Generic optimization in infinitely large search domains

In 1995 and the following few years, Wolpert and Macready published a technical reports and a journal article on what has come to be called the No Free Lunch theorems. These theorems say that all optimization algorithms yield the same average performance in optimizing a completely random function.

Ice block stunt

An isreali illusionist, Hezi Dean, celebrated New Year by breaking the world record for staying inside a block of ice. New record: an impressive 66 hours.

Questionable military research on gauge invariance

Checking for new eprints, three papers by P. J. Cote and M. A. Johnson caught my attention: (A) New perspectives on classical electromagnetism (arXiv:0903.4104v2), (B) On the peculiarity of the Coulomb gauge (arXiv:0906.4752v1), (C) Groupthink and the blunder of the gauges (arXiv:0912.2977v1). The main point of these papers seems to be that the exploitation of gauge freedom in electromagnetism is questionable. Specific objections are raised to steps in standard derivations using Lorentz and Coulomb gauges. For example, the title of paper B refers to the non-locality of potentials in the Coulomb gauge and some remarks on the matter quoted from J. D. Jackson’s classic textbook.

The papers themselves border on crankery, which is a bit alarming since the authors are affiliated with the army, but are at least wrong in instructive ways. Electromagnetism is manifestly gauge invariant and there are no genuine causality problems in the Coulomb gauge since the non-local potentials are not physically meaningful quantities in and of themselves.

A condensed matter interpretation of the Standard Model

Condensed matter physics provides mathematical analogies with particle physics. Quasiparticles, i.e. particle-like excitations of a given ground state, often share many physical properties with more fundamental particles. Among the many attempts to find deeper insights into the Standard Model is the exportation of analogies in the other direction: from condensed-matter physics to fundamental physics. Here’s one such line of work:

I. Schmelzer. A condensed matter interpretation of SM fermions and gauge fields. arXiv:0908.0591

Abstract: We present the bundle Aff(3) x C x /(R^3), with a geometric Dirac equation on it, as a three-dimensional geometric interpretation of the SM fermions. Each C x /(R^3) describes an electroweak doublet. The Dirac equation has a doubler-free staggered spatial discretization on the lattice space Aff(3) x C (Z^3). This space allows a simple physical interpretation as a phase space of a lattice of cells in R^3. We find the SM SU(3)_c x SU(2)_L x U(1)_Y action on Aff(3) x C x /(R^3) to be a maximal anomaly-free special gauge action preserving E(3) symmetry and symplectic structure, which can be constructed using two simple types of gauge-like lattice fields: Wilson gauge fields and correction terms for lattice deformations. The lattice fermion fields we propose to quantize as low energy states of a canonical quantum theory with Z_2-degenerated vacuum state. We construct anticommuting fermion operators for the resulting Z_2-valued (spin) field theory. A metric theory of gravity compatible with this model is presented too.

A recent follow up paper focuses on neutrinos:

I. Schmelzer. Neutrinos as pseudo-acoustic ether phonons. arXiv:0912.3892

Abstract: Recently [arXiv:0908.0591] the author has proposed a condensed matter model which gives all fermions and gauge fields of the standard model of particle physics. In the model, the inertness of right-handed neutrinos is explained by an association with translational symmetry. We argue that this association may be used as well to explain the small neutrino masses. They appear to be pseudo-Goldstone particles associated with an approximate translational symmetry of a subsystem. Then we propose to explain the masslessness of SU(3)_c x U(1)_em with an unbroken SU(3)x U(1) gauge symmetry of the model. We also detect a violation of a necessary symmetry property in the lattice Dirac equation and present a fix for this problem.

Restart

After a busy few months it’s time to resurrect this blog from the blog graveyard. Like I hope at the end of every year, I hope I’ll have more spare time next year!

Convex analysis and thermodynamics

A previous post briefly reviewed convex analysis. Here I’ll review the application of convexity in basic thermodynamics.

Equilibrium states

The concept of thermodynamic equilibrium is a generalization of mechanical equilibrium, where all forces and torques cancel each other. Informally, the idea is that a system in thermodynamic equilibrium has stable, unchanging macroscopic properties, which may be characterized by an n-tuple of extensive variables.

I, AIXI

A new cool eprint has appeared:

J. Veness, et al. A Monte Carlo AIXI Approximation. eprint: 0909.0801

The question that defines the context for this article is: How should probabilities be assigned? One way, with much to recommend itself, is take them to be algorithmic probabilities or universal priors. Suppose one has observed the first N values of a discrete time series, maybe a byte stream, and wishes to predict or make a bet about the next value. Is there a general probability measure appropriate for all cases that fit this abstract setting and, if so, which one?

A priori bias in the Dembski-Marks representation

Dembski and Marks (2009b) recently published a minimalistic (and simplistic) representation of a search problem consisting of a search space and a distinguished target set (blogger reactions: 1, 2, 3, 4, 5, 7). From the discussion in the paper and two other articles it is clear that the authors’ object of study is related to, though not equivalent to, the issues raised by Wolpert and Macready’s No Free Lunch theorems. Despite its minimalistic character, the Dembski-Marks representation is not less restrictive than the Wolpert-Macready representation of a search/optimization problem. The Dembski-Marks representation, i.e., a distinguished target in a search space, can easily be introduced as an extra feature in the Wolpert-Macready representation. However, it is not possible to introduce the full Wolpert-Macready representation within the Dembski-Marks representation. Indeed, the absence of a constant, distinguished target set is a prerequisite for all No Free Lunch theorems. It is therefore interesting to estimate how much of a restriction it is to make the Wolpert-Macready representation conform to the Dembski-Marks representation.

Clusters and tree structure from genealogical data

Modern biology provides a wealth of interesting mathematical challenges in the modeling and reconstruction of evolution. A new eprint explores the theoretical prospects for defining a phylogenetic tree structure, despite complications like lateral gene transfer, hybdridization and the difference between gene and species trees:

A. Dress, et al. Species, Clusters and the ‘Tree of Life’: A graph-theoretic perspective. eprint: 0908.2885

Entropy decrease results in memory loss

I found a fun arrow-of-time paper in PRL, arguing that dynamical decreases in the entropy of an isolated system are not at all impossible, just impossible to remember!

L. Maccone. Quantum Solution to the Arrow-of-Time Dilemma, Phys. Rev. Lett. 103:080401 (2009), arXiv:0802.0438

Abstract: The arrow-of-time dilemma states that the laws of physics are invariant for time inversion, whereas the familiar phenomena we see everyday are not (i.e., entropy increases). I show that, within a quantum mechanical framework, all phenomena which leave a trail of information behind (and hence can be studied by physics) are those where entropy necessarily increases or remains constant. All phenomena where the entropy decreases must not leave any information of their having happened. This situation is completely indistinguishable from their not having happened at all. In the light of this observation, the second law of thermodynamics is reduced to a mere tautology: physics cannot study those processes where entropy has decreased, even if they were commonplace.

While interesting, I wonder if this approach is really promising. Basically, the author proves his Eq. (2), stating that the sum of entropy changes in a system A containing the observer and another system C equals the entropy change in a reservoir plus the change in the total amount of correlations (i.e., mutual information) between A and C. In the most interesting case when the entropy of the reservoir is constant, any entropy decreases in A and C must come at the expense of decreasing the amount/strength of correlation between A and C. This, according to the author, means that an entropy decrease in C automatically results in the observer in system A losing any memories or records (a memory/record is a kind of correlation) of C’s previous higher entropy state. However, very little correlation (a very small memory/record) is needed to retain, say, just the numerical values of C’s entropy at different times. Entropy decreases in C could, for all we know, come at the expense of other, more detailed, correlations between A and C, while leaving memories of measured entropies intact. Thus, it seems that much more work is needed to actually establish that entropy decreases are unobservable (due to being impossible to remember). The Phys. Rev. Focus commentary also hints at this problem.