How Does Dimension Defect Decompose for Sums of Finitely Many Subspaces?

Why ordinary inclusion–exclusion fails beyond two subspaces

The classical Grassmann formula handles two subspaces exactly: dim(V1 + V2) = dim(V1) + dim(V2) − dim(V1 ∩ V2). It measures the loss of dimension caused by overlap. For three or more subspaces, the direct inclusion–exclusion analogue obtained by alternating dimensions of ordinary intersections fails, because the lattice of subspaces is modular but generally not distributive. For subspaces U, V, W, the intersection U ∩ (V + W) need not equal (U ∩ V) + (U ∩ W). A subspace can therefore intersect V and W trivially while intersecting their sum nontrivially, an overlap that ordinary inclusion–exclusion cannot see.

The paper's running example makes this concrete. Take V = F² with standard basis {e1, e2} and the three distinct lines V1 = span{e1}, V2 = span{e2}, V3 = span{e1 + e2}. Every pairwise intersection is {0}, yet Vi ∩ (Vj + Vk) = Vi for each permutation (i, j, k) of (1, 2, 3). The symmetric formula then gives dim(V1 + V2 + V3) = 3 − (1/3)(1 + 1 + 1) = 2, recovering exactly the information that pairwise inclusion–exclusion misses.

A symmetric, nonrecursive formula for the dimension of a sum

The paper's first main result is a symmetric, nonrecursive Grassmann-type formula. Averaging the ordered form of the two-subspace Grassmann formula over all n! orderings yields dim(Σ Vi) = Σ dim(Vi) − Σ Σ c_{n,|S|+1} dim(Vi ∩ Σ_{j∈S} Vj), where the inner sum runs over nonempty S ⊆ [n] \ {i} and c_{n,k} = (1/k) / C(n, k). Each correction term records the part of a distinguished subspace already contained in a partial sum of the others, and its coefficient depends only on the number of subspaces in that sum. No recursive application of a formula for fewer subspaces is needed, and the expression is invariant under relabeling.

The coefficients are exactly the Shapley coefficients |S|!(n − |S| − 1)!/n!, so the formula is the Shapley efficiency identity for the rank function r(S) = dim(Σ_{i∈S} Vi), which is the rank function of a linearly representable integral polymatroid. For n = 2 the formula reduces to the classical Grassmann identity. For n = 3 it recovers dim(V1 + V2 + V3) = Σ dim(Vi) − (1/3) Σ_{i<j} dim(Vi ∩ Vj) − (1/3) Σ_i dim(Vi ∩ Σ_{j≠i} Vj), where the first correction records pairwise overlap and the second captures the genuinely non-distributive phenomenon.

Weighted orderings and a dual formula for intersections

The uniform average is not essential to the identity. Any probability distribution μ on the orderings produces a valid, generally nonsymmetric Grassmann-type expansion in which each intersection term dim(Vi ∩ Σ_{j∈S} Vj) is weighted by the probability p_μ(i, S) that exactly the elements of S precede i. The uniform distribution is the canonical label-symmetric choice and recovers the symmetric formula. A worked example with V = F², V1 = span{e1}, V2 = V, and ordering probabilities 3/4 and 1/4 gives dim(V1 + V2) = 1 + 2 − (3/4)(1) − (1/4)(1) = 2: the weights shift the average contributions of V1 and V2 to 3/4 and 5/4, while their sum stays 2.

Passing to annihilators gives a dual counterpart for intersections: dim(∩ Vi) = Σ dim(Vi) − Σ Σ c_{n,|S|+1} dim(Vi + ∩_{j∈S} Vj). The proof applies the symmetric formula in the algebraic dual V∨ to the annihilators V1°, …, Vn°, using (Σ_{i∈S} Vi)° = ∩_{i∈S} Vi° and dim(U°) = dim(V) − dim(U). The sum of all coefficients in the double correction term equals n − 1, so the ambient-dimension terms cancel. For n = 2 this reduces again to the classical Grassmann identity.

The defect profile is monotone in the number of participating subspaces

Define the dimension defect def(V1, …, Vn) = Σ dim(Vi) − dim(Σ Vi). Grouping the correction terms of the symmetric formula by the number of participating subspaces gives a canonical decomposition def = Σ_{k=2}^{n} Ak, where Ak is the average overlap between one distinguished subspace and the sum of k − 1 other subspaces, taken over all such choices. The paper proves that this profile is nondecreasing: A2 ≤ A3 ≤ … ≤ An. Here A2 is the average pairwise intersection dimension, while An is the average dimension of the intersection of a distinguished subspace with the sum of the remaining subspaces.

The levels are indexed by the number of participating subspaces, but they do not isolate interactions involving exactly that many subspaces, since overlap among fewer subspaces may also contribute at higher levels. The successive gaps Bk = Ak+1 − Ak, for 2 ≤ k ≤ n − 1, refine the hierarchy. Writing Rk for the average dimension of the sum of a uniformly chosen collection of k subspaces (with R0 = 0) and Δk = Rk − Rk−1, the paper proves Ak = R1 − Δk and Bk = Δk − Δk+1. Thus Δk is the average marginal dimension contributed when the number of selected subspaces grows from k − 1 to k, Ak measures its loss relative to the first step, and Bk = 2Rk − Rk−1 − Rk+1 measures the discrete curvature of the average-rank profile, which is discretely concave.

Endpoint bounds and the common-core equality condition

The decomposition and monotonicity yield the two-sided bounds (n − 1)A2 ≤ def(V1, …, Vn) ≤ (n − 1)An, and the successive gaps give exact remainder formulas for both estimates. The paper characterizes the arrangements attaining either bound: with D = ∩_{i∈[n]} Vi, equality in either bound holds precisely when the quotient subspaces Vi/D form an internal direct sum in V/D. This common-core characterization identifies the arrangements whose dependence is entirely accounted for by their shared intersection. Annihilators yield a parallel hierarchy for intersections.

Kinser slack decomposes into quotient dimensions and recovers curvature

Ingleton's inequality and Kinser's inequalities constrain linearly representable polymatroids beyond submodularity. The paper gives an exact geometric decomposition of the Kinser slack: retaining the nonnegative remainders in Kinser's proof expresses the slack as a sum of explicit quotient dimensions and yields necessary and sufficient geometric conditions for equality. This decomposition is related to Friedman and Tootooni's study of failures of distributivity through discoordination.

The geometric decomposition connects to the defect curvatures. Averaged over all orderings, the Kinser slack equals (n − 2)B2. More generally, quotienting by the sum of k − 2 selected subspaces and averaging over all such selections and all orderings of the remaining subspaces gives (n − k)Bk for 2 ≤ k ≤ n − 2. Consequently, for these k, Bk = 0 if and only if every such ordered quotient arrangement saturates the corresponding Kinser inequality. Because every contracted Kinser slack is nonnegative, the average vanishes only when every summand vanishes individually, so Bk = 0 forces each contracted ordering to attain equality, not merely on average. The final gap Bn−1 has no counterpart in this contraction hierarchy, since the Kinser hierarchy begins with four subspaces. Full symmetrization therefore recovers the submodular curvature Bk ≥ 0 and yields no additional symmetric inequality; the representability-specific information remains in the individual ordered slacks.

Entropy analogues express defect levels as mutual information

The average-rank identities have an information-theoretic counterpart. For arbitrary finite-valued random variables, the entropy analogues of Ak and Bk are averages of mutual information and conditional mutual information, respectively: AH_k = c_{n,k} Σ_{|T|=k} Σ_{i∈T} I(Xi; X_{T\{i}}), and BH_k = (c_{n,k}/(n − k)) Σ_i Σ_{|S|=k−1} Σ_{j∉S∪{i}} I(Xi; Xj | XS). Summing the entropy analogues of Ak over k gives the total correlation Σ H(Xi) − H(X1, …, Xn), and the average entropy profile is discretely concave.

For a linear source over a finite field Fq with q elements, the entropy analogues are exactly Ak log q and Bk log q. The construction takes Z uniformly random in Fq^d, chooses a matrix Gi whose row space is Vi, and sets Xi = GiZ; then H(XS) = dim(Σ_{i∈S} Vi) log q. The identities for arbitrary finite-valued random variables still hold even when no such linear representation exists.

The final curvature captures dependence the contraction hierarchy omits

The paper's own example separates the two hierarchies cleanly. For n ≥ 4, take e1, …, e_{n−1} the standard basis of F^{n−1} and set Vi = span{ei} for 1 ≤ i ≤ n − 1 and Vn = span{e1 + ⋯ + e_{n−1}}. Every proper selection of these lines is independent, while the full selection has rank n − 1. Hence Rk = min{k, n − 1}, A2 = ⋯ = A_{n−1} = 0, An = 1, and B2 = ⋯ = B_{n−2} = 0 while B_{n−1} = 1. Every contracted Kinser slack appearing in the averaging identity vanishes, even though the total dimension defect is 1.

This is the paper's most decision-relevant split. The averaged Kinser identities recover Bk only for 2 ≤ k ≤ n − 2; the last curvature Bn−1 lies outside the contraction construction and is exactly where this arrangement's dependence shows up. Readers comparing symmetric defect quantities with representability-specific linear-rank inequalities should therefore treat Bn−1 as the level that the averaged Kinser hierarchy cannot see, and should look to the individual ordered slacks rather than their symmetrization when the additional representability information matters. The paper also notes that its results describe the geometric slack and the symmetric component of the Kinser hierarchy rather than settling the completeness problem for linear-rank inequalities.

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