Hybrid Implicit Signals

• Hybrid implicit signals are a class of inter-agent measurements that fuse implicit sensing, communication, and modeling through latent, continuous field-valued variables.
• They enable robust information exchange across diverse domains including neural implicit modeling, multi-modal behavioral inference, and millimeter-wave MIMO beamforming.
• These signals underpin formal system theories and multi-object modeling, facilitating compositional reasoning, probabilistic importance sampling, and collision-free reconstructions.

Hybrid implicit signals denote a class of signal representations, system variables, or inter-agent measurements that combine implicit sensing, communication, or modeling mechanisms—often mediated by environmental fields, latent neural codes, or cross-modality features—without explicit structured transmission. Such signals feature prominently in formal system theory, neural implicit modeling, multi-modal behavioral inference, and millimeter-wave communication design. Their salient property is that crucial information is exchanged, fused, or conditioned through latent, continuous, or field-valued means, facilitating compositional reasoning, efficient inference, or robust control in complex systems.

1. Formal System-Theoretic Foundations

Hybrid implicit signals originate in the theory of Hybrid Input/Output Automata (HIOA) and their extensions, notably with world variables for implicit communication (Capiluppi et al., 2012, Capiluppi et al., 2013). In this context, standard HIOA is generalized to assign special variables whose values are functions of both time and space: world variables $w(t, p)$. These variables represent physical fields—such as pressure, color, signal strength—perturbed by agents and sensed by others, serving as the main conduit for implicit, non-message-based information exchange.

Formally, a HIOAW or World Automaton (WA) comprises disjoint sets of automaton variables $(U_a, X_a, Y_a)$, world variables $(U_w, X_w, Y_w)$, action sets $(I,H,O)$, state space, transitions, and trajectories. Each world variable $w: J \times M \to B_w$ dynamically encodes a space-time field over a metric domain (e.g., $M=\mathbb R^n$). Parallel composition crucially sums world outputs:

$w(t, p) = w_1(t, p) + w_2(t, p)$

enforcing superposition of environmental perturbations (Capiluppi et al., 2012, Capiluppi et al., 2013). Hierarchical nesting ("inplacement") allows one WA to reside within another, manipulating world input/output variables across levels while preserving trace inclusion and bisimulation properties.

2. Hybrid Implicit Signals in Neural Representations

In geometric inference and neural implicit modeling, hybrid implicit signals materialize as joint representations combining distinct implicit fields and explicit geometric priors. For large-scale 3D reconstruction, "hybrid implicit surface learning" (e.g., ViiNeuS/SCILLA (Djeghim et al., 2024)) employs both a volumetric density field $\sigma(x) \geq 0$ and a signed distance field (SDF) $f(x) \in \mathbb R$, each parameterized by geometry MLPs with multi-resolution hash embeddings. Volume rendering leverages both regimes:

• Volumetric: classical NeRF-style $\alpha^v$ transitions.
• SDF: NeuS-style $\alpha^f$ using $\Phi_s(f(x))$ for sharp zero-level set localization.

The hybrid approach uses a self-supervised proposal network for probabilistic density-based importance sampling, rapidly bootstraps SDF accuracy via staged transitions—first learning $\sigma$, then combining $\alpha^v$ and $\alpha^f$, finally fully refining $f(x)$—all without requiring external geometric priors. Typical pipelines optimize composite losses:

$$\mathcal{L} = \mathcal{L}_{rgb} + \lambda_1\mathcal{L}_{dssim} + \lambda_2\mathcal{L}_N + \lambda_3\mathcal{L}_{eik} + \lambda_4\mathcal{L}_s + \mathcal{L}_{prop} + \mathcal{L}_{sky}$$

demonstrating superior efficiency and mesh fidelity (Djeghim et al., 2024).

"Hybrid representations" in neural implicit surface modeling also appear as iterative extraction of explicit iso-points from the zero-level set of $f(p;\theta)$, which are then fed back as regularization and sampling targets in the loss (Yifan et al., 2020). This alternating implicit-to-explicit loop leverages on-surface samples $\{p \mid f(p) = 0\}$, geometric constraints (normal consistency, edge-aware upsampling), and importance sampling tuned to surface curvature or loss, yielding accelerated convergence and robust topology (Yifan et al., 2020).

3. Multi-Modality Fusion and Behavioral Signal Analysis

Behavioral inference systems frequently deploy hybrid implicit signals across sensor modalities—e.g., combining EEG and eye-tracking for non-identifying gender and emotion recognition (Bilalpur et al., 2017). Here, implicit user signals from low-cost devices are cleaned (band-pass filtering, ICA), reduced (PCA), and concatenated (early fusion) or probabilistically fused (late fusion):

$$\mathbf{x}^{EF} = [\mathbf{x}^{EEG} \| \mathbf{x}^{EYE}]$$

or

$S(c) = \alpha_1 F_1 p_1(c) + \alpha_2 F_2 p_2(c)$

with optimized weights. Analysis reveals modality-dependent discriminability:

• EEG features outperform eye-tracking for gender recognition ($\sim$0.71 AUC for high-intensity anger/disgust conditions).
• Eye-tracking surpasses EEG for coarse valence recognition ($\sim$0.64 AUC). Early fusion consistently outperforms elaborate decision-level fusion due to limited complementary signal content or data scarcity. Temporal analysis shows that discriminative EEG signatures for gender are distributed across the entire post-stimulus epoch, mapping to frontal electrodes (Bilalpur et al., 2017).

4. Communication Systems: Millimeter-Wave Hybrid Beamforming via Implicit Signals

In millimeter-wave MIMO, hybrid implicit signals refer to the system's use of coupling coefficients between pairs of analog beamformers as proxies for channel state information (CSI) (Chiang et al., 2017, Chiang et al., 2018). Rather than estimating full channel matrices $H[k]$, only scalar beam-pair measurements

$c_{i,j}[k] = w_i^H H[k] f_j$

are collected by correlating pilots. These coefficients are assembled into small coupling matrices $C[k]$ feeding directly into selection metrics:

• Frobenius norm: $\|C[k]\|_F^2$
• Determinant: $|\det C[k]|^2$

Candidate analog beam sets are ranked on these metrics, dramatically lowering training overhead and computational burden. Final digital beamformers are constructed from SVDs of low-dimensional $C[k]$, achieving performance nearly matching or surpassing full-CSI methods (e.g., $<5\%$ throughput loss in simulation) (Chiang et al., 2017, Chiang et al., 2018).

5. Cross-Category Signaling and Multi-Object Implicit Fields

In multi-object modeling, hybrid implicit signals describe the integration of per-object deformation fields and cross-category refinement networks. MODIF (Liu et al., 2023) encodes each category via an SDF $s'_{i,j}$ with a per-instance code $\alpha_{i,j}$ and deformation-correction pipeline (rigid + non-rigid warp, local SDF residuals, feature summaries). To enforce collision-free, consistent reconstructions, object-specific features $\gamma_{i,1} \dots \gamma_{i,m}$ are concatenated and transduced by an MLP $U$:

$$\Delta s_{O_i} = U([\gamma_{i,1} \dots \gamma_{i,m}])$$

yielding global corrections that "push-pull" object boundaries, further shaped by an attraction–repulsion contact loss. The joint loss aggregates per-object reconstructions and cross-category regularizations, optimizing both latent codes and network weights for high-fidelity, non-interpenetrating anatomical models (Liu et al., 2023).

6. Applications, Properties, and Future Directions

Hybrid implicit signal frameworks enable robust compositional modeling, multi-agent coordination, multi-modal inference, and scalable neural surface reconstruction, with the following persistent themes:

• Summation or fusion of field-based signals in parallel composition enables modular design and verification of large systems (Capiluppi et al., 2012, Capiluppi et al., 2013).
• Probabilistic or geometry-aware importance sampling via hybrid representations accelerates convergence and improves accuracy in neural 3D inference (Djeghim et al., 2024, Yifan et al., 2020).
• Low-dimensional implicit measurements reduce computational and estimation overhead in wireless communications (Chiang et al., 2017, Chiang et al., 2018).
• Cross-object signaling and hybrid loss design empower collision-free, semantically consistent multi-object models in medical and biological settings (Liu et al., 2023).
• Multi-modal fusion of behavioral signals supports privacy-preserving recognition pipelines adaptable to workload estimation and recommendation tasks (Bilalpur et al., 2017).

A plausible implication is that further work may extend hybrid implicit frameworks to richer agent creation/destruction, multi-level environmental hierarchies, and advanced graph or attention layers for cross-category signaling (Capiluppi et al., 2013, Liu et al., 2023).

7. Synthesis: Structural and Computational Patterns

The following table summarizes core manifestations of hybrid implicit signals across research domains:

Domain	Signal/Variable Type	Core Mechanisms
Formal automata	World variables $w(t, p)$	Field summation, hierarchical inplacement
Neural representation	$\sigma(x)$, $f(x)$, iso-points	Multi-field, explicit/implicit loop, importance
Behavior inference	EEG/artifact-reduced, oculomotor	PCA/fusion, modality matching
MIMO beamforming	Coupling coefficients $c_{i,j}[k]$	Pilot correlations, small-matrix precoding
Multi-object modeling	Per-object SDF, cross-feature MLP	Latent decomposition, cross-category corrections

Hybrid implicit signals drive a transition from explicit, message-centric designs to architectures leveraging implicit, latent, or field-valued interdependencies, achieving scalable inference, effective multi-agent communication, efficient reconstruction, and robust fusion in complex environments.