The basic idea of MCMC-UMS is find one valid sample on the manifold,
then choose a direction to move along a 1-dimensional line
in the linear subspace spanned by the columns of matrix B.
If the manifold set is convex (it is in our case),
the valid points on the line are a compact interval,
the endpoints of which can be easily solved for, and
a new sample is chosen by uniformly sampling on the interval.
The process repeats by choosing a new direction, and repeating.
We consider two ways of choosing directions:
Here is a summary of the MCMC-UMS algorithm.
- the random-directions approach in which
a new random direction is chosen each time.
The random direction algorithm, also called
hit-and-run is suggested by Smith
The random direction is chosen by choosing a point
uniformly distributed on a hypersphere (by choosing a vector
of independent Gaussian random variables and normalizing
it to have length 1). Smith makes the claim in 2011 :
``This version of hit-and-run
is considered to be the most efficient algorithm
for generating an asymptotic uniform point if the
set under consideration is convex".
- the systematic way of moving along the axes, by simply
changing each element of one at a time.
We see our systematic approach as a
simplification of slice sampling in multiple dimensions
, in which the target distribution is uniform.
Indeed, in slice sampling, the uniform samples
are drawn within an ``axis-aligned" hyper-rectangle.
We can stop after a number of iterations
and let be our generated sample of
There is naturally a burn-in period required
to ``forget" initial conditions.
- As a starting point, we will need a vector that is valid,
i.e. produces an that lies in
. Let be a starting point,
which maps back to using (5.10).
- We now seek a direction vector
, of unit norm.
Clearly must lie in the column space of which span
the space in which we have freedom to move.
In the systematic method, we let be a column of matrix .
Each iteration, we choose the next column, in order,
returning back to the first column after iterations.
In the random directions algorithm, we choose
to be a random direction within the column space of .
To do this, we generate an -dimensional vector
of independent standard normal variates, then normalize so that
We then let
- We then create a new as follows:
where must lie in a compact interval of the real line
zero (to produce the current value ).
We can easily find these limits as follows.
Define the vector
calculated from as
Then, is -1 times the reciprocal of
the largest positive value of ,
and is -1 times the reciprocal of the most
negative value of .
- We select by drawing a uniform
random variable in
As long as is sampled uniformly in this range,
the resulting (new) value of will be valid.
- We then set
, and repeat (go to step 2).