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No-cover rank growth on arbitrary finite indices

Lax342547.IndexedRankGrowth · concepts/Lax342547/IndexedRankGrowth.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    The componentwise no-cover bound depends on the finite sample cardinality and applies to any index type of the prescribed dyadic size, including even moment samples.

    Concept map
    51 concepts
    100%
    Actual projected span deficitSampling after exposed coordinatesAdaptive sampling on disjoint coordinatesetsUnion over adaptive component coversRank of a lifted tensor sumCombined column and row mode exposureRows of diagonal tensor mapsComponentwise mode spaces and diagonaltensorsExact conditioning costs and recovery offinite probability massesEntropy progress for residual pair lawsIndependence of distinct sample positionsJoin-stable classes of mode coversCount tensors killed by exposureActual exposure cover projectionRank of a diagonal familyDyadic span deficit estimatesA small dyadic tail scaleActual dyadic deficit recurrenceEntropy along feasible mixture lines,including new supportFull feasible support and finite informationprojectionProjection removes the exposed termsCount tested index occurrencesFinite independent sampling and vertexexception tailsGreedy mode space exposureSmall actual greedy exposure tailsNo-cover rank growth on arbitrary finiteindicesActual deficits indexed by a distinct listFinite list tail statisticsDimension deficits after an arbitrary modemapExposed mode dimension budgetBoolean point moments with restricted basecoordinatesA low rank sum supplies an actual adaptivecoverRank growth without an adaptive modecoverSquared restriction cost for independent uniteventsFinite exposure partitionsProjected nonzero terms in the actualremaining sumDimensions of projected mode spacesWalsh bounds for independent image lawsand separated phasesRank of a tensor killed in two quotientspacesRank loss under two restrictionsOriginal retained-cell laws from finite PMFsFinite relative entropy and support costsRank loss under restriction of a bilinear formImage caps inside original retained cellsOne exposure controls both modesMonotonicity of span deficitsMode space span deficitsNonzero tensor count from span deficitsRank of an actual linear map sumAdmissible pair lawsOrthogonality and finite Walsh correlationbounds
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.NoCoverRank
    2import Lax342547.ComponentDuals
    3
    4/-!
    5---
    6title: No-cover rank growth on arbitrary finite indices
    7type: lemma
    8---
    9The componentwise no-cover bound depends on the finite sample cardinality and applies to any index type of the prescribed dyadic size, including even moment samples.
    10-/
    11
    12namespace Lax342547.IndexedRankGrowth
    13
    14open Lax342547.ComponentSpaces Lax342547.ComponentDuals Lax342547.CoverProjection
    15open Lax342547.RelativeEntropy Lax342547.FiniteSampling Lax342547.RetainedImages
    16open scoped BigOperators
    17
    18axiom component_no_cover_rank_indexed {K e Ω ι : Type} [Field K] [Fintype e] [DecidableEq e] [Fintype Ω] [Fintype ι] [DecidableEq ι]
    19 {V W : e → Type} [∀ i, AddCommGroup (V i)] [∀ i, Module K (V i)]
    20 [∀ i, AddCommGroup (W i)] [∀ i, Module K (W i)]
    21 [∀ i, FiniteDimensional K (V i)] [∀ i, FiniteDimensional K (W i)]
    22 (μ : Ω → ℝ) (M : Ω → ∀ i, V i →ₗ[K] W i) (r k : ℕ) (p : ℝ)
    23 (hμ : Probability μ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hr : 1 ≤ r)
    24 (hM : ∀ x, ∑ i, Module.finrank K (LinearMap.range (M x i)) ≤ r)
    25 (hk : 100*r+10 ≤ k) (hcard : Fintype.card ι = 2^k)
    26 (hcover : ∀ S : Submodule K (∀ i, W i), ∀ T : Submodule K (Module.Dual K (∀ i, V i)),
    27 Componentwise S → DualComponentwise T →
    28 (Module.finrank K S : ℝ) ≤ (r : ℝ)*Fintype.card ι →
    29 (Module.finrank K T : ℝ) ≤ (r : ℝ)*Fintype.card ι →
    30 cellMass μ (fun x => projection (LinearMap.piMap (M x)) S T = 0) ≤ p) :
    31 cellMass (productLaw (fun _ : ι => μ))
    32 (fun sample => ((∑ i, Module.finrank K (LinearMap.range (∑ j : ι, M (sample j) i))) : ℝ) <
    33 (2^(k-(100*r+10)) : ℝ)/4) ≤
    34 (4 : ℝ)^Fintype.card ι*p^(2^(k-(100*r+10)))
    35
    36end Lax342547.IndexedRankGrowth
    37
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