Hybrid RSP+NS Scheme: Design Philosophy

• Hybrid RSP+NS scheme is a design philosophy coupling distinct modules—such as randomized sketch-and-project with Newton–Schulz refinement or RIS scheduling with network allocation—to overcome individual limitations.
• It is implemented in diverse fields, including quaternion linear algebra, wireless RIS systems, and RSMA-based secure and sensing communications, with each context tailoring the approach to specific objectives.
• Evaluations reveal improved runtime accuracy, energy savings, and enhanced signal security, while also highlighting challenges like non-convex optimization and idealized assumptions.

“Hybrid RSP+NS Scheme” is used in recent arXiv literature as a context-dependent label for constructions that deliberately combine two distinct modules, layers, or physical mechanisms. In quaternion numerical linear algebra, it denotes a solver that interleaves randomized sketch-and-project steps with hyperpower Newton–Schulz refinement for the Moore–Penrose pseudoinverse (Leplat et al., 23 Aug 2025). In wireless communications, it denotes joint RIS-element mode switching and network-level power–time scheduling, RSMA-based layered retransmission through a common stream, RSMA-enhanced artificial-noise transmission, and RSMA-enabled near-field sensing (Yuan et al., 2024, Loli et al., 2023, Zhou et al., 30 Nov 2025, Zhou et al., 2024). In AdS$_3$ string theory, an analogous use points to the hybrid formalism that mixes NS–NS and RR-sensitive ingredients in a supergroup sigma model (Eberhardt et al., 2018). This suggests that the phrase functions as a structural descriptor for hybridized designs rather than as a uniquely standardized term.

1. Terminological scope and domain-specific meanings

Across the cited works, the paired components called “RSP” and “NS” do not have a single invariant expansion. In the wireless-powered RIS setting, the first component is explicitly the RIS-side choice of element modes and amplification factors, while the second is network-level transmit-power and time allocation. In quaternion computation, the pair is randomized sketch-and-project plus Newton–Schulz. In RSMA-based systems, the pair is rate-splitting signaling plus a shared retransmission, secrecy, or sensing layer. In the AdS$_3$ literature, the “NS” side refers directly to NS–NS flux, while the companion component is the RR-sensitive sector of the hybrid formalism (Leplat et al., 23 Aug 2025, Yuan et al., 2024, Loli et al., 2023, Zhou et al., 30 Nov 2025, Zhou et al., 2024, Eberhardt et al., 2018).

Context	First hybrid component	Second hybrid component
Quaternion pseudoinverse computation	Randomized sketch-and-project	Hyperpower Newton–Schulz refinement
Wireless-powered hybrid RIS uplink	RIS element mode switching and amplification design	User transmit-power and time allocation
RSMA-HARQ	Common/private RSMA signaling structure	Layered retransmissions through the common stream
Near-field secure RSMA	RSMA common/private signaling	Common stream as artificial-noise-like interference plus hybrid beamfocusing
Near-field ISAC	RSMA communication beams	Near-field sensing with receive filtering and sensing constraints
AdS$_3$ hybrid formalism	NS–NS/WZW-like sector	RR-sensitive supergroup sigma model and ghost sector

The shared feature is not a common acronym dictionary but a common design pattern: two subsystems with different operating roles are optimized or coupled so that one compensates for the limitations of the other.

2. Quaternion-native randomized projection plus Newton–Schulz refinement

The most literal algorithmic use of the label appears in quaternion numerical linear algebra, where the target is the Moore–Penrose pseudoinverse $A^\dagger\in\mathbb{H}^{n\times m}$ of $A\in\mathbb{H}^{m\times n}$, defined by

$$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$

For the full-column-rank case $m\ge n$, the scheme is organized around the right identity constraint $XA\to I_n$ and the projector relation $AX\to P=AA^\dagger$. The initialization

$$X_0=\alpha A^H,\qquad \alpha\in\Bigl(0,\frac{2}{\|A\|_2^2}\Bigr)$$

ensures spectral contraction of the right deviation $_3$0.

The randomized stage, denoted RSP–Q, draws a sketch $_3$1, forms $_3$2, and projects $_3$3 onto the affine set $_3$4 through

$_3$5

For i.i.d. Gaussian sketches, the paper states the standard sketch-and-project contraction

$_3$6

so the randomized phase supplies global linear progress in expectation.

The Newton–Schulz stage then exploits the residual $_3$7. In the order-$_3$8 hyperpower form, the update is

$_3$9

with the exact residual recurrence

$_3$0

This delivers order-$_3$1 local convergence once $_3$2. The hybrid cycle performs $_3$3 RSP–Q steps, computes $_3$4, applies one hyperpower Newton–Schulz correction, and checks convergence by a test sketch $_3$5 through

$_3$6

Polynomial evaluation is accelerated by Paterson–Stockmeyer or binary factorization, so the NS correction does not require naive summation of powers. The method is matrix-free, operates directly in $_3$7 without real or complex embeddings, and preserves left/right multiplication order required by noncommutativity. In the reported benchmark on random tall matrices, damped order-2 NS is the fastest and most accurate baseline, while the hybrid RSP+NS method attains good accuracy with competitive runtime; RSP–Q alone is slower because repeated thin QR factorizations or SPD Gram solves for $_3$8 dominate cost (Leplat et al., 23 Aug 2025).

3. Wireless-powered RIS scheduling and active relaying formulations

In wireless-powered hybrid RIS-aided uplink, the phrase is used in a fully explicit scheduling sense. The system comprises $_3$9 single-antenna users, a single-antenna BS, a hybrid RIS with $A^\dagger\in\mathbb{H}^{n\times m}$0 elements, a co-located energy harvesting surface with $A^\dagger\in\mathbb{H}^{n\times m}$1 elements, and an energy station. Each RIS element has an idle indicator $A^\dagger\in\mathbb{H}^{n\times m}$2, an active/passive indicator $A^\dagger\in\mathbb{H}^{n\times m}$3, and an active-mode amplification factor $A^\dagger\in\mathbb{H}^{n\times m}$4. For user $A^\dagger\in\mathbb{H}^{n\times m}$5, the reflection matrix is

$A^\dagger\in\mathbb{H}^{n\times m}$6

so the architecture mixes active, passive, and idle elements within the same frame. The time-splitting protocol has a direct-only phase and a direct-plus-RIS-assisted phase, and the optimization target is

$A^\dagger\in\mathbb{H}^{n\times m}$7

subject to per-user QoS and an RIS energy constraint supplied by the energy station. The paper identifies the mode variables $A^\dagger\in\mathbb{H}^{n\times m}$8 as the RSP part and the user powers and transmission times $A^\dagger\in\mathbb{H}^{n\times m}$9 as the NS part. Because these variables are coupled nonlinearly through the SNR and RIS energy expressions, the resulting problem is a mixed-integer program. The proposed solution is hierarchical: PPO solves the outer RIS configuration problem, while convex optimization solves the inner network allocation problem. The action-space size is reduced from $A\in\mathbb{H}^{m\times n}$0 to $A\in\mathbb{H}^{m\times n}$1 by action composition with a logarithmic transformation, and the reported simulations show that the scheme can reduce user energy consumption by $A\in\mathbb{H}^{m\times n}$2 relative to the baseline without RIS (Yuan et al., 2024).

A related wireless interpretation appears in active RIS-assisted multi-antenna WPCN, where the active RIS assists both wireless energy transfer and wireless information transmission within a harvest-then-transmit protocol. Here the hybrid aspect is not named RSP+NS in the paper title, but the technical structure is closely aligned: RIS phase shifts and amplitude reflection coefficients are optimized separately for WET and WIT, together with PS transmit beamforming, RS receive beamforming, user powers, and time allocation. The uplink SNR explicitly includes amplified RIS noise,

$A\in\mathbb{H}^{m\times n}$3

and the solution uses alternating optimization with LMMSE, SDR, and SCA, with the SDR relaxation proved tight. With 10 reflecting elements and 4 antennas, the reported gains are $A\in\mathbb{H}^{m\times n}$4 over the corresponding single-antenna active-RIS scheme and $A\in\mathbb{H}^{m\times n}$5 over a passive-RIS scheme with 100 reflecting elements (Jiang et al., 2023). This suggests a broader RIS-centered meaning of hybridization in which relay physics, beamforming, and network resources are co-designed rather than optimized in isolation.

4. RSMA-based retransmission, secrecy, and near-field ISAC variants

In downlink RSMA with HARQ, the hybrid construction is a layered RSMA-HARQ mechanism in which retransmissions are scheduled through the common stream. The physical layer uses one common stream $A\in\mathbb{H}^{m\times n}$6 and $A\in\mathbb{H}^{m\times n}$7 private streams $A\in\mathbb{H}^{m\times n}$8, with transmit signal

$A\in\mathbb{H}^{m\times n}$9

The advanced design places retransmission bits for failed common and private packets into the common stream, which is decoded first under SIC and therefore offers higher success probability. This produces an L-HARQ architecture combining new data and retransmission bits in the same block and enabling backtrack decoding of earlier packets. The paper reports that the proposed scheme outperforms RSMA with conventional HARQ, helps close the throughput gap between HARQ and AMC in the high-SNR regime, and also achieves decreased packet error rate and lower latency. It also reports a regime split: at high SNR, the advanced scheme is superior, while at low SNR the baseline can have higher throughput because the advanced scheme uses retransmission lengths fixed at 15% of the baseline length, which may be inadequate under larger CSIT error; a fallback strategy is therefore suggested (Loli et al., 2023).

In near-field secure communication with hybrid analog–digital beamfocusing, the hybrid RS-plus-NS construction takes a different form. Each message is split into common and private components, but now the common stream is designed for dual purposes: it carries decodable information for legitimate users while acting as artificial noise for the eavesdropper. The secrecy structure is explicit: $$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$0 The design objective is to maximize the minimum secrecy rate by jointly optimizing the analog beamfocuser $$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$1, digital beamfocuser $$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$2, and common secrecy-rate allocation. The resulting non-convex problem is handled by a penalty-based alternating optimization algorithm that partitions the variables into three blocks, solves one by surrogate optimization, and updates the others in closed form. The reported simulations show that the transmit scheme approaches fully digital beamfocusing with substantially fewer RF chains, outperforms conventional beamfocusing-only and far-field security schemes, and preserves secrecy without significantly compromising communication rates (Zhou et al., 30 Nov 2025).

In near-field ISAC, the same RSMA framework is merged with sensing rather than secrecy. The BS employs a hybrid analog–digital architecture, the common and private streams form the communication side, and sensing performance is imposed through target-rate constraints and receive filters. The joint design maximizes the minimum communication rate while optimizing receive filters, analog and digital beamformers, common-rate allocation, and the number of dedicated sensing beams. The central theoretical result is a rank-zero solution reconstruction showing that dedicated sensing beams are unnecessary for near-field multi-target detection: the optimal solution has $$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$3, so sensing is embedded directly into the RSMA communication covariance. The algorithmic stack is a PDD-based double loop with WMMSE reformulation for communication rates and quadratic transforms for sensing rates. The reported simulations indicate performance comparable to fully digital beamforming with fewer RF chains, maintenance of near-field multi-target detection without compromising communication rates, and significant gains over conventional multiple-access schemes and far-field ISAC systems (Zhou et al., 2024).

5. Mixed-flux worldsheet hybrid formalism in AdS$$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$4

A distinct theoretical use of the phrase arises in AdS$$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$5 string theory, where the hybrid formalism of Berkovits–Vafa–Witten is used for mixed NS–NS/RR backgrounds on

$$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$6

The worldsheet dynamics are encoded in a sigma model on $$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$7 with action

$$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$8

where $$AA^\dagger A = A,\qquad A^\dagger A A^\dagger = A^\dagger,\qquad (AA^\dagger)^H = AA^\dagger,\qquad (A^\dagger A)^H = A^\dagger A.$$9 and

$m\ge n$0

The pure NS–NS WZW point is $m\ge n$1. In this setting, the hybrid construction mixes the NS–NS/WZW sector with RR-sensitive and ghost sectors in a single exact CFT.

The paper uses this formalism to revisit long strings and gaps in the BPS spectrum. At the pure NS–NS point, continuous $m\ge n$2 representations with

$m\ge n$3

describe long strings that can reach the AdS boundary at finite energy. Away from the WZW point, the conformal weights of charged excitations in these continuous representations become complex, so the corresponding states cannot remain in a unitary spectrum. The hybrid formalism therefore shows directly that continuous long-string representations are only allowed at $m\ge n$4. It also revises the BPS analysis: the $m\ge n$5 bound $m\ge n$6, responsible at the WZW point for missing chiral primaries, dissolves in mixed flux, and previously missing BPS multiplets reappear. The same analysis yields the unitarity range

$m\ge n$7

In this domain, the hybrid scheme is not a scheduling or numerical algorithm at all, but a representation-theoretic and worldsheet-CFT interpolation between pure NS–NS and mixed NS–NS/RR regimes (Eberhardt et al., 2018).

6. Shared structural patterns, limitations, and interpretive cautions

Across these otherwise unrelated literatures, the repeated motif is the coupling of two subsystems with sharply different operating roles. In quaternion computation, randomized projection supplies cheap global linear contraction and Newton–Schulz supplies high-order local correction. In wireless-powered RIS, element-level mode switching determines the feasible SNR–energy geometry and network-level allocation exploits it. In RSMA-HARQ, the common stream supplies a robust control layer for retransmissions; in secure RSMA it doubles as artificial noise; in near-field ISAC it coexists with sensing constraints that can be absorbed into the communication covariance. In the AdS$m\ge n$8 hybrid formalism, NS–NS solvability is coupled to RR-sensitive sectors that remove long-string continua and fill BPS gaps. This suggests that “hybrid RSP+NS” is best understood as a design philosophy of complementary coupling, not as a single algorithmic template (Leplat et al., 23 Aug 2025, Yuan et al., 2024, Loli et al., 2023, Zhou et al., 30 Nov 2025, Zhou et al., 2024, Eberhardt et al., 2018, Jiang et al., 2023).

The limitations are equally domain-specific. The wireless-powered RIS formulation assumes perfect CSI, a linear energy-harvesting model, single-antenna BS and users, and offline DRL training; the active-RIS WPCN model is likewise built on idealized channel knowledge and amplifier-noise models. In quaternion computation, the bottleneck of RSP–Q is the repeated formation of $m\ge n$9 through thin QR or SPD Gram solves. In RSMA-HARQ, the advanced common-stream retransmission strategy can underperform at low SNR, and the paper explicitly suggests fallback to the baseline there. In the near-field secure and ISAC settings, the optimization is still non-convex and the reported algorithms target stationary points under unit-modulus hybrid-architecture constraints. In AdS$XA\to I_n$0, the spectral conclusions are tied to the mixed-flux hybrid sigma model and the associated unitarity range $XA\to I_n$1 (Yuan et al., 2024, Leplat et al., 23 Aug 2025, Loli et al., 2023, Zhou et al., 30 Nov 2025, Zhou et al., 2024, Eberhardt et al., 2018, Jiang et al., 2023).

A common misconception is to read “RSP” and “NS” as fixed, universal acronyms. The cited literature does not support that interpretation. The same label can mean randomized sketch-and-project plus Newton–Schulz, RIS planning plus network scheduling, rate-splitting plus shared retransmission or artificial-noise behavior, RSMA plus near-field sensing, or NS–NS plus RR-sensitive hybridization. The technically correct reading is therefore always local: one must identify the underlying state variables, constraints, and objective function of the specific paper in which the phrase appears.