Neural Precoding in Complex Projective Spaces

This paper proposes a deep learning framework for MU-MISO precoding that leverages complex projective space parameterizations to eliminate global phase redundancies, thereby achieving superior sum-rate performance and generalization compared to conventional real or complex-valued representations.

The Gist

The Big Picture: The "Radio Orchestra" Problem

Imagine a modern wireless network (like 5G or future 6G) as a conductor leading an orchestra.

• The Conductor: The Base Station (the cell tower).
• The Musicians: The users (your phone, your laptop, your smart fridge).
• The Music: The data signals being sent.

The problem is that the "hall" (the air) is noisy and echoey. If the conductor just shouts at everyone at once, the sound waves crash into each other, creating a mess of noise (interference). To fix this, the conductor needs to precoding: adjusting the volume and timing of each musician's signal so that when they reach the audience, they blend perfectly without canceling each other out.

Doing this perfectly is like solving a massive, impossible math puzzle in real-time. It's so hard that even supercomputers struggle to do it fast enough for your phone call.

The Old Way: Teaching a Robot with "Confusing" Instructions

To make this faster, engineers started using Deep Learning (AI). They tried to teach a computer (a neural network) to act like the conductor.

However, they were teaching the AI using a confusing language.

• The Old Method: They described the signals using Real and Imaginary numbers (like $3 + 4i$) or Amplitude and Phase (like "loudness" and "timing").
• The Flaw: In physics, if you rotate a signal's timing by a full circle (360 degrees), it sounds exactly the same. It's like spinning a clock hand all the way around; the time is still 12:00.
• The AI's Struggle: Because the old method treated "12:00" and "12:00 + 360 degrees" as two different numbers, the AI had to waste its brainpower learning that these two things are actually the same. It was like trying to learn a language where "Hello" and "Hello-rotated-360-degrees" were treated as different words. The AI got confused, learned slowly, and didn't generalize well to new situations.

The New Idea: The "Projective Space" Solution

The authors of this paper say: "Stop teaching the AI the confusing version. Teach it the version that matches reality."

They propose using Complex Projective Spaces (CPS).

• The Analogy: Imagine a globe. On a globe, the North Pole is the same point whether you approach it from the left or the right. You don't need to tell the AI, "Go to the North Pole from the left" and "Go to the North Pole from the right." You just say, "Go to the North Pole."
• What they did: They stripped away the "global phase" (the redundant rotation) from the math. They mapped the signals onto a shape where every physically unique signal has only one mathematical representation.

By doing this, they removed the "noise" from the training data. The AI no longer has to waste time learning that $A$ is the same as $B$; it just learns the relationship between the unique shapes of the signals.

Two Ways to Describe the Shape

The paper tests two ways to describe these "clean" shapes to the AI:

1. Real-Valued Embeddings (The "Flat Map"): This is like flattening the globe onto a piece of paper. It's simple, direct, and the AI learns it very quickly.
2. Complex Hyperspherical Coordinates (The "3D Globe"): This is like describing the globe using latitude and longitude. It's mathematically elegant, but the paper found it made the AI's job slightly harder because the math gets "twisty" and confusing for the computer to process.

The Winner: The "Flat Map" (Real-Valued Embeddings) won. It gave the best results with the least amount of computer power.

The Results: Faster, Smarter, and Stronger

When they tested this new method against the old ones:

• Speed: The AI learned much faster because it wasn't distracted by redundant information.
• Accuracy: The AI made better decisions, resulting in higher data speeds (sum-rate) for users.
• Generalization: This is the most important part. If you train a driver on a sunny day, can they drive in the rain?
  • The old AI (trained on the confusing method) struggled when the signal conditions changed (e.g., low signal strength).
  • The new AI (trained on the "clean" geometry) handled bad conditions almost as well as the perfect, slow supercomputer method. It understood the essence of the problem, not just the surface details.

The Bottom Line

Think of this paper as the difference between teaching a child to draw a circle by giving them a list of 100 coordinate points (the old way) versus teaching them the concept of "roundness" (the new way).

By realizing that the "rotation" of a signal doesn't change what it is, the authors cleaned up the math. They built an AI that understands the geometry of wireless signals, allowing it to beam data to your phone much more efficiently, even when the connection is weak. It's a smarter way to teach the computer how to be the conductor of the wireless orchestra.

Technical Summary

1. Problem Statement

In Multi-User Multiple-Input Single-Output (MU-MISO) wireless systems, the base station (BS) must design precoding vectors to mitigate inter-user interference and maximize the weighted sum-rate (WSR).

• The Challenge: Traditional deep learning (DL) approaches for precoding treat complex-valued channel and precoder coefficients as either real/imaginary pairs or amplitude/phase pairs.
• The Core Issue: Precoding performance depends on the magnitude of inner products between channel and precoding vectors ($|h^H w|$). These magnitudes are invariant to global phase rotations (i.e., multiplying a vector by $e^{j\psi}$ does not change the physical outcome).
• Consequence: Conventional representations force the DL model to implicitly learn this phase invariance, introducing redundant degrees of freedom (DoF). This leads to inefficient learning, degraded generalization, and increased difficulty in training, as the model must distinguish between physically identical states that are mathematically distinct in standard representations.

2. Methodology

The authors propose a Deep Learning framework based on Complex Projective Space ($CP^n$) parameterizations to eliminate global phase redundancies.

A. Theoretical Foundation: Complex Projective Space ($CP^n$)

• $CP^n$ is the space of complex lines in $\mathbb{C}^{n+1}$ passing through the origin.
• In this space, two vectors differing only by a non-zero complex scalar (specifically a global phase rotation for unit-norm vectors) are considered the same point.
• By mapping channel and precoder vectors to $CP^{N-1}$, the model learns directly on the manifold of physically distinct channel-precoder pairs.

B. Proposed Framework

The framework decomposes the learning problem into two components:

1. Precoder Direction: Learning the shape/direction of the precoder in $CP^{N-1}$ (phase-invariant).
2. Power Allocation: Learning the power split ratios among users.

The authors investigate two specific parameterizations for the $CP^{N-1}$ space:

1. $P_{cps}$ (Real-valued Embeddings):
   • Normalizes vectors to unit length.
   • Removes global phase by setting the first element's phase to zero (using the first element as a reference).
   • Represents the vector using $2N-1$ real coefficients.
2. $P_{hsc}$ (Complex Hyperspherical Coordinates):
   • Parameterizes the unit-norm, phase-removed vector using hyperspherical coordinates ($\theta$ for amplitude angles, $\phi$ for phase angles).
   • Uses $\sin/\cos$ encoding for phases to avoid discontinuities at $\pm \pi$.

C. Network Architecture & Training

• Model: A standard feed-forward Deep Neural Network (DNN) with 5 fully connected layers.
• Loss Function: A sum-rate aware loss is used, combining:
  • Power Split Loss: Kullback-Leibler (KL) divergence for power ratios.
  • Direction/Angle Loss: Measures the distance between predicted and target vectors (using inner product magnitude for $P_{cps}$ or circular distance for $P_{hsc}$).
  • Rate Penalty: A weighting factor based on the ratio of the DNN's achieved sum-rate to the optimal WMMSE sum-rate to prioritize high-performance samples.
• Benchmarks: The proposed methods are compared against:
  • $P_{ri}$: Classical real/imaginary representation (no phase removal).
  • $P_{ncv}$: Normalized complex vectors (unit length) but without global phase removal.

3. Key Contributions

1. Geometry-Aware Learning: Introduction of a DL framework that maps MU-MISO precoding tasks to $CP^n$, explicitly removing phase-related ambiguities inherent in conventional representations.
2. Two Novel Parameterizations: Development of $P_{cps}$ (real-valued embeddings) and $P_{hsc}$ (hyperspherical coordinates) to represent the intrinsic $2N-2$ degrees of freedom of the system.
3. End-to-End Constrained Learning: A decomposition strategy separating direction learning (in $CP^n$) from power allocation, trained with a loss function sensitive to sum-rate performance.
4. Comprehensive Benchmarking: Demonstration that geometry-aware parameterizations significantly outperform standard baselines in both accuracy and generalization, often with the same or fewer model parameters.

4. Results

Simulations were conducted for a system with $N=4$ antennas and $K=4$ users, comparing the proposed methods against WMMSE (optimal benchmark) and DL baselines.

• Convergence & Accuracy:

  • The $P_{cps}$ parameterization achieved the highest accuracy (76.6% of WMMSE sum-rate at convergence), outperforming the classical $P_{ri}$ (68.6%) and the normalized $P_{ncv}$ (73%).
  • Notably, $P_{cps}$ achieved this with the exact same number of trainable parameters as the classical $P_{ri}$, proving the gain comes from the representation, not model complexity.
  • The hyperspherical approach ($P_{hsc}$) performed well but slightly worse than $P_{cps}$, likely due to ill-conditioned gradients caused by nested trigonometric functions.

• Generalization (SNR Robustness):

  • Low SNR: $P_{cps}$ achieved 94% of the WMMSE rate at 0 dB, compared to 82.5% for $P_{ri}$.
  • High SNR: The gap widened. At 18 dB, $P_{cps}$ outperformed $P_{ri}$ by 1 bit/s/Hz and $P_{ncv}$ by 0.9 bit/s/Hz.
  • The $P_{ncv}$ benchmark (normalized but phase-ambiguous) showed good performance at low SNR but degraded significantly at high SNR, confirming that phase invariance is critical for high-performance regimes.

• Inference Latency:

  • All DL methods were significantly faster (milliseconds) than the iterative WMMSE algorithm (133ms).
  • $P_{cps}$ had a negligible latency increase compared to $P_{ri}$ (3.77ms vs 2.87ms on CPU), justified by the substantial performance gains.

5. Significance

This paper establishes that representation matters more than architecture in DL-based wireless communications. By aligning the input/output representations with the underlying physical geometry of the problem (specifically, the phase-invariance of MU-MISO precoding), the authors achieve:

• Higher Efficiency: Better learning efficiency and faster convergence.
• Superior Generalization: Models trained on $CP^n$ generalize much better across varying Signal-to-Noise Ratios (SNR).
• Practicality: The approach achieves near-optimal performance with negligible computational overhead compared to standard DL methods, making it highly suitable for real-time deployment in 6G and future wireless systems.

The work suggests that future DL designs for physical layer tasks should prioritize geometry-aware parameterizations over simply increasing network depth or width.