast, allows players to move at some limited velocity (consider 2 cases, acceleration is limited and acceleration is infinite) through a "strategy space" s.t. R+P+S = 1, and (0.5, 0.5, 0) vs. (0,1,0) returns 0.5 to player one and -0.5 to player 2, while (1,0,0) returns 1 to player one, -1 to player 2. To avoid the "go directly to the middle" strategy, it's fine to remove (1/3,1/3,1/3) or some disk around it from the strategy plane.

So, is continuous RPS effectively a game of perfect information?

For a more dramatic example, consider the stock market as a game. If it were continuous, would randomness essentially be removed? Would a player also need to explicitly know the strategies of all other players as individuals, or only the end result of those strategies (i.e. value of stocks at a given point in time) in order to play perfectly?

For a more realistic example, consider a hunt between a dog and hare. Strategies for them are the direction they choose to run in the pursuit. The dog has reflexes r, the time it takes him to notice the hare's change in direction. The rabbit has acceleration a. Ignore the dog's acceleration for now. As r*a becomes extremely small (i.e. the dog's reflexes are swift relative to the hare's acceleration), does this effectively converge to a continuous, perfect-information game (specifically the game of the homicidal chaffeur), or is the difference still important? Specifically, suppose that the dog can only make decisions on pursuit directions at increments equal to r - I don't want it to be a continuous game with a lagging signal.

Learning statistical mechanics for non-particle phenomena

2010-05-23T06:54:17Z

I'm interested in various areas of complex systems, and I often come across articles like these:

http://arxiv.org/PS_cache/cond-mat/pdf/0106/0106096v1.pdf

http://arxiv.org/abs/cond-mat/9804180

The main points are accessible in each (much less so the 2nd one though), but I'd like to be able to understand this sort of writing deeply, or even be able to do it myself.

What sort of studies would I need to undertake? Would a standard thermal/statistical physics class do it, or do I need something more drastic? Are there any resources along the lines "statistical physics for the social scientist" that are still rigorous and high-level?

(There's a question about "statistical physics for the mathematician", but this is almost exactly the opposite of what I need, funnily enough").

Answer by DoubleJay for Major mathematical advances past age fifty

2010-05-23T07:51:28Z

Christos Papadimitriou is in his late 50's now (I can't find his exact age, which is a little strange), and in just the past few years he's done major work in algorithmic game theory, a field at least somewhat removed from the one he made his career in. Technically, he's a theoretical computer scientist - I say this is close enough though.

Comment by DoubleJay

2012-04-11T00:49:07Z

Do np-complete problems include the stable polyamorous marriage problem?

Comment by DoubleJay

2011-10-24T23:05:31Z

You're right, it's kn x n, or some small factor difference depending on how I deal with edge effects. I'm using iterative methods, like Iterative Shrinkage and Thresholding, and I've found a paper that uses a greedy method. None of this takes advantage of the Toeplitz structure except through the relative efficiency of applying the convolution. I'll take a look at that link, thanks.

Comment by DoubleJay

2011-07-29T16:47:01Z

This is interesting because I originally learned about Quaternions solely in their context as an interesting mathematical construct, a non-commutative algebra of historical interest.

Comment by DoubleJay

2011-05-27T15:49:57Z

This is a neat answer. In its current form it is waaaay too complicated for my application, but the idea of using power series approximation is a good one. Thanks!

Comment by DoubleJay

2011-05-27T03:47:03Z

Unfortunately, that will probably have to be determined empirically. The application, however, is interpolating between additive units and multiplicative units for a neural network. Igor's suggestion has the virtue of simplicity, and would probably work well for a sparse vector of ts, actually. Upvoting it.

Comment by DoubleJay

2011-05-27T03:19:10Z

To be quite honest, any easonable-looking function that's smooth and differentiable will suit my purposes. I want to interpolate between additive and multiplicative weights for a machine learning application.

Comment by DoubleJay

2011-05-27T03:11:42Z

I don't expect there to be just one, but perhaps there is a particular one that is particularly useful. The Gamma function is not the only function interpolating between factorials, but it is by far the most used, and in many senses that "natural" one.

Comment by DoubleJay

2011-01-22T02:11:06Z

Even though this topic was closed, this answer is exactly what I was looking for. Thanks!

Comment by DoubleJay

2011-01-21T20:53:37Z

I would disagree that it's off-topic. Many topics here are professors asking how they should teach; why not a student asking how he should learn?

Comment by DoubleJay

2010-07-23T16:29:43Z

The drift of the responses so far is a lot more "dive in head first, then read while you work" than I expected. That's actually really nice to know - I've been reading myself to death on a problem, and still feeling like I'm missing out on a vast amount of important facts. You guys have inspired me to get my hands dirty!

Comment by DoubleJay

2010-07-20T14:15:55Z

I glanced at the course you mentioned, and it seems very interesting. Can you tell me in what circumstances these techniques are competitive with heuristics like particle swarm optimization for applications?

Comment by DoubleJay

2010-07-20T04:38:30Z

Nvm the edit - it's null.

Comment by DoubleJay

2010-07-19T21:16:43Z

Yes, or if we can't do that, then can we at least make it easier to iterate through local minima, to smooth our surface in a productive way, or guarantee bounds on how densely we need to sample from the search space?

Comment by DoubleJay

2010-07-19T20:53:07Z

Alright, I changed a couple things - these changes actually do make the question more clear, so thanks for suggesting that.

Comment by DoubleJay

2010-07-19T19:47:19Z

You might want to look into something called discrete differential geometry, used mainly for computer graphics. It doesn't provide a formula, but it could help you measure your surface. In general, something like bregman divergence from a sphere would be good, I think. (Though I don't know exactly how you'd measure it).