Dual-Space Propagation Approach

• Dual-Space Propagation Approach is a design principle that couples two complementary spaces, each providing unique information to improve learning and inference.
• It is applied across various domains such as DDoS detection, domain adaptation, graph contrastive learning, and massive-MIMO CSI compression to boost performance.
• The method employs diverse instantiations—from dual metrics and RKHS pairings to operator-space formulations—to reconcile ambiguous information and reduce computational complexity.

Dual-Space Propagation Approach denotes a family of constructions in which learning, inference, or numerical solution is carried out by coupling two spaces and propagating information between them rather than operating in a single representation alone. In the cited literature, these paired spaces take markedly different forms: geometric and angular similarity spaces in prototypical DDoS detection, image and routing feature spaces in open-set domain adaptation, local and fusion reproducing-kernel Hilbert spaces in distributed estimation, dominant-path and NLOS propagation spaces in massive-MIMO CSI compression, angle–range and space–time domains in near-field ISAC, Euclidean and hyperbolic manifolds in graph contrastive learning, and dyadic activity–difference state spaces in single-phase contrastive learning (Martinez et al., 2024, Du et al., 2023, Raghavan et al., 2024, Zhang et al., 2023, Hussain et al., 15 Jan 2025, Yang et al., 2022, Høier et al., 2023). Across these works, the approach serves to transmit class structure, physical constraints, or adjoint information through coupled domains that are individually specialized and jointly more informative than either space alone.

1. Conceptual scope and recurrent architecture

The literature shows that “dual-space” is not a single mathematical object but a recurring architectural pattern. In compressed sensing, it refers to simultaneous sampling in two mutually incoherent domains, such as real space and $k$-space, with information fed-forward from one space to the other to simplify the reconstruction problem (Lv et al., 2022). In fast convolution, it refers to physical space and Fourier space, where hierarchical local interactions are propagated through adaptive box structures while each scale is diagonalized by a short Fourier transform (Jiang et al., 2023). In deep generative sampling, the phrase denotes a one-dimensional dual divergence space induced by the Donsker–Varadhan variational representation of KL divergence, together with walks in the original data space guided by gradients of the dual function (Garg et al., 2024). In measurement-induced entanglement studies, the dual space is a doubled Hilbert space $\rho_D=\rho\otimes\rho$ in which second Rényi entropy is extracted through a SWAP operator after dual-space propagation under a replicated Lindblad equation (Belov et al., 2024). In renormalon subtraction, dual space is obtained by a Laplace transform in the hard scale of an observable, and propagation back to the physical scale is performed by an inverse Laplace integral analyzed by expansion-by-regions (Hayashi et al., 2023). In wave propagation, the term appears in a space-time dual-pairing summation-by-parts framework in which forward and adjoint discrete operators are paired through space-time inner products (Wiratama et al., 17 Mar 2026).

This diversity rules out an overly narrow interpretation. A Dual-Space Propagation Approach may couple two metrics on one latent space, two feature extractors, two RKHSs, two physical propagation domains, two replicated quantum systems, or a primal–adjoint pair of discrete operators. A common structural feature is that one space supplies information that is incomplete or ambiguous in the other, and the propagation mechanism is the device that transfers and reconciles this information.

2. Latent-space metric coupling and cross-space classification

A prototypical latent-space instantiation appears in DDoS attack detection, where traffic flow $x$ is embedded as $f_\phi(x)\in\mathbb{R}^d$, and “dual-space” means that the same latent representation is endowed with two simultaneous metrics: a normalized Euclidean distance and a cosine distance to class prototypes (Martinez et al., 2024). For a query embedding $\mathbf{q}_i=f_\phi(x_i)$ and prototype $\mathbf{p}_j$,

$$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$

and the combined distance is

$$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$

Classification is then performed by

$\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$

while training minimizes a prototypical negative log-likelihood over the dual-space distance. The support set is first aggregated into prototypes

$$\mathbf{p}_c = \frac{1}{|S_c|} \sum_{(x_i, y_i) \in S_c} f_\phi(x_i),$$

and class information is then propagated from support samples to prototypes and from prototypes to queries. The paper does not name this mechanism “propagation” formally, but explicitly interprets it as support $\rho_D=\rho\otimes\rho$0 prototype $\rho_D=\rho\otimes\rho$1 query $\rho_D=\rho\otimes\rho$2 backpropagation (Martinez et al., 2024).

A second latent-space example is open set domain adaptation with mixture-of-experts. There the two spaces are the image feature space $\rho_D=\rho\otimes\rho$3 and the routing feature space $\rho_D=\rho\otimes\rho$4, the latter extracted from the Graph Router rather than from routing scores (Du et al., 2023). Source prototypes $\rho_D=\rho\otimes\rho$5 are maintained by momentum updates, target instances are stored in memory banks, and target pseudo-labels are assigned independently in both spaces via nearest known-class prototypes. Unknown detection is a discrete inconsistency criterion: $\rho_D=\rho\otimes\rho$6 followed by

$\rho_D=\rho\otimes\rho$7

Unknown samples are then clustered in image feature space to form unknown prototypes, and a contrastive objective propagates these prototype assignments back into the embedding. The result is a threshold-free dual-space detection mechanism.

Graph representation learning provides a geometric variant of the same idea. In Dual Space Graph Contrastive Learning, a graph is propagated in Euclidean space and in hyperbolic space, with Euclidean embeddings mapped into the Poincaré ball through the exponential map and aligned to native hyperbolic embeddings by a hyperbolic contrastive objective (Yang et al., 2022). The two spaces are thus not two coordinate systems over the same features, but two distinct geometries with complementary inductive biases: Euclidean relative-distance structure and hyperbolic hierarchy.

3. Function-space, operator-space, and adjoint pairings

In distributed estimation by two agents with different feature spaces, the paired spaces are explicit function spaces. Each agent $\rho_D=\rho\otimes\rho$8 is endowed with linearly independent features $\rho_D=\rho\otimes\rho$9 and builds a local knowledge space

$x$0

with reproducing kernel

$x$1

The fusion center constructs a second RKHS,

$x$2

and information is propagated by two linear bounded operators: the upload $x$3 from $x$4 to $x$5, and the download

$x$6

from $x$7 back to the local knowledge space (Raghavan et al., 2024). The fusion space is not merely a bookkeeping device. It is the space in which meta-learning is carried out, after which the fused estimate is downloaded into each agent’s own RKHS. This is a bidirectional dual-space propagation in the strict operator-theoretic sense.

A related, but discretization-centric, pairing appears in the space-time dual-pairing summation-by-parts framework for wave equations (Wiratama et al., 17 Mar 2026). Spatial DP-SBP operators $x$8 satisfy

$x$9

and the self-adjoint second derivative is

$f_\phi(x)\in\mathbb{R}^d$0

In time, penalized operators $f_\phi(x)\in\mathbb{R}^d$1 and their adjoints $f_\phi(x)\in\mathbb{R}^d$2 are paired by

$f_\phi(x)\in\mathbb{R}^d$3

The fully discrete forward problem and the fully discrete adjoint are then linked by a space-time Lagrangian and by time reversal. This yields adjoint consistent fully discrete approximations, meaning that the adjoint of the discrete forward operator is itself a consistent discretization of the continuous adjoint wave equation. Here, dual-space propagation means propagation in the primal and dual operator spaces associated with the space-time inner product $f_\phi(x)\in\mathbb{R}^d$4.

4. Physical-domain and multiscale dual spaces

In massive-MIMO CSI compression, the two spaces are physically motivated propagation-feature spaces within the same angular–delay representation (Zhang et al., 2023). The input CSI image

$f_\phi(x)\in\mathbb{R}^d$5

contains dominant propagation-path features, which appear as a few sharp, high-amplitude localized pixels, and NLOS propagation-path features, which appear as many smooth, low-amplitude spread-out pixels. DuffinNet therefore uses a ConvNet branch to extract NLOS features, an attention-empowered branch to emphasize dominant paths, and an FNet fusion module to combine both. In the encoder, the fusion output is compressed; in the decoder, cascaded dual-feature blocks reconstruct the CSI. This is dual-space propagation in the sense of propagating two physically distinct channel structures through separate branches and a learned fusion operator.

Near-field ISAC provides a dual physical-domain formulation of a different kind. Near-field propagation depends jointly on angle and range, while clutter mitigation and target detection are performed in space-time through STAP (Hussain et al., 15 Jan 2025). The method couples a DFT-based codebook for coarse estimation with a polar codebook for refinement, then propagates the resulting angle–range estimates into a reduced-dimension near-field STAP stage. The complexity is reduced from

$f_\phi(x)\in\mathbb{R}^d$6

to

$f_\phi(x)\in\mathbb{R}^d$7

with an approximate reduction factor

$f_\phi(x)\in\mathbb{R}^d$8

This is not merely two stages of processing. The dual-purpose codebooks and the space-time processor are co-designed so that information obtained in the angle–range domain directly constrains propagation in the space-time adaptive filter.

Dual-space compressed sensing uses two mutually incoherent sampling spaces for the same sparse object, typically real space and Fourier space (Lv et al., 2022). The central constrained problem is

$f_\phi(x)\in\mathbb{R}^d$9

A partial reconstruction from $\mathbf{q}_i=f_\phi(x_i)$0-space is used to select targeted $\mathbf{q}_i=f_\phi(x_i)$1-space samples, which identify known nonzero entries and induce the decomposition $\mathbf{q}_i=f_\phi(x_i)$2. The second CS problem is then solved only for the truncated unknown part: $\mathbf{q}_i=f_\phi(x_i)$3 The paper formalizes the speedup condition through the x-space success probability $\mathbf{q}_i=f_\phi(x_i)$4 and the k-space success probability $\mathbf{q}_i=f_\phi(x_i)$5, with dual-space CS advantageous when $\mathbf{q}_i=f_\phi(x_i)$6.

The multilevel kernel-splitting framework offers a multiscale analogue. DMK operates simultaneously in physical space and Fourier space. For the 3D Laplace kernel,

$\mathbf{q}_i=f_\phi(x_i)$7

where $\mathbf{q}_i=f_\phi(x_i)$8 is a windowed mollified kernel, $\mathbf{q}_i=f_\phi(x_i)$9 are difference kernels, and $\mathbf{p}_j$0 is a localized residual (Jiang et al., 2023). Outgoing Fourier expansions, diagonal translations, and local polynomial expansions are propagated up and down an adaptive tree. The essential feature is that each scale is diagonalized in a short Fourier basis while near-field interactions remain local in physical space.

5. Dual states, replica spaces, and divergence spaces

In single-phase contrastive learning, dual propagation replaces phase differences by state differences inside each neuron (Høier et al., 2023). A dyadic neuron carries two intrinsic states,

$\mathbf{p}_j$1

with mean

$\mathbf{p}_j$2

and difference

$\mathbf{p}_j$3

The mean acts as activity, the difference as an error-like quantity. Weight updates are local: $\mathbf{p}_j$4 The original formulation shows that a single inference phase with layerwise closed-form updates can match backpropagation accuracy and runtime on common benchmarks (Høier et al., 2023). The later adjoint-state reformulation introduces a related objective

$\mathbf{p}_j$5

revealing a direct connection to adjoint-state methods and yielding stable asymmetric nudging (Høier et al., 2024). In this setting, “dual-space” is effectively the coupled mean–difference state space.

Quantum measurement-induced phase transitions use a replica-space counterpart. The doubled density matrix

$\mathbf{p}_j$6

evolves under a dual-space Lindblad equation, and the second Rényi entropy is extracted by a SWAP operator $\mathbf{p}_j$7: $\mathbf{p}_j$8 The doubled system allows the calculation of the relevant trajectory-averaged purity without conflating quantum entanglement and classical measurement randomness (Belov et al., 2024).

Deep generative sampling in the dual divergence space uses a different replica-free construction. A neural dual function $\mathbf{p}_j$9 is trained through the Donsker–Varadhan representation

$$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$0

where $$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$1 is the product of marginals of the data distribution $$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$2 (Garg et al., 2024). The learned scalar $$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$3 embeds each sample into a one-dimensional dual divergence space. Localized kNN divergences detect cut points, producing clusters between two endpoints on the real line, and new samples are generated by gradient-based walks in the original data dimensions that interpolate between clusters. The paper explicitly states that, in theory, any sample embedded between those two end points is in-distribution with respect to the data distribution.

6. Dual-space analytic continuation and renormalon separation

In high-precision QCD and the $$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$4 nonlinear sigma model, dual space is obtained through a Laplace transform in the hard scale rather than through a second feature extractor or metric (Hayashi et al., 2023). For a one-scale observable $$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$5, with $$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$6, the dual representation is

$$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$7

with inverse

$$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$8

The dual Wilson coefficient $$D_{\text{E}, i, j} = \frac{\lVert \mathbf{q}_i - \mathbf{p}_j \rVert_2}{\sum_{k=1}^{C} \lVert \mathbf{q}_i - \mathbf{p}_k \rVert_2}, \qquad D_{\text{C}, i, j} = 1 - \frac{\mathbf{q}_i \cdot \mathbf{p}_j}{\lVert \mathbf{q}_i \rVert_2 \,\lVert \mathbf{p}_j \rVert_2},$$9 becomes a renormalon-free perturbative series in $$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$0, and infrared renormalons are suppressed because a contribution of the form

$$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$1

vanishes whenever $$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$2 with $$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$3. This allows simultaneous suppression of multiple infrared renormalons by a suitable choice of $$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$4.

The inverse Laplace representation is then analyzed by expansion-by-regions. Hard contributions come from the dual-space OPE $$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$5, while soft contributions arise by expanding the kernel $$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$6 at small $$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$7. In the sigma model, the cancellation of renormalon ambiguities and UV/IR divergences is encoded in the exact contour identity

$$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$8

This analytic version of dual-space propagation is distinct from machine-learning uses of the phrase, yet it follows the same general pattern: a transformed space removes pathological structure locally, and propagation back to the original observable is carried out by a controlled integral operator.

7. Empirical behavior, clarifications, and open directions

The empirical record across these works is domain-specific but broadly favorable. In reduced-data DDoS detection, the MLP with Attention trained in the dual-space prototypical regime achieves average accuracy $$D_{\text{E+C}, i, j} = \alpha \cdot D_{\text{E}, i, j} + (1-\alpha) \cdot D_{\text{C}, i, j}.$$9, F1-score $\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$0, precision $\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$1, and recall $\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$2 with only 100 labeled training samples, outperforming both offline supervised training and a traditional prototypical baseline under data scarcity (Martinez et al., 2024). In CSI compression, Duffin-CsiNet reaches indoor NMSE $\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$3 dB at $\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$4 and $\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$5 dB at $\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$6, outperforming ACRNet and CLNet in the reported settings; at $\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$7, it requires only 2 bits per codeword element to reach NMSE $\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$8 dB (Zhang et al., 2023). In dual propagation, performance is essentially on par with backpropagation on VGG16 for CIFAR-10, CIFAR-100, and ImageNet32x32, while retaining a local single-phase update rule (Høier et al., 2023). In the central-spin MIPT study, the dual-space second Rényi analysis yields a critical measurement rate $\hat{y}_j = \arg\min_{c} D_{\text{E+C}, j, c},$9, significantly different from the $$\mathbf{p}_c = \frac{1}{|S_c|} \sum_{(x_i, y_i) \in S_c} f_\phi(x_i),$$0 obtained from the mutual entropy proxy (Belov et al., 2024). In near-field ISAC, the proposed codebook-plus-NF-STAP framework reduces STAP complexity by three orders of magnitude (Hussain et al., 15 Jan 2025).

A recurring misconception is that dual-space methods must involve two physically distinct coordinate systems. The literature shows otherwise. In some cases the two spaces are merely two metrics on one latent embedding (Martinez et al., 2024). In others they are local and fusion RKHSs linked by bounded operators (Raghavan et al., 2024), or Euclidean and hyperbolic manifolds coupled by log/exp maps and a contrastive loss (Yang et al., 2022). This suggests that “dual-space” should be read structurally rather than geometrically: the essential point is the existence of two coupled representational regimes with a nontrivial transfer mechanism between them.

Open directions are explicit in the cited works. Distributed estimation points to more than two agents, iterative collaborative learning, and infinite-dimensional RKHS extensions (Raghavan et al., 2024). CSI feedback suggests temporal dual-space modeling, multiuser or cell-free extensions, and more explicit physics-informed partitioning of propagation components (Zhang et al., 2023). Graph contrastive learning suggests adding further manifolds or curvatures beyond the Euclidean–hyperbolic pair (Yang et al., 2022). Dual propagation points toward analog and neuromorphic implementations, asymmetric feedback, and richer compartmental models (Høier et al., 2024). DMK highlights high-frequency kernels, non-translation-invariant kernels, and fast inversion and preconditioning as unresolved directions (Jiang et al., 2023). The wave-equation framework indicates immediate applicability to adjoint-based inverse problems in higher dimensions and more complex boundary treatments (Wiratama et al., 17 Mar 2026).

Taken together, these results suggest that Dual-Space Propagation Approach is best understood not as a domain-specific algorithm but as a general design principle: isolate complementary structure in two spaces, define a mathematically controlled transfer between them, and use that transfer to stabilize inference, sharpen separation, or reduce complexity.