The most erasure-robust measurement directions
Choose n nonzero vectors in R^d. For every subset of size d, take the squared determinant of the matrix with those vectors as columns, divided by the product of their squared norms; maximize the minimum of this normalized volume over all subsets. Zero means some d surviving measurements cannot recover the space at all.
Formal definition
- ContainerReal d-space R^d; an answer is n measurement directions
- SubmissionExactly n nonzero vectors with d coordinates each
- ObjectiveMaximize min det(V_S)²/Π|v_i|² over all C(n,d) subsets; determinants and norms are rational, compared by cross-multiplication
- WordingThe normalized volume is a robustness proxy aligned with numerical stability, not the optimal reconstruction error under every noise model; this problem claims maximin volume and nothing more
- ScoringThe record is floor(min normalized volume · 10¹⁸), rounded against the submitter; the page shows that volume, rounded down at the twelfth decimal
Getting a feel for it
What it defends against
Redundant measurements exist so the signal survives losing a few. A full-spark frame demands that any d survivors span the space; this problem goes further and asks how far the worst set of survivors is from degenerate — the concern of sparse signal processing, erasure-robust transmission and phase retrieval.
Where the frontier is
Existence and constructions of full-spark frames are well studied, but maximizing the worst subset volume at fixed (n, d) has no published table of optima. Every sub-problem is open.
SourceCurrent best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
Discussion (0) ↓Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
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