Posts with this logo use 
June 28, 2007
Kernels in Machine Learning II
Posted by David Corfield
We’ve had a slight whiff of the idea that groupoids and groupoidication might have something to do with the kernel methods we discussed last time. It would surely help if we understood better what a kernel is. First off, why is it called a ‘kernel’?
This must relate to the kernels appearing in integral transforms,
where , a function on the domain, is mapped to , a function on the domain. Examples include those used in the Fourier and Laplace transforms. Do these kernels get viewed as kinds of matrix?
In our case these two domains are the same and is symmetric. Our trained classifier has the form:
So the is . The kernel has allowed the transform of a weighted set of points in , or, more precisely, a weighted sum of delta functions at those points, to a function on .
Conference on Categories in Geometry and Physics
Posted by Urs Schreiber
In September, the following conference will take place:
Categories in Geometry and Physics
September 24-28, 2007
at MedILS, Split, Croatia,
organized by Zoran Škoda and Igor Baković.
Requests for possible attendance should be promptly emailed to zskoda at irb.hr.
This Week’s Finds in Mathematical Physics (Week 253)
Posted by John Baez
In week253, read about mysterious relations between the Standard Model, the SU(5) and SO(10) grand unified theories, the exceptional group E6, the complexified octonionic projective plane… and maybe even E8!
June 26, 2007
Colorings of Graphs
Posted by Urs Schreiber
Just heard a talk by Dmitry Kozlov on combinatorial algebraic topology, concerned mainly with the problem of coloring graphs.
I didn’t fully understand many of the points and unfortunately I didn’t have the chance of asking the speaker about more details afterwards. Hence this here is not a report of the talk, but rather a vague mentioning of some aspects and a couple of questions. It feels like much of what Kozlov had to say would be of quite some interest to an -categorical audience, if maybe just because there might be room to make that -categorical structure more explicit.
The main point is this:
given a graph , we are looking for colorings of its vertices such that no two vertices connected by an edge have the same color. What is the minimum number of colors, the chromatic number such that such a coloring exists?
If the graph is the dual of a triangulation drawn on a plane, the answer is given by the famous four color theorem to be The proof of that is not easy and in particular not elegant. This is symptomatic for the coloring problem more generally: exact answers are hard to come by. What one aims for instead are good estimates, lower bounds in particular, of .
One nice way to reformulate the problem of graph colorings, which all of Kozlov’s machinery is based on, amounts to realizing that a consistent (i.e. no equal-colored neighbours) coloring of a graph with -colors is the same as a graph morphism Here is what graph theorists apparently call the “complete” graph on vertices, which here I call the codiscrete graph on vertices: it has precisely one edge from each of its vertices to every other.
A morphism of graphs is much like a functor of the corresponding categories, but with the important constraint that every edge has to go to an edge: as opposed to the categories generated from them, the graphs have no “identity edges”. (And, by the way, I think Kozlov assumes throughout all graphs to be oriented, to have at most one edge for every ordered pair of vertices, and to have no edges from a vertex to itself.)
This is what makes the above reformulation possible: a morphism from a graph to a codiscrete graph will label each vertex of with one of the colors, and cannot assign the same color to two neighbouring vertices.
So then, the next step is to get a handle on the Hom-spaces in the category of graphs.
The crucial point of Kozlov’s work is that he realizes these Hom spaces as simplicial complexes. I don’t quite understand the precise definition in detail yet. You can find it for instance as definition 2.1.5 on p. 11 of
Dmitry N. Kozlov
Chromatic numbers, morphism complexes, and Stiefel-Whitney characteristic classes
arXiv:math/0505563v2.
So this looks like we are secretly talking about an -category of graphs now, or something like this. I don’t know.
The point is that using these simplicial complexes, the space of all possible graph colorings becomes more accessible. Kozlov proves powerful theorems using these complexes, in particular the Lovász conjecture:
For a graph , such that the complex is -connected for some integers and , we have
(Here is the graph corresponding to the -gon.)
The general strategy is to map these simplicial complexes functorially to topological spaces, and then use known obstruction theory of these spaces to determine if can be nontrivial at all.
In this context I was quite intrigued by what Kozlov said about spin structures. As described in section 3 of the above paper, there is a way to use the theory of Stiefel-Whitney characteristic classes to study the Hom-complexes of graphs. That sounds fascinating, since it seems to indicate a relation of the abstract coloring problem – which has the appearance of being “of academic interest” only (if you know what I mean) – to interesting geometric and maybe even physical questions. I’d like to better understand this.
But not right now. I need to call it a day.
June 25, 2007
Why Theoretical Physics is Hard…
Posted by Urs Schreiber
… while mathematical physics is easy, but much slower? Because for the former you need an oracle, without being able to strictly confirm what counts as an oracle.
In his latest piece #, E. Witten comments on the role of these oracles as follows:
We make at each stage the most optimistic possible assumption. Decisive arguments in favor of the proposals made here are still lacking. The literature on three-dimensional gravity is filled with claims (including some by the present author) that in hindsight seem less than fully satisfactory. Hopefully, future work will clarify things.
The proposal in question is about
[…] the problem of identifying the [2-dimensional quantum field theories] that may be dual to pure gravity in three dimensions with negative cosmological constant.
Jacques Distler has a summary here.
It is hard to find a precise definition even of what the word “dual” here – meant is holographic duality – is supposed to refer to.
Last time somebody guessed what the axiomatics behind this might be, he was told that this is the wrong axiomatics for what physicsists had in mind. Which may be true. But then it would be good to try to find the right axiomatics.
So I have my own proposal for what the holographic principle really is.
If a -dimensional QFT is a -functor from -dimensional extended cobordisms to -vector spaces, then a -dimensional QFT and a -dimensional QFT are “holographically related” if the former is the component map of a pseudonatural transformation “trivializing” the latter :
Looks pretty through the binocular like this, but turns out to be an intimidating mountain height as one approaches it.
With a little luck Jens Fjelstad will be vising Hamburg in a month, and we’ll continue to scale that rock.
Kernels in Machine Learning I
Posted by David Corfield
I keep feeling small twinges of familiarity between some of what goes on here at the Café and what I read in the machine learning literature. I’ll jot down a sketch of what’s going on and see if I can get the connection clearer in my head.
One of the most important developments in machine learning over the past 15 years has been the introduction of kernel methods. A kernel on a set is a function , which is symmetric and positive definite, in the sense that for any and any , the matrix is positive definite, i.e., for all . (Complex-valued kernels are possible.)
Another way of looking at this situation is to reformulate it as a mapping , where is a reproducing kernel Hilbert space, a function space in which pointwise evaluation is a continuous linear functional.
June 23, 2007
Some Recreational Thoughts on Super-Riemannian Cobordisms
Posted by Urs Schreiber
Recently, in the discussions about QFTs and representations of cobordisms categories, once again the following question came up:
What is the best way (the right way?) to conceive categories of Riemannian and super-Riemannian cobordisms? How is that extra metric structure best encoded?
Maybe we need to know: what is a Riemannian metric, really?
A priori, there seem to be a couple of alternative choices.
Is it an isomorphism of the tangent space with its dual? Or, more generally (also applicable to superspaces), an isomorphism of the algebra of derivations with its dual?
Or is it most naturally a spectral triple? We sure do expect to get spectral triples (for “target space”) from representations of our cobordisms (the Hilbert space coming from those over the objects, the Laplace or Dirac operato coming from the cylinder) – but do we also want to equip the cobordisms themselves with spectral triple data? How do we compose cobordisms equipped with spectral triples?
Or is it maybe less systematic? A special presciption best suitable for each dimension? A mere (super) 1-form on 1-dimensional cobordisms, for instance? A square root of the canonical line bundle in two dimensions? Something yet to be thought of in three dimensions?
Or is it maybe best thought of as a connection on the cobordisms, with values in the Poincaré Lie algebra (or one of its super versions)?
June 21, 2007
Faith and Reason
Posted by David Corfield
Back last October I mentioned a book – The Good Life in the Scientific Revolution: Descartes, Pascal, Leibniz, and the Cultivation of Virtue – which describes
how three key early modern scientists, René Descartes, Blaise Pascal, and Gottfried Leibniz, envisioned their new work as useful for cultivating virtue and for pursuing a good life. Their scientific and philosophical innovations stemmed in part from their understanding of mathematics and science as cognitive and spiritual exercises that could create a truer mental and spiritual nobility.
Someone closer to our times who considered the relationship between scientific and spiritual enquiry was the chemist turned philosopher Michael Polanyi. In Faith and Reason, The Journal of Religion, Vol. 41, No. 4 (Oct., 1961), pp. 237-247, Polanyi establishes the central concept of his epistemology:
Understanding, comprehension – this is the cognitive faculty cast aside by a positivistic theory of knowledge, which refuses to acknowledge the existence of comprehensive entities as distinct from their particulars; and this is the faculty which I recognize as the central act of knowing. For comprehension can never be absent from any process of knowing and is indeed the ultimate sanction of any such act. What is not understood cannot be said to be known. (p. 240)
June 20, 2007
Curvature, the Atiyah Sequence and Inner Automorphisms
Posted by Urs Schreiber
I am still trying to better understand -curvature and its relation to inner automorphism -groups.
A while ago David Roberts emphasized that the parallel transport functor of a principal -bundle with connection can be thought of as inducing a morphism of short exact sequences of groupoids: from the sequence of path groupoids of to the integrated Atiyah sequence of .
Here I take a fresh look at the curvature 2-functor of a parallel transport 1-functor from this point of view, emphasizing the role played by inner automorphisms in this construction.
Curvature 2-Transport, the Atiyah Sequence and Inner Automorphisms
June 19, 2007
Degeneracy
Posted by David Corfield
Eugenia Cheng and Nick Gurski have just put on the ArXiv a sequel to their
The periodic table in low dimensions I: degenerate categories and degenerate bicategories,
entitled
The periodic table of -categories for low dimensions II: degenerate tricategories.
Cohomology and Computation (Week 27)
Posted by John Baez
In the last of this year’s classes on Cohomology and Computation, we sketched a few of the simplest consequences of the bar construction:
- Week 27 (June 7) - Cohomology of algebraic gadgets. The bar construction "puffs up" any algebraic gadget, replacing equations by edges, syzygies by triangles and so on, with the result being a simplicial object with one contractible component for each element of the original gadget. Examples: Ext and Tor, group cohomology and homology, Lie algebra cohomology and homology. How Ext and Tor arise from the adjoint functors between the category of abelian groups and the category of modules of a ring. Free resolutions. Group cohomology as a special case of Ext. Group cohomology as the cohomology of the the classifying space .
Last week’s notes are here.