bpchesney.org

We know about sparsity for vectors, which is the property of having a lot of 0′s (or elements close enough to 0).  The 0′s in a sparse matrix or vector, generally, make computation easier.  This has been a boon to sampling systems in the form of compressive sampling, which allows recovery of sampled data by using computationally-efficient algorithms that exploit sparsity.

The opposite of sparsity is density.  A vector whose entries are mostly nonzero is said to be “dense,” both in the sense that it is not sparse, as well as being slow and difficult to muddle through calculations on this vector if it is really large.  In general, dense vectors are not very useful.  Or are they?  In this post, I introduce the notion of diverse vectors, a subset of dense vectors, as a more interesting foil to sparsity than simply dense vectors.

This post explores a measure of diversity called the peak-to-sum ratio (PSR). We can use PSR to find the most diverse solution in a linear program, but what can it be used for? We’ll find this linear program optimally distributes quantized, positive units, such as currency or genes. Maximizing diversity, by minimizing PSR, is desirable in several applications:

• Investing
• Workload Distribution
• Product Distribution
• Sensor Data Fusion and Machine Learning

In contrast to some systems which allow dense vectors, these systems actually desire the most diverse solution.

Consider the following system of equations:

x₁ + x₂ + x₃ + x₄ = 12

x₂ - x₃     = 0

x₄ = 0

The following table gives a partial list of possible solutions (for x₁, x₂, and x₃, since x₄ is already given).

The sparsest solution is highlighted in green, (12, 0, 0, 0).  It has the most zeroes in it.  It has everything concentrated in one element of the vector, the first element.  All other elements are zero, making several computations on that vector faster and easier.

Notice the solution highlighted in yellow, (4,4,4,0).  It has very few zeros in it, in fact, none except the one variable that was explicitly set to 0.  But there are other solutions that have only one zero, or are, as they say, 1-sparse.  For instance, (10,1,1,0) is also 1-sparse.  The vector (4,4,4,0) is highlighted and (10,1,1,0) is not because (4,4,4,0) is more diverse.  What makes it more diverse?  The (4,4,4,0) vector has the most equitable distribution among its elements, which is, conceptually, the opposite of being sparse.  Whereas the most sparse solution has its energy concentrated in the fewest elements, the most diverse solution has its energy distributed to the most elements.  In fact, (10,1,1,0) could be considered to be more sparse than (4,4,4,0) as it is closer to having its energy concentrated in one vector (first posited in [1]).

So if we’re talking about more or less diverse/sparse, this implies that we can quantify it.  One way to quantify it is to look at the peak-to-sum  ratio (PSR).  The peak to sum ratio is defined as the ratio of the largest element in x to the sum of all elements of x.

PSR can also be expressed as the ratio of two norms, the infinity norm and the first norm.  Let’s see what this ratio looks like for the solutions we considered above, plus a few more.

The (4,4,4,0) solution appears to have the lowest PSR, until we proceed to solution (13,13,-14,0).  As we proceed past (13,13,-14,0), PSR becomes vanishingly small.  In fact, we see that (4,4,4,0) was just a local minima for PSR.

This is unsatisfying, for a couple of reasons.  The first is linear programming.  We can’t use a linear program to minimize PSR and arrive at the most diverse solution because PSR is not convex, that is, we can’t be sure we’re on a path to smaller and smaller PSR because we could encounter a local minima, like we did at (4,4,4,0).

The second way this is unsatisfying is that (13,13,-14,0) does not seem more diverse than (4,4,4,0).  If we’re considering the elements of x to be bins of energy, or some other thing that is to be distributed, then having an element that is -14 does not make any sense.  Instead of distributing, or deciding how things are to be allocated, we are actually taking them away, which defeats the whole purpose of what we are trying to do. So let’s add in a constraint that x has to be greater than 0.

For this particular problem, we can proceed in a linear fashion from the first candidate solution (12,0,0,0) to the most diverse one (4,4,4,0). However, in general, is this problem convex? That is, could we choose another set of equations and be able to march towards the most diverse answer?

Consider the definition of convexity for a feasible set of solutions.  A set is convex if, when proceeding from one point to the next, you stay in that set [2].  It helps to rewrite our problem in terms of linear algebra.  We seek the most diverse solution to a set of linear equations,

where Ax=b represents our equations we’re trying to solve.  Since x cannot be negative, we know that,

Is this convex?  The shaded region in the following graph shows the x’s that satisfy the quantity (PSR) to be minimized when it’s less than or equal to 1.

In the example in the graph above, there are only two elements in x.  The possible values of x outside of the shaded region, when either x1 or x2 or both are negative, are not in the set of feasible solutions.  So do the answers in this shaded region form a convex set? Yes, because we can pick two points in the shaded region, and draw a straight line between them that has to stay in the shaded region.

Notice if we restrict values of the elements of x to the integers, we still have a convex set!  This result is satisfying because we’ll want to distribute units of energy, currency, etc…, anyway.

In the language of linear algebra,

where Z represents the set of integers.  So the two constraints we’ve added to our convex optimization problem to find the most diverse solution:

1. Must not contain a negative distribution (x greater than or equal to 0)
2. Must have a base unit of distribution (x must contain integers)

We can now use an integer program to perform the optimization.  Restricted to positive, integer values, when we minimize PSR, our linear program will march algorithmically, inexorably to the most diverse solution to a system of linear equations.

References

[1]  Zonoobi et al. “Gini Index as Sparsity Measure for Signal Reconstruction from Compressive Samples,”  IEEE Journal of Selected Topics in Signal Processing.  vol 5, no. 5, Sept. 2011.

[2]  Strang, Gilbert.  Computational Science and Engineering.  Wellesley, MA: Wellesley-Cambridge Press, 2007.