Exclusivity Partitions: Theory and Applications

• Exclusivity partitions are defined as disjoint regions induced by exclusion conditions across various fields such as decision theory, quantum information, and combinatorics.
• They delineate zones where no single estimator or assignment can achieve simultaneous optimality, reflecting intrinsic differences in loss functions or event exclusivity.
• Applications span metric search, conic optimization, and integer partition theory, demonstrating practical techniques for uniform pruning and effective problem decomposition.

Exclusivity partitions are partition structures induced by exclusivity or exclusion conditions. In statistical decision theory, an exclusivity partition is a collection of pairwise-disjoint exclusivity regions in a loss space, organized so that no estimator can be optimal across different regions (Halkiewicz, 16 Jul 2025). In quantum contextuality and Bell theory, the same phrase denotes either an assignment of exclusivity edges in a graph-based Bell scenario or a collection of pairwise-exclusive events imposing linear inequalities on probability vectors (Sadiq et al., 2011, Amaral, 2015). Related constructions also occur in additive combinatorics, metric indexing, integer partition theory, and conic optimization, where partition blocks encode uniform exclude distributions, static pruning regions, forbidden local configurations, or refined complementarity classes (Thornburgh, 2024, Connor, 2022, Ramírez et al., 2018).

1. Decision-theoretic definition

In the decision-theoretic framework, let $\Theta\subseteq\mathbb{R}$ be the parameter space and $\mathcal{X}$ the sample space. A loss function is

$L:\Theta\times\Theta\to[0,\infty),$

with $L(\theta,\theta)=0$ for all $\theta$ and $L(\theta,a)>0$ for $a\neq\theta$. The ambient family $\mathcal{L}$ is taken to be a large cone of admissible losses, continuous in both arguments, with suitable growth and smoothness near the diagonal $\theta=a$. For each exponent $p>0$, the canonical power-type loss is

$\mathcal{X}$0

More generally, a loss $\mathcal{X}$1 is a power-type loss of order $\mathcal{X}$2 if, as $\mathcal{X}$3,

$\mathcal{X}$4

with uniform $\mathcal{X}$5 in $\mathcal{X}$6 on compacts. The class of all such losses is denoted $\mathcal{X}$7.

An estimator is a measurable mapping $\mathcal{X}$8. Its frequentist risk under loss $\mathcal{X}$9 is

$L:\Theta\times\Theta\to[0,\infty),$0

and the worst-case risk is

$L:\Theta\times\Theta\to[0,\infty),$1

An estimator $L:\Theta\times\Theta\to[0,\infty),$2 is minimax under $L:\Theta\times\Theta\to[0,\infty),$3 if

$L:\Theta\times\Theta\to[0,\infty),$4

Fixing an optimality notion $L:\Theta\times\Theta\to[0,\infty),$5, here $L:\Theta\times\Theta\to[0,\infty),$6, a subset $L:\Theta\times\Theta\to[0,\infty),$7 is an exclusivity region if no single estimator can be $L:\Theta\times\Theta\to[0,\infty),$8-optimal on one loss $L:\Theta\times\Theta\to[0,\infty),$9 and simultaneously on any $L(\theta,\theta)=0$0. For a given estimator $L(\theta,\theta)=0$1, an exclusivity class $L(\theta,\theta)=0$2 satisfies two conditions: $L(\theta,\theta)=0$3 is an exclusivity region, and $L(\theta,\theta)=0$4 is $L(\theta,\theta)=0$5-optimal for at least one $L(\theta,\theta)=0$6. Every estimator admits the trivial exclusivity class $L(\theta,\theta)=0$7, but the framework is designed to isolate nontrivial classes determined by intrinsic loss geometry rather than by a single estimator. A collection $L(\theta,\theta)=0$8 of pairwise-disjoint exclusivity regions is a realizable exclusivity partition if for each $L(\theta,\theta)=0$9 there exists some estimator $\theta$0 that is $\theta$1-optimal for at least one $\theta$2; if additionally $\theta$3, the partition is total (Halkiewicz, 16 Jul 2025).

2. Power-type minimax exclusivity and conic geometry

Under standard regularity and smoothness assumptions, including dominated densities $\theta$4, Gâteaux differentiability of $\theta$5, and two-sided local comparability of losses, the power-type classes form disjoint minimax exclusivity regions: for any $\theta$6, $\theta$7 and $\theta$8 are disjoint minimax exclusivity regions, and no estimator can be simultaneously minimax for a loss in $\theta$9 and a loss in $L(\theta,a)>0$0 (Halkiewicz, 16 Jul 2025). The paper’s intuition is that an $L(\theta,a)>0$1 criterion balances small deviations differently from an $L(\theta,a)>0$2 criterion: when $L(\theta,a)>0$3, $L(\theta,a)>0$4 penalizes small errors more heavily, whereas $L(\theta,a)>0$5 trades off large deviations more strongly.

The proof proceeds by perturbation. First, by the positive-scalar cone structure of each $L(\theta,a)>0$6, rescaling $L(\theta,a)>0$7 scales $L(\theta,a)>0$8 but does not change the minimizers, so one may reduce to canonical losses $L(\theta,a)>0$9 and $a\neq\theta$0. If $a\neq\theta$1 is minimax for $a\neq\theta$2, then $a\neq\theta$3 is Fréchet differentiable at $a\neq\theta$4 and $a\neq\theta$5. For $a\neq\theta$6, one shows $a\neq\theta$7. Setting $a\neq\theta$8, taking $a\neq\theta$9, and defining $\mathcal{L}$0, a Danskin-type directional derivative and Taylor expansion yield, for small $\mathcal{L}$1,

$\mathcal{L}$2

while

$\mathcal{L}$3

Thus the worst-case $\mathcal{L}$4-risk strictly decreases while the $\mathcal{L}$5-risk changes only at second order, contradicting simultaneous minimaxity.

The ambient loss space $\mathcal{L}$6 is a convex cone in the Banach space $\mathcal{L}$7, closed under addition and nonnegative scalar multiplication. Each power-class $\mathcal{L}$8 is also a convex cone, but not a vector subspace: it contains no negatives, and sums of two $\mathcal{L}$9-losses need not preserve the same leading behavior. If $\theta=a$0, then $\theta=a$1, each $\theta=a$2 is closed in the sup-norm, and no sequence in $\theta=a$3 converges uniformly on compacts to a limit in $\theta=a$4. The partition $\theta=a$5 is therefore nontrivial and realizable, since each $\theta=a$6 is nonempty and for each $\theta=a$7 one can exhibit a minimax $\theta=a$8 under $\theta=a$9, such as the sample mean or median in well-known models. The same framework raises the further questions recorded in the paper: whether admissibility or Bayes optimality produce different exclusivity classes, whether finite-sample and asymptotic partitions coincide, whether all exclusivity classes are convex cones or may instead form more intricate strata such as manifolds or polyhedral cones, and whether there exists a nontrivial, total, realizable exclusivity partition of the full admissible-loss space $p>0$0.

3. Exclusivity graphs, Bell scenarios, and the Exclusivity Principle

In the graph-theoretic and quantum-information literature, an exclusivity graph $p>0$1 is a simple graph whose vertices represent elementary yes/no events, and two vertices are joined by an edge if and only if the corresponding events are exclusive. In a bipartite Bell scenario, exclusivity can arise because Alice’s outcomes are exclusive, because Bob’s outcomes are exclusive, or because one event’s exclusivity comes from Alice and the other from Bob; in the pentagon construction of the relevant Bell paper, the last possibility does not occur. An exclusivity partition is then the assignment of each edge of $p>0$2 to A-type or B-type exclusivity. For the pentagon $p>0$3 with vertices $p>0$4 and cycle edges $p>0$5, the Bell-pentagon admits the unique edge pattern $p>0$6 around the cycle. Its classical bound is the independence number $p>0$7, its quantum bound is at most the Lovász $p>0$8-number $p>0$9, and the fractional packing number $\mathcal{X}$00 appears as the linear-programming relaxation (Sadiq et al., 2011).

With that partition fixed, exactly three bipartite Bell inequalities arise. They all have local hidden-variable bound $\mathcal{X}$01:

$\mathcal{X}$02

$\mathcal{X}$03

and

$\mathcal{X}$04

Here $\mathcal{X}$05 is algebraically equivalent, up to an affine rescaling, to the standard CHSH form, $\mathcal{X}$06 appears as a primitive five-term building block in the eight-term CHSH written on the circulant graph $\mathcal{X}$07, and $\mathcal{X}$08 is induced inside the $\mathcal{X}$09 inequality as a five-event substructure. The experimental implementation used polarization-entangled photon pairs via type-II SPDC in BBO, narrowband filters, single-mode fibers, half-wave plates, and PBS analyzers, with $\mathcal{X}$10 per setting; the reported values were $\mathcal{X}$11, $\mathcal{X}$12, and $\mathcal{X}$13.

A closely related but distinct formalization appears in the Exclusivity Principle literature. There, an exclusivity partition is a collection $\mathcal{X}$14 of events such that every pair in $\mathcal{X}$15 is exclusive. Each such set imposes the linear constraint

$\mathcal{X}$16

together with $\mathcal{X}$17, and the intersection of all these half-spaces defines the polytope $\mathcal{X}$18. The framework exhibits activation effects: a distribution can satisfy all single-copy exclusivity constraints and still fail them after composition with another distribution on the same graph. For $\mathcal{X}$19, the uniform assignment $\mathcal{X}$20 satisfies the single-copy constraints, but in the OR-product graph $\mathcal{X}$21 the events $\mathcal{X}$22 form a clique of size $\mathcal{X}$23, and

$\mathcal{X}$24

Applying the Exclusivity Principle to two copies of $\mathcal{X}$25 yields $\mathcal{X}$26, hence $\mathcal{X}$27 and $\mathcal{X}$28, reproducing the quantum maximum of the KCBS inequality. Under additional assumptions, the same formalism recovers the Lovász $\mathcal{X}$29-bound and, for self-complementary graphs, the full quantum set (Amaral, 2015).

4. Uniform exclude distributions of Sidon sets

In additive combinatorics over $\mathcal{X}$30, let $\mathcal{X}$31 be a Sidon set, meaning that whenever $\mathcal{X}$32 are pairwise distinct, the equation $\mathcal{X}$33 has no solutions; equivalently,

$\mathcal{X}$34

for all $\mathcal{X}$35. The exclude-point set is

$\mathcal{X}$36

and the exclude-multiplicity, or exclude distribution, is the function

$\mathcal{X}$37

defined by

$\mathcal{X}$38

For disjoint subsets $\mathcal{X}$39 of the same size, $\mathcal{X}$40 is locally equivalent on $\mathcal{X}$41 if there exists a bijection $\mathcal{X}$42 such that $\mathcal{X}$43 for all $\mathcal{X}$44. If $\mathcal{X}$45 is a partition of some $\mathcal{X}$46 into equal-sized blocks, then $\mathcal{X}$47 is uniform on $\mathcal{X}$48 if every pair of blocks is locally equivalent.

When $\mathcal{X}$49 is the graph of a function $\mathcal{X}$50, the natural partition is

$\mathcal{X}$51

with each $\mathcal{X}$52 of size $\mathcal{X}$53. For APN plateaued maps with all nonzero component functions unbalanced, Carlet’s identity and a Walsh-transform triple-counting lemma imply

$\mathcal{X}$54

If $\mathcal{X}$55 and $\mathcal{X}$56, then

$\mathcal{X}$57

so the map $\mathcal{X}$58 establishes local equivalence of $\mathcal{X}$59 with $\mathcal{X}$60. The resulting theorem states that if $\mathcal{X}$61 is APN, plateaued, and all its nonzero component functions are unbalanced, then $\mathcal{X}$62 is uniform on the partition $\mathcal{X}$63 (Thornburgh, 2024).

For even $\mathcal{X}$64, the Gold and Kasami power maps provide explicit exclude-distribution formulas. Let

$\mathcal{X}$65

If $\mathcal{X}$66 is either a Gold map $\mathcal{X}$67 or a Kasami map $\mathcal{X}$68 with $\mathcal{X}$69, then exactly two exclude-multiplicities occur, namely $\mathcal{X}$70 and $\mathcal{X}$71. There are

$\mathcal{X}$72

points $\mathcal{X}$73 with value $\mathcal{X}$74, and

$\mathcal{X}$75

points with value $\mathcal{X}$76. The image of $\mathcal{X}$77 is therefore $\mathcal{X}$78. The paper further records the conjecture that if an APN map has uniform $\mathcal{X}$79 on $\mathcal{X}$80, then $\mathcal{X}$81 must be a maximal Sidon set in $\mathcal{X}$82.

5. Integer partitions, excludants, and exclusion conditions

In partition theory, one use of the term is highly specific: a partition $\mathcal{X}$83 of $\mathcal{X}$84 is an exclusivity-partition, or a $\mathcal{X}$85-sequence-free, distinct-parts partition, if its parts are all distinct and no two successive integers both appear. Equivalently, if $\mathcal{X}$86, then $\mathcal{X}$87 for all $\mathcal{X}$88. Writing $\mathcal{X}$89 for the number of such partitions of $\mathcal{X}$90 with exactly $\mathcal{X}$91 parts, the two-variable generating function is

$\mathcal{X}$92

and for $\mathcal{X}$93, $\mathcal{X}$94 one has

$\mathcal{X}$95

Setting $\mathcal{X}$96 recovers

$\mathcal{X}$97

the Rogers–Ramanujan product for partitions into parts differing by at least $\mathcal{X}$98. As $\mathcal{X}$99, the saddle-point analysis uses

$L:\Theta\times\Theta\to[0,\infty),$00

whose unique maximum occurs at purely imaginary $L:\Theta\times\Theta\to[0,\infty),$01 with $L:\Theta\times\Theta\to[0,\infty),$02, $L:\Theta\times\Theta\to[0,\infty),$03. Ingham’s Tauberian theorem then gives

$L:\Theta\times\Theta\to[0,\infty),$04

with

$L:\Theta\times\Theta\to[0,\infty),$05

The same paper establishes the inequalities

$L:\Theta\times\Theta\to[0,\infty),$06

and hence $L:\Theta\times\Theta\to[0,\infty),$07 (Bringmann et al., 2015).

A different exclusion statistic is the maximal excludant. For a nonempty partition $L:\Theta\times\Theta\to[0,\infty),$08 with largest part $L:\Theta\times\Theta\to[0,\infty),$09,

$L:\Theta\times\Theta\to[0,\infty),$10

If

$L:\Theta\times\Theta\to[0,\infty),$11

then its generating function is closely related to a mock theta function. The paper proves

$L:\Theta\times\Theta\to[0,\infty),$12

and equivalently

$L:\Theta\times\Theta\to[0,\infty),$13

If $L:\Theta\times\Theta\to[0,\infty),$14 denotes the sum of largest parts over partitions of $L:\Theta\times\Theta\to[0,\infty),$15, then $L:\Theta\times\Theta\to[0,\infty),$16 as $L:\Theta\times\Theta\to[0,\infty),$17, and the expectation

$L:\Theta\times\Theta\to[0,\infty),$18

converges to $L:\Theta\times\Theta\to[0,\infty),$19 (Chern, 2019).

A third partition-theoretic usage studies intersections of three classical conditions. A partition is $L:\Theta\times\Theta\to[0,\infty),$20-regular if no part is divisible by $L:\Theta\times\Theta\to[0,\infty),$21, $L:\Theta\times\Theta\to[0,\infty),$22-distinct if each part-size appears with multiplicity $L:\Theta\times\Theta\to[0,\infty),$23, and $L:\Theta\times\Theta\to[0,\infty),$24-flat if consecutive parts differ by less than $L:\Theta\times\Theta\to[0,\infty),$25 and the smallest part is $L:\Theta\times\Theta\to[0,\infty),$26. For the simultaneous $L:\Theta\times\Theta\to[0,\infty),$27-regular and $L:\Theta\times\Theta\to[0,\infty),$28-distinct condition,

$L:\Theta\times\Theta\to[0,\infty),$29

an eta-quotient that yields Ramanujan-type congruences such as

$L:\Theta\times\Theta\to[0,\infty),$30

For the simultaneous $L:\Theta\times\Theta\to[0,\infty),$31-regular and $L:\Theta\times\Theta\to[0,\infty),$32-flat condition,

$L:\Theta\times\Theta\to[0,\infty),$33

By contrast, no closed product-form generating function is known in general for the simultaneous $L:\Theta\times\Theta\to[0,\infty),$34-distinct and $L:\Theta\times\Theta\to[0,\infty),$35-flat family. The special case $L:\Theta\times\Theta\to[0,\infty),$36 consists of the triangular partitions $L:\Theta\times\Theta\to[0,\infty),$37 with generating function $L:\Theta\times\Theta\to[0,\infty),$38, while for $L:\Theta\times\Theta\to[0,\infty),$39 one has a finite-order linear recurrence in the largest part. For the triple intersection, the $L:\Theta\times\Theta\to[0,\infty),$40 case gives a Jacobi theta series, the $L:\Theta\times\Theta\to[0,\infty),$41 case coincides with the fifth-order mock theta function $L:\Theta\times\Theta\to[0,\infty),$42, and for $L:\Theta\times\Theta\to[0,\infty),$43 no closed form is known (Keith, 2019).

6. Partitioning by exclusion in metric search and conic optimization

In exact metric search, exclusion partitions arise as static decompositions of a finite metric space enabling whole classes of data points to be pruned at query time. For a Ptolemaic metric space $L:\Theta\times\Theta\to[0,\infty),$44, Ptolemy’s inequality yields the lower bound

$L:\Theta\times\Theta\to[0,\infty),$45

Writing

$L:\Theta\times\Theta\to[0,\infty),$46

this becomes

$L:\Theta\times\Theta\to[0,\infty),$47

Choosing pivots $L:\Theta\times\Theta\to[0,\infty),$48 and a parameter $L:\Theta\times\Theta\to[0,\infty),$49, one defines the three-way partition

$L:\Theta\times\Theta\to[0,\infty),$50

The corresponding exclusion tests are

$L:\Theta\times\Theta\to[0,\infty),$51

$L:\Theta\times\Theta\to[0,\infty),$52

$L:\Theta\times\Theta\to[0,\infty),$53

If the condition for a class holds, that subset contains no solutions to the radius query and can be skipped entirely. The mechanism is always at least as strong as hyperplane exclusion, weaker than Hilbert exclusion in isolation, and combinable with Hilbert exclusion without additional expensive distance calls. The reported empirical results for uniform Euclidean data with $L:\Theta\times\Theta\to[0,\infty),$54 points, $L:\Theta\times\Theta\to[0,\infty),$55 queries, and $L:\Theta\times\Theta\to[0,\infty),$56NN search radius give, for $L:\Theta\times\Theta\to[0,\infty),$57 pivots, approximately $L:\Theta\times\Theta\to[0,\infty),$58–$L:\Theta\times\Theta\to[0,\infty),$59 exclusion for hyperplane, $L:\Theta\times\Theta\to[0,\infty),$60–$L:\Theta\times\Theta\to[0,\infty),$61 for Ptolemaic alone up to $L:\Theta\times\Theta\to[0,\infty),$62D, $L:\Theta\times\Theta\to[0,\infty),$63–$L:\Theta\times\Theta\to[0,\infty),$64 for Hilbert alone, and $L:\Theta\times\Theta\to[0,\infty),$65–$L:\Theta\times\Theta\to[0,\infty),$66 for the combined method; with $L:\Theta\times\Theta\to[0,\infty),$67 or $L:\Theta\times\Theta\to[0,\infty),$68, combined exclusion reaches approximately $L:\Theta\times\Theta\to[0,\infty),$69–$L:\Theta\times\Theta\to[0,\infty),$70 even in $L:\Theta\times\Theta\to[0,\infty),$71–$L:\Theta\times\Theta\to[0,\infty),$72D (Connor, 2022).

In multifold conic optimization, a different partitioning problem concerns complementarity classes of blocks in a primal-dual conic pair. Let $L:\Theta\times\Theta\to[0,\infty),$73, let $L:\Theta\times\Theta\to[0,\infty),$74 be closed regular convex cones, and consider the KKT complementarity system

$L:\Theta\times\Theta\to[0,\infty),$75

Bonnans–Ramírez define the four-set partition $L:\Theta\times\Theta\to[0,\infty),$76 by whether some solution has $L:\Theta\times\Theta\to[0,\infty),$77, some solution has $L:\Theta\times\Theta\to[0,\infty),$78, or some solution has $L:\Theta\times\Theta\to[0,\infty),$79. Peña–Roshchina define the six-set partition $L:\Theta\times\Theta\to[0,\infty),$80 using the “always zero” sets

$L:\Theta\times\Theta\to[0,\infty),$81

Their intersection yields the refined seven-set partition

$L:\Theta\times\Theta\to[0,\infty),$82

The paper further proves that these partitions are preserved under blockwise nonsingular linear transformations and, in particular, under the standard second-order-cone to semidefinite-program conversion via the arrow map

$L:\Theta\times\Theta\to[0,\infty),$83

for which $L:\Theta\times\Theta\to[0,\infty),$84 if and only if $L:\Theta\times\Theta\to[0,\infty),$85 (Ramírez et al., 2018).

Across these literatures, exclusivity partitions organize spaces into blocks on which incompatibility, optimality, or feasibility has a uniform structural form. The specific objects differ—loss functions, events, excluded points, integer partitions, metric sublists, or conic blocks—but the common operation is a decomposition into regions that cannot be merged without losing a decisive exclusivity or exclusion property.