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Compression that fixes pin/key vectors and preserves target matrices

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

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    Natural Language Statement

    Lemma

    Block maps are chosen from the actual projected pin/key spaces and the row/column observations of each residual matrix. They fix all required vectors, preserve every target functional, and have the paper block-rank bound independent of the base dimension.

    Concept map
    81 concepts
    100%
    Exact-image bounds for independent affinecolumnsOrdered atom products in the actualgradient formPoint atoms and finite flavor distributionsSparse unselected primal vectors in barredspaces lie in the pinsBarred response spaces and the actualobstruction rank budgetCompression preserving allowed pointmoments and the actual cut domainThe exact space of binary base momentsRank control for the frozen baseline onprimal inputsFormal ordered product bits realizesymmetric correctionsBounded allowed derivatives preserving theactual linearized responseAllowed channel changes and the actual tableinjection testsConcrete cut-space testers and the orderedmixer formConcrete coordinates, quadratic testers, andthe self-Gram formNumerical recipes prescribe the concretegradients on witness atomsThe affine minus-column law at a fixed plusframeCompression that fixes pin/key vectors andpreserves target matricesCut profiles and the constant kernelThe full linearized response on pairs ofactual cut profilesFull response obstructions are effectiveprofiles plus selected atomsExact image pins in nominal coefficientspacesFinite linear images and their uniform-lawdensity boundsAmbient symmetries and frame marginalsFresh key directions are independent modulotable spacesBaseline bilinear extensions retaining allfrozen rows and columnsBaseline contractions on the selected atomsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesThe binary hole relationPaying the reference-image conditioning anddimension costsUniform injective frames and channeltranspose failureFresh point-ray spans are disjoint from thenominal table spacesLow-rank Boolean moments have boundedlabel supportTriangle relations separate into individuallabel blocksBoolean point moments with restricted basecoordinatesThe 3K+28 baseline bound in the actualnominal coordinatesPrimal blocks of the nominal spaces andtheir bounded table partEndpoint projections of paired pure-responseannihilatorsActual paired-witness key spaces satisfy thebaseline hypothesesBinary prescriptions at all endpoints of apaired scalar recipeFull paired witness lists and scalar recipeequationsSparse pin exclusions with arbitrary basecoefficientsSparse residual contractions belong to theactual primal pinsRetractions with bounded rank on theprimal inputsExact images mixing independent injectiveframesJoint minus images after exposing severalplus framesPrimal projections and preservation ofeffective spacesPure obstructions on the actual pair of cutprofilesCompression on the actual barred nominalquotientsExtracting fresh label coefficients throughpin quotientsRank of a tensor killed in two quotientspacesRank-controlled pure forms on the actualbarred quotientsBounded baselines for both actual crossorientationsActual frame observations realize thenominal channel contractionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawGradient residuals vanish on effective profilesand selected atomsCoupled scalar recipes give consistent atomgradientsJoint reference image caps across both signsand all drawsThe full reference cap for exact pin eventsRemoving selected atoms leaves onlyunselected component labelsThe bounded pure remainder of an actualscalar recipeMatrix representations and the boundedresidual rank ingredientsUnrestricted linearized solutions for actualscalar recipesRank loss under restriction of a bilinear formSelected tensor blocks of actual pureobstructionsA selected affine ray determines its momentblockSubtracting selected atoms preserves pureannihilationSelected label coefficients agree across thecut profileInterpolation of finitely many binary selectorlabelsNumerical cross tables, injection flags, andunary admissibilityUnselected sparse vectors cannot concealfresh key coefficientsA uniform label budget for all sparse pinvectorsConsistent symmetric binary prescriptions ontwo witness listsTable contractions on effective profiles andtheir full extensionsTable coordinates and private channelcomplementsMajority intersections in the cyclic taggeometryPure bilinear responses detect quotienttensorsQuotient extractors isolate individual tensorlabel blocksWell-defined channel contractions onprojected tensor spacesWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 9 statements. Each proof establishes one of them relative to its assumptions.

    1 actual_block_compression proven

    2 exists_block_compression proven

    3 fixedSlices_mem proven

    4 fixedSlices_rank proven

    5 lifted_decomposition proven

    6 observations_rank proven

    7 pi_range_rank proven

    8 preserved_matrix proven

    9 vector_decomposition proven

    Lean source view on GitHub

    1import Lax342547.QuotientCompression
    2
    3/-!
    4---
    5title: Compression that fixes pin/key vectors and preserves target matrices
    6type: lemma
    7---
    8Block maps are chosen from the actual projected pin/key spaces and the row/column observations of each residual matrix. They fix all required vectors, preserve every target functional, and have the paper block-rank bound independent of the base dimension.
    9-/
    10
    11namespace Lax342547.ConstrainedCompression
    12
    13open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    14open Lax342547.BaseCompression
    15open Lax342547.PairedWitnesses Lax342547.TableSpaces Lax342547.ProjectedPins
    16open Lax342547.ConcreteCut Lax342547.ResponseMatrices
    17open Lax342547.CutProfiles Lax342547.ExactPins Lax342547.BarredSpaces
    18
    19def baseAt {k n : ℕ} (q : Block k) (j : Fin n) : Base k n := match q with
    20 | Sum.inl (d, t) => Sum.inl ((d, t), j)
    21 | Sum.inr q => Sum.inr (q, j)
    22
    23def blockSlice {k n b degree : ℕ} (s : SelectorCoordinates b degree) (q : Block k) :
    24 Vector k n b degree →ₗ[Binary] (Fin n → Binary) where
    25 toFun v j := v (s, some (baseAt q j))
    26 map_add' _ _ := rfl
    27 map_smul' _ _ := rfl
    28
    29noncomputable def blockEmbed {k n b degree : ℕ} (s : SelectorCoordinates b degree) (q : Block k) :
    30 (Fin n → Binary) →ₗ[Binary] Vector k n b degree := by
    31 classical
    32 exact
    33 { toFun := fun v c => if c.1 = s then match c.2 with
    34 | none => 0
    35 | some (Sum.inl ((d, t), j)) => if q = Sum.inl (d, t) then v j else 0
    36 | some (Sum.inr (q', j)) => if q = Sum.inr q' then v j else 0
    37 else 0
    38 map_add' := by
    39 intro v w
    40 ext ⟨a, c⟩
    41 cases c with
    42 | none => simp
    43 | some c =>
    44 rcases c with ⟨⟨d, t⟩, j⟩ | ⟨q', j⟩ <;>
    45 simp only [Pi.add_apply] <;> split_ifs <;> simp_all
    46 map_smul' := by
    47 intro c v
    48 ext ⟨a, q'⟩
    49 cases q' with
    50 | none => simp
    51 | some q' =>
    52 rcases q' with ⟨⟨d, t⟩, j⟩ | ⟨q', j⟩ <;>
    53 simp only [Pi.smul_apply] <;> split_ifs <;> simp_all }
    54
    55def constantPart {k n b degree : ℕ} (v : Vector k n b degree) : Vector k n b degree :=
    56 fun c => match c.2 with
    57 | none => v (c.1, none)
    58 | some _ => 0
    59
    60axiom vector_decomposition {k n b degree : ℕ} (v : Vector k n b degree) :
    61 v = constantPart v + ∑ s, ∑ q, blockEmbed s q (blockSlice s q v)
    62
    63axiom lifted_decomposition {k n b degree : ℕ}
    64 (A : Block k → (Fin n → Binary) →ₗ[Binary] (Fin n → Binary)) (v : Vector k n b degree) :
    65 liftBase (blockMap A) v = constantPart v + ∑ s, ∑ q, blockEmbed s q (A q (blockSlice s q v))
    66
    67axiom pi_range_rank {I V X : Type} [Fintype I] [AddCommGroup V] [Module Binary V]
    68 [AddCommGroup X] [Module Binary X] [FiniteDimensional Binary V] [FiniteDimensional Binary X]
    69 (L : I → V →ₗ[Binary] X) :
    70 Module.finrank Binary (LinearMap.range (LinearMap.pi L)) ≤
    71 ∑ i, Module.finrank Binary (LinearMap.range (L i))
    72
    73noncomputable def fixedSlices {I : Type} [Fintype I] {k n b degree : ℕ}
    74 (S : I → Submodule Binary (Vector k n b degree)) (q : Block k) :
    75 (∀ j : I × SelectorCoordinates b degree, S j.1) →ₗ[Binary] (Fin n → Binary) := by
    76 classical
    77 exact LinearMap.lsum Binary _ Binary (fun j => (blockSlice j.2 q).comp (S j.1).subtype)
    78
    79axiom fixedSlices_mem {I : Type} [Fintype I] {k n b degree : ℕ}
    80 (S : I → Submodule Binary (Vector k n b degree)) (q : Block k)
    81 (i : I) (s : SelectorCoordinates b degree) (v : Vector k n b degree) (hv : v ∈ S i) :
    82 blockSlice s q v ∈ LinearMap.range (fixedSlices S q)
    83
    84axiom fixedSlices_rank {I : Type} [Fintype I] {k n b degree B : ℕ}
    85 (S : I → Submodule Binary (Vector k n b degree))
    86 (hS : ∀ i, Module.finrank Binary (S i) ≤ B) (q : Block k) :
    87 Module.finrank Binary (LinearMap.range (fixedSlices S q)) ≤
    88 Fintype.card I * Fintype.card (SelectorCoordinates b degree) * B
    89
    90noncomputable def observations {I X : Type} {k n b degree : ℕ} [AddCommGroup X] [Module Binary X]
    91 (T : I → Vector k n b degree →ₗ[Binary] X) (q : Block k) :
    92 (Fin n → Binary) →ₗ[Binary] (I × SelectorCoordinates b degree → X) :=
    93 LinearMap.pi (fun j => (T j.1).comp (blockEmbed j.2 q))
    94
    95axiom observations_rank {I X : Type} [Fintype I] [AddCommGroup X] [Module Binary X]
    96 [FiniteDimensional Binary X] {k n b degree R : ℕ}
    97 (T : I → Vector k n b degree →ₗ[Binary] X)
    98 (hT : ∀ i, Module.finrank Binary (LinearMap.range (T i)) ≤ R) (q : Block k) :
    99 Module.finrank Binary (LinearMap.range (observations T q)) ≤
    100 Fintype.card I * Fintype.card (SelectorCoordinates b degree) * R
    101
    102axiom exists_block_compression {I X : Type} [Fintype I] [AddCommGroup X] [Module Binary X]
    103 [FiniteDimensional Binary X] {k n b degree B R : ℕ}
    104 (S : I → Submodule Binary (Vector k n b degree))
    105 (hS : ∀ i, Module.finrank Binary (S i) ≤ B)
    106 (T : I → Vector k n b degree →ₗ[Binary] X)
    107 (hT : ∀ i, Module.finrank Binary (LinearMap.range (T i)) ≤ R) :
    108 ∃ A : Block k → (Fin n → Binary) →ₗ[Binary] (Fin n → Binary),
    109 (∀ q, Module.finrank Binary (LinearMap.range (A q)) ≤
    110 Fintype.card I * Fintype.card (SelectorCoordinates b degree) * (B + R)) ∧
    111 (∀ i v, v ∈ S i → liftBase (blockMap A) v = v) ∧
    112 ∀ i, (T i).comp (liftBase (blockMap A)) = T i
    113
    114axiom preserved_matrix {B : Type} [Fintype B]
    115 (f : (B → Binary) →ₗ[Binary] (B → Binary)) (M : Matrix B B Binary)
    116 (hrow : M.mulVecLin.comp f = M.mulVecLin)
    117 (hcol : M.transpose.mulVecLin.comp f = M.transpose.mulVecLin) : by
    118 classical
    119 exact (LinearMap.toMatrix' f).transpose * M * LinearMap.toMatrix' f = M
    120
    121axiom actual_block_compression {k n b degree r K R : ℕ} {hr : 2 * r ≤ n}
    122 {H N : Type} [Fintype H] (W : Lists k n b degree r hr)
    123 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    124 (hK : P.rank ≤ K) (i : Fin 2) (M : Component (Tag k) → Moment k n b degree)
    125 (hM : ∀ e, (M e).rank ≤ R) :
    126 ∃ A : Block k → (Fin n → Binary) →ₗ[Binary] (Fin n → Binary),
    127 (∀ q, Module.finrank Binary (LinearMap.range (A q)) ≤
    128 2 * Fintype.card (SelectorCoordinates b degree) * Fintype.card (Component (Tag k)) *
    129 (K + 14 + R)) ∧
    130 (∀ a v, v ∈ projected P i a ⊔ individualKeys W i a.1 → liftBase (blockMap A) v = v) ∧
    131 ∀ e, by
    132 classical
    133 exact
    134 (LinearMap.toMatrix' (liftBase (Coord := SelectorCoordinates b degree) (blockMap A))).transpose *
    135 M e * LinearMap.toMatrix' (liftBase (blockMap A)) = M e
    136
    137end Lax342547.ConstrainedCompression
    138
    Show ProofShow ProofShow ProofShow ProofShow ProofShow ProofShow ProofShow ProofShow Proof
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