HoloNorm: Normalization for Transformers & MIMO

• HoloNorm is a unified framework offering mathematically rigorous, invertible normalization methods for both transformer neural networks and holographic MIMO systems.
• It preserves vector geometry by maintaining direction, orthogonality, and contraction in transformer models, ensuring robust activation dynamics.
• In MIMO systems, HoloNorm aligns channel matrix scaling with actual physical antenna gains, correcting conventional normalization errors and accurately reflecting capacity.

HoloNorm is a term applied to two distinct but mathematically rigorous normalization methods: (1) vector normalization for transformer neural networks, providing stable, geometry-preserving nonlinearities, and (2) electromagnetic normalization of channel matrices in holographic MIMO communications to correctly reflect realized physical gain and ensure meaningful capacity analysis. Both incarnations introduce principled, invertible normalization schemes replacing ad hoc or componentwise approaches, with demonstrated mathematical and empirical advantages in their respective domains. The following synthesizes the major results, theoretical underpinnings, and application guidelines for each HoloNorm setting.

1. Mathematical Definitions

HoloNorm for Vector Normalization (Transformers)

Let $x \in \mathbb{R}^d$ be a feature vector. The HoloNorm operator with parameter $p$ (typically $p=2$) is

$\mathrm{HN}_p(x) = \frac{x}{1 + \|x\|_p}$

where $\|x\|_p$ is the $L_p$ norm, most commonly Euclidean ($\|x\|_2$). Key properties include:

• Inverse mapping: For $y = \mathrm{HN}_p(x)$ with $\|y\|_p < 1$, the exact inverse is

$x = \mathrm{HN}_p^{-1}(y) = \frac{y}{1 - \|y\|_p}$

• Jacobian: The derivative is everywhere finite, and for $p$0 reads

$p$1

• 1-Lipschitz contraction: The operation strictly contracts vector lengths, preventing norm explosion.

HoloNorm for Electromagnetic MIMO Normalization

For a channel matrix $p$2 in a holographic MIMO system, HoloNorm normalization enforces

$p$3

where $p$4 is the number of transmit antennas, and $p$5 is the physically realized far-field or near-field gain of the receive aperture, integrating antenna topology, spatial correlations, and loss factors. For near-field, $p$6 captures additional losses via dyadic Green's functions: $p$7

2. Geometry-Preserving Properties and Rationale

HoloNorm (vector version) applies a scalar transformation to the full vector, thereby ensuring:

• Direction preservation: $p$8.
• Orthogonality: For orthogonal $p$9, $p=2$0, their images under HoloNorm remain orthogonal.
• Invertibility: The mapping is bijective on the open unit ball ($p=2$1). By contrast, $p=2$2 and Softsign, applied elementwise, destroy both directionality and orthogonality in vector spaces of dimension $p=2$3, and lack global invertibility. LayerNorm, although stabilizing, alters the angular geometry due to mean subtraction and componentwise variance rescaling (Yongueng et al., 13 Nov 2025).

In electromagnetic MIMO, HoloNorm ensures the physical scaling of channel matrices matches the realized array gain rather than simply antenna count, eliminating artificial SNR or capacity inflation in dense or unconventional arrays (Yuan et al., 2024).

3. Comparative Analysis with Existing Schemes

Method	Orthogonality Preserved	Invertible	Geometry Respected	Contraction
HoloNorm	Yes	Yes	Yes	Yes
Tanh	No	No (global)	No	Yes
Softsign	No	No (vector)	No	Yes
LayerNorm	No	Conditional	No	No

Componentwise functions (Tanh, Softsign) disrupt vector relationships. LayerNorm eliminates mean and rescales by variance, but distorts vector geometry and is not always invertible. Only HoloNorm guarantees all key vector-wise properties with computational simplicity (involving only a norm and rescaling) (Yongueng et al., 13 Nov 2025).

For MIMO, conventional normalization enforces $p=2$4, which misreports channel strength in topologies deviating from idealized planar arrays. HoloNorm normalization (matching to $p=2$5 or full near-field gain) corrects this, providing physically meaningful metrics for advanced holographic arrays (Yuan et al., 2024).

4. Application in Transformers and MIMO Systems

Transformers

HoloNorm is incorporated as a pre-normalization step before each sublayer: $\mathrm{HN}_p(x) = \frac{x}{1 + \|x\|_p}$0 It preserves residual connections, shrinks activations into the open unit ball, and avoids introduction of statistical or geometric distortions. The pre-norm application ensures improved gradient flow and stability in deep architectures (Yongueng et al., 13 Nov 2025).

Holographic MIMO

HoloNorm normalization connects the channel matrix scaling to actual electromagnetic gain. Far-field scenarios use analytical, physical area-based, or full-wave simulation-derived gains. Near-field HoloNorm employs dyadic Green's tensors to account for polarization, illumination, and beamforming losses, normalizing each polarization submatrix to the realized gain at the target distance and scan sector (Yuan et al., 2024).

5. Empirical Results and Practical Outcomes

In transformer experiments (MusicCaps audio–text, $p=2$6 orthogonal vectors):

• With HoloNorm: Zero distortion in cosine similarity throughout training, perfect preservation of orthogonality.
• With $p=2$7: Similarity drifted into 40–80% range, indicating geometry destruction.
• Efficiency: HoloNorm exhibited faster computation (avoiding exponentials) and lower energy consumption than $p=2$8 in optimization tasks (Yongueng et al., 13 Nov 2025).

For holographic MIMO:

• Traditional normalization yields nonphysical SNR/capacity scaling with dense/volumetric arrays.
• HoloNorm normalization exposes capacity saturation for planar arrays at $p=2$9 spacing and reveals true capacity advantage of volumetric topologies (15–20% over planar) when capturing additional effective aperture. Near-field application correctly penalizes omitted beamforming and polarization loss, providing accurate performance bounds in regimes where coupling and electromagnetic interaction dominate (Yuan et al., 2024).

6. Theoretical and Design Implications

HoloNorm advances both neural network and electromagnetic system design:

• In transformer models, it eliminates vanishing/exploding activations while preserving vector geometry, supporting invertible and robust architectures.
• In MIMO, it enables meaningful optimization of aperture geometry, element count, and spacing (e.g., showing over-densification in 2D is counterproductive). For volumetric arrays, HoloNorm calibration demonstrates the gain of extra elevation degrees-of-freedom.
• The 1-Lipschitz property of the vector form provides compositional stability across deep layers; the physical normalization in MIMO averts spectral efficiency runaway and correctly surfaces array topology advantages.

7. Summary and Outlook

HoloNorm provides dimension-agnostic, invertible, and geometry-respecting normalization in both deep neural transformers and electromagnetic holographic MIMO. Its vector normalization form is unique in guaranteeing directionality, orthogonality, and contraction with computational efficiency. Its electromagnetic version anchors channel matrix scaling to physically realizable array gain, correcting deficiencies in conventional approaches for dense and nonplanar topologies. Empirical evaluations validate both superior structural properties in learning systems and key performance insights for physical communications arrays, setting a standard for normalization in both machine learning and signal processing contexts (Yongueng et al., 13 Nov 2025, Yuan et al., 2024).