Trilinear Hilbert transform: mainRange_wholeRanges #
Blueprint lem:whole-ranges (lines 1777-1789) in vector form on the main columns: group vectors
that are sums of elements of main ranges at certified active cells, with the group's ball masks at
pads at most H_max, are almost orthogonal across the groups of one color,
‖Σ_G Z_G‖² ≤ (1 + τ N colBudget²) Σ_G ‖Z_G‖², where at most N group-label pairs are present
at each point.
The global depth family of the ball projections of the slot l for the group G, the ball
level j and the pad index p (blueprint line 1859).
Instances For
At an active cell the ball depth projection after the restriction to the cell is the cell projection of the main tree.
The relabeled ball cell projection is the main ball projection of the cell.
The relabeled ball rows of one interval on a union of active right-endpoint cells: the sum over the cells of the main ball projection at the cell minus the main ball projection at its left-endpoint ancestor applied to the cell restriction.
The relabeled right-endpoint ball rows on a union of active cells of their depth.
eq:whole-ranges for the interval rows of the variation on the main columns (blueprint lines
1875: "their adjoints lie in sums of whole ball ranges at the two endpoints; a right certificate
certifies every needed earlier endpoint ... Lemma whole-ranges makes the rows jointly Bessel across
groups").
eq:whole-ranges for the outer ball rows at a common right endpoint on the main columns
(blueprint line 1877: "sum the outer ball rows at each common right endpoint using
eq:whole-ranges").
The restriction to a union of cells of the setup vanishes off the root.
The interval rows of the variation split orthogonally over the depth partition of the main tree at the left depth.
Blueprint lem:variation (eq:variation, lines 1861-1877) on the main columns for one ball
level and color of the attempt, with a common input z: the global ball outputs
Z_{G,s} = L_{G,s} B'_s z of the groups, restricted at the right endpoints to unions of certified
active cells, have square-summable variation over disjoint depth intervals,
Σ_{G,[r,t]} ‖1_E (Z_{G,t} - Z_{G,r})‖² ≤ 2 (1 + τ N colBudget²) (C_{|ls|}² + |ls| Σ_m C_m²) ‖z‖²
with the complement compression constants C_m of the main tree.
Blueprint lines 1859, 1879 for a rank-zero innermost factor: rows that are Bessel for all inputs and split orthogonally over the cells of the depth partition of a basic column family remain Bessel after inserting its depth projection through a contraction commuting with restrictions, at the cost of one compression constant.
Blueprint lem:variation (eq:variation) for histories acting on a rank-zero basic projection
through a contraction commuting with restrictions: if the history rows S_i (T_i(t_i) - T_i(r_i)) are Bessel for all inputs and split over the basic partition at the left depths, then
the variation of the outputs T_i(h) (E P_h Z) is bounded by the compressed row constant plus the
joint size J of the outputs on the jumps of the basic projections.
The embedding of the basic family (l, j) into the relevant main labels: the basic column
embedding followed by the adjoint of the relabeling.
Equations
- bf.famEmbR Mc s l j = ContinuousLinearMap.adjoint (TrilinearHT.Auto.colRelabel (bf.rv Mc s)) ∘SL bf.famEmb Mc l j
Instances For
The global ball output of the slot l for the group G at the ball level j and the depth
h: the global ball filter after the global shared history (blueprint lines 1859, 1871:
Z_{G,h} = L_{G,h} B'_h).
Instances For
The global ball outputs at depth h commute with the restriction to every dyadic cell of
depth at most the clamped depth h.
The restricted jump rows of the global ball outputs between two depths split orthogonally over the depth partition of a basic family at the left depth.
Blueprint eq:whole-ranges (line 1875) for the global ball outputs of one depth: the
outputs of the groups, restricted to unions of certified cells of that depth, are jointly Bessel
with constant 1 + τ N colBudget².
Blueprint lem:variation (eq:variation, lines 1861-1877) on the main columns for a basic slot
and one of its basic families j_b: the histories act on the basic depth projection
P_{l,j_b,h} F (the input of the cells of depth h), and the certified variation of the global
ball outputs over disjoint depth intervals is bounded by
2 (B C₁² + (1 + τ N colBudget²)) ‖F‖², with B the constant of main_variation and C₁ the
one-factor compression constant of the basic tree.
The complement compression constants increase with the number of factors.
compression_loss_le with the basic projection inserted as a one-layer history.
Blueprint lem:variation, "the compression loss is G₀" (lines 1859, 1877: "the O(log L)
terms, signed expansions and averages are absorbed in G₀"): the constant of
main_variation_basic for histories of length n ≤ c (log k + 1), column budgets with
log (2 budget) ≤ k^c and a whole-range constant at most 2 is at most
G₀ = exp(C_q (1 + log 2k)^2), C_q = 4096 (c+1) (1024 (c+1) + c + 7) + c² + 3.
The global coarse output of a basic slot (blueprint eq:localized, lines 1841, 1859): the
global ball output of the group at the depth h acting on the global basic input, in the main
column space. Its restriction to an active cell of depth h is the leaf output
(basic_leaf_out).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The restriction of the basic coarse output to an active cell of its depth is the leaf output
L_{l,G,s} B_{l,s} B_l of the cell (blueprint eq:localized and line 1859).
The basic families reach the depth of the attempt root.
On unions of active cells, the global ball outputs of a basic family depend only on the
restriction of the family input to the attempt root (blueprint line 1865: ‖1_R F‖).
Blueprint lem:variation (eq:variation, lines 1861-1877) on the main columns for a basic
coarse output: for depth intervals [r_i, t_i] (offsets below the attempt root) that are
disjoint, and unions E_{G,i} of active cells of depth t_i carrying downstream actual-use
certificates of G,
Σ_{G,i} ‖1_E (Z_{G,t_i} - Z_{G,r_i})‖² ≤ Σ_{j_b} 2 V_{j_b} ‖1_R F_{l,j_b}‖², where
Z_{G,h} = L_{G,h} B'_h B_l is the global coarse output (basicOut) and
V_{j_b} = 2 (2 C_w S) C_{1,j_b}² + 2 C_w with the whole-range constant
C_w = 1 + τ N colBudget², the history compression sum S and the one-factor compression
constant C_{1,j_b} of the basic tree.
Sums over a finite family colored with M colors are at most M times a bound valid on every
color class (blueprint line 1877: "color the depth intervals to handle bounded overlap").
The global shared histories commute with the restriction to every dyadic cell of depth at most the clamped depth.
The restricted jump rows of the global shared histories split orthogonally over the depth partition of a basic family at the left depth.
On unions of active cells, the global shared histories of a basic family depend only on the restriction of the family input to the attempt root.
The global shared history of a basic slot acting on its global basic input, in the main
column space; its restriction to an active cell of depth h is B_{l,s} B_l (basic_history).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Blueprint lem:variation, shared-history assertion ("without an outer ball the same assertion
holds for a shared history, omitting G"): over disjoint depth intervals and unions E_i of
active cells of depth t_i,
Σ_i ‖1_{E_i} (B'_{t_i} B_l - B'_{r_i} B_l)‖² ≤ Σ_{j_b} 2 (2 S C_{1,j_b}² + 2) ‖1_R F_{l,j_b}‖²
with S = |ls| Σ_m C_m² the history jump constant.
The basic inputs have energy at most |R| on the root: ‖1_R F_{l,j}‖² ≤ |R| (blueprint
line 2075: ‖F_l‖₂² ≤ |R|).
Blueprint lem:variation, intervals crossing the attempt start: with the zero convention the
outputs before the attempt vanish, so such an interval contributes the one-time projection
estimate at its right endpoint,
Σ_G ‖1_{E_G} Z_{G,t}‖² ≤ Σ_{j_b} 2 (1 + τ N colBudget²) ‖1_R F_{l,j_b}‖².
Blueprint lem:variation (eq:variation, lines 1861-1866) with the compression loss G₀:
under log (2 D_m) ≤ k^c, log (2 colBudget) ≤ k^c, histories of O(log k) layers and a pair
threshold with τ N colBudget² ≤ 1 (chosen after the counts, line 1854),
Σ_{G,[r,t]} ‖1_E (Z_{G,t} - Z_{G,r})‖² ≤ G₀ Σ_{j_b} 2 ‖1_R F_{l,j_b}‖² ≤ 4 G₀ |R| with
G₀ = exp(C_q (1 + log 2k)²).