# Linear map of unit cube (generalized)

This question is a slight generalization of this question.

Let M be a fixed rational m $$\times$$ n matrix, e.g., M = Rationalize[RandomReal[{-1, 1}, {m, n}], 0], and X = Table[Symbol["x" <> ToString[i]], {i, n}] be a vector of n variables.

Assuming that 0 <= X <= 1, I would like to compute the linear map M.X. This can be expressed with Reduce:

Y = Table[Symbol["y" <> ToString[i]], {i, m}]
formula = Exists[Evaluate[X], And@@Thread[Y == M.X] && And@@Thread[0 <= X <= 1]]
Reduce[formula, Y, Reals]

The solution Y forms a polyhedron. How can this polyhedron be efficiently computed with Mathematica? The solution above does not scale.

Another way of stating the problem is to compute a matrix $$A$$ and a vector $$b$$ such that $$\forall x: ~ x \in [0,1]^n \Leftrightarrow A M x \leq b$$.

• What do you mean by "it does not scale"? What is the typical size for m and n? Is m greater or smaller than n. And what do you want to do with the result? – Henrik Schumacher Nov 10 '19 at 20:28
• If I understood it correctly, for a vector Y, you simply have to test whether PseudoInverse[M].Y lies in the unit cube. Since the pseudoinverse PseudoInverse[M] may look pretty complicated when M is given symbolically, I suggest to do the computation of PseudoInverse[M] numerically. – Henrik Schumacher Nov 10 '19 at 20:31
• @HenrikSchumacher I think the question is about finding the convex polyhedron of all vectors Y for which PseudoInverse[M].Y lies in the unit hypercube. How do you parametrize the faces of this convex polyhedron as inequalities? – Roman Nov 10 '19 at 20:39
• @HenrikSchumacher also you need to be careful with the pseudoinverse: there are infinitely many pseudoinverses, and the question is: for which vectors Y does there exist any pseudoinverse P such that P.Y lies in the unit-hypercube. So you need to search over the space of pseudo-inverses. – Roman Nov 10 '19 at 21:06
• You could of course calculate all images of the vertices of the unit hypercube with P = Tuples[{0, 1}, n].Transpose[M] and then try to find the convex hull of these points with ConvexHullMesh[P]. But this seems like a terrible idea. – Roman Nov 10 '19 at 21:17

Here's a method that is a generalization of this one and should work for arbitrary $$n$$ and $$m$$. It is very efficient, as long as you don't want to do a graphics plot as below.

Let's do a random example where $$m=3$$ so we can visualize the polytope in 3D:

n = 5;
m = 3;
M = RandomVariate[NormalDistribution[], {m, n}]
(*    {{-0.304994, -0.392532, 0.823638, -0.201907, 0.108934},
{-0.100295, -0.66419, -2.50501, -0.483007, -0.982172},
{0.00520204, -0.601936, 1.99376, 1.07407, -0.579136}}    *)

To calculate the facet normals, we take all length-$$(m-1)$$ tuples of columns of $$M$$, and for each such tuple calculate the vector that is perpendicular to all of them (via NullSpace):

W = Join @@ NullSpace /@ Subsets[Transpose[M], {m - 1}]
(*    {{0.250025, -0.727159, 0.639318},
{-0.176108, 0.576911, 0.797596},
{-0.287587, 0.89256, 0.34732},
{0.174337, -0.485727, 0.856549},
{-0.876301, 0.0887525, 0.473518},
{-0.878533, 0.475208, 0.0485508},
{-0.355215, -0.50372, 0.787457},
{0.739474, 0.55098, 0.386781},
{-0.968459, -0.197213, 0.152295},
{0.982782, 0.0000529691, 0.184769}}    *)

In this case, there are $$\binom{n}{m-1}=\binom{5}{2}=10$$ such facet normals. For each facet we use the linear programming trick of your previous question to get the minimum and maximum values of the scalar product with this facet normal, to find the boundaries of the polytope:

w = {#.LinearProgramming[#, -IdentityMatrix[n, SparseArray], ConstantArray[-1,n]],
#.LinearProgramming[-#, -IdentityMatrix[n, SparseArray], ConstantArray[-1,n]]} & /@ (W.M)
(*    {{0., 4.66071}, {-1.84188, 0.613583}, {-3.57839, 0.},
{-0.261406, 4.1875}, {-0.456861, 0.903485}, {-2.40775, 0.220539},
{0., 3.86301}, {-1.85243, 0.}, {0., 1.18978},
{-0.795815, 1.17771}}    *)

We can now assemble the inequalities that describe the polytope's facets, using the facet normals W and the associated scalar-product ranges w:

Y = Array[y, m];
F = And @@ MapThread[#2[[1]] <= #1.Y <= #2[[2]] &, {W, w}]
(*    0. <= 0.250025 y[1] - 0.727159 y[2] + 0.639318 y[3] <= 4.66071 &&
-1.84188 <= -0.176108 y[1] + 0.576911 y[2] + 0.797596 y[3] <= 0.613583 &&
-3.57839 <= -0.287587 y[1] + 0.89256 y[2] + 0.34732 y[3] <= 0. &&
-0.261406 <= 0.174337 y[1] - 0.485727 y[2] + 0.856549 y[3] <= 4.1875 &&
-0.456861 <= -0.876301 y[1] + 0.0887525 y[2] + 0.473518 y[3] <= 0.903485 &&
-2.40775 <= -0.878533 y[1] + 0.475208 y[2] + 0.0485508 y[3] <= 0.220539 &&
0. <= -0.355215 y[1] - 0.50372 y[2] + 0.787457 y[3] <= 3.86301 &&
-1.85243 <= 0.739474 y[1] + 0.55098 y[2] + 0.386781 y[3] <= 0. &&
0. <= -0.968459 y[1] - 0.197213 y[2] + 0.152295 y[3] <= 1.18978 &&
-0.795815 <= 0.982782 y[1] + 0.0000529691 y[2] + 0.184769 y[3] <= 1.17771    *)

To show the polytope:

J = ImplicitRegion[F, Evaluate@Y];
RegionPlot3D[J, PlotPoints -> 100]

To show off, let's do this for n=20 and m=3: we get $$\binom{20}{2}=190$$ facet normals, 380 inequalities, and an almost ellipsoidal polytope: