Description
1) Simulation:
a) Write a function to simulate N uniformly-drawn points within bound polyhedra. The function
SimPolyHedra would proceed by a 1st stage of uniformly drawing (possibly >N) points within
a box that is bounding the desired polyhedra, followed by a 2nd stage of filtering, so that the
function outputs only the first N of those points that fall within the polyhedra. The 2nd stage
would retain a point iff it is in any of p1 convex polyhedra that are provided as input. The input
arguments are:
N: The number of points to simulate
Bounds: A real D×2 matrix, each of its rows specifying the (lower,upper) bounds of a Ddimensional box from which all points are drawn at the 1st stage
Polyhedra: A cell-array of real matrices M
1
,…,Mp
. M
i
is of size fi×(D+1). It defines a convex
polyhedron of fi faces, as all the vectors x in R
D
such that
[
] ⃗ . Each face
is thus defined by a row of M
i
interpreted as a hyperplane in R
D
. Note that M
i
may be unbounded, by devfined faces, as we only consider its intersection with
the Bounds box.
Output:
X: A real N×D matrix, each of its rows specifying a vector that is inside the
Bounds box and inside at least one of the polyhedra M
i
. To generate those you
draw potential D dimensional vectors x whose transpose could serve as rows of X
. You would then check whether to include each row, i.e. whether any for each
such x there exists at least one M
i
such that the
[
] ⃗ condition is satisfied.
[20 points]
b) Use the above to write a function SimTanzania() that simulate points of particular colors in
the Tanzanian flag (see attached Tanzania.pdf). This flag spans the axis-bounded rectangle
between (0,0) and (1.5,1.0), and has 5 regions in green, yellow, black, yellow and cyan, separated
by the lines
Draw points inside the
planar rectangle of the flag, and save 4 text files, each of N=50 rows and D=2 columns of
numbers in text, specifying 50 points of the appropriate color: Tanzania_green.txt,
Tanzania_cyan.txt,Tanzania_black.txt and Tanzania_yellow.txt .
[5 points]
2) Probabilistic interpretation:
a) Consider SimPoly of Assignment2, Question 1 as defining a probability space of potential
outputs y. As such, it defines a probability density function f (y) over possible values of y= y .
Of course, this probability space is different for each input, so f (y) depends on the inputs
RealThetas, sigma and x. Denote it fθ,,x (y) for RealThetas=θ, sigma= and
x=x.Prove that the least-squares regression resultθ=θ* maximizes fθ,,x (y) for any ,x and y.
[15 points]
b) Consider SimLogistic of Assignment2, Question 3 , with zero noise, as defining a probability
space of potential outputs y. As such, it defines a probability function P(y) over possible values
of y= y . Of course, this probability space is different for each input, so P (y) depends on the
inputs RealThetas and x. Denote it Pθ, x (y) for RealThetas=θ and x=x. Prove that
the logistic regression resultθ=θ* maximizes Pθ,x (y) for any x and y.
[15 points]
Guidance: Neither 2a nor 2b requires computing derivatives. Both can be solved by the definition of
θ
*
as an ERM, so you are welcome to just use that.
3) Optional:
Prepare a single sided, single page, 12-font English-letter (with potential notations in Greek) cheat
sheet for the quiz scheduled for Feb 19th. This would be the only allowed material.
[0 points]
Good luck!
