Exact Optimal Half-Space (EOHS) Overview

• EOHS is a context-dependent framework that defines exact or optimal half-space constructions across active learning, antenna systems, statistical depth, and kinetic equations.
• The methodology employs bounded candidate directions and parallel binary search techniques to achieve query complexities of Θ(D + log n) in active learning applications.
• Applications range from adaptive label recovery without synthetic queries to precise geometric port selection and quasi-optimal numerical solvers with high computational accuracy.

Searching arXiv for the named topic and directly related papers. arxiv_search query: "^^^^^^^^4Exact Optimal Half-Space EOHS4^^^^^^^^" arxiv_search query: "^^^^^^^^4\4 Optimal Half-Space^^^^^^^^4\4 OR ^^^^^^^^4\4 Learning Halfspaces without Synthetic Data^^^^^^^^4\4 OR ^^^^^^^^4\4 Port Selection in CUMA Systems^^^^^^^^4\4 Exact Optimal Half-Space (EOHS) is used in the supplied literature as a context-dependent label rather than a single canonical method. In the cited sources, it denotes an exact active-learning framework for halfspaces without synthetic queries, a geometric port-selection scheme for compact ultra-massive antenna-array systems, and summary labels for exact halfspace-depth computation and for numerical solution of linear half-space kinetic equations. The shared terminology reflects a common reliance on half-space structure, but the formal objects, optimization criteria, and guarantees differ substantially across these settings (&&&^^^^4Exact Optimal Half-Space EOHS4^^^^&&&, &&&^^^^4\4^^^^&&&, &&&^^^^4 OR \4^^^^&&&, &&&^^^^4 OR \4^^^^&&&).

^^^^4\4^^^^. Terminological range and problem classes

The term EOHS appears in four distinct technical settings in the supplied sources. In active learning, the object is an unknown halfspace over a finite point set PRESERVED_PLACEHOLDER_^^^^4Exact Optimal Half-Space EOHS4^^^^, and the task is exact or approximate recovery of labels under query access restricted to points in PRESERVED_PLACEHOLDER_^^^^4\4^^^^. In compact ultra-massive antenna-array (CUMA) systems, EOHS is a port-selection rule that chooses a projection direction in PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^ to maximize desired-signal build-up. In multivariate statistics, the label is attached to exact computation of Tukey halfspace depth. In kinetic theory, it is attached to an exact and quasi-optimal half-space Galerkin framework (&&&^^^^4Exact Optimal Half-Space EOHS4^^^^&&&, &&&^^^^4\4^^^^&&&, &&&^^^^4 OR \4^^^^&&&, &&&^^^^4 OR \4^^^^&&&).

Context	Core object	Main task
Active learning	PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^, halfspace $f$, candidate normals $V$	Recover labels without point synthesis
CUMA systems	Unit vector $\mathbf c\in\mathbb R^2$	Maximize instantaneous desired-signal build-up
Halfspace depth	Data cloud $X$, query point $z$	Compute exact Tukey depth
Kinetic equations	Distribution $f(x,v)$	Solve linear half-space problem via damping and recovery

A recurrent source of confusion is to treat EOHS as if it named one standardized algorithmic paradigm. The supplied sources do not support that reading. Instead, they document several non-equivalent uses of the same label.

^^^^4 OR \4^^^^. EOHS in active learning: model, bounded directions, and exact complexity

In “Actively Learning Halfspaces without Synthetic Data” (&&&^^^^4Exact Optimal Half-Space EOHS4^^^^&&&), the EOHS setting is the point location problem on an arbitrary finite set PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS4^^^^ of PRESERVED_PLACEHOLDER_^^^^4\4\4^^^^ points, with query access to an unknown halfspace

PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^

The learner may adaptively query points PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^ and observe PRESERVED_PLACEHOLDER_^^^^4\44^^^^, but points outside PRESERVED_PLACEHOLDER_^^^^4\4^^^^5 may not be queried. The paper introduces a bounded-direction assumption: the learner knows a finite candidate set

PRESERVED_PLACEHOLDER_^^^^4\4^^^^6

such that PRESERVED_PLACEHOLDER_^^^^4\4^^^^7. Thus, PRESERVED_PLACEHOLDER_^^^^4\4^^^^8 is the size of the candidate normal-vector set.

The central exact-learning theorem states that there is a deterministic membership-query algorithm that exactly recovers PRESERVED_PLACEHOLDER_^^^^4\4^^^^9 on all of PRESERVED_PLACEHOLDER_^^^^4 OR \4Exact Optimal Half-Space EOHS4^^^^ using

PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^

label queries. Moreover, any algorithm without point-synthesis requires

PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^

queries, so the complexity is PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^ whenever PRESERVED_PLACEHOLDER_^^^^4 OR \44^^^^ (&&&^^^^4Exact Optimal Half-Space EOHS4^^^^&&&). The lower-bound structure is additive: PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^5 is needed even in one dimension to locate the threshold among PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^6 points, while PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^7 reflects the need to rule out incorrect directions.

The bounded-direction hypothesis is the mechanism by which the paper circumvents the PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^8 lower bound that applies when point synthesis is disallowed and the direction class is unbounded. The source states explicitly that bounding PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^9 is the only way to beat PRESERVED_PLACEHOLDER_^^^^4 OR \4Exact Optimal Half-Space EOHS4^^^^ in that setting.

A particularly important specialization is axis-aligned halfspaces, or decision stumps, in PRESERVED_PLACEHOLDER_^^^^4 OR \4\4^^^^. There the normal vectors are the PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^ vectors PRESERVED_PLACEHOLDER_^^^^4 OR \4 OR \4^^^^, so PRESERVED_PLACEHOLDER_^^^^4 OR \44^^^^. The exact-learning result then yields

PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^5

matching the PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^6 lower bound and closing the previously known gap between an PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^7 upper bound and the PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^8 information-theoretic lower bound (&&&^^^^4Exact Optimal Half-Space EOHS4^^^^&&&).

^^^^4 OR \4^^^^. Deterministic exact-learning mechanism: permutations, invariants, and parallel binary search

The exact algorithm in (&&&^^^^4Exact Optimal Half-Space EOHS4^^^^&&&) is presented through a more abstract problem: learning a Boolean function over PRESERVED_PLACEHOLDER_^^^^4 OR \4^^^^9 elements that is monotone under at least one of $f$^^^^4Exact Optimal Half-Space EOHS4^^^^ provided orderings. The halfspace setting is obtained by associating to each $f$^^^^4\4^^^^ the permutation $f$^^^^4 OR \4^^^^ obtained by sorting $f$^^^^4 OR \4^^^^ in ascending order of $f$4. The unknown target then corresponds to some index $f$5 and threshold location $f$6, so the label function is non-decreasing along $f$7.

The algorithm, denoted LSMF in the summary, maintains three sets: $f$8 Its invariant is that if $f$9, then the true boundary points $V$^^^^4Exact Optimal Half-Space EOHS4^^^^; otherwise the correct halfspace has already been placed into $V$^^^^4\4^^^^. This separation between active directions and accumulated candidate hypotheses is essential to the algorithm’s deterministic progress guarantee.

A main iteration begins by choosing a direction $V$^^^^4 OR \4^^^^, selecting four points $V$^^^^4 OR \4^^^^ whose ranks under $V$4 split the order into gaps of size less than $V$5, and querying their labels. If $V$6 or $V$7, then $V$8 and that direction is removed. Otherwise, the queried labels define an interval $V$9 of length less than $\mathbf c\in\mathbb R^2$^^^^4Exact Optimal Half-Space EOHS4^^^^ in which the unknown threshold must lie. Repeating this construction over active directions yields a family of intervals $\mathbf c\in\mathbb R^2$^^^^4\4^^^^, after which the subroutine

$\mathbf c\in\mathbb R^2$^^^^4 OR \4^^^^

is invoked.

Lemma ^^^^4 OR \4^^^^.^^^^4 OR \4^^^^, as summarized in the source, guarantees that in at most $\mathbf c\in\mathbb R^2$^^^^4 OR \4^^^^ further queries the subroutine either removes a wrong direction, identifies a boundary pair $\mathbf c\in\mathbb R^2$4 and adds the corresponding halfspace to $\mathbf c\in\mathbb R^2$5, or discards at least $\mathbf c\in\mathbb R^2$6 points from $\mathbf c\in\mathbb R^2$7 whose labels are thereby determined. The proof is described as an exhaustive eight-case analysis based on interval geometry and a coloring of points by relative position. Each call reduces either $\mathbf c\in\mathbb R^2$8 by one or $\mathbf c\in\mathbb R^2$9 by a factor of $X$^^^^4Exact Optimal Half-Space EOHS4^^^^, yielding $X$^^^^4\4^^^^ total calls. A final cleanup phase handles the constant-size residual instance, and an additional $X$^^^^4 OR \4^^^^ queries through Claim ^^^^4 OR \4^^^^.^^^^4\4^^^^ (“Contender”) select the single correct halfspace from $X$^^^^4 OR \4^^^^ (&&&^^^^4Exact Optimal Half-Space EOHS4^^^^&&&).

The paper’s stated technical insight is to perform a binary search in parallel across the candidate orderings rather than considering each ordering sequentially. A plausible implication is that the method is best understood not as $X$4 independent threshold searches but as a structured joint search over a family of order-compatible monotonicity constraints.

4. PAC and tolerant extensions derived from the exact learner

The exact learner in (&&&^^^^4Exact Optimal Half-Space EOHS4^^^^&&&) is used as a black box to obtain nearly optimal active-PAC guarantees. In the realizable setting, if $X$5, then with probability $X$6 one can find a hypothesis $X$7 with

$X$8

using

$X$9

queries. When

$z$^^^^4Exact Optimal Half-Space EOHS4^^^^

the source states that this can even be made deterministic.

The tolerant result allows adversarial corruption on up to a $z$^^^^4\4^^^^-fraction of $z$^^^^4 OR \4^^^^, for a sufficiently small constant $z$^^^^4 OR \4^^^^, while still obtaining a hypothesis of error at most $z$4 using

$z$5

queries. The bound is stated to be optimal up to a $z$6 factor, including in the realizable setting (&&&^^^^4Exact Optimal Half-Space EOHS4^^^^&&&).

The key subroutines are also explicit. Lemma 4.^^^^4\4^^^^ provides a randomized binary search for one-dimensional tolerant monotone-function learning using

$z$7

queries to output a monotone $z$8 with $z$9. This is combined with a voting or test step over the $f(x,v)$^^^^4Exact Optimal Half-Space EOHS4^^^^ directions to identify which coordinate is most consistent with monotonicity on a random subsample, followed by a final call to the randomized one-dimensional subroutine on the chosen direction.

The paper summarizes the overall picture as paying an additive $f(x,v)$^^^^4\4^^^^ for the bounded-direction assumption, plus the usual $f(x,v)$^^^^4 OR \4^^^^ or $f(x,v)$^^^^4 OR \4^^^^ cost associated with locating a threshold.

5. EOHS in CUMA systems: exact geometric port selection

In “Geometric Port Selection in CUMA Systems” (&&&^^^^4\4^^^^&&&), EOHS denotes an adaptive single-RF port-selection scheme for fluid antenna systems. Conventional CUMA partitions the $f(x,v)$4 ports by the sign of the real part of the desired-signal gains and activates the half-space with the larger rectified sum. EOHS replaces that fixed real-axis rule with a search over all unit projection directions $f(x,v)$5.

For user $f(x,v)$6, the per-port complex channel gains are $f(x,v)$7, and the corresponding real two-dimensional vectors are

$f(x,v)$8

A unit direction $f(x,v)$9 induces the partition

PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS4Exact Optimal Half-Space EOHS4^^^^

with rectified sums

PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS4\4^^^^

EOHS chooses the half-space with the larger sum, giving instantaneous desired-signal power

PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS4 OR \4^^^^

and solves

PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS4 OR \4^^^^

The exact solution is derived by parameterizing PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS44^^^^ and expressing each PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS4^^^^5 in polar form: PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS4^^^^6 Then

PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS4^^^^7

so the objective becomes

PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS4^^^^8

Each term changes sign when PRESERVED_PLACEHOLDER_^^^^4\4Exact Optimal Half-Space EOHS4^^^^9 crosses one of the PRESERVED_PLACEHOLDER_^^^^4\4\4Exact Optimal Half-Space EOHS4^^^^ boundary angles

PRESERVED_PLACEHOLDER_^^^^4\4\4\4^^^^

On any interval PRESERVED_PLACEHOLDER_^^^^4\4\4 OR \4^^^^, the active set is fixed and

PRESERVED_PLACEHOLDER_^^^^4\4\4 OR \4^^^^

which is unimodal on that interval. Consequently, the global maximizer lies in the finite candidate set consisting of all critical angles PRESERVED_PLACEHOLDER_^^^^4\4\44^^^^ together with interior stationary points PRESERVED_PLACEHOLDER_^^^^4\4\4^^^^5 that fall inside their intervals. Enumerating these PRESERVED_PLACEHOLDER_^^^^4\4\4^^^^6 candidates and evaluating each in PRESERVED_PLACEHOLDER_^^^^4\4\4^^^^7 time yields an exact PRESERVED_PLACEHOLDER_^^^^4\4\4^^^^8 algorithm (&&&^^^^4\4^^^^&&&).

The paper makes an important qualification: “Strictly speaking, EOHS is not optimal for SIR since it ignores interference, but it is statistically optimal for maximizing the desired-signal build-up.” The output SIR is

PRESERVED_PLACEHOLDER_^^^^4\4\4^^^^9

Thus, the optimization target is desired-signal aggregation rather than direct SIR maximization.

The same paper also introduces a PCA-based alternative with PRESERVED_PLACEHOLDER_^^^^4\4 OR \4Exact Optimal Half-Space EOHS4^^^^ complexity and reports that it achieves performance close to EOHS at a fraction of the computational cost. Simulation settings include square arrays PRESERVED_PLACEHOLDER_^^^^4\4 OR \4\4^^^^, aperture PRESERVED_PLACEHOLDER_^^^^4\4 OR \4 OR \4^^^^, rich-scattering Rayleigh channels at PRESERVED_PLACEHOLDER_^^^^4\4 OR \4 OR \4^^^^, and PRESERVED_PLACEHOLDER_^^^^4\4 OR \44^^^^ Monte-Carlo realizations. The reported metrics are per-user average rate, overall sum rate, BER, and outage. The source states that EOHS and the PCA-based scheme outperform conventional CUMA across user densities, port counts, and aperture sizes, while theoretical SIR PDFs match empirical histograms closely (&&&^^^^4\4^^^^&&&).

6. Other technical uses: exact depth computation and half-space kinetic solvers

In “Exact computation of the halfspace depth” (&&&^^^^4 OR \4^^^^&&&), EOHS is used in the supplied summary for a framework centered on Tukey halfspace depth. For a data cloud PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^5 and a query point PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^6, the halfspace depth is

PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^7

After translating to PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^8, the integer version is

PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^9

The key structural lemma states that if an optimal PRESERVED_PLACEHOLDER_^^^^4\4 OR \4Exact Optimal Half-Space EOHS4^^^^ has only PRESERVED_PLACEHOLDER_^^^^4\4 OR \4\4^^^^ data points on the boundary PRESERVED_PLACEHOLDER_^^^^4\4 OR \4 OR \4^^^^, then PRESERVED_PLACEHOLDER_^^^^4\4 OR \4 OR \4^^^^ can be tilted to bring an additional point onto the boundary without increasing the negative-side count. This leads to the exact reduction theorem

PRESERVED_PLACEHOLDER_^^^^4\4 OR \44^^^^

where PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^5 ranges over linearly independent PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^6-subsets, PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^7 spans the orthogonal complement of PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^8, and PRESERVED_PLACEHOLDER_^^^^4\4 OR \4^^^^9 spans the selected subspace. The framework yields recursive, PRESERVED_PLACEHOLDER_^^^^4\44Exact Optimal Half-Space EOHS4^^^^-dimensional-projection, and simple combinatorial variants with complexities PRESERVED_PLACEHOLDER_^^^^4\44\4^^^^, PRESERVED_PLACEHOLDER_^^^^4\44 OR \4^^^^, and PRESERVED_PLACEHOLDER_^^^^4\44 OR \4^^^^, respectively, while explicitly handling ties and degeneracies without a general-position assumption (&&&^^^^4 OR \4^^^^&&&).

In “A convergent method for linear half-space kinetic equations” (&&&^^^^4 OR \4^^^^&&&), the supplied EOHS exposition concerns a linear half-space problem

PRESERVED_PLACEHOLDER_^^^^4\444^^^^

posed on PRESERVED_PLACEHOLDER_^^^^4\44^^^^5, with incoming data at PRESERVED_PLACEHOLDER_^^^^4\44^^^^6 for velocities satisfying PRESERVED_PLACEHOLDER_^^^^4\44^^^^7 and an end-state condition as PRESERVED_PLACEHOLDER_^^^^4\44^^^^8. The method introduces a damped operator PRESERVED_PLACEHOLDER_^^^^4\44^^^^9 by adding mode-dependent damping terms to restore coercivity, obtaining

PRESERVED_PLACEHOLDER_^^^^4\4^^^^5^^^^4Exact Optimal Half-Space EOHS4^^^^

The analysis then uses an even-odd decomposition shifted about PRESERVED_PLACEHOLDER_^^^^4\4^^^^5^^^^4\4 a weighted trial space PRESERVED_PLACEHOLDER_^^^^4\4^^^^5^^^^4 OR \4^^^^, and a bilinear form PRESERVED_PLACEHOLDER_^^^^4\4^^^^5^^^^4 OR \4^^^^ satisfying boundedness and an inf-sup condition. Galerkin approximations PRESERVED_PLACEHOLDER_^^^^4\4^^^^54 satisfy the quasi-optimality estimate

PRESERVED_PLACEHOLDER_^^^^4\4^^^^55

Finally, the undamped solution is recovered from the damped one by solving a finite-dimensional linear system for correction coefficients and forming

PRESERVED_PLACEHOLDER_^^^^4\4^^^^56

The supplied summary characterizes this as exact, in the sense of being correct up to machine precision in the chosen basis, and optimal, in the sense of quasi-optimality in the PRESERVED_PLACEHOLDER_^^^^4\4^^^^57-norm (&&&^^^^4 OR \4^^^^&&&).

Taken together, these usages show that EOHS is best read as a family of exact or optimal half-space constructions whose content is determined entirely by context: exact active learning under bounded directions, exact geometric port selection, exact halfspace-depth reduction, or exact and quasi-optimal numerical treatment of half-space kinetic equations.