--- title: Exact Optimal Half-Space (EOHS) Overview url: https://www.emergentmind.com/topics/exact-optimal-half-space-eohs type: topic --- # Exact Optimal Half-Space (EOHS) Overview Searching arXiv for the named topic and directly related papers. arxiv_search query: "Exact Optimal Half-Space EOHS" arxiv_search query: "\"Exact Optimal Half-Space\" OR \"Actively Learning Halfspaces without Synthetic Data\" OR \"Geometric Port Selection in CUMA Systems\"" 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 [2509.20848] [2509.20299] [1411.6927] [1408.6630]. ## 1. 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 \(X\subset\mathbb R^d\), and the task is exact or approximate recovery of labels under query access restricted to points in \(X\). In compact ultra-massive antenna-array (CUMA) systems, EOHS is a port-selection rule that chooses a projection direction in \(\mathbb R^2\) 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 [2509.20848] [2509.20299] [1411.6927] [1408.6630]. | Context | Core object | Main task | |---|---|---| | Active learning | \(X\subset\mathbb R^d\), 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. ## 2. EOHS in active learning: model, bounded directions, and exact complexity In “Actively Learning Halfspaces without Synthetic Data” [2509.20848], the EOHS setting is the point location problem on an arbitrary finite set \(X\subset\mathbb R^d\) of \(n\) points, with query access to an unknown halfspace \[ f(x)=\mathbf 1(\langle u^\star,x\rangle>t^\star), \qquad u^\star\in\mathbb R^d,\; t^\star\in\mathbb R. \] The learner may adaptively query points \(x\in X\) and observe \(f(x)\in\{0,1\}\), but points outside \(X\) may not be queried. The paper introduces a bounded-direction assumption: the learner knows a finite candidate set \[ V=\{u_1,\dots,u_D\}\subset S^{d-1} \] such that \(u^\star\in V\). Thus, \(D\) 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 \(f\) on all of \(X\) using \[ Q_{\rm exact}=O\bigl(\min\{D+\log n,\;n\}\bigr) \] label queries. Moreover, any algorithm without point-synthesis requires \[ \Omega(\min\{D+\log n,n\}) \] queries, so the complexity is \(\Theta(D+\log n)\) whenever \(D+\log n\ll n\) [2509.20848]. The lower-bound structure is additive: \(\Omega(\log n)\) is needed even in one dimension to locate the threshold among \(n\) points, while \(\Omega(D)\) reflects the need to rule out incorrect directions. The bounded-direction hypothesis is the mechanism by which the paper circumvents the \(\Omega(n)\) lower bound that applies when point synthesis is disallowed and the direction class is unbounded. The source states explicitly that bounding \(D\) is the only way to beat \(\Omega(n)\) in that setting. A particularly important specialization is axis-aligned halfspaces, or decision stumps, in \(\mathbb R^d\). There the normal vectors are the \(2d\) vectors \(\pm e_i\), so \(D=2d\). The exact-learning result then yields \[ Q_{\rm exact}(DS_d)=O(d+\log n), \] matching the \(\Omega(d+\log n)\) lower bound and closing the previously known gap between an \(O(d\log n)\) upper bound and the \(\Omega(d+\log n)\) information-theoretic lower bound [2509.20848]. ## 3. Deterministic exact-learning mechanism: permutations, invariants, and parallel binary search The exact algorithm in [2509.20848] is presented through a more abstract problem: learning a Boolean function over \(n\) elements that is monotone under at least one of \(D\) provided orderings. The halfspace setting is obtained by associating to each \(u_i\in V\) the permutation \(\sigma_i\colon X\to[n]\) obtained by sorting \(X\) in ascending order of \(\langle u_i,x\rangle\). The unknown target then corresponds to some index \(i^\star\) and threshold location \(t^\star\), so the label function is non-decreasing along \(\sigma_{i^\star}\). The algorithm, denoted LSMF in the summary, maintains three sets: \[ S=\text{active directions},\qquad Z=\text{active points},\qquad C=\text{candidate hypotheses}. \] Its invariant is that if \(i^\star\in S\), then the true boundary points \(x^\star,y^\star\in Z\); otherwise the correct halfspace has already been placed into \(C\). 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 \(k\in S\), selecting four points \(z_1,z_2,z_3,z_4\in Z\) whose ranks under \(\sigma_k\) split the order into gaps of size less than \(n/3\), and querying their labels. If \(f(z_1)=1\) or \(f(z_4)=0\), then \(k\neq i^\star\) and that direction is removed. Otherwise, the queried labels define an interval \(W_k\) of length less than \(n/3\) in which the unknown threshold must lie. Repeating this construction over active directions yields a family of intervals \(\{W_i\}\), after which the subroutine \[ \mathsf{RemoveOrReduce}(S,Z,\{W_i\}) \] is invoked. Lemma 3.3, as summarized in the source, guarantees that in at most \(3\) further queries the subroutine either removes a wrong direction, identifies a boundary pair \((x,y)\) and adds the corresponding halfspace to \(C\), or discards at least \(n/3\) points from \(Z\) 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 \(|S|\) by one or \(|Z|\) by a factor of \(2/3\), yielding \(O(D+\log n)\) total calls. A final cleanup phase handles the constant-size residual instance, and an additional \(O(D)\) queries through Claim 2.1 (“Contender”) select the single correct halfspace from \(C\) [2509.20848]. 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 \(D\) 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 [2509.20848] is used as a black box to obtain nearly optimal active-PAC guarantees. In the realizable setting, if \(f\in(V,X)\), then with probability \(1-\delta\) one can find a hypothesis \(h\in(V,X)\) with \[ \Pr_{x\in X}[f(x)\ne h(x)]\le\varepsilon \] using \[ Q_{\rm PAC}=O\!\Bigl(\min\{D+\log(1/\varepsilon),\tfrac{\log D}{\varepsilon}\}\Bigr) \] queries. When \[ D+\log(1/\varepsilon)\le \frac{\log D}{\varepsilon}, \] the source states that this can even be made deterministic. The tolerant result allows adversarial corruption on up to a \(c\varepsilon\)-fraction of \(X\), for a sufficiently small constant \(c\), while still obtaining a hypothesis of error at most \(\varepsilon\) using \[ Q_{\rm tol}=O\Bigl(\min\{D+\log(1/\varepsilon),1/\varepsilon\}\cdot \log D\Bigr) \] queries. The bound is stated to be optimal up to a \(\log D\) factor, including in the realizable setting [2509.20848]. The key subroutines are also explicit. Lemma 4.1 provides a randomized binary search for one-dimensional tolerant monotone-function learning using \[ O(\log(1/\varepsilon)+\log(1/\delta)) \] queries to output a monotone \(\hat g\) with \(\|f-\hat g\|_1=O(\varepsilon)\). This is combined with a voting or test step over the \(D\) 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 \(\Theta(D)\) for the bounded-direction assumption, plus the usual \(\Theta(\log n)\) or \(\Theta(\log(1/\varepsilon))\) cost associated with locating a threshold. ## 5. EOHS in CUMA systems: exact geometric port selection In “Geometric Port Selection in CUMA Systems” [2509.20299], EOHS denotes an adaptive single-RF port-selection scheme for fluid antenna systems. Conventional CUMA partitions the \(N\) 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 \(\mathbf c\in\mathbb R^2\). For user \(u\), the per-port complex channel gains are \(h_{u,u}^{(k)}\in\mathbb C\), and the corresponding real two-dimensional vectors are \[ \mathbf g_{u,u,k}= \begin{bmatrix} \Re\{h_{u,u}^{(k)}\}\\[6pt] \Im\{h_{u,u}^{(k)}\} \end{bmatrix}\in\mathbb R^2. \] A unit direction \(\mathbf c\) induces the partition \[ \mathcal K_1(\mathbf c)=\{k:\mathbf g_{u,u,k}^T\mathbf c\ge 0\},\qquad \mathcal K_2(\mathbf c)=\{1,\dots,N\}\setminus \mathcal K_1(\mathbf c), \] with rectified sums \[ S_1(\mathbf c)=\sum_{k=1}^N\max\{\mathbf g_{u,u,k}^T\mathbf c,0\}, \qquad S_2(\mathbf c)=\sum_{k=1}^N\max\{-\mathbf g_{u,u,k}^T\mathbf c,0\}. \] EOHS chooses the half-space with the larger sum, giving instantaneous desired-signal power \[ \alpha_u(\mathbf c)=\bigl[\max\{S_1(\mathbf c),S_2(\mathbf c)\}\bigr]^2, \] and solves \[ \max_{\|\mathbf c\|=1}\; f(\mathbf c)=\sum_{k=1}^N\max\{\mathbf g_{u,u,k}^T\mathbf c,0\}. \] The exact solution is derived by parameterizing \(\mathbf c=[\cos\theta,\sin\theta]^T\) and expressing each \(\mathbf g_{u,u,k}\) in polar form: \[ \mathbf g_{u,u,k}=w_k \begin{bmatrix} \cos\phi_k\\ \sin\phi_k \end{bmatrix}, \qquad w_k=\|\mathbf g_{u,u,k}\|, \qquad \phi_k=\arg(\mathbf g_{u,u,k}). \] Then \[ \mathbf g_{u,u,k}^T\mathbf c=w_k\cos(\theta-\phi_k), \] so the objective becomes \[ f(\theta)=\sum_{k=1}^N\max\{w_k\cos(\theta-\phi_k),0\}. \] Each term changes sign when \(\theta\) crosses one of the \(2N\) boundary angles \[ b_k^+=\phi_k+\tfrac{\pi}{2},\qquad b_k^-=\phi_k-\tfrac{\pi}{2}\pmod{2\pi}. \] On any interval \((b_i,b_{i+1})\), the active set is fixed and \[ f(\theta)=C_i\cos\theta+D_i\sin\theta=R_i\cos(\theta-\psi_i), \] which is unimodal on that interval. Consequently, the global maximizer lies in the finite candidate set consisting of all critical angles \(b_i\) together with interior stationary points \(\psi_i\) that fall inside their intervals. Enumerating these \(O(N)\) candidates and evaluating each in \(O(N)\) time yields an exact \(O(N^2)\) algorithm [2509.20299]. 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 \[ z_u= \frac{\left|\sum_{k\in\mathcal K_u}\mathbf g_{u,u,k}^T\mathbf c^\star\right|^2} {\sum_{v\neq u}\left|\sum_{k\in\mathcal K_u}\mathbf g_{v,u,k}^T\mathbf c^\star\right|^2}. \] Thus, the optimization target is desired-signal aggregation rather than direct SIR maximization. The same paper also introduces a PCA-based alternative with \(O(N)\) complexity and reports that it achieves performance close to EOHS at a fraction of the computational cost. Simulation settings include square arrays \(N=N_1\times N_2\), aperture \(W\lambda\times W\lambda\), rich-scattering Rayleigh channels at \(f_c=15\,\mathrm{GHz}\), and \(10^5\) 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 [2509.20299]. ## 6. Other technical uses: exact depth computation and half-space kinetic solvers In “Exact computation of the halfspace depth” [1411.6927], EOHS is used in the supplied summary for a framework centered on Tukey halfspace depth. For a data cloud \(X=\{x_1,\dots,x_n\}\subset\mathbb R^d\) and a query point \(z\in\mathbb R^d\), the halfspace depth is \[ D(z\mid X)=\min_{u\in S^{d-1}}\frac{1}{n}\#\{i:u^T x_i\ge u^T z\}. \] After translating to \(z=0\), the integer version is \[ HD(0\mid X)=nD(z\mid X)=\min_{p\neq 0}\#\{i:p^T x_i\ge 0\}. \] The key structural lemma states that if an optimal \(p\) has only \(\ell0\) and an end-state condition as \(x\to\infty\). The method introduces a damped operator \(\mathcal L_d\) by adding mode-dependent damping terms to restore coercivity, obtaining \[ \langle f,\mathcal L_d f\rangle_{x,v}\ge \sigma_1\|f\|_{L_x^2L^2(a\,dv)}^2, \qquad \sigma_1>0. \] The analysis then uses an even-odd decomposition shifted about \(v_1=-u\), a weighted trial space \(\Gamma\), and a bilinear form \(\mathcal B(\cdot,\cdot)\) satisfying boundedness and an inf-sup condition. Galerkin approximations \(f_N\in\Gamma_N\) satisfy the quasi-optimality estimate \[ \|f-f_N\|_\Gamma\le C\inf_{w\in\Gamma_N}\|f-w\|_\Gamma. \] Finally, the undamped solution is recovered from the damped one by solving a finite-dimensional linear system for correction coefficients and forming \[ f_\phi(x,v)=f(x,v)-g(x,v)+\Phi(v). \] 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 \(\Gamma\)-norm [1408.6630]. 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. Source: https://www.emergentmind.com/topics/exact-optimal-half-space-eohs