Permutation symmetry and finding a `noise model’

If we have a time series of data points \{ x_i\}, how can we tell if these come from some stationary distribution or if there is some secular process coupled to an uncertainty inducing step? Many statistical theorems start out by assuming stationarity. Suppose we ask for the probability of a model P(f,\eta|x_i) where x_i = f(t_i) + \eta_i,  t_i is the `time coordinate’ of the i^{\rm th} data point, and \eta comes from a stationary distribution. How do we disentangle f and \eta? By definition, the f likelihood should be invariant under replacing \eta_i \rightarrow \eta_{\sigma(i)}, \sigma \in P_N, the permutation group on N data points. In other words, P(f,\eta|D) = P(f,\eta_\sigma|D), but then this is going to require P(D|f,\eta) = P(D|f,\eta_\sigma) which suggests the trivial solution of just defining P(D|f,\langle\eta\rangle) as the product over \sigma, which seems a bit heavy-handed. It is easily achieved, of course, but this will be a computational nightmare. If this could be done, I think it would be a pretty strong constraint on what \eta and f could be.

One way to try to do this would be to do Monte-Carlo sampling over P_N in calculating the cost function at each step of the MCMC for finding f, \eta. We really want to evaluate something like

- \log P(x|f,\langle\eta\rangle) \equiv \sum_{\sigma\in P_N} \sum_i (x_i - f(t_i) - \eta_{\sigma(i)})^2,

and I think that for N large enough the sum over \sigma could be approximated by a probabilistic sampling over some permutations. The worry here is that \eta will come out all constant, so the noise model preferred will be the trivial distribution, so all the variation will try to be fit by f. The other extreme is that f will be a constant and all the data will be fit by \eta.

The quick way might be to use something like singular spectrum analysis on each permuted summand and demand that the SSA interpolations all agree: Pick \eta, compute SSA for some sample of permutations, vary \eta in a standard MCMC to sample over possible \eta. The end result should be a de-noised prediction for the actual functional dependence f.


Ray Solomonoff’s universal distribution

Everyone knows about Kolmogorov complexity. Kolmogorov addressed randomness and came up with the Kolmogorov complexity Cof a string  (defined as the length of the shortest program that can reproduce the string) as a measure of its randomness. Solomonoff, two years before Kolmogorov, looked at the problem from a much deeper perspective in my opinion. He asked, in drastic paraphrase: How should you weight  different programs that could generate a given string? And the two are related because it turns out that the function that Solomonoff picked is C. [So why isn’t it called Solomonoff complexity, though Kolmogorov referenced Solomonoff in his paper? Because Kolmogorov was already famous.]

What I would like to understand is the following: How do you go about generating programs that could reproduce the string? Induction is a hoary field, but most of the work is done assuming that you have to decide between different programs, and very little, in proportion, addresses the question of how to go about generating such programs. That is actually a problem because the search space is, to put it mildly, large. Is there a fundamental process that is the optimal manner to induce programs given data? This must itself be a probabilistic process. In other words, the program output match to the desired string must be allowed to be imperfect, so the search space summation is not just over programs that reproduce a given string S

\sum_{{\rm programs\ reproducing\ } S} \exp(-{\rm length}({\rm program}))

but rather weighted in some way to balance fitting S and reducing program length. Why does everything involve a balance between energy and entropy? Because there is nothing more fundamental in all of science.

So I think there should be a way to write something like

\sum_{S',{\rm programs\ producing\ } S'} \exp(-{\rm length}({\rm program})) \exp(- {\rm mismatch}(S,S'))

and then the question is: What is the universal form of the mismatch function? An interesting point here is that this process might work even with noisy S since if the correct string is 0101010101 but you read S= 0111010101, then you’ll find a very short program that can reproduce the string S' = 0101010101, and since the mismatch is only one bit this process would autocorrect your initial misread string.

We want symbols and rules for manipulation such that the resulting stream contains the known sequence of symbols. In other words, an inverse Godel problem: Given a stream of symbols, find a set of rules so that this stream is a proof.

Living on the edge

Given a finite set of points, \{x_i\}, a standard problem in Bayesian inference is to figure out likely probability distributions, if any, these points could have been drawn from. In particular, supposing that we actually know that these points have no sequence dependent variation (to which I’ll come back later), is the distribution likely to have finite or infinite support? I don’t know the statistical literature very well, but whatever I can find on density estimation  assumes the density has infinite support and then transforms to finite support before inferring the likely density. This cannot be correct: A priori we don’t know if the density has finite or infinite support. If it has finite support and we assume infinite support, then our inferred density will be non-zero in regions where it should be zero. This support determination subproblem is a problem which seems to me to require a balance between smoothness and complexity, but somehow all the action is really taking place at the edges of the distribution, i.e. in the regions where there are very few data points. There is no point to studying this problem as a perturbation of the infinite number of data points limit as with an infinite number of data points, we certainly know the distribution’s support.

What do we expect? The probability density is a field confined to a certain domain. The energy of a field configuration is a measure of how well it conforms to the observed data points, and the entropy is the a priori expectation of smoothness or continuity that translates into a term in the Radon-Nikodym derivative proportional to some measure of how big derivatives are. Then in a region with no data points this entropy term is essentially weighting vacuum fluctuations. In other words, is there something like a Casimir energy trying to push the boundaries apart, with the potential well set up by the data points responsible for keeping the boundaries closer? And for an infinite distribution, there should be a marginal mode allowing the boundary to fluctuate without changing the free energy.

Suppose we think in terms of a Thomas-Fermi picture. The data points are fixed nuclei and the density is a cloud of electrons with kinetic energy. The question is: What is the extent of the cloud? And how does it depend on the distribution of the fixed nuclei? The Thomas-Fermi energy functional takes into account the kinetic energy of the electrons, normalized so \int \rho = N_e:

E_{TF} = \int d^3r c \rho(r)^{5/3} - {1\over 2}(N/N_e)^2 \int\int \rho(x) G(x,y) \rho(y) + (N/N_e) \int\sum_i^N \rho(x) G(x,x_i)

where the last term is the attraction to data-points and the middle term is the mutual repulsion of electrons. G is defined so that G(x,y) increases as |x-y| increases. What is interesting here is that we can naturally consider two limits: N = number of data points, and N_e = the number of `bins’ or the resolution of the inferred pdf because we have the freedom to change the ratio of the electron charge to the data point charge, and the TF theory becomes exact in the limit of large numbers of data points, which is in fact what we want.

Another amusing thing about the TF theory is that it actually predicts that molecules will fly apart. This would seem to be a deal breaker because in the way I’m trying to apply it here, that would make the pdf a sum of disjoint pdfs around each of the data points. However, this is only when you put in a term involving nucleus-nucleus repulsion, which I did not do here. (Little nugget buried in the dim recesses from Barry Simon’s or Elliot Lieb’s lectures, misspent youth and all that.) Here there is no reason to put in such a term and therefore the TF pdf should not fly apart. Of course, the dimension dependence of the Green function G is amusing. For D=1, the Coulomb potential leads to a constant electric field. So if we’re thinking in terms of electrostatic analogies, we have perfect screening provided enough electrons are around each data point, and the only thing spreading them out is the density \rho^{1+2/D} term, which doesn’t want accumulations. This is not electrostatic repulsion, just the Pauli exclusion principle.

Varying \rho after scaling by N_e, we find at leading order in N, that \rho = {1\over N} \sum \delta(x-x_i). The question is: What about the term proportional to N_e^{1+2/D}? This term is a smoothing term trying to lessen accumulations of electrons, acting in concert with the electrostatic repulsion. The electrostatic repulsion can be screened, but this term cannot so it plays a crucial role in actually getting a smooth cloud instead of lumping electrons to minimize electrostatic energy. Nevertheless, we do want the leading result to stay valid so it brings us to the question of finding the appropriate N dependence of N_e. Fluctuations about the expected distribution will be O(N), so we will try N_e^{1+2/D} = 1, and see if the term smooths out fluctuations. If you make this term dominant then we would expect that the uniform distribution would dominate the actual presence of data points. On the other hand, too small and we would expect sharp peaks about each data point, close to the sum of delta function limit. This choice is actually dictated by the desire that our probability of a certain \rho should be independent of how many data points we expect. In other words, it is perfectly reasonable to alter the constant in front of E_{TF} because that is all dependent on our notion of how closely we want to match the data but it is not reasonable to alter the a priori probability of \rho.

It turns out that the Green function that seems to work reasonably well is actually the logarithm, which is the inverse of the Laplacian in 2d.

PCA on projective space

Projective spaces are spaces of lines in some ambient space. For example, {\mathbf R}P^2 is the space of straight lines in {\mathbf R}^3. What is this space? Well, it is the space you get when you identify points in {\mathbf R}^3\backslash{0} up to the multiplicative group x \rightarrow \lambda x, \lambda\neq 0. In this particular case, it is the upper hemisphere of {\mathbf S}^2, with the boundary circle points identified. All good stuff which Brock Fuller and Jack Conn taught me a long long time ago.

But now we’re dealing with trying to do something like PCA on projective space, so we want to ask: What is PCA in a linear space and why do we care? We’re trying to find the directions along which some cloud of points is most spread out. In particular, the rotation symmetry in this space is what underlies wanting to diagonalize \langle (X_i - \langle X_i\rangle)(X_j - \langle X_j\rangle)\rangle. So on a projective space, we still have lots of symmetry, and we should be using this symmetry to find something like a great circle along which the cloud of points in projective space is most spread out. The distance measure inherited from the underlying ambient space has the same rotation symmetry (except for a minor caveat about the identification of antipodal points, which affects the group by modding out a discrete Z_2).

It’s not obvious to me how to do this as some sort of simple matrix manipulation, but as a maximization problem, we want to find a rotation in the ambient space such that the points in the projective space have the largest possible range of \phi\in [0,\pi). (Remember the identification of x and  -x.) Then the other directions have to be determined by rotations that don’t change the inverse image of this half-circle in the ambient space and so on. (Might be better to have some different orientation standard but this is more a matter of notation and calculation.)

Each line is represented by a direction vector. A 2-plane in the original space defines a circle on the projective space. A 2-plane could be defined by two data points in the original space that maximize the projective distance between pairs of data points. Obviously, picking two specific data points is too susceptible to outliers so we want something like two orthonormal vectors n_1, n_2 such that the mean distance of the data points from the plane defined by these vectors (which is the angle between the data point and its projection on the plane) is minimized and the standard deviation of the angles between the projections of the points in this plane is maximized. Just implement the second cost:

n_{1j}n_{1k} +n_{2j}n_{2k}

Of course, there are other sets of charts on the projective space, but the metric distances are not so trivially calculated. It is interesting that this projective PCA would reduce the dynamic range. Is this good? There are instances in image processing where I imagine that the important thing is not the absolute intensity but the relative intensity. I wonder if one would get different results from the Pearson correlation. The Pearson correlation normalizes the mean and the standard deviation before computing the cosine of the angle between the two vectors. The projective distance is scale-independent but does take the mean of the two vectors into account. PCA can be generalized in many ways. There are nonlinear manifold PCA methods, neural network PCA methods and so on.