Zero-Shot Parseval-Frame Equalizers

• The paper introduces a closed-form, zero-shot equalizer that uses Parseval frames to align heterogeneous transmitter and receiver latent spaces without retraining.
• It leverages anchor set encoding and a whitening process to construct numerically stable, training-free equalizers, ensuring semantic consistency over both digital and physical channels.
• Empirical results reveal that these equalizers nearly match supervised aligners, achieving 90–95% performance gap closure under various benchmarks and operational conditions.

Zero-shot Parseval-frame-based equalizers are closed-form, training-free operators for aligning mismatched latent spaces in AI-native semantic communication systems, particularly where transmitter (TX) and receiver (RX) models are heterogeneously trained and cannot be jointly adapted. By leveraging Parseval frames constructed on shared anchor data, these equalizers provide robust, numerically-stable signal alignment and semantic consistency over both digital and physical (including over-the-air) channels, without requiring explicit retraining or pilot exchange at runtime (Fiorellino et al., 23 Jul 2025, Pannacci et al., 6 Oct 2025, Pandolfo et al., 5 Dec 2025, Au-Yeung et al., 2013).

1. Mathematical and Theoretical Foundations

A frame $\{f_k\}_{k=1}^N$ for a real or complex Hilbert space $\mathcal{H}$ (of dimension $d$) satisfies frame bounds $0<A\leq B<\infty$ so that for all $x\in\mathcal{H}$

$$A \|x\|^2 \leq \sum_{k=1}^N |\langle x, f_k\rangle|^2 \leq B \|x\|^2.$$

A Parseval frame is a frame with $A=B=1$, implying perfect numerical conditioning: $F F^* = I_{d}$ for analysis operator $F$. This property guarantees that analysis-synthesis chains preserve norm and avoid noise amplification, which is central to zero-shot equalization (Fiorellino et al., 23 Jul 2025, Au-Yeung et al., 2013).

Given a non-tight frame (arbitrary $A$ and $B$), there exists an explicit procedure for whitening it into a Parseval frame via the eigen-decomposition of the frame operator $S = F F^*$: setting $g_i = S^{-1/2} f_i$ yields a Parseval frame $\{g_i\}_{i=1}^N$ (Au-Yeung et al., 2013). This procedure remains applicable in both digital latent spaces and their physical (e.g., wireless) implementations.

2. Zero-Shot Parseval-Frame Equalizer Construction

The zero-shot Parseval-frame equalizer is formed as follows:

• Anchor Set Encoding: TX and RX agree on a reference set $\mathcal{S}$ (no labels/gradients needed). Each computes latent embeddings of these anchors $\tilde G$ (TX) and $\tilde F$ (RX).
• Parseval Frame Construction: Both sides whiten their anchor matrices (e.g., $G = \tilde G (\tilde G^T \tilde G)^{-1/2}$) so columns of $G$ (TX) and $F$ (RX) form Parseval frames.
• Analysis: TX projects new latent $x$ as $c = G x$.
• (Over-the-air/physical channel) Transmission: $c$ is transmitted (possibly after compression/quantization).
• Synthesis: RX reconstructs $\hat y = F^T c$ in its own latent space.

The overall equalizer mapping is $\mathcal{E}(x) = F^T G x$. No model retraining or pilot-based adaptation is required. Robust alignment is enabled for heterogeneous encoder/decoder pairs under the assumption that latent embedding spaces differ by approximately unitary transforms (Fiorellino et al., 23 Jul 2025, Pannacci et al., 6 Oct 2025).

Algorithmic steps and summary:

Stage	TX Operation	RX Operation
Anchor set	Compute $\tilde G$ from anchors	Compute $\tilde F$ from anchors
Frame whitening	$G = \tilde G (\tilde G^T \tilde G)^{-1/2}$	$F = \tilde F (\tilde F^T \tilde F)^{-1/2}$
Analysis	$c = Gx$
Synthesis		$\hat y= F^T c$

3. Implementation and Physical Realization

Digital domain: The analysis-synthesis operations are lightweight, involving matrix-vector products and inner products only. The process is efficiently implementable at the edge, with computational complexity dominated by the offline whitening ($O(d^3)$ per agent) and runtime scaling as $O(|\mathcal{S}| (d+m))$ for $d$-, $m$-dimensional TX/RX latent spaces (Pannacci et al., 6 Oct 2025).

Over-the-air realization: Stacked intelligent metasurfaces (SIM) can emulate zero-shot Parseval-frame equalizers in the physical layer. Here, a multi-layer SIM (with parameterizable phase shift coefficients) is optimized by gradient descent to match the digital Parseval equalizer (e.g., $A = F_R^H F_T$) in Frobenius norm. This approach achieves direct latent-space alignment at the wave level, substantially reducing device-side computational load and maintaining robustness properties intrinsic to Parseval mapping (Pandolfo et al., 5 Dec 2025).

4. Dynamic Optimization and Multi-Agent Coordination

For multi-agent scenarios with time-varying network, task, and device conditions, dynamic resource coordination is critical. Parseval-frame equalizers are embedded in a Lyapunov-based multi-objective control framework that jointly optimizes:

• Communication resources: Number of frame coefficients $N$, quantization bits, bandwidth allocation.
• Computation resources: CPU frequencies at TX/RX.
• Learning/prototype resources: Anchor set size or prototype selection.

This framework introduces virtual queues for latency and accuracy constraints, and solves a quadratic (or integer-convex) per-slot optimization to minimize energy and latency while respecting task-driven accuracy. The Parseval equalizer provides consistent semantic alignment, enabling stable queue behavior and compliance with long-term performance guarantees (Fiorellino et al., 23 Jul 2025).

5. Empirical Performance and Comparative Analysis

On standard benchmarks (e.g., CIFAR-10/100, Tiny-ImageNet), zero-shot Parseval-frame equalizers (PFE) close approximately 90–95% of the performance gap to supervised unitary (Procrustes) aligners, even under strong compression:

• On Tiny-ImageNet, PFE with 128 coefficients reaches ~38% accuracy vs. 35% for unwhitened frame equalizers and 39% for supervised unitary aligners.
• In three-user edge inference setups, prototype-based PFE (Proto-PFE) matches long-term latency and accuracy targets while consuming 0.12–0.15 W, outperforming non-anchor-based frame equalizers (0.18 W) (Fiorellino et al., 23 Jul 2025).

In over-the-air implementations, 64×64 SIMs emulating PFEs attain ≈90% downstream accuracy at high SNR, maintaining strong performance (60–80%) even at SNR = –10 dB, and consistently outperform supervised-linear aligners in the low-SNR regime (Pandolfo et al., 5 Dec 2025).

6. Deployment Guidelines, Limitations, and Applicability

PFEs are most effective where:

• Model retraining, gradient-based adaptation, or semantic pilot exchange are infeasible.
• SNR is moderate-to-high; PFE maintains fidelity in these regimes even as other non-adaptive aligners degrade.
• Channel-agnostic operation is desired; the same equalizers function under varying SNR and channel models (AWGN, Rayleigh fading) without reconfiguration.

Key practical considerations:

• Anchor set distribution: Sharing a fixed set of anchors up-front is sufficient (modest overhead).
• Compression vs. fidelity trade-off: Varying the number of frame coefficients adjusts the rate-reliability curve. Dimensionality mismatches ($d\neq m$) are handled by appropriate frame size selection.
• Extensions: For non-unitary mismatches, small pilot sets or sample-efficient inversion may be required.
• Limitations: At very low SNR, CNN-based or supervised aligners may surpass PFE in bandwidth efficiency or sample efficiency (Pannacci et al., 6 Oct 2025).

7. Connections to Frame Theory and Signal Reconstruction

Parseval-frame-based equalizers are grounded in frame theory, specifically in the structure and properties of tight and Parseval frames. The closed-form conversion of arbitrary frames to Parseval frames via whitening (eigen-decomposition of the frame operator) ensures numerically-stable, norm-preserving reconstruction, inherently robust to noise and channel impairment (Au-Yeung et al., 2013). Error analysis reveals truncation and additive noise incur bounded and non-amplified losses. In practice, these desirable properties translate to stable, robust semantic communication and alignment in both digital and physical AI-native communications contexts.