boltzmann

Convolutional Boltzmann machines have been developed, and neural networks trained with Backpropagation can be pre-trained with generative models.

Since a deep belief networks are two-way like restricted boltzmann machines you can make hidden inputs generate valid visual inputs.

The simplest example is the RNN-RBM (restricted boltzmann machine) from the Theano deep learning tutorials [1].

Learning algorithms like Boltzmann Machines will learn a distribution over the inputs they see.

I used it to write a restricted Boltzmann machine in less than a page of code...

> He began by rewriting the complicated Boltzmann equation as the sum of a series of decreasing terms.

Maybe Ludwig Boltzmann or the entropy formula?

If you are interested in some of the details: The Navier-Stokes equations can be derived from the Boltzmann equation by applying a slight perturbation, expanding the result as a series, and taking the moments.

There may be some news in the generalized understanding of how to turn the atomic-level Boltzmann equation into an appropriate macroscopic equation, but the incompleteness of the Navier-Stokes equation is just not all that interesting.

I think it is unfair to give Shannon too much credit, Boltzmann and Gibbs came up with the exact definition Shannon gave S = -k \Sigma p_i ln(p_i), including a suitable information theoretic interpretation.

> You can get a lot of the way (just not to `exp((E_old - E_new) / Temp)`) by explaining the origin of the algorithm The exp is from the so-called Boltzmann factor [1] which (in somewhat simplified terms) describes the likelihood that a physical system in equilibrium at temperature T will transition from a state with energy E_old to one with energy E_new.

Basically you want to generate a Boltzmann-distribution of states in the function of "energy".

Even then, as was pointed out to me recently, it's not clear that the "always accept a lower energy, accept a higher energy proportional to the Boltzmann factor" yields a steady-state Boltzmann distribution -- especially because that would imply that the result is independent of the number of configurations with a given energy (the "density of states"), which seems surprising.

In either case, if each event has a chance of occurring during a particle collision, there must be some finite number of collisions beyond which some event such as "a fragment of Boltzmann Bread[0] appears in a critical component of the LHC" becomes more likely than "no world-ending event occurring in X collisions".

On a related note, "Dangerous Knowledge" is a great documentary "about four of the most brilliant mathematicians of all time, Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing, their genius, their tragic madness and their ultimate suicides".

You only simulate the Boltzmann-distribution and you measure interesting mean values and correlations.

Perhaps a more sensible system is one that uses \beta = -1/kT to measure temperature, where k is Boltzmann's constant.

Summary: Navier-Stokes cannot translate to Boltzmann, because Navier-Stokes is incomplete...