⚡ electrical engineering

SEMIKHORN: Globally balanced affinities for mmWave Localization in MU mMIMO systems

This paper proposes SEMIKHORN, a semisupervised channel charting framework for mmWave localization in multi-user mMIMO systems that utilizes the globally balanced t-SNEkhorn algorithm to fuse local channel state information dissimilarities, achieving a mean localization error of 6.86% with less than 15% labeled data.

Original authors: Abhisha Garg, Raghav Shukla, Suraj Srivastava, Aditya K. Jagannatham

Published 2026-06-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Abhisha Garg, Raghav Shukla, Suraj Srivastava, Aditya K. Jagannatham

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to draw a map of a giant, foggy city, but you don't have a GPS. You only have a group of people (the users) standing in the city, and a few radio towers (the base stations) that can "hear" their voices. The towers can tell you how loud or clear a voice is, and from that, they can guess how far apart two people are.

This paper introduces a new, smarter way to draw that map, called SEMIKHORN. Here is how it works, broken down into simple concepts:

1. The Problem: The "Distorted" Map

Usually, when we try to map these users based on radio signals, we use a standard tool called t-SNE. Think of t-SNE as a very enthusiastic tour guide who tries to keep friends close together on a map. However, this guide has a flaw: they are a bit self-centered. They make sure their immediate friends are close, but they don't care if the whole group ends up squashed into a tiny corner or stretched out weirdly.

In technical terms, standard t-SNE creates "unbalanced" neighborhoods. Some areas of the map get crowded, while others get empty, distorting the true shape of the city.

2. The Solution: The "Fair" Guide (t-SNEkhorn)

The authors created a new tool called t-SNEkhorn. Imagine this as a super-fair tour guide who uses a mathematical rule called "Entropic Optimal Transport."

The Analogy: Instead of just looking at one person's friends, this guide looks at the entire crowd at once. They ensure that every single person has the exact same number of "neighbors" on the map.
The Result: This creates a globally balanced map. No one gets squashed, and no one gets lost in the void. The relationships between all the users are treated equally, preserving the true shape of the city much better than the old method.

3. Gathering Clues from Multiple Towers

In the real world, one radio tower can't see the whole city.

The Old Way: A single tower draws a local map, which might be wrong for people far away.
The New Way (SEMIKHORN): The system asks five different towers to draw their own local maps. Then, it acts like a wise editor, combining all five maps into one Global Map. It weighs the clues based on how clear the signal is (like trusting a loud, clear voice over a whisper).

4. The "Semi-Supervised" Trick (The Anchors)

To make the map accurate, the system needs a few "anchors."

The Analogy: Imagine you are drawing a map of a forest, but you don't know where the North Pole is. If you place 14% of the hikers at spots where you already know the exact coordinates (like a famous tree or a signpost), the system can use those fixed points to pull the rest of the map into the correct orientation.
The Paper's Claim: The system only needs these labeled "anchors" for a small fraction of users (less than 15%) to figure out where everyone else is.

5. Measuring Distance: The "Manhattan" vs. "Euclidean"

When the system calculates how far apart two users are based on their radio signals, it usually uses a straight-line measurement (Euclidean).

The Paper's Twist: The authors found that in this specific high-tech environment, a "Manhattan" measurement (like counting city blocks: go 3 blocks east, then 4 blocks north) works better. It's less sensitive to static and noise, giving a clearer picture of who is actually close to whom.

6. Tuning the Engine (Bayesian Optimization)

Building this map requires tuning many dials (like how many neighbors to look at, or how many anchors to use). Instead of guessing and checking, the system uses Bayesian Optimization.

The Analogy: Think of this as a smart robot that plays a game of "Hot and Cold." It tries a setting, sees how good the map is, and then intelligently guesses the next setting to get even closer to the perfect map, rather than randomly spinning the dials.

The Final Result

The paper tested this system in a simulated outdoor city with a radius of 100 meters.

Accuracy: The system achieved a very low error rate (about 6.86% error in a circular area).
Efficiency: It did all this while using less than 15% of the users as known "anchors."
Comparison: The new SEMIKHORN map was more accurate and kept the neighborhood relationships (continuity) and trustworthiness of the map much better than the standard t-SNE or other older mapping techniques.

In short: The paper presents a smarter, fairer, and more collaborative way to turn radio signals into a location map, ensuring that the "neighbors" on the map are actually neighbors in real life, even with very little prior knowledge of where people are standing.

Technical Summary: SEMIKHORN for mmWave Localization

Problem Statement
Accurate user localization is a critical prerequisite for harnessing the full potential of millimeter-wave (mmWave) massive MIMO systems. While absolute positions are not always strictly required for location-based services, preserving relative neighborhood relationships (pseudo-locations) is essential. Existing Channel Charting (CC) frameworks, which learn latent representations of radio geometry without ground-truth data, face two primary limitations:

Limited Spatial Coverage: Single-base station (BS) approaches capture only limited spatial relationships, leading to distorted neighborhood representations for users in different cells.
Algorithmic Bias in Manifold Learning: Standard t-distributed Stochastic Neighbor Embedding (t-SNE), often used for CC, performs asymmetric affinity normalization. This results in unbalanced neighborhood representations and distorts global geometry. Furthermore, existing semi-supervised approaches often rely on extensive labeled data or complex deep learning models that require substantial computational resources and careful hyperparameter tuning.

Methodology
The paper proposes SEMIKHORN, a semi-supervised channel charting framework designed for multi-user (MU) mmWave massive MIMO systems with distributed base stations. The methodology proceeds through the following stages:

Channel Modeling and Feature Extraction:
- The system considers $B$ distributed BSs, each equipped with $N$ antennas, serving $K$ single-antenna users.
- Instead of instantaneous channel coefficients, the framework utilizes the second raw moment of the channel state information (CSI), $R^b_k$ , as it evolves more smoothly with user mobility.
- Features are extracted by scaling the second raw moment to account for path loss and transforming them into the beamspace domain via a Fourier transform matrix.
- A Manhattan (L1) distance metric is employed to compute the dissimilarity between users. The authors argue that in high-dimensional spaces, the L1 metric provides a stronger correlation between feature dissimilarity and true physical separation compared to the conventional Euclidean (L2) metric, which can amplify noise.
Dissimilarity Matrix Construction:
- Each BS constructs a local dissimilarity matrix based on its observed CSI.
- These local matrices are fused into a single global dissimilarity matrix using a weighted average. The weights are determined by the Signal-to-Noise Ratio (SNR) of the links, ensuring that more reliable CSI contributions from specific BSs are prioritized.
t-SNEkhorn and Entropic Optimal Transport:
- The core innovation replaces standard t-SNE with t-SNEkhorn, which utilizes entropic optimal transport to construct pairwise similarities.
- Unlike standard t-SNE, which normalizes affinities independently per data point (row-stochastic), t-SNEkhorn enforces doubly stochastic affinities (symmetric and both row and column sums equal to 1) using the Sinkhorn-Knopp algorithm.
- Theoretical Justification: The paper analytically demonstrates that non-doubly stochastic affinity matrices in standard t-SNE introduce a positive lower bound on the Kullback-Leibler (KL) divergence objective, limiting optimization. By enforcing double stochasticity, t-SNEkhorn removes this lower bound, allowing for more effective minimization of the KL objective and producing balanced neighborhood graphs that align with the assumption of uniformly distributed users.
Semi-Supervised Learning and Optimization:
- The framework incorporates a small set of labeled user positions as "hard anchors" to infer absolute geographical positions for unlabeled users.
- The embedding is optimized by minimizing the KL divergence between the high-dimensional affinity matrix ( $G^*$ ) and the low-dimensional affinity matrix ( $B^*$ ), subject to the constraint that labeled points remain fixed at their ground-truth coordinates.
- Bayesian Optimization is employed to automatically tune critical hyperparameters (specifically perplexity and the number of hard anchors) to minimize the Mean Localization Error (MLE), avoiding manual tuning.

Key Contributions

First Application of Doubly Stochastic Affinities: The paper introduces the first application of symmetric, doubly stochastic entropic affinities (via t-SNEkhorn) for semi-supervised wireless localization, replacing the Euclidean projection onto symmetric matrices used in standard t-SNE.
Theoretical Improvement in Optimization: It provides an analytical proof that enforcing double stochasticity relaxes the positive lower bound on the KL divergence objective, thereby improving embedding fidelity.
Metric Reformulation: The framework reformulates the dissimilarity measure using the Manhattan (L1) metric, demonstrating its superior correlation with spatial proximity compared to the Euclidean metric in high-dimensional channel feature spaces.
Automated Hyperparameter Tuning: The integration of Bayesian optimization to determine optimal embedding parameters, yielding substantial gains in trustworthiness and reduced localization error.

Experimental Results
The framework was evaluated in a simulated outdoor environment with 5 distributed mmWave BSs and 2048 uniformly distributed users within a 100m radius.

Performance: SEMIKHORN achieved a Mean Localization Error (MLE) of 6.86% within the circular vicinity.
Data Efficiency: The framework required less than 15% of labeled CSI samples to achieve these results.
Comparative Analysis: Compared to state-of-the-art techniques (standard t-SNE, Laplacian Eigenmaps, Kernel PCA, Sammon Mapping), SEMIKHORN demonstrated superior continuity (0.95–0.98) and trustworthiness (0.95–0.97). The results indicate that the doubly stochastic formulation effectively mitigates the bias toward dense regions found in standard t-SNE, preserving both global topology and local neighborhood continuity.
Metric Validation: Visualizations and correlation plots confirmed that the Manhattan distance metric provided clearer discrimination of spatially distinct CSI samples than the Euclidean metric.

Significance
The paper claims that SEMIKHORN offers a robust, scalable solution for mmWave localization that does not rely on extensive labeled datasets or computationally heavy deep learning models. By leveraging globally balanced affinities through entropic optimal transport, the framework ensures consistent neighborhood representation across the entire network, addressing the limitations of single-BS approaches and the geometric distortions of standard manifold learning techniques. The use of Bayesian optimization further enhances the framework's practicality by automating the tuning process to minimize localization error.