ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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A two-dimensional three-vector full-spark real frame

Example

In $R^{2}$ take $Φ = (a_{1}, a_{2}, a_{3})$ with $a_{1} = (1, 0)$ , $a_{2} = (0, 1)$ , and $a_{3} = (1, 1)$ . Its intensity map on classes modulo $x \sim - x$ is $[x] \mapsto (x_{0}^{2}, x_{1}^{2}, (x_{0} + x_{1})^{2})$ . Every pair is linearly independent, so every pair spans $R^{2}$ ; consequently, for every partition one side spans $R^{2}$ . Equality of the displayed triples therefore forces two signals to be equal up to a global sign.

Example references

Real intensity map and global-sign quotient

Full-spark condition

Sharp full-spark measurement count

Radu Balan, Pete Casazza, and Dan Edidin, full-spark threshold, Corollary 2.6

Radu Balan, full-spark threshold, Corollary 3.3

Facts & Assumptions

Given: The family $a_{1} = (1, 0)$ , $a_{2} = (0, 1)$ , $a_{3} = (1, 1)$ in $R^{2}$ and the real sign-quotient intensity map.

[F1]

Full-spark real measurement families, Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i < n} λ_{i} v_{i} = 0_{V}$ forces every $λ_{i} = 0_{F}$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent: a three-vector family in $R^{2}$ is full spark when every two vectors are linearly independent.

[F2]

Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ , Linear subspace of a vector space: every span is a subspace containing the listed vectors and is closed under addition and scalar multiplication.

[F3]

The sharp real phase-retrieval count for full-spark measurements: a full-spark real family with at least $2 d - 1$ measurements performs phase retrieval.

Verification

1.1givenF1F2algebra

For every $(u, v) \in R^{2}$ , $(u, v) = u a_{1} + v a_{2} = (u - v) a_{1} + v a_{3} = u a_{3} + (v - u) a_{2},$ so each pair spans $R^{2}$ by [F2]. For independence, a vanishing combination of $a_{1}, a_{2}$ has coordinates $(α, β) = (0, 0)$ ; a vanishing combination $α a_{1} + β a_{3}$ has coordinates $(α + β, β) = (0, 0)$ , forcing $β = α = 0$ ; and a vanishing combination $α a_{2} + β a_{3}$ has coordinates $(β, α + β) = (0, 0)$ , again forcing $α = β = 0$ . Thus every pair is linearly independent, so [F1] gives full spark.

2.1step 1.1F3algebra∎

Here $d = 2$ and $m = 3 = 2 d - 1$ . Applying [F3] to step 1.1 proves phase retrieval on the stated global-sign quotient; the intensity coordinates are exactly the three squared inner products displayed in the example.

Depends on

Used by

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Dependency tree · two levels

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Sources

Radu Balan, Pete Casazza, and Dan Edidin, full-spark phase retrieval threshold, Corollary 2.6 (standard reference, not scraped)
Radu Balan, Frames and Phaseless Reconstruction, full lecture notes (2015), Corollary 3.3 (standard reference, not scraped)