Matrix Cumulative Coherence: Fourier Bases, Random and Sensing Matrices

Compressive sampling (CS) is revolutionizing the way we process analog to digital conversion, our understanding of linear systems and the limits of information theory. One of the key concept in CS is that a signal can be represented in a sparse bases or it is already sparse. The novelty of sparsity bases is that when signal is randomly sampled, in a nutshell it can be recovered (or a solution can be found to a linear problem) with fewer samples. A land mark example is being sparse MRI.

The concept of choosing "more" sparse bases or CS sensing matrix lies in the measure of mutual coherence. It is defined as follows for a given matrix at order k.
$$  M(A, k) = max_{p} max_{p \ne q, q \in \Omega } \sum_{q} | <a_{p}, a_{q}> | / ( |a_{p}| |a_{q}|)$$ When k=1, it is easy to understand what it means. Basically we find the largest inner product among columns of the given matrix. Lower the value better the sparsity i.e. incoherence. However a single number may not be so informative, after all how low is better.  With the definition of David Donoho and Joel Tropp,  if M is slowly increasing then matrix said to be enchances sparsity. Larger value of k forms a set of columns $\Omega$, and the second colums are selected from this set i.e. second max argument in the above definition.

In a recent post I have shortly reviewed my R package for CS called R1magic. Its recent version 0.2 contains a functionality to compute $M(A, k)$.  Also now there is a public Github repository of the package. mutualCoherence function is written fully functional way. All operations for computing $M(A,k)$ performed in vectorial fashion in R, using function closures and apply. However, for much larger matrices, a low level implementation may be required.

Example

Here we shortly investigate coherence of Fourier, random and mixed bases in R.

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require("R1magic") set.seed(42) A <- DFTMatrix0(10) # Fourier Bases B <- matrix(rnorm(100), 10, 10) # Gaussian Random Matrix C <- A %*% B # A sensing matrix with A and B as above aa<-mutualCoherence(A, 8) bb<-mutualCoherence(A, 8) bb<-mutualCoherence(A, 8) aa [1] 1 1 1 1 1 1 1 1 bb [1] 0.6784574 1.2011489 1.7001046 2.1713561 2.4664608 2.7302690 2.7908302 [8] 2.9623327 cc [1] 0.7506222 1.3448452 1.8047043 2.1105348 2.3350516 2.4703822 2.5898766 [8] 2.6882250

We observe that mixed bases, where we define so called a CS matrix, matrix $C$ in above notation, with Fourier ($A$) and Random basis ($B$). Its mutual coherence increases slower than the pure random matrix with unit measure. This may signify a better incoherent basis.  However, since this is a "syntetic data', it does not tell any universal behaviour at all. A real data or signal measurement matrix  shall be tested for a better judgement of which bases is better to use in sparse recovery.  A recent paper in optical tomography uses this concept to identify a better sensing matrix in tomographic image reconstruction.

Tensor Algebra: Efficient Operations on Multidimensional Arrays with R

 Multidimensional arrays are ubiquitous. Any complex problem having multivariate observables would easily generate a need to represent corresponding data in multidimensional arrays. Most of the practitioners would choose to apply operations on these objects with an index notation.  For example a 3-D array would have 3 indices A[i, j, k] or A[i][j][k] depending on the syntax. However there are languages which takes array operations as their native component, for example APL. More familiar examples are matrix operations in MATLAB, Fortran 90 and alike. Of course this concept is noting more than vectorisation.

Mathematical objects called tensors can be used to represent multidimensional objects. In reality a scalar is rank 0 tensor,  so scalar is the simplest tensor. Rank of a tensor can be thought as objects spatial dimension.  Let's consider matrix multiplication, where there are two indices, tensor of rank 2. If we have two matrices $A_{ij}$ and $B_{ij}$. We can multiply them with index notation as follows using an obscure language R with a naive algorithm:

> set.seed(42)
> A <- matrix(rnorm(4), 2, 2)
> B <- matrix(rnorm(4), 2, 2)
> C <- matrix(0, 2, 2)
> for(i in 1:2) {
+  for(j in 1:2) {
+   for(k in 1:2) {
+     C[i,j] = C[i,j] + A[i, k] * B[k,j]
+    }
+   }
+ }
> C
          [,1]       [,2]
[1,]  6.413566 -2.5178879
[2,] -2.249789  0.4908884
> A %*% B
           [,1]       [,2]
[1,]  0.5156982  2.0378605
[2,] -0.2954518 -0.9134599

You should have noticed that we introduce a third index. That is called a dummy index which we actually do the sum. This approach is technically called Einstein summation rule. So we represent the multiplication as follows: $C_{ij} = \sum_{k=1}^{m} A_{ik} B_{kj}$. Normally sum is dropped for short notation. It is quite intuitive from this notation that number of columns of A should be equal to number of rows in B to perform the sum consistently.

There are software packages that can do symbolic and numerical tensor algebra . Here we demonstrate an R package called tensorA developed by Professor van den Boogaart of TU Freiberg. We can repeat the matrix multiplication by using this package as follows.

> library("tensorA")
> set.seed(42)
> At <- to.tensor(rnorm(4), c(i=2, k=2))
> Bt <- to.tensor(rnorm(4), c(k=2, j=2))
> At %e% Bt
      j
i            [,1]       [,2]
  [1,]  0.5156982  2.0378605
  [2,] -0.2954518 -0.9134599
attr(,"class")
[1] "tensor" "matrix"

Note that here $\%e\%$ implies tensor multiplication with Einstein summation rule. This is pretty trivial example but if you imagine higher dimensional objects, tracking indices would be cumbersome so multi-linear algebra makes life easy.

In conclusion,  I think,  using tensor arithmetic for multidimensional arrays looks more compacts and efficient (~ 2-3 times).  A discussion related to this appeared in R help list.

Why Open Access?

Open access is one of the hot topics discussed these days. I have recently commented about the issue in the context of ACM's position [link].  Editor Prof Vardi was kind enough to give me an opportunity to express my views. Apart from the views I express there, I also would like to point out the parallels with the open source software. Open access publishing currently viewed as just an access to article's content. Probably PLoS is the only model that goes beyond this, that one can leave comments on it after peer-review. I think this signifies future tendencies on peer-review, why a paper should be a static document? For example, largest preprint service arXiv supports versions on the articles. I wouldn't surprise if we adapt similar practices of open source movement under the open science umbrella; living peer-review and access to every details of the research. Well, it opens a big debate as well but looks like industry is embracing the value of openness, slowly.

Football Goal Distributions and Statistical Mechanics

With the advent of complex network science and its allied approaches in the last decade or so with the data driven research, using statistical mechanical techniques out side of materials or atomic physics becomes a standard and quite a popular practice. One of the most interesting of this usage was in football (or soccer for Americans). This is probably because of mass football-mania in the UK and rest of the Europe and the new-world of course. However, using statistics is not a new thing at all but finding similarities with the atomic systems. Specially quantitative approach to sports is very well known, a recent film featuring Brad Pitt, Moneyball, shows this. Every major sports club (merchandise in the US) is now running a statistics division, considering the sky high salaries of players. There are interesting works in the goal statistics, being non-Gaussian [link] [link] [link] and passing network for football strategies [link] among other works. This kind of research is classified as econophysics.