Timeline for Linear map of unit cube (generalized)
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Nov 17, 2019 at 22:59 | vote | accept | Mathtrix | ||
| Nov 12, 2019 at 8:34 | answer | added | Roman | timeline score: 3 | |
| Nov 10, 2019 at 21:50 | comment | added | Mathtrix | A reference can be found in the answer here | |
| Nov 10, 2019 at 21:31 | comment | added | Mathtrix | @Roman: Thanks for your suggestion. I will wait a bit more for answers here before posting the question at math.stackexchange.com. BTW, I agree with your convex hull solution but I am looking for a more efficient way. | |
| Nov 10, 2019 at 21:28 | history | edited | Mathtrix | CC BY-SA 4.0 |
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| Nov 10, 2019 at 21:28 | comment | added | Roman | Maybe ask at the math.stackexchange.com ? | |
| Nov 10, 2019 at 21:21 | comment | added | Mathtrix | @Roman: Yes, I am looking for a matrix $A$ and a vector $b$ such that $\forall x: x \in [0,1]^n \Leftrightarrow A M x \leq b$ | |
| Nov 10, 2019 at 21:17 | comment | added | Roman |
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.
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| Nov 10, 2019 at 21:11 | history | edited | Mathtrix | CC BY-SA 4.0 |
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| Nov 10, 2019 at 21:06 | comment | added | Roman |
@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.
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| Nov 10, 2019 at 20:39 | comment | added | Roman |
@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?
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| Nov 10, 2019 at 20:31 | comment | added | Henrik Schumacher |
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.
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| Nov 10, 2019 at 20:28 | comment | added | Henrik Schumacher |
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?
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| Nov 10, 2019 at 19:39 | history | edited | Mathtrix | CC BY-SA 4.0 |
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| Nov 10, 2019 at 19:33 | history | asked | Mathtrix | CC BY-SA 4.0 |